[091424] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[d92713] | 2 | version="$Id: primdec.lib,v 1.145 2009-03-18 14:17:33 Singular Exp $"; |
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[0ae4ce] | 3 | category="Commutative Algebra"; |
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[5480da] | 4 | info=" |
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[8942a5] | 5 | LIBRARY: primdec.lib Primary Decomposition and Radical of Ideals |
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[f3c6e5] | 6 | AUTHORS: Gerhard Pfister, pfister@mathematik.uni-kl.de (GTZ)@* |
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| 7 | Wolfram Decker, decker@math.uni-sb.de (SY)@* |
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| 8 | Hans Schoenemann, hannes@mathematik.uni-kl.de (SY)@* |
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| 9 | Santiago Laplagne, slaplagn@dm.uba.ar (GTZ) |
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[f34c37c] | 10 | |
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[b9b906] | 11 | OVERVIEW: |
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[07c623] | 12 | Algorithms for primary decomposition based on the ideas of |
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[367e88] | 13 | Gianni, Trager and Zacharias (implementation by Gerhard Pfister), |
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| 14 | respectively based on the ideas of Shimoyama and Yokoyama (implementation |
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[7f7c25e] | 15 | by Wolfram Decker and Hans Schoenemann).@* |
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| 16 | The procedures are implemented to be used in characteristic 0.@* |
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| 17 | They also work in positive characteristic >> 0.@* |
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| 18 | In small characteristic and for algebraic extensions, primdecGTZ |
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| 19 | may not terminate.@* |
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[b9b906] | 20 | Algorithms for the computation of the radical based on the ideas of |
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[7f7c25e] | 21 | Krick, Logar, Laplagne and Kemper (implementation by Gerhard Pfister and Santiago Laplagne). |
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| 22 | They work in any characteristic. |
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[8942a5] | 23 | |
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[f34c37c] | 24 | PROCEDURES: |
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[24f458] | 25 | Ann(M); annihilator of R^n/M, R=basering, M in R^n |
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[8942a5] | 26 | primdecGTZ(I); complete primary decomposition via Gianni,Trager,Zacharias |
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| 27 | primdecSY(I...); complete primary decomposition via Shimoyama-Yokoyama |
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[7f7c25e] | 28 | minAssGTZ(I); the minimal associated primes via Gianni,Trager,Zacharias (with modifications by Laplagne) |
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[8942a5] | 29 | minAssChar(I...); the minimal associated primes using characteristic sets |
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| 30 | testPrimary(L,k); tests the result of the primary decomposition |
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[7f7c25e] | 31 | radical(I); computes the radical of I via Krick/Logar (with modifications by Laplagne) and Kemper |
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| 32 | radicalEHV(I); computes the radical of I via Eisenbud,Huneke,Vasconcelos |
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[8942a5] | 33 | equiRadical(I); the radical of the equidimensional part of the ideal I |
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| 34 | prepareAss(I); list of radicals of the equidimensional components of I |
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| 35 | equidim(I); weak equidimensional decomposition of I |
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| 36 | equidimMax(I); equidimensional locus of I |
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| 37 | equidimMaxEHV(I); equidimensional locus of I via Eisenbud,Huneke,Vasconcelos |
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| 38 | zerodec(I); zerodimensional decomposition via Monico |
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[326dba] | 39 | absPrimdecGTZ(I); the absolute prime components of I |
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[8942a5] | 40 | "; |
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[e801fe] | 41 | |
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| 42 | LIB "general.lib"; |
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[67bd4c] | 43 | LIB "elim.lib"; |
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[e801fe] | 44 | LIB "poly.lib"; |
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| 45 | LIB "random.lib"; |
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[8afd58] | 46 | LIB "inout.lib"; |
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[7f24dd7] | 47 | LIB "matrix.lib"; |
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[24f458] | 48 | LIB "triang.lib"; |
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[6fa3af] | 49 | LIB "absfact.lib"; |
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[d6db1f2] | 50 | /////////////////////////////////////////////////////////////////////////////// |
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[ebecf83] | 51 | // |
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[091424] | 52 | // Gianni/Trager/Zacharias |
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[ebecf83] | 53 | // |
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| 54 | /////////////////////////////////////////////////////////////////////////////// |
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| 55 | |
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[07c623] | 56 | static proc sat1 (ideal id, poly p) |
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[d2b2a7] | 57 | "USAGE: sat1(id,j); id ideal, j polynomial |
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[d6db1f2] | 58 | RETURN: saturation of id with respect to j (= union_(k=1...) of id:j^k) |
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| 59 | NOTE: result is a std basis in the basering |
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[d2b2a7] | 60 | " |
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[d6db1f2] | 61 | { |
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[70ab73] | 62 | int @k; |
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| 63 | ideal inew=std(id); |
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| 64 | ideal iold; |
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| 65 | intvec op=option(get); |
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| 66 | option(returnSB); |
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| 67 | while(specialIdealsEqual(iold,inew)==0 ) |
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| 68 | { |
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| 69 | iold=inew; |
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| 70 | inew=quotient(iold,p); |
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| 71 | @k++; |
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| 72 | } |
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| 73 | @k--; |
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| 74 | option(set,op); |
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| 75 | list L =inew,p^@k; |
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| 76 | return (L); |
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[d6db1f2] | 77 | } |
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| 78 | |
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| 79 | /////////////////////////////////////////////////////////////////////////////// |
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| 80 | |
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[07c623] | 81 | static proc sat2 (ideal id, ideal h) |
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[d2b2a7] | 82 | "USAGE: sat2(id,j); id ideal, j polynomial |
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[d6db1f2] | 83 | RETURN: saturation of id with respect to j (= union_(k=1...) of id:j^k) |
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| 84 | NOTE: result is a std basis in the basering |
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[d2b2a7] | 85 | " |
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[d6db1f2] | 86 | { |
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[70ab73] | 87 | int @k,@i; |
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| 88 | def @P= basering; |
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| 89 | if(ordstr(basering)[1,2]!="dp") |
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| 90 | { |
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| 91 | execute("ring @Phelp=("+charstr(@P)+"),("+varstr(@P)+"),(C,dp);"); |
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| 92 | ideal inew=std(imap(@P,id)); |
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| 93 | ideal @h=imap(@P,h); |
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| 94 | } |
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| 95 | else |
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| 96 | { |
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| 97 | ideal @h=h; |
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| 98 | ideal inew=std(id); |
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| 99 | } |
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| 100 | ideal fac; |
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[d6db1f2] | 101 | |
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[70ab73] | 102 | for(@i=1;@i<=ncols(@h);@i++) |
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| 103 | { |
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| 104 | if(deg(@h[@i])>0) |
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| 105 | { |
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| 106 | fac=fac+factorize(@h[@i],1); |
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| 107 | } |
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| 108 | } |
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| 109 | fac=simplify(fac,6); |
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| 110 | poly @f=1; |
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| 111 | if(deg(fac[1])>0) |
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| 112 | { |
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| 113 | ideal iold; |
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| 114 | for(@i=1;@i<=size(fac);@i++) |
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| 115 | { |
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| 116 | @f=@f*fac[@i]; |
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| 117 | } |
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| 118 | intvec op = option(get); |
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| 119 | option(returnSB); |
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| 120 | while(specialIdealsEqual(iold,inew)==0 ) |
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| 121 | { |
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| 122 | iold=inew; |
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| 123 | if(deg(iold[size(iold)])!=1) |
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[d6db1f2] | 124 | { |
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[70ab73] | 125 | inew=quotient(iold,@f); |
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[d6db1f2] | 126 | } |
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[70ab73] | 127 | else |
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| 128 | { |
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| 129 | inew=iold; |
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| 130 | } |
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| 131 | @k++; |
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| 132 | } |
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| 133 | option(set,op); |
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| 134 | @k--; |
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| 135 | } |
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[d6db1f2] | 136 | |
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[70ab73] | 137 | if(ordstr(@P)[1,2]!="dp") |
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| 138 | { |
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| 139 | setring @P; |
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| 140 | ideal inew=std(imap(@Phelp,inew)); |
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| 141 | poly @f=imap(@Phelp,@f); |
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| 142 | } |
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| 143 | list L =inew,@f^@k; |
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| 144 | return (L); |
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[d6db1f2] | 145 | } |
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| 146 | |
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| 147 | /////////////////////////////////////////////////////////////////////////////// |
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| 148 | |
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[24f458] | 149 | |
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| 150 | proc minSat(ideal inew, ideal h) |
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[d6db1f2] | 151 | { |
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[70ab73] | 152 | int i,k; |
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| 153 | poly f=1; |
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| 154 | ideal iold,fac; |
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| 155 | list quotM,l; |
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[d6db1f2] | 156 | |
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[70ab73] | 157 | for(i=1;i<=ncols(h);i++) |
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| 158 | { |
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| 159 | if(deg(h[i])>0) |
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| 160 | { |
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| 161 | fac=fac+factorize(h[i],1); |
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| 162 | } |
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| 163 | } |
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| 164 | fac=simplify(fac,6); |
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| 165 | if(size(fac)==0) |
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| 166 | { |
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| 167 | l=inew,1; |
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| 168 | return(l); |
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| 169 | } |
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| 170 | fac=sort(fac)[1]; |
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| 171 | for(i=1;i<=size(fac);i++) |
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| 172 | { |
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| 173 | f=f*fac[i]; |
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| 174 | } |
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| 175 | quotM[1]=inew; |
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| 176 | quotM[2]=fac; |
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| 177 | quotM[3]=f; |
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| 178 | f=1; |
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| 179 | intvec op = option(get); |
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| 180 | option(returnSB); |
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| 181 | while(specialIdealsEqual(iold,quotM[1])==0) |
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| 182 | { |
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| 183 | if(k>0) |
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| 184 | { |
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| 185 | f=f*quotM[3]; |
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| 186 | } |
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| 187 | iold=quotM[1]; |
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| 188 | quotM=quotMin(quotM); |
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| 189 | k++; |
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| 190 | } |
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| 191 | option(set,op); |
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| 192 | l=quotM[1],f; |
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| 193 | return(l); |
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[18dd47] | 194 | } |
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[d6db1f2] | 195 | |
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[07c623] | 196 | static proc quotMin(list tsil) |
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[d6db1f2] | 197 | { |
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[70ab73] | 198 | int i,j,k,action; |
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| 199 | ideal verg; |
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| 200 | list l; |
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| 201 | poly g; |
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[d6db1f2] | 202 | |
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[70ab73] | 203 | ideal laedi=tsil[1]; |
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| 204 | ideal fac=tsil[2]; |
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| 205 | poly f=tsil[3]; |
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[3939bc] | 206 | |
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[70ab73] | 207 | ideal star=quotient(laedi,f); |
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[b1d1e8c] | 208 | |
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[70ab73] | 209 | if(specialIdealsEqual(star,laedi)) |
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| 210 | { |
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| 211 | l=star,fac,f; |
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| 212 | return(l); |
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| 213 | } |
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[b9b906] | 214 | |
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[70ab73] | 215 | action=1; |
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[18dd47] | 216 | |
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[70ab73] | 217 | while(action==1) |
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| 218 | { |
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| 219 | if(size(fac)==1) |
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| 220 | { |
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| 221 | action=0; |
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| 222 | break; |
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| 223 | } |
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| 224 | for(i=1;i<=size(fac);i++) |
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| 225 | { |
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| 226 | g=1; |
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| 227 | verg=laedi; |
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| 228 | for(j=1;j<=size(fac);j++) |
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[d6db1f2] | 229 | { |
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[70ab73] | 230 | if(i!=j) |
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| 231 | { |
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| 232 | g=g*fac[j]; |
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| 233 | } |
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[d6db1f2] | 234 | } |
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[70ab73] | 235 | verg=quotient(laedi,g); |
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[3939bc] | 236 | |
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[70ab73] | 237 | if(specialIdealsEqual(verg,star)==1) |
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| 238 | { |
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| 239 | f=g; |
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| 240 | fac[i]=0; |
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| 241 | fac=simplify(fac,2); |
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| 242 | break; |
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[d6db1f2] | 243 | } |
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[70ab73] | 244 | if(i==size(fac)) |
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| 245 | { |
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| 246 | action=0; |
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| 247 | } |
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| 248 | } |
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| 249 | } |
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| 250 | l=star,fac,f; |
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| 251 | return(l); |
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[d6db1f2] | 252 | } |
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| 253 | |
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[091424] | 254 | /////////////////////////////////////////////////////////////////////////////// |
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| 255 | |
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[07c623] | 256 | static proc testFactor(list act,poly p) |
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[d6db1f2] | 257 | { |
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[e801fe] | 258 | poly keep=p; |
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[3939bc] | 259 | |
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[70ab73] | 260 | int i; |
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| 261 | poly q=act[1][1]^act[2][1]; |
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| 262 | for(i=2;i<=size(act[1]);i++) |
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| 263 | { |
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| 264 | q=q*act[1][i]^act[2][i]; |
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| 265 | } |
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| 266 | q=1/leadcoef(q)*q; |
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| 267 | p=1/leadcoef(p)*p; |
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| 268 | if(p-q!=0) |
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| 269 | { |
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| 270 | "ERROR IN FACTOR, please inform the authors"; |
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| 271 | } |
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[d6db1f2] | 272 | } |
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[091424] | 273 | /////////////////////////////////////////////////////////////////////////////// |
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[d6db1f2] | 274 | |
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[07c623] | 275 | static proc factor(poly p) |
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[d2b2a7] | 276 | "USAGE: factor(p) p poly |
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[d6db1f2] | 277 | RETURN: list=; |
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[18dd47] | 278 | NOTE: |
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[d6db1f2] | 279 | EXAMPLE: example factor; shows an example |
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[d2b2a7] | 280 | " |
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[d6db1f2] | 281 | { |
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| 282 | ideal @i; |
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| 283 | list @l; |
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| 284 | intvec @v,@w; |
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| 285 | int @j,@k,@n; |
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| 286 | |
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| 287 | if(deg(p)<=1) |
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| 288 | { |
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[70ab73] | 289 | @i=ideal(p); |
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| 290 | @v=1; |
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| 291 | @l[1]=@i; |
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| 292 | @l[2]=@v; |
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| 293 | return(@l); |
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[d6db1f2] | 294 | } |
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| 295 | if (size(p)==1) |
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| 296 | { |
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[70ab73] | 297 | @w=leadexp(p); |
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| 298 | for(@j=1;@j<=nvars(basering);@j++) |
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| 299 | { |
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| 300 | if(@w[@j]!=0) |
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| 301 | { |
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| 302 | @k++; |
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| 303 | @v[@k]=@w[@j]; |
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| 304 | @i=@i+ideal(var(@j)); |
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| 305 | } |
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| 306 | } |
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| 307 | @l[1]=@i; |
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| 308 | @l[2]=@v; |
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| 309 | return(@l); |
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[d6db1f2] | 310 | } |
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[091424] | 311 | // @l=factorize(p,2); |
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[e801fe] | 312 | @l=factorize(p); |
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| 313 | // if(npars(basering)>0) |
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| 314 | // { |
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[70ab73] | 315 | for(@j=1;@j<=size(@l[1]);@j++) |
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| 316 | { |
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| 317 | if(deg(@l[1][@j])==0) |
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| 318 | { |
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| 319 | @n++; |
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| 320 | } |
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| 321 | } |
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| 322 | if(@n>0) |
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| 323 | { |
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| 324 | if(@n==size(@l[1])) |
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| 325 | { |
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| 326 | @l[1]=ideal(1); |
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| 327 | @v=1; |
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| 328 | @l[2]=@v; |
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| 329 | } |
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| 330 | else |
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| 331 | { |
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| 332 | @k=0; |
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| 333 | int pleh; |
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| 334 | for(@j=1;@j<=size(@l[1]);@j++) |
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[d6db1f2] | 335 | { |
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[70ab73] | 336 | if(deg(@l[1][@j])!=0) |
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| 337 | { |
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| 338 | @k++; |
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| 339 | @i=@i+ideal(@l[1][@j]); |
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| 340 | if(size(@i)==pleh) |
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| 341 | { |
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| 342 | "//factorization error"; |
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| 343 | @l; |
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| 344 | @k--; |
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| 345 | @v[@k]=@v[@k]+@l[2][@j]; |
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| 346 | } |
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| 347 | else |
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| 348 | { |
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| 349 | pleh++; |
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| 350 | @v[@k]=@l[2][@j]; |
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| 351 | } |
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| 352 | } |
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[d6db1f2] | 353 | } |
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[70ab73] | 354 | @l[1]=@i; |
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| 355 | @l[2]=@v; |
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| 356 | } |
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| 357 | } |
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| 358 | // } |
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[d6db1f2] | 359 | return(@l); |
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| 360 | } |
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| 361 | example |
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| 362 | { "EXAMPLE:"; echo = 2; |
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| 363 | ring r = 0,(x,y,z),lp; |
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| 364 | poly p = (x+y)^2*(y-z)^3; |
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| 365 | list l = factor(p); |
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| 366 | l; |
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| 367 | ring r1 =(0,b,d,f,g),(a,c,e),lp; |
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| 368 | poly p =(1*d)*e^2+(1*d*f^2*g); |
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| 369 | list l = factor(p); |
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| 370 | l; |
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| 371 | ring r2 =(0,b,f,g),(a,c,e,d),lp; |
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| 372 | poly p =(1*d)*e^2+(1*d*f^2*g); |
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| 373 | list l = factor(p); |
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| 374 | l; |
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| 375 | } |
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| 376 | |
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[091424] | 377 | /////////////////////////////////////////////////////////////////////////////// |
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[d6db1f2] | 378 | |
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[50cbdc] | 379 | proc idealsEqual( ideal k, ideal j) |
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[18dd47] | 380 | { |
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[70ab73] | 381 | return(stdIdealsEqual(std(k),std(j))); |
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[d6db1f2] | 382 | } |
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| 383 | |
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[07c623] | 384 | static proc specialIdealsEqual( ideal k1, ideal k2) |
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[d6db1f2] | 385 | { |
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[70ab73] | 386 | int j; |
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[d6db1f2] | 387 | |
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[70ab73] | 388 | if(size(k1)==size(k2)) |
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| 389 | { |
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| 390 | for(j=1;j<=size(k1);j++) |
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| 391 | { |
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| 392 | if(leadexp(k1[j])!=leadexp(k2[j])) |
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[d6db1f2] | 393 | { |
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[70ab73] | 394 | return(0); |
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[d6db1f2] | 395 | } |
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[70ab73] | 396 | } |
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| 397 | return(1); |
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| 398 | } |
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| 399 | return(0); |
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[d6db1f2] | 400 | } |
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| 401 | |
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[07c623] | 402 | static proc stdIdealsEqual( ideal k1, ideal k2) |
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[d6db1f2] | 403 | { |
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[70ab73] | 404 | int j; |
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[d6db1f2] | 405 | |
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[70ab73] | 406 | if(size(k1)==size(k2)) |
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| 407 | { |
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| 408 | for(j=1;j<=size(k1);j++) |
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| 409 | { |
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| 410 | if(leadexp(k1[j])!=leadexp(k2[j])) |
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[d6db1f2] | 411 | { |
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[70ab73] | 412 | return(0); |
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[d6db1f2] | 413 | } |
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[70ab73] | 414 | } |
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| 415 | attrib(k2,"isSB",1); |
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| 416 | if(size(reduce(k1,k2,1))==0) |
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| 417 | { |
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| 418 | return(1); |
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| 419 | } |
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| 420 | } |
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| 421 | return(0); |
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[d6db1f2] | 422 | } |
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[091424] | 423 | /////////////////////////////////////////////////////////////////////////////// |
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[d6db1f2] | 424 | |
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[50cbdc] | 425 | proc primaryTest (ideal i, poly p) |
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[d6db1f2] | 426 | { |
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[70ab73] | 427 | int m=1; |
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| 428 | int n=nvars(basering); |
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| 429 | int e,f; |
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| 430 | poly t; |
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| 431 | ideal h; |
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| 432 | list act; |
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[d6db1f2] | 433 | |
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[70ab73] | 434 | ideal prm=p; |
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| 435 | attrib(prm,"isSB",1); |
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[d6db1f2] | 436 | |
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[70ab73] | 437 | while (n>1) |
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| 438 | { |
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| 439 | n--; |
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| 440 | m++; |
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[d6db1f2] | 441 | |
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[70ab73] | 442 | //search for i[m] which has a power of var(n) as leading term |
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| 443 | if (n==1) |
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| 444 | { |
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| 445 | m=size(i); |
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| 446 | } |
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| 447 | else |
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| 448 | { |
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| 449 | while (lead(i[m])/var(n-1)==0) |
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[d6db1f2] | 450 | { |
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[70ab73] | 451 | m++; |
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[d6db1f2] | 452 | } |
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[70ab73] | 453 | m--; |
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| 454 | } |
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| 455 | //check whether i[m] =(c*var(n)+h)^e modulo prm for some |
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| 456 | //h in K[var(n+1),...,var(nvars(basering))], c in K |
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| 457 | //if not (0) is returned, else var(n)+h is added to prm |
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| 458 | |
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| 459 | e=deg(lead(i[m])); |
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| 460 | if(char(basering)!=0) |
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| 461 | { |
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| 462 | f=1; |
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| 463 | if(e mod char(basering)==0) |
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[d6db1f2] | 464 | { |
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[70ab73] | 465 | if ( voice >=15 ) |
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[d6db1f2] | 466 | { |
---|
[70ab73] | 467 | "// WARNING: The characteristic is perhaps too small to use"; |
---|
| 468 | "// the algorithm of Gianni/Trager/Zacharias."; |
---|
| 469 | "// This may result in an infinte loop"; |
---|
| 470 | "// loop in primaryTest, voice:",voice;""; |
---|
| 471 | } |
---|
| 472 | while(e mod char(basering)==0) |
---|
| 473 | { |
---|
| 474 | f=f*char(basering); |
---|
| 475 | e=e/char(basering); |
---|
[a3432c] | 476 | } |
---|
[971ba6f] | 477 | } |
---|
[70ab73] | 478 | t=leadcoef(i[m])*e*var(n)^f+(i[m]-lead(i[m]))/var(n)^((e-1)*f); |
---|
| 479 | i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m]; |
---|
| 480 | if (reduce(i[m]-t^e,prm,1) !=0) |
---|
| 481 | { |
---|
| 482 | return(ideal(0)); |
---|
| 483 | } |
---|
| 484 | if(f>1) |
---|
| 485 | { |
---|
| 486 | act=factorize(t); |
---|
| 487 | if(size(act[1])>2) |
---|
| 488 | { |
---|
| 489 | return(ideal(0)); |
---|
| 490 | } |
---|
| 491 | if(deg(act[1][2])>1) |
---|
| 492 | { |
---|
| 493 | return(ideal(0)); |
---|
| 494 | } |
---|
| 495 | t=act[1][2]; |
---|
| 496 | } |
---|
| 497 | } |
---|
| 498 | else |
---|
| 499 | { |
---|
| 500 | t=leadcoef(i[m])*e*var(n)+(i[m]-lead(i[m]))/var(n)^(e-1); |
---|
| 501 | i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m]; |
---|
| 502 | if (reduce(i[m]-t^e,prm,1) !=0) |
---|
[a3432c] | 503 | { |
---|
[70ab73] | 504 | return(ideal(0)); |
---|
[a3432c] | 505 | } |
---|
[70ab73] | 506 | } |
---|
[6ffa84] | 507 | |
---|
[70ab73] | 508 | h=interred(t); |
---|
| 509 | t=h[1]; |
---|
[d6db1f2] | 510 | |
---|
[70ab73] | 511 | prm = prm,t; |
---|
| 512 | attrib(prm,"isSB",1); |
---|
| 513 | } |
---|
| 514 | return(prm); |
---|
[d6db1f2] | 515 | } |
---|
| 516 | |
---|
[d12f079] | 517 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 518 | proc gcdTest(ideal act) |
---|
| 519 | { |
---|
| 520 | int i,j; |
---|
| 521 | if(size(act)<=1) |
---|
| 522 | { |
---|
[70ab73] | 523 | return(0); |
---|
[d12f079] | 524 | } |
---|
| 525 | for (i=1;i<=size(act)-1;i++) |
---|
| 526 | { |
---|
[70ab73] | 527 | for(j=i+1;j<=size(act);j++) |
---|
| 528 | { |
---|
| 529 | if(deg(std(ideal(act[i],act[j]))[1])>0) |
---|
| 530 | { |
---|
| 531 | return(0); |
---|
| 532 | } |
---|
| 533 | } |
---|
[d12f079] | 534 | } |
---|
| 535 | return(1); |
---|
| 536 | } |
---|
[d6db1f2] | 537 | |
---|
| 538 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 539 | static proc splitPrimary(list l,ideal ser,int @wr,list sact) |
---|
[d6db1f2] | 540 | { |
---|
[70ab73] | 541 | int i,j,k,s,r,w; |
---|
| 542 | list keepresult,act,keepprime; |
---|
| 543 | poly @f; |
---|
| 544 | int sl=size(l); |
---|
| 545 | for(i=1;i<=sl/2;i++) |
---|
| 546 | { |
---|
| 547 | if(sact[2][i]>1) |
---|
| 548 | { |
---|
| 549 | keepprime[i]=l[2*i-1]+ideal(sact[1][i]); |
---|
| 550 | } |
---|
| 551 | else |
---|
| 552 | { |
---|
| 553 | keepprime[i]=l[2*i-1]; |
---|
| 554 | } |
---|
| 555 | } |
---|
| 556 | i=0; |
---|
| 557 | while(i<size(l)/2) |
---|
| 558 | { |
---|
| 559 | i++; |
---|
| 560 | if((size(ser)>0)&&(size(reduce(ser,l[2*i-1],1))==0)) |
---|
| 561 | { |
---|
| 562 | l[2*i-1]=ideal(1); |
---|
| 563 | l[2*i]=ideal(1); |
---|
| 564 | continue; |
---|
| 565 | } |
---|
| 566 | |
---|
| 567 | if(size(l[2*i])==0) |
---|
| 568 | { |
---|
| 569 | if(homog(l[2*i-1])==1) |
---|
[d6db1f2] | 570 | { |
---|
[70ab73] | 571 | l[2*i]=maxideal(1); |
---|
| 572 | continue; |
---|
[d6db1f2] | 573 | } |
---|
[70ab73] | 574 | j=0; |
---|
| 575 | /* |
---|
| 576 | if(i<=sl/2) |
---|
[d6db1f2] | 577 | { |
---|
[70ab73] | 578 | j=1; |
---|
[d6db1f2] | 579 | } |
---|
[70ab73] | 580 | */ |
---|
| 581 | while(j<size(l[2*i-1])) |
---|
[d6db1f2] | 582 | { |
---|
[70ab73] | 583 | j++; |
---|
| 584 | act=factor(l[2*i-1][j]); |
---|
| 585 | r=size(act[1]); |
---|
| 586 | attrib(l[2*i-1],"isSB",1); |
---|
| 587 | if((r==1)&&(vdim(l[2*i-1])==deg(l[2*i-1][j]))) |
---|
| 588 | { |
---|
| 589 | l[2*i]=std(l[2*i-1],act[1][1]); |
---|
| 590 | break; |
---|
| 591 | } |
---|
| 592 | if((r==1)&&(act[2][1]>1)) |
---|
| 593 | { |
---|
| 594 | keepprime[i]=interred(keepprime[i]+ideal(act[1][1])); |
---|
| 595 | if(homog(keepprime[i])==1) |
---|
| 596 | { |
---|
[d6db1f2] | 597 | l[2*i]=maxideal(1); |
---|
[70ab73] | 598 | break; |
---|
| 599 | } |
---|
| 600 | } |
---|
| 601 | if(gcdTest(act[1])==1) |
---|
| 602 | { |
---|
| 603 | for(k=2;k<=r;k++) |
---|
| 604 | { |
---|
| 605 | keepprime[size(l)/2+k-1]=interred(keepprime[i]+ideal(act[1][k])); |
---|
| 606 | } |
---|
| 607 | keepprime[i]=interred(keepprime[i]+ideal(act[1][1])); |
---|
| 608 | for(k=1;k<=r;k++) |
---|
| 609 | { |
---|
| 610 | if(@wr==0) |
---|
| 611 | { |
---|
| 612 | keepresult[k]=std(l[2*i-1],act[1][k]^act[2][k]); |
---|
| 613 | } |
---|
| 614 | else |
---|
| 615 | { |
---|
| 616 | keepresult[k]=std(l[2*i-1],act[1][k]); |
---|
| 617 | } |
---|
| 618 | } |
---|
| 619 | l[2*i-1]=keepresult[1]; |
---|
| 620 | if(vdim(keepresult[1])==deg(act[1][1])) |
---|
| 621 | { |
---|
| 622 | l[2*i]=keepresult[1]; |
---|
| 623 | } |
---|
| 624 | if((homog(keepresult[1])==1)||(homog(keepprime[i])==1)) |
---|
| 625 | { |
---|
| 626 | l[2*i]=maxideal(1); |
---|
| 627 | } |
---|
| 628 | s=size(l)-2; |
---|
| 629 | for(k=2;k<=r;k++) |
---|
| 630 | { |
---|
| 631 | l[s+2*k-1]=keepresult[k]; |
---|
| 632 | keepprime[s/2+k]=interred(keepresult[k]+ideal(act[1][k])); |
---|
| 633 | if(vdim(keepresult[k])==deg(act[1][k])) |
---|
| 634 | { |
---|
| 635 | l[s+2*k]=keepresult[k]; |
---|
| 636 | } |
---|
| 637 | else |
---|
| 638 | { |
---|
| 639 | l[s+2*k]=ideal(0); |
---|
| 640 | } |
---|
| 641 | if((homog(keepresult[k])==1)||(homog(keepprime[s/2+k])==1)) |
---|
| 642 | { |
---|
| 643 | l[s+2*k]=maxideal(1); |
---|
| 644 | } |
---|
| 645 | } |
---|
| 646 | i--; |
---|
| 647 | break; |
---|
| 648 | } |
---|
| 649 | if(r>=2) |
---|
| 650 | { |
---|
| 651 | s=size(l); |
---|
| 652 | @f=act[1][1]; |
---|
| 653 | act=sat1(l[2*i-1],act[1][1]); |
---|
| 654 | if(deg(act[1][1])>0) |
---|
| 655 | { |
---|
| 656 | l[s+1]=std(l[2*i-1],act[2]); |
---|
| 657 | if(homog(l[s+1])==1) |
---|
| 658 | { |
---|
| 659 | l[s+2]=maxideal(1); |
---|
| 660 | } |
---|
| 661 | else |
---|
| 662 | { |
---|
| 663 | l[s+2]=ideal(0); |
---|
[d6db1f2] | 664 | } |
---|
[70ab73] | 665 | keepprime[s/2+1]=interred(keepprime[i]+ideal(@f)); |
---|
| 666 | if(homog(keepprime[s/2+1])==1) |
---|
[18dd47] | 667 | { |
---|
[70ab73] | 668 | l[s+2]=maxideal(1); |
---|
[d6db1f2] | 669 | } |
---|
[70ab73] | 670 | keepprime[i]=act[1]; |
---|
| 671 | l[2*i-1]=act[1]; |
---|
| 672 | attrib(l[2*i-1],"isSB",1); |
---|
| 673 | if(homog(l[2*i-1])==1) |
---|
[d6db1f2] | 674 | { |
---|
[70ab73] | 675 | l[2*i]=maxideal(1); |
---|
[d6db1f2] | 676 | } |
---|
[70ab73] | 677 | i--; |
---|
| 678 | break; |
---|
| 679 | } |
---|
| 680 | } |
---|
[d6db1f2] | 681 | } |
---|
[70ab73] | 682 | } |
---|
| 683 | } |
---|
| 684 | if(sl==size(l)) |
---|
| 685 | { |
---|
| 686 | return(l); |
---|
| 687 | } |
---|
| 688 | for(i=1;i<=size(l)/2;i++) |
---|
| 689 | { |
---|
| 690 | attrib(l[2*i-1],"isSB",1); |
---|
[3939bc] | 691 | |
---|
[70ab73] | 692 | if((size(ser)>0)&&(size(reduce(ser,l[2*i-1],1))==0)&&(deg(l[2*i-1][1])>0)) |
---|
| 693 | { |
---|
| 694 | "Achtung in split"; |
---|
[3939bc] | 695 | |
---|
[70ab73] | 696 | l[2*i-1]=ideal(1); |
---|
| 697 | l[2*i]=ideal(1); |
---|
| 698 | } |
---|
| 699 | if((size(l[2*i])==0)&&(specialIdealsEqual(keepprime[i],l[2*i-1])!=1)) |
---|
| 700 | { |
---|
| 701 | keepprime[i]=std(keepprime[i]); |
---|
| 702 | if(homog(keepprime[i])==1) |
---|
| 703 | { |
---|
| 704 | l[2*i]=maxideal(1); |
---|
[d6db1f2] | 705 | } |
---|
[70ab73] | 706 | else |
---|
| 707 | { |
---|
| 708 | act=zero_decomp(keepprime[i],ideal(0),@wr,1); |
---|
| 709 | if(size(act)==2) |
---|
| 710 | { |
---|
| 711 | l[2*i]=act[2]; |
---|
| 712 | } |
---|
| 713 | } |
---|
| 714 | } |
---|
| 715 | } |
---|
| 716 | return(l); |
---|
[d6db1f2] | 717 | } |
---|
| 718 | example |
---|
| 719 | { "EXAMPLE:"; echo=2; |
---|
| 720 | ring r = 32003,(x,y,z),lp; |
---|
| 721 | ideal i1=x*(x+1),yz,(z+1)*(z-1); |
---|
| 722 | ideal i2=xy,yz,(x-2)*(x+3); |
---|
| 723 | list l=i1,ideal(0),i2,ideal(0),i2,ideal(1); |
---|
| 724 | list l1=splitPrimary(l,ideal(0),0); |
---|
| 725 | l1; |
---|
| 726 | } |
---|
[651953] | 727 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 728 | static proc splitCharp(list l) |
---|
[651953] | 729 | { |
---|
| 730 | if((char(basering)==0)||(npars(basering)>0)) |
---|
| 731 | { |
---|
[70ab73] | 732 | return(l); |
---|
[651953] | 733 | } |
---|
| 734 | def P=basering; |
---|
[24f458] | 735 | int i,j,k,m,q,d,o; |
---|
[651953] | 736 | int n=nvars(basering); |
---|
| 737 | ideal s,t,u,sact; |
---|
| 738 | poly ni; |
---|
| 739 | string minp,gnir,va; |
---|
[24f458] | 740 | list sa,keep,rp,keep1; |
---|
[651953] | 741 | for(i=1;i<=size(l)/2;i++) |
---|
| 742 | { |
---|
| 743 | if(size(l[2*i])==0) |
---|
| 744 | { |
---|
[70ab73] | 745 | if(deg(l[2*i-1][1])==vdim(l[2*i-1])) |
---|
| 746 | { |
---|
| 747 | l[2*i]=l[2*i-1]; |
---|
| 748 | } |
---|
[651953] | 749 | } |
---|
| 750 | } |
---|
| 751 | for(i=1;i<=size(l)/2;i++) |
---|
| 752 | { |
---|
| 753 | if(size(l[2*i])==0) |
---|
| 754 | { |
---|
[24f458] | 755 | s=factorize(l[2*i-1][1],1); //vermeiden!!! |
---|
[651953] | 756 | t=l[2*i-1]; |
---|
| 757 | m=size(t); |
---|
| 758 | ni=s[1]; |
---|
| 759 | if(deg(ni)>1) |
---|
| 760 | { |
---|
| 761 | va=varstr(P); |
---|
| 762 | j=size(va); |
---|
| 763 | while(va[j]!=","){j--;} |
---|
| 764 | va=va[1..j-1]; |
---|
[24f458] | 765 | gnir="ring RL=("+string(char(P))+","+string(var(n))+"),("+va+"),lp;"; |
---|
[651953] | 766 | execute(gnir); |
---|
| 767 | minpoly=leadcoef(imap(P,ni)); |
---|
| 768 | ideal act; |
---|
| 769 | ideal t=imap(P,t); |
---|
[24f458] | 770 | |
---|
[651953] | 771 | for(k=2;k<=m;k++) |
---|
[b9b906] | 772 | { |
---|
[70ab73] | 773 | act=factorize(t[k],1); |
---|
| 774 | if(size(act)>1){break;} |
---|
[651953] | 775 | } |
---|
| 776 | setring P; |
---|
| 777 | sact=imap(RL,act); |
---|
[24f458] | 778 | |
---|
[651953] | 779 | if(size(sact)>1) |
---|
| 780 | { |
---|
[70ab73] | 781 | sa=sat1(l[2*i-1],sact[1]); |
---|
| 782 | keep[size(keep)+1]=std(l[2*i-1],sa[2]); |
---|
| 783 | l[2*i-1]=std(sa[1]); |
---|
| 784 | l[2*i]=primaryTest(sa[1],sa[1][1]); |
---|
[651953] | 785 | } |
---|
[24f458] | 786 | if((size(sact)==1)&&(m==2)) |
---|
| 787 | { |
---|
[70ab73] | 788 | l[2*i]=l[2*i-1]; |
---|
| 789 | attrib(l[2*i],"isSB",1); |
---|
[24f458] | 790 | } |
---|
| 791 | if((size(sact)==1)&&(m>2)) |
---|
| 792 | { |
---|
[70ab73] | 793 | setring RL; |
---|
| 794 | option(redSB); |
---|
| 795 | t=std(t); |
---|
| 796 | |
---|
| 797 | list sp=zero_decomp(t,0,0); |
---|
| 798 | |
---|
| 799 | setring P; |
---|
| 800 | rp=imap(RL,sp); |
---|
| 801 | for(o=1;o<=size(rp);o++) |
---|
| 802 | { |
---|
| 803 | rp[o]=interred(simplify(rp[o],1)+ideal(ni)); |
---|
| 804 | } |
---|
| 805 | l[2*i-1]=rp[1]; |
---|
| 806 | l[2*i]=rp[2]; |
---|
| 807 | rp=delete(rp,1); |
---|
| 808 | rp=delete(rp,1); |
---|
| 809 | keep1=keep1+rp; |
---|
| 810 | option(noredSB); |
---|
[24f458] | 811 | } |
---|
| 812 | kill RL; |
---|
[651953] | 813 | } |
---|
| 814 | } |
---|
| 815 | } |
---|
| 816 | if(size(keep)>0) |
---|
| 817 | { |
---|
| 818 | for(i=1;i<=size(keep);i++) |
---|
| 819 | { |
---|
[70ab73] | 820 | if(deg(keep[i][1])>0) |
---|
| 821 | { |
---|
| 822 | l[size(l)+1]=keep[i]; |
---|
| 823 | l[size(l)+1]=primaryTest(keep[i],keep[i][1]); |
---|
| 824 | } |
---|
[651953] | 825 | } |
---|
| 826 | } |
---|
[24f458] | 827 | l=l+keep1; |
---|
[651953] | 828 | return(l); |
---|
| 829 | } |
---|
[d6db1f2] | 830 | |
---|
[091424] | 831 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d6db1f2] | 832 | |
---|
[24f458] | 833 | proc zero_decomp (ideal j,ideal ser,int @wr,list #) |
---|
[d2b2a7] | 834 | "USAGE: zero_decomp(j,ser,@wr); j,ser ideals, @wr=0 or 1 |
---|
[7f7c25e] | 835 | (@wr=0 for primary decomposition, @wr=1 for computation of associated |
---|
[d6db1f2] | 836 | primes) |
---|
| 837 | RETURN: list = list of primary ideals and their radicals (at even positions |
---|
| 838 | in the list) if the input is zero-dimensional and a standardbases |
---|
| 839 | with respect to lex-ordering |
---|
| 840 | If ser!=(0) and ser is contained in j or if j is not zero-dimen- |
---|
| 841 | sional then ideal(1),ideal(1) is returned |
---|
[7b3971] | 842 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
[d6db1f2] | 843 | EXAMPLE: example zero_decomp; shows an example |
---|
[d2b2a7] | 844 | " |
---|
[d6db1f2] | 845 | { |
---|
| 846 | def @P = basering; |
---|
[20057b] | 847 | int uytrewq; |
---|
[d6db1f2] | 848 | int nva = nvars(basering); |
---|
[e801fe] | 849 | int @k,@s,@n,@k1,zz; |
---|
[a39a07] | 850 | list primary,lres0,lres1,act,@lh,@wh; |
---|
[e801fe] | 851 | map phi,psi,phi1,psi1; |
---|
| 852 | ideal jmap,jmap1,jmap2,helpprim,@qh,@qht,ser1; |
---|
[d6db1f2] | 853 | intvec @vh,@hilb; |
---|
| 854 | string @ri; |
---|
| 855 | poly @f; |
---|
| 856 | if (dim(j)>0) |
---|
| 857 | { |
---|
[70ab73] | 858 | primary[1]=ideal(1); |
---|
| 859 | primary[2]=ideal(1); |
---|
| 860 | return(primary); |
---|
[d6db1f2] | 861 | } |
---|
[a90eb0] | 862 | intvec save=option(get); |
---|
| 863 | option(redSB); |
---|
[3939bc] | 864 | j=interred(j); |
---|
[0bcebab] | 865 | |
---|
[d6db1f2] | 866 | attrib(j,"isSB",1); |
---|
[24f458] | 867 | |
---|
[d6db1f2] | 868 | if(vdim(j)==deg(j[1])) |
---|
[3939bc] | 869 | { |
---|
[70ab73] | 870 | act=factor(j[1]); |
---|
| 871 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 872 | { |
---|
| 873 | @qh=j; |
---|
| 874 | if(@wr==0) |
---|
| 875 | { |
---|
| 876 | @qh[1]=act[1][@k]^act[2][@k]; |
---|
| 877 | } |
---|
| 878 | else |
---|
| 879 | { |
---|
| 880 | @qh[1]=act[1][@k]; |
---|
| 881 | } |
---|
| 882 | primary[2*@k-1]=interred(@qh); |
---|
| 883 | @qh=j; |
---|
| 884 | @qh[1]=act[1][@k]; |
---|
| 885 | primary[2*@k]=interred(@qh); |
---|
| 886 | attrib( primary[2*@k-1],"isSB",1); |
---|
[3939bc] | 887 | |
---|
[70ab73] | 888 | if((size(ser)>0)&&(size(reduce(ser,primary[2*@k-1],1))==0)) |
---|
| 889 | { |
---|
| 890 | primary[2*@k-1]=ideal(1); |
---|
| 891 | primary[2*@k]=ideal(1); |
---|
| 892 | } |
---|
| 893 | } |
---|
[a90eb0] | 894 | option(set,save); |
---|
[70ab73] | 895 | return(primary); |
---|
[d6db1f2] | 896 | } |
---|
| 897 | |
---|
[a90eb0] | 898 | option(set,save); |
---|
[d6db1f2] | 899 | if(homog(j)==1) |
---|
| 900 | { |
---|
[70ab73] | 901 | primary[1]=j; |
---|
| 902 | if((size(ser)>0)&&(size(reduce(ser,j,1))==0)) |
---|
| 903 | { |
---|
| 904 | primary[1]=ideal(1); |
---|
| 905 | primary[2]=ideal(1); |
---|
| 906 | return(primary); |
---|
| 907 | } |
---|
| 908 | if(dim(j)==-1) |
---|
| 909 | { |
---|
| 910 | primary[1]=ideal(1); |
---|
| 911 | primary[2]=ideal(1); |
---|
| 912 | } |
---|
| 913 | else |
---|
| 914 | { |
---|
| 915 | primary[2]=maxideal(1); |
---|
| 916 | } |
---|
| 917 | return(primary); |
---|
[d6db1f2] | 918 | } |
---|
[18dd47] | 919 | |
---|
[d6db1f2] | 920 | //the first element in the standardbase is factorized |
---|
| 921 | if(deg(j[1])>0) |
---|
| 922 | { |
---|
| 923 | act=factor(j[1]); |
---|
| 924 | testFactor(act,j[1]); |
---|
| 925 | } |
---|
| 926 | else |
---|
| 927 | { |
---|
[70ab73] | 928 | primary[1]=ideal(1); |
---|
| 929 | primary[2]=ideal(1); |
---|
| 930 | return(primary); |
---|
[d6db1f2] | 931 | } |
---|
| 932 | |
---|
[9050ca] | 933 | //with the factors new ideals (hopefully the primary decomposition) |
---|
[d6db1f2] | 934 | //are created |
---|
| 935 | if(size(act[1])>1) |
---|
| 936 | { |
---|
[70ab73] | 937 | if(size(#)>1) |
---|
| 938 | { |
---|
| 939 | primary[1]=ideal(1); |
---|
| 940 | primary[2]=ideal(1); |
---|
| 941 | primary[3]=ideal(1); |
---|
| 942 | primary[4]=ideal(1); |
---|
| 943 | return(primary); |
---|
| 944 | } |
---|
| 945 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 946 | { |
---|
| 947 | if(@wr==0) |
---|
| 948 | { |
---|
| 949 | primary[2*@k-1]=std(j,act[1][@k]^act[2][@k]); |
---|
| 950 | } |
---|
| 951 | else |
---|
| 952 | { |
---|
| 953 | primary[2*@k-1]=std(j,act[1][@k]); |
---|
| 954 | } |
---|
| 955 | if((act[2][@k]==1)&&(vdim(primary[2*@k-1])==deg(act[1][@k]))) |
---|
| 956 | { |
---|
[a36e78] | 957 | primary[2*@k] = primary[2*@k-1]; |
---|
[70ab73] | 958 | } |
---|
| 959 | else |
---|
| 960 | { |
---|
| 961 | primary[2*@k] = primaryTest(primary[2*@k-1],act[1][@k]); |
---|
| 962 | } |
---|
| 963 | } |
---|
[d6db1f2] | 964 | } |
---|
| 965 | else |
---|
[3939bc] | 966 | { |
---|
[70ab73] | 967 | primary[1]=j; |
---|
| 968 | if((size(#)>0)&&(act[2][1]>1)) |
---|
| 969 | { |
---|
| 970 | act[2]=1; |
---|
| 971 | primary[1]=std(primary[1],act[1][1]); |
---|
| 972 | } |
---|
| 973 | if(@wr!=0) |
---|
| 974 | { |
---|
| 975 | primary[1]=std(j,act[1][1]); |
---|
| 976 | } |
---|
| 977 | if((act[2][1]==1)&&(vdim(primary[1])==deg(act[1][1]))) |
---|
| 978 | { |
---|
| 979 | primary[2]=primary[1]; |
---|
| 980 | } |
---|
| 981 | else |
---|
| 982 | { |
---|
| 983 | primary[2]=primaryTest(primary[1],act[1][1]); |
---|
| 984 | } |
---|
[d6db1f2] | 985 | } |
---|
[50cbdc] | 986 | |
---|
[d6db1f2] | 987 | if(size(#)==0) |
---|
| 988 | { |
---|
[70ab73] | 989 | primary=splitPrimary(primary,ser,@wr,act); |
---|
[d6db1f2] | 990 | } |
---|
[24f458] | 991 | |
---|
| 992 | if((voice>=6)&&(char(basering)<=181)) |
---|
| 993 | { |
---|
[70ab73] | 994 | primary=splitCharp(primary); |
---|
[24f458] | 995 | } |
---|
| 996 | |
---|
| 997 | if((@wr==2)&&(npars(basering)>0)&&(voice>=6)&&(char(basering)>0)) |
---|
| 998 | { |
---|
| 999 | //the prime decomposition of Yokoyama in characteristic p |
---|
[70ab73] | 1000 | list ke,ek; |
---|
| 1001 | @k=0; |
---|
| 1002 | while(@k<size(primary)/2) |
---|
| 1003 | { |
---|
| 1004 | @k++; |
---|
| 1005 | if(size(primary[2*@k])==0) |
---|
| 1006 | { |
---|
| 1007 | ek=insepDecomp(primary[2*@k-1]); |
---|
| 1008 | primary=delete(primary,2*@k); |
---|
| 1009 | primary=delete(primary,2*@k-1); |
---|
| 1010 | @k--; |
---|
| 1011 | } |
---|
| 1012 | ke=ke+ek; |
---|
| 1013 | } |
---|
| 1014 | for(@k=1;@k<=size(ke);@k++) |
---|
| 1015 | { |
---|
| 1016 | primary[size(primary)+1]=ke[@k]; |
---|
| 1017 | primary[size(primary)+1]=ke[@k]; |
---|
| 1018 | } |
---|
[24f458] | 1019 | } |
---|
| 1020 | |
---|
[0266ac] | 1021 | if(voice>=8){primary=extF(primary);}; |
---|
[24f458] | 1022 | |
---|
[d6db1f2] | 1023 | //test whether all ideals in the decomposition are primary and |
---|
| 1024 | //in general position |
---|
| 1025 | //if not after a random coordinate transformation of the last |
---|
| 1026 | //variable the corresponding ideal is decomposed again. |
---|
[24f458] | 1027 | if((npars(basering)>0)&&(voice>=8)) |
---|
| 1028 | { |
---|
[70ab73] | 1029 | poly randp; |
---|
| 1030 | for(zz=1;zz<nvars(basering);zz++) |
---|
| 1031 | { |
---|
| 1032 | randp=randp |
---|
[24f458] | 1033 | +(random(0,5)*par(1)^2+random(0,5)*par(1)+random(0,5))*var(zz); |
---|
[70ab73] | 1034 | } |
---|
| 1035 | randp=randp+var(nvars(basering)); |
---|
[24f458] | 1036 | } |
---|
[d6db1f2] | 1037 | @k=0; |
---|
[67bd4c] | 1038 | while(@k<(size(primary)/2)) |
---|
[d6db1f2] | 1039 | { |
---|
| 1040 | @k++; |
---|
| 1041 | if (size(primary[2*@k])==0) |
---|
| 1042 | { |
---|
[70ab73] | 1043 | for(zz=1;zz<size(primary[2*@k-1])-1;zz++) |
---|
| 1044 | { |
---|
| 1045 | attrib(primary[2*@k-1],"isSB",1); |
---|
| 1046 | if(vdim(primary[2*@k-1])==deg(primary[2*@k-1][zz])) |
---|
| 1047 | { |
---|
| 1048 | primary[2*@k]=primary[2*@k-1]; |
---|
| 1049 | } |
---|
| 1050 | } |
---|
[67bd4c] | 1051 | } |
---|
| 1052 | } |
---|
[3939bc] | 1053 | |
---|
[67bd4c] | 1054 | @k=0; |
---|
[e801fe] | 1055 | ideal keep; |
---|
[67bd4c] | 1056 | while(@k<(size(primary)/2)) |
---|
| 1057 | { |
---|
| 1058 | @k++; |
---|
| 1059 | if (size(primary[2*@k])==0) |
---|
| 1060 | { |
---|
[70ab73] | 1061 | jmap=randomLast(100); |
---|
| 1062 | jmap1=maxideal(1); |
---|
| 1063 | jmap2=maxideal(1); |
---|
| 1064 | @qht=primary[2*@k-1]; |
---|
| 1065 | if((npars(basering)>0)&&(voice>=10)) |
---|
| 1066 | { |
---|
| 1067 | jmap[size(jmap)]=randp; |
---|
| 1068 | } |
---|
[67bd4c] | 1069 | |
---|
[70ab73] | 1070 | for(@n=2;@n<=size(primary[2*@k-1]);@n++) |
---|
| 1071 | { |
---|
| 1072 | if(deg(lead(primary[2*@k-1][@n]))==1) |
---|
| 1073 | { |
---|
| 1074 | for(zz=1;zz<=nva;zz++) |
---|
[d6db1f2] | 1075 | { |
---|
[70ab73] | 1076 | if(lead(primary[2*@k-1][@n])/var(zz)!=0) |
---|
| 1077 | { |
---|
| 1078 | jmap1[zz]=-1/leadcoef(primary[2*@k-1][@n])*primary[2*@k-1][@n] |
---|
[a36e78] | 1079 | +2/leadcoef(primary[2*@k-1][@n])*lead(primary[2*@k-1][@n]); |
---|
[70ab73] | 1080 | jmap2[zz]=primary[2*@k-1][@n]; |
---|
| 1081 | @qht[@n]=var(zz); |
---|
| 1082 | } |
---|
[d6db1f2] | 1083 | } |
---|
[70ab73] | 1084 | jmap[nva]=subst(jmap[nva],lead(primary[2*@k-1][@n]),0); |
---|
| 1085 | } |
---|
| 1086 | } |
---|
| 1087 | if(size(subst(jmap[nva],var(1),0)-var(nva))!=0) |
---|
| 1088 | { |
---|
| 1089 | // jmap[nva]=subst(jmap[nva],var(1),0); |
---|
| 1090 | //hier geaendert +untersuchen!!!!!!!!!!!!!! |
---|
| 1091 | } |
---|
| 1092 | phi1=@P,jmap1; |
---|
| 1093 | phi=@P,jmap; |
---|
| 1094 | for(@n=1;@n<=nva;@n++) |
---|
| 1095 | { |
---|
| 1096 | jmap[@n]=-(jmap[@n]-2*var(@n)); |
---|
| 1097 | } |
---|
| 1098 | psi=@P,jmap; |
---|
| 1099 | psi1=@P,jmap2; |
---|
| 1100 | @qh=phi(@qht); |
---|
[24f458] | 1101 | |
---|
| 1102 | //=================== the new part ============================ |
---|
| 1103 | |
---|
[8992ed] | 1104 | if (npars(basering)>1) { @qh=groebner(@qh,"par2var"); } |
---|
| 1105 | else { @qh=groebner(@qh); } |
---|
[24f458] | 1106 | |
---|
| 1107 | //============================================================= |
---|
| 1108 | // if(npars(@P)>0) |
---|
| 1109 | // { |
---|
| 1110 | // @ri= "ring @Phelp =" |
---|
| 1111 | // +string(char(@P))+", |
---|
| 1112 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 1113 | // } |
---|
| 1114 | // else |
---|
| 1115 | // { |
---|
| 1116 | // @ri= "ring @Phelp =" |
---|
| 1117 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 1118 | // } |
---|
| 1119 | // execute(@ri); |
---|
| 1120 | // ideal @qh=homog(imap(@P,@qht),@t); |
---|
| 1121 | // |
---|
| 1122 | // ideal @qh1=std(@qh); |
---|
| 1123 | // @hilb=hilb(@qh1,1); |
---|
| 1124 | // @ri= "ring @Phelp1 =" |
---|
| 1125 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 1126 | // execute(@ri); |
---|
| 1127 | // ideal @qh=homog(imap(@P,@qh),@t); |
---|
| 1128 | // kill @Phelp; |
---|
| 1129 | // @qh=std(@qh,@hilb); |
---|
| 1130 | // @qh=subst(@qh,@t,1); |
---|
| 1131 | // setring @P; |
---|
| 1132 | // @qh=imap(@Phelp1,@qh); |
---|
| 1133 | // kill @Phelp1; |
---|
| 1134 | // @qh=clearSB(@qh); |
---|
| 1135 | // attrib(@qh,"isSB",1); |
---|
| 1136 | //============================================================= |
---|
| 1137 | |
---|
[70ab73] | 1138 | ser1=phi1(ser); |
---|
| 1139 | @lh=zero_decomp (@qh,phi(ser1),@wr); |
---|
[18dd47] | 1140 | |
---|
[70ab73] | 1141 | kill lres0; |
---|
| 1142 | list lres0; |
---|
| 1143 | if(size(@lh)==2) |
---|
| 1144 | { |
---|
| 1145 | helpprim=@lh[2]; |
---|
| 1146 | lres0[1]=primary[2*@k-1]; |
---|
| 1147 | ser1=psi(helpprim); |
---|
| 1148 | lres0[2]=psi1(ser1); |
---|
| 1149 | if(size(reduce(lres0[2],lres0[1],1))==0) |
---|
| 1150 | { |
---|
| 1151 | primary[2*@k]=primary[2*@k-1]; |
---|
| 1152 | continue; |
---|
| 1153 | } |
---|
| 1154 | } |
---|
| 1155 | else |
---|
| 1156 | { |
---|
| 1157 | lres1=psi(@lh); |
---|
| 1158 | lres0=psi1(lres1); |
---|
| 1159 | } |
---|
[d6db1f2] | 1160 | |
---|
[24f458] | 1161 | //=================== the new part ============================ |
---|
[d6db1f2] | 1162 | |
---|
[70ab73] | 1163 | primary=delete(primary,2*@k-1); |
---|
| 1164 | primary=delete(primary,2*@k-1); |
---|
| 1165 | @k--; |
---|
| 1166 | if(size(lres0)==2) |
---|
| 1167 | { |
---|
[a36e78] | 1168 | lres0[2]=groebner(lres0[2]); |
---|
[70ab73] | 1169 | } |
---|
| 1170 | else |
---|
| 1171 | { |
---|
| 1172 | for(@n=1;@n<=size(lres0)/2;@n++) |
---|
| 1173 | { |
---|
| 1174 | if(specialIdealsEqual(lres0[2*@n-1],lres0[2*@n])==1) |
---|
[d6db1f2] | 1175 | { |
---|
[a36e78] | 1176 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
[70ab73] | 1177 | lres0[2*@n]=lres0[2*@n-1]; |
---|
| 1178 | attrib(lres0[2*@n],"isSB",1); |
---|
[d6db1f2] | 1179 | } |
---|
[70ab73] | 1180 | else |
---|
| 1181 | { |
---|
[a36e78] | 1182 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
| 1183 | lres0[2*@n]=groebner(lres0[2*@n]); |
---|
[70ab73] | 1184 | } |
---|
| 1185 | } |
---|
| 1186 | } |
---|
| 1187 | primary=primary+lres0; |
---|
[18dd47] | 1188 | |
---|
[24f458] | 1189 | //============================================================= |
---|
| 1190 | // if(npars(@P)>0) |
---|
| 1191 | // { |
---|
| 1192 | // @ri= "ring @Phelp =" |
---|
| 1193 | // +string(char(@P))+", |
---|
| 1194 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 1195 | // } |
---|
| 1196 | // else |
---|
| 1197 | // { |
---|
| 1198 | // @ri= "ring @Phelp =" |
---|
| 1199 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 1200 | // } |
---|
| 1201 | // execute(@ri); |
---|
| 1202 | // list @lvec; |
---|
| 1203 | // list @lr=imap(@P,lres0); |
---|
| 1204 | // ideal @lr1; |
---|
| 1205 | // |
---|
| 1206 | // if(size(@lr)==2) |
---|
| 1207 | // { |
---|
| 1208 | // @lr[2]=homog(@lr[2],@t); |
---|
| 1209 | // @lr1=std(@lr[2]); |
---|
| 1210 | // @lvec[2]=hilb(@lr1,1); |
---|
| 1211 | // } |
---|
| 1212 | // else |
---|
| 1213 | // { |
---|
| 1214 | // for(@n=1;@n<=size(@lr)/2;@n++) |
---|
| 1215 | // { |
---|
| 1216 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 1217 | // { |
---|
| 1218 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 1219 | // @lr1=std(@lr[2*@n-1]); |
---|
| 1220 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 1221 | // @lvec[2*@n]=@lvec[2*@n-1]; |
---|
| 1222 | // } |
---|
| 1223 | // else |
---|
| 1224 | // { |
---|
| 1225 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 1226 | // @lr1=std(@lr[2*@n-1]); |
---|
| 1227 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 1228 | // @lr[2*@n]=homog(@lr[2*@n],@t); |
---|
| 1229 | // @lr1=std(@lr[2*@n]); |
---|
| 1230 | // @lvec[2*@n]=hilb(@lr1,1); |
---|
| 1231 | // |
---|
| 1232 | // } |
---|
| 1233 | // } |
---|
| 1234 | // } |
---|
| 1235 | // @ri= "ring @Phelp1 =" |
---|
| 1236 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 1237 | // execute(@ri); |
---|
| 1238 | // list @lr=imap(@Phelp,@lr); |
---|
| 1239 | // |
---|
| 1240 | // kill @Phelp; |
---|
| 1241 | // if(size(@lr)==2) |
---|
| 1242 | // { |
---|
| 1243 | // @lr[2]=std(@lr[2],@lvec[2]); |
---|
| 1244 | // @lr[2]=subst(@lr[2],@t,1); |
---|
| 1245 | // } |
---|
| 1246 | // else |
---|
| 1247 | // { |
---|
| 1248 | // for(@n=1;@n<=size(@lr)/2;@n++) |
---|
| 1249 | // { |
---|
| 1250 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 1251 | // { |
---|
| 1252 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 1253 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 1254 | // @lr[2*@n]=@lr[2*@n-1]; |
---|
| 1255 | // attrib(@lr[2*@n],"isSB",1); |
---|
| 1256 | // } |
---|
| 1257 | // else |
---|
| 1258 | // { |
---|
| 1259 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 1260 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 1261 | // @lr[2*@n]=std(@lr[2*@n],@lvec[2*@n]); |
---|
| 1262 | // @lr[2*@n]=subst(@lr[2*@n],@t,1); |
---|
| 1263 | // } |
---|
| 1264 | // } |
---|
| 1265 | // } |
---|
| 1266 | // kill @lvec; |
---|
| 1267 | // setring @P; |
---|
| 1268 | // lres0=imap(@Phelp1,@lr); |
---|
| 1269 | // kill @Phelp1; |
---|
| 1270 | // for(@n=1;@n<=size(lres0);@n++) |
---|
| 1271 | // { |
---|
| 1272 | // lres0[@n]=clearSB(lres0[@n]); |
---|
| 1273 | // attrib(lres0[@n],"isSB",1); |
---|
| 1274 | // } |
---|
| 1275 | // |
---|
| 1276 | // primary[2*@k-1]=lres0[1]; |
---|
| 1277 | // primary[2*@k]=lres0[2]; |
---|
| 1278 | // @s=size(primary)/2; |
---|
| 1279 | // for(@n=1;@n<=size(lres0)/2-1;@n++) |
---|
| 1280 | // { |
---|
| 1281 | // primary[2*@s+2*@n-1]=lres0[2*@n+1]; |
---|
| 1282 | // primary[2*@s+2*@n]=lres0[2*@n+2]; |
---|
| 1283 | // } |
---|
| 1284 | // @k--; |
---|
| 1285 | //============================================================= |
---|
[70ab73] | 1286 | } |
---|
[d6db1f2] | 1287 | } |
---|
| 1288 | return(primary); |
---|
| 1289 | } |
---|
| 1290 | example |
---|
| 1291 | { "EXAMPLE:"; echo = 2; |
---|
| 1292 | ring r = 0,(x,y,z),lp; |
---|
| 1293 | poly p = z2+1; |
---|
| 1294 | poly q = z4+2; |
---|
| 1295 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 1296 | i=std(i); |
---|
| 1297 | list pr= zero_decomp(i,ideal(0),0); |
---|
| 1298 | pr; |
---|
| 1299 | } |
---|
[24f458] | 1300 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1301 | proc extF(list l,list #) |
---|
| 1302 | { |
---|
| 1303 | //zero_dimensional primary decomposition after finite field extension |
---|
[70ab73] | 1304 | def R=basering; |
---|
| 1305 | int p=char(R); |
---|
[24f458] | 1306 | |
---|
[70ab73] | 1307 | if((p==0)||(p>13)||(npars(R)>0)){return(l);} |
---|
[24f458] | 1308 | |
---|
[70ab73] | 1309 | int ex=3; |
---|
| 1310 | if(size(#)>0){ex=#[1];} |
---|
[24f458] | 1311 | |
---|
[70ab73] | 1312 | list peek,peek1; |
---|
| 1313 | while(size(l)>0) |
---|
| 1314 | { |
---|
| 1315 | if(size(l[2])==0) |
---|
| 1316 | { |
---|
| 1317 | peek[size(peek)+1]=l[1]; |
---|
| 1318 | } |
---|
| 1319 | else |
---|
| 1320 | { |
---|
| 1321 | peek1[size(peek1)+1]=l[1]; |
---|
| 1322 | peek1[size(peek1)+1]=l[2]; |
---|
| 1323 | } |
---|
| 1324 | l=delete(l,1); |
---|
| 1325 | l=delete(l,1); |
---|
| 1326 | } |
---|
| 1327 | if(size(peek)==0){return(peek1);} |
---|
| 1328 | |
---|
| 1329 | string gnir="ring RH=("+string(p)+"^"+string(ex)+",a),("+varstr(R)+"),lp;"; |
---|
| 1330 | execute(gnir); |
---|
| 1331 | string mp="minpoly="+string(minpoly)+";"; |
---|
| 1332 | gnir="ring RL=("+string(p)+",a),("+varstr(R)+"),lp;"; |
---|
| 1333 | execute(gnir); |
---|
| 1334 | execute(mp); |
---|
| 1335 | list L=imap(R,peek); |
---|
| 1336 | list pr, keep; |
---|
| 1337 | int i; |
---|
| 1338 | for(i=1;i<=size(L);i++) |
---|
| 1339 | { |
---|
| 1340 | attrib(L[i],"isSB",1); |
---|
| 1341 | pr=zero_decomp(L[i],0,0); |
---|
| 1342 | keep=keep+pr; |
---|
| 1343 | } |
---|
| 1344 | for(i=1;i<=size(keep);i++) |
---|
| 1345 | { |
---|
| 1346 | keep[i]=simplify(keep[i],1); |
---|
| 1347 | } |
---|
| 1348 | mp="poly pp="+string(minpoly)+";"; |
---|
[24f458] | 1349 | |
---|
[70ab73] | 1350 | string gnir1="ring RS="+string(p)+",("+varstr(R)+",a),lp;"; |
---|
| 1351 | execute(gnir1); |
---|
| 1352 | execute(mp); |
---|
| 1353 | list L=imap(RL,keep); |
---|
[24f458] | 1354 | |
---|
[70ab73] | 1355 | for(i=1;i<=size(L);i++) |
---|
| 1356 | { |
---|
| 1357 | L[i]=eliminate(L[i]+ideal(pp),a); |
---|
| 1358 | } |
---|
| 1359 | i=0; |
---|
| 1360 | int j; |
---|
| 1361 | while(i<size(L)/2-1) |
---|
| 1362 | { |
---|
| 1363 | i++; |
---|
| 1364 | j=i; |
---|
| 1365 | while(j<size(L)/2) |
---|
| 1366 | { |
---|
| 1367 | j++; |
---|
| 1368 | if(idealsEqual(L[2*i-1],L[2*j-1])) |
---|
| 1369 | { |
---|
| 1370 | L=delete(L,2*j-1); |
---|
| 1371 | L=delete(L,2*j-1); |
---|
| 1372 | j--; |
---|
[24f458] | 1373 | } |
---|
[70ab73] | 1374 | } |
---|
| 1375 | } |
---|
| 1376 | setring R; |
---|
| 1377 | list re=imap(RS,L); |
---|
| 1378 | re=re+peek1; |
---|
[24f458] | 1379 | |
---|
[70ab73] | 1380 | return(extF(re,ex+1)); |
---|
[24f458] | 1381 | } |
---|
| 1382 | |
---|
| 1383 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1384 | proc zeroSp(ideal i) |
---|
| 1385 | { |
---|
| 1386 | //preparation for the separable closure |
---|
| 1387 | //decomposition into ideals of special type |
---|
| 1388 | //i.e. the minimal polynomials of every variable mod i are irreducible |
---|
| 1389 | //returns a list of 2 lists: rr=pe,qe |
---|
| 1390 | //the ideals in pe[l] are special, their special elements are in qe[l] |
---|
| 1391 | //pe[l] is a dp-Groebnerbasis |
---|
| 1392 | //the radical of the intersection of the pe[l] is equal to the radical of i |
---|
| 1393 | |
---|
[70ab73] | 1394 | def R=basering; |
---|
[24f458] | 1395 | |
---|
[70ab73] | 1396 | //i has to be a reduced groebner basis |
---|
| 1397 | ideal F=finduni(i); |
---|
[24f458] | 1398 | |
---|
[70ab73] | 1399 | int j,k,l,ready; |
---|
| 1400 | list fa; |
---|
| 1401 | fa[1]=factorize(F[1],1); |
---|
| 1402 | poly te,ti; |
---|
| 1403 | ideal tj; |
---|
| 1404 | //avoid factorization of the same polynomial |
---|
| 1405 | for(j=2;j<=size(F);j++) |
---|
| 1406 | { |
---|
| 1407 | for(k=1;k<=j-1;k++) |
---|
| 1408 | { |
---|
| 1409 | ti=F[k]; |
---|
| 1410 | te=subst(ti,var(k),var(j)); |
---|
| 1411 | if(te==F[j]) |
---|
[24f458] | 1412 | { |
---|
[70ab73] | 1413 | tj=fa[k]; |
---|
| 1414 | fa[j]=subst(tj,var(k),var(j)); |
---|
| 1415 | ready=1; |
---|
| 1416 | break; |
---|
[24f458] | 1417 | } |
---|
[70ab73] | 1418 | } |
---|
| 1419 | if(!ready) |
---|
| 1420 | { |
---|
| 1421 | fa[j]=factorize(F[j],1); |
---|
| 1422 | } |
---|
| 1423 | ready=0; |
---|
| 1424 | } |
---|
| 1425 | execute( "ring P=("+charstr(R)+"),("+varstr(R)+"),(C,dp);"); |
---|
| 1426 | ideal i=imap(R,i); |
---|
| 1427 | if(npars(basering)==0) |
---|
| 1428 | { |
---|
| 1429 | ideal J=fglm(R,i); |
---|
| 1430 | } |
---|
| 1431 | else |
---|
| 1432 | { |
---|
[a36e78] | 1433 | ideal J=groebner(i); |
---|
[70ab73] | 1434 | } |
---|
| 1435 | list fa=imap(R,fa); |
---|
| 1436 | list qe=J; //collects a dp-Groebnerbasis of the special ideals |
---|
| 1437 | list keep=ideal(0); //collects the special elements |
---|
[24f458] | 1438 | |
---|
[70ab73] | 1439 | list re,em,ke; |
---|
| 1440 | ideal K,L; |
---|
[24f458] | 1441 | |
---|
[70ab73] | 1442 | for(j=1;j<=nvars(basering);j++) |
---|
| 1443 | { |
---|
| 1444 | for(l=1;l<=size(qe);l++) |
---|
| 1445 | { |
---|
| 1446 | for(k=1;k<=size(fa[j]);k++) |
---|
[24f458] | 1447 | { |
---|
[70ab73] | 1448 | L=std(qe[l],fa[j][k]); |
---|
| 1449 | K=keep[l],fa[j][k]; |
---|
| 1450 | if(deg(L[1])>0) |
---|
| 1451 | { |
---|
| 1452 | re[size(re)+1]=L; |
---|
| 1453 | ke[size(ke)+1]=K; |
---|
| 1454 | } |
---|
[24f458] | 1455 | } |
---|
| 1456 | } |
---|
[70ab73] | 1457 | qe=re; |
---|
| 1458 | re=em; |
---|
| 1459 | keep=ke; |
---|
| 1460 | ke=em; |
---|
| 1461 | } |
---|
| 1462 | |
---|
| 1463 | setring R; |
---|
| 1464 | list qe=imap(P,keep); |
---|
| 1465 | list pe=imap(P,qe); |
---|
| 1466 | for(l=1;l<=size(qe);l++) |
---|
| 1467 | { |
---|
| 1468 | qe[l]=simplify(qe[l],2); |
---|
| 1469 | } |
---|
| 1470 | list rr=pe,qe; |
---|
| 1471 | return(rr); |
---|
[24f458] | 1472 | } |
---|
| 1473 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1474 | |
---|
| 1475 | proc zeroSepClos(ideal I,ideal F) |
---|
| 1476 | { |
---|
| 1477 | //computes the separable closure of the special ideal I |
---|
| 1478 | //F is the set of special elements of I |
---|
| 1479 | //returns the separable closure sc(I) of I and an intvec v |
---|
| 1480 | //such that sc(I)=preimage(frobenius definde by v) |
---|
| 1481 | //i.e. var(i)----->var(i)^(p^v[i]) |
---|
| 1482 | |
---|
[70ab73] | 1483 | if(homog(I)==1){return(maxideal(1));} |
---|
[24f458] | 1484 | |
---|
[70ab73] | 1485 | //assume F[i] irreducible in I and depending only on var(i) |
---|
[24f458] | 1486 | |
---|
[70ab73] | 1487 | def R=basering; |
---|
| 1488 | int n=nvars(R); |
---|
| 1489 | int p=char(R); |
---|
| 1490 | intvec v; |
---|
| 1491 | v[n]=0; |
---|
| 1492 | int i,k; |
---|
| 1493 | list l; |
---|
[24f458] | 1494 | |
---|
[70ab73] | 1495 | for(i=1;i<=n;i++) |
---|
| 1496 | { |
---|
| 1497 | l[i]=sep(F[i],i); |
---|
| 1498 | F[i]=l[i][1]; |
---|
| 1499 | if(l[i][2]>k){k=l[i][2];} |
---|
| 1500 | } |
---|
[24f458] | 1501 | |
---|
[70ab73] | 1502 | if(k==0){return(list(I,v));} //the separable case |
---|
| 1503 | ideal m; |
---|
[24f458] | 1504 | |
---|
[70ab73] | 1505 | for(i=1;i<=n;i++) |
---|
| 1506 | { |
---|
| 1507 | m[i]=var(i)^(p^l[i][2]); |
---|
| 1508 | v[i]=l[i][2]; |
---|
| 1509 | } |
---|
| 1510 | map phi=R,m; |
---|
| 1511 | ideal J=preimage(R,phi,I); |
---|
| 1512 | return(list(J,v)); |
---|
[24f458] | 1513 | } |
---|
| 1514 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1515 | |
---|
| 1516 | proc insepDecomp(ideal i) |
---|
| 1517 | { |
---|
| 1518 | //decomposes i into special ideals |
---|
| 1519 | //computes the prime decomposition of the special ideals |
---|
| 1520 | //and transforms it back to a decomposition of i |
---|
| 1521 | |
---|
[70ab73] | 1522 | def R=basering; |
---|
| 1523 | list pr=zeroSp(i); |
---|
| 1524 | int l,k; |
---|
| 1525 | list re,wo,qr; |
---|
| 1526 | ideal m=maxideal(1); |
---|
| 1527 | ideal K; |
---|
| 1528 | map phi=R,m; |
---|
| 1529 | int p=char(R); |
---|
| 1530 | intvec op=option(get); |
---|
| 1531 | |
---|
| 1532 | for(l=1;l<=size(pr[1]);l++) |
---|
| 1533 | { |
---|
| 1534 | wo=zeroSepClos(pr[1][l],pr[2][l]); |
---|
| 1535 | for(k=1;k<=nvars(basering);k++) |
---|
| 1536 | { |
---|
| 1537 | m[k]=var(k)^(p^wo[2][k]); |
---|
| 1538 | } |
---|
| 1539 | phi=R,m; |
---|
| 1540 | qr=decomp(wo[1],2); |
---|
[24f458] | 1541 | |
---|
[70ab73] | 1542 | option(redSB); |
---|
| 1543 | for(k=1;k<=size(qr)/2;k++) |
---|
| 1544 | { |
---|
| 1545 | K=qr[2*k]; |
---|
| 1546 | K=phi(K); |
---|
| 1547 | K=groebner(K); |
---|
| 1548 | re[size(re)+1]=zeroRad(K); |
---|
| 1549 | } |
---|
| 1550 | option(noredSB); |
---|
| 1551 | } |
---|
| 1552 | option(set,op); |
---|
| 1553 | return(re); |
---|
[24f458] | 1554 | } |
---|
| 1555 | |
---|
| 1556 | |
---|
[67bd4c] | 1557 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1558 | |
---|
[07c623] | 1559 | static proc clearSB (ideal i,list #) |
---|
[d2b2a7] | 1560 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
---|
[d6db1f2] | 1561 | RETURN: ideal = minimal SB |
---|
[18dd47] | 1562 | NOTE: |
---|
[d6db1f2] | 1563 | EXAMPLE: example clearSB; shows an example |
---|
[d2b2a7] | 1564 | " |
---|
[d6db1f2] | 1565 | { |
---|
| 1566 | int k,j; |
---|
| 1567 | poly m; |
---|
| 1568 | int c=size(i); |
---|
[18dd47] | 1569 | |
---|
[d6db1f2] | 1570 | if(size(#)==0) |
---|
| 1571 | { |
---|
| 1572 | for(j=1;j<c;j++) |
---|
| 1573 | { |
---|
| 1574 | if(deg(i[j])==0) |
---|
| 1575 | { |
---|
| 1576 | i=ideal(1); |
---|
| 1577 | return(i); |
---|
[18dd47] | 1578 | } |
---|
[d6db1f2] | 1579 | if(deg(i[j])>0) |
---|
| 1580 | { |
---|
| 1581 | m=lead(i[j]); |
---|
| 1582 | for(k=j+1;k<=c;k++) |
---|
| 1583 | { |
---|
[70ab73] | 1584 | if(size(lead(i[k])/m)>0) |
---|
| 1585 | { |
---|
| 1586 | i[k]=0; |
---|
| 1587 | } |
---|
[d6db1f2] | 1588 | } |
---|
| 1589 | } |
---|
| 1590 | } |
---|
| 1591 | } |
---|
| 1592 | else |
---|
| 1593 | { |
---|
| 1594 | j=0; |
---|
| 1595 | while(j<c-1) |
---|
| 1596 | { |
---|
| 1597 | j++; |
---|
| 1598 | if(deg(i[j])==0) |
---|
| 1599 | { |
---|
| 1600 | i=ideal(1); |
---|
| 1601 | return(i); |
---|
[18dd47] | 1602 | } |
---|
[d6db1f2] | 1603 | if(deg(i[j])>0) |
---|
| 1604 | { |
---|
| 1605 | m=lead(i[j]); |
---|
| 1606 | for(k=j+1;k<=c;k++) |
---|
| 1607 | { |
---|
[70ab73] | 1608 | if(size(lead(i[k])/m)>0) |
---|
| 1609 | { |
---|
| 1610 | if((leadexp(m)!=leadexp(i[k]))||(#[j]<=#[k])) |
---|
| 1611 | { |
---|
| 1612 | i[k]=0; |
---|
| 1613 | } |
---|
| 1614 | else |
---|
| 1615 | { |
---|
| 1616 | i[j]=0; |
---|
| 1617 | break; |
---|
| 1618 | } |
---|
| 1619 | } |
---|
[d6db1f2] | 1620 | } |
---|
| 1621 | } |
---|
| 1622 | } |
---|
| 1623 | } |
---|
| 1624 | return(simplify(i,2)); |
---|
| 1625 | } |
---|
| 1626 | example |
---|
| 1627 | { "EXAMPLE:"; echo = 2; |
---|
| 1628 | ring r = (0,a,b),(x,y,z),dp; |
---|
| 1629 | ideal i=ax2+y,a2x+y,bx; |
---|
| 1630 | list l=1,2,1; |
---|
| 1631 | ideal j=clearSB(i,l); |
---|
| 1632 | j; |
---|
| 1633 | } |
---|
| 1634 | |
---|
[f54c83] | 1635 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1636 | static proc clearSBNeu (ideal i,list #) |
---|
| 1637 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
---|
| 1638 | RETURN: ideal = minimal SB |
---|
| 1639 | NOTE: |
---|
| 1640 | EXAMPLE: example clearSB; shows an example |
---|
| 1641 | " |
---|
| 1642 | { |
---|
[a36e78] | 1643 | int k,j; |
---|
| 1644 | intvec m,n,v,w; |
---|
| 1645 | int c=size(i); |
---|
| 1646 | w=leadexp(0); |
---|
| 1647 | v[size(i)]=0; |
---|
| 1648 | |
---|
| 1649 | j=0; |
---|
| 1650 | while(j<c-1) |
---|
| 1651 | { |
---|
| 1652 | j++; |
---|
| 1653 | if(deg(i[j])>=0) |
---|
| 1654 | { |
---|
[f54c83] | 1655 | m=leadexp(i[j]); |
---|
| 1656 | for(k=j+1;k<=c;k++) |
---|
| 1657 | { |
---|
| 1658 | n=leadexp(i[k]); |
---|
| 1659 | if(n!=w) |
---|
| 1660 | { |
---|
[a36e78] | 1661 | if(((m==n)&&(#[j]>#[k]))||((teilt(n,m))&&(n!=m))) |
---|
| 1662 | { |
---|
| 1663 | i[j]=0; |
---|
| 1664 | v[j]=1; |
---|
| 1665 | break; |
---|
| 1666 | } |
---|
| 1667 | if(((m==n)&&(#[j]<=#[k]))||((teilt(m,n))&&(n!=m))) |
---|
| 1668 | { |
---|
| 1669 | i[k]=0; |
---|
| 1670 | v[k]=1; |
---|
| 1671 | } |
---|
[f54c83] | 1672 | } |
---|
| 1673 | } |
---|
| 1674 | } |
---|
| 1675 | } |
---|
| 1676 | return(v); |
---|
| 1677 | } |
---|
| 1678 | |
---|
| 1679 | static proc teilt(intvec a, intvec b) |
---|
| 1680 | { |
---|
[70ab73] | 1681 | int i; |
---|
| 1682 | for(i=1;i<=size(a);i++) |
---|
| 1683 | { |
---|
| 1684 | if(a[i]>b[i]){return(0);} |
---|
| 1685 | } |
---|
| 1686 | return(1); |
---|
[f54c83] | 1687 | } |
---|
[d6db1f2] | 1688 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1689 | |
---|
[07c623] | 1690 | static proc independSet (ideal j) |
---|
[d2b2a7] | 1691 | "USAGE: independentSet(i); i ideal |
---|
[d6db1f2] | 1692 | RETURN: list = new varstring with the independent set at the end, |
---|
| 1693 | ordstring with the corresponding block ordering, |
---|
| 1694 | the integer where the independent set starts in the varstring |
---|
[18dd47] | 1695 | NOTE: |
---|
[d6db1f2] | 1696 | EXAMPLE: example independentSet; shows an example |
---|
[d2b2a7] | 1697 | " |
---|
[d6db1f2] | 1698 | { |
---|
[70ab73] | 1699 | int n,k,di; |
---|
| 1700 | list resu,hilf; |
---|
| 1701 | string var1,var2; |
---|
| 1702 | list v=indepSet(j,1); |
---|
[18dd47] | 1703 | |
---|
[70ab73] | 1704 | for(n=1;n<=size(v);n++) |
---|
| 1705 | { |
---|
| 1706 | di=0; |
---|
| 1707 | var1=""; |
---|
| 1708 | var2=""; |
---|
| 1709 | for(k=1;k<=size(v[n]);k++) |
---|
| 1710 | { |
---|
| 1711 | if(v[n][k]!=0) |
---|
| 1712 | { |
---|
| 1713 | di++; |
---|
| 1714 | var2=var2+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1715 | } |
---|
| 1716 | else |
---|
| 1717 | { |
---|
[70ab73] | 1718 | var1=var1+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1719 | } |
---|
[70ab73] | 1720 | } |
---|
| 1721 | if(di>0) |
---|
| 1722 | { |
---|
| 1723 | var1=var1+var2; |
---|
| 1724 | var1=var1[1..size(var1)-1]; |
---|
| 1725 | hilf[1]=var1; |
---|
| 1726 | hilf[2]="lp"; |
---|
| 1727 | //"lp("+string(nvars(basering)-di)+"),dp("+string(di)+")"; |
---|
| 1728 | hilf[3]=di; |
---|
| 1729 | resu[n]=hilf; |
---|
| 1730 | } |
---|
| 1731 | else |
---|
| 1732 | { |
---|
| 1733 | resu[n]=varstr(basering),ordstr(basering),0; |
---|
| 1734 | } |
---|
| 1735 | } |
---|
| 1736 | return(resu); |
---|
[d6db1f2] | 1737 | } |
---|
| 1738 | example |
---|
| 1739 | { "EXAMPLE:"; echo = 2; |
---|
| 1740 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
---|
| 1741 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
---|
| 1742 | i=std(i); |
---|
| 1743 | list l=independSet(i); |
---|
| 1744 | l; |
---|
| 1745 | i=i,g; |
---|
| 1746 | l=independSet(i); |
---|
| 1747 | l; |
---|
| 1748 | |
---|
| 1749 | ring s=0,(x,y,z),lp; |
---|
| 1750 | ideal i=z,yx; |
---|
| 1751 | list l=independSet(i); |
---|
| 1752 | l; |
---|
| 1753 | |
---|
| 1754 | |
---|
| 1755 | } |
---|
| 1756 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1757 | |
---|
[07c623] | 1758 | static proc maxIndependSet (ideal j) |
---|
[d2b2a7] | 1759 | "USAGE: maxIndependentSet(i); i ideal |
---|
[d6db1f2] | 1760 | RETURN: list = new varstring with the maximal independent set at the end, |
---|
| 1761 | ordstring with the corresponding block ordering, |
---|
| 1762 | the integer where the independent set starts in the varstring |
---|
[18dd47] | 1763 | NOTE: |
---|
[d6db1f2] | 1764 | EXAMPLE: example maxIndependentSet; shows an example |
---|
[d2b2a7] | 1765 | " |
---|
[d6db1f2] | 1766 | { |
---|
[70ab73] | 1767 | int n,k,di; |
---|
| 1768 | list resu,hilf; |
---|
| 1769 | string var1,var2; |
---|
| 1770 | list v=indepSet(j,0); |
---|
[18dd47] | 1771 | |
---|
[70ab73] | 1772 | for(n=1;n<=size(v);n++) |
---|
| 1773 | { |
---|
| 1774 | di=0; |
---|
| 1775 | var1=""; |
---|
| 1776 | var2=""; |
---|
| 1777 | for(k=1;k<=size(v[n]);k++) |
---|
| 1778 | { |
---|
| 1779 | if(v[n][k]!=0) |
---|
| 1780 | { |
---|
| 1781 | di++; |
---|
| 1782 | var2=var2+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1783 | } |
---|
| 1784 | else |
---|
| 1785 | { |
---|
[70ab73] | 1786 | var1=var1+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1787 | } |
---|
[70ab73] | 1788 | } |
---|
| 1789 | if(di>0) |
---|
| 1790 | { |
---|
| 1791 | var1=var1+var2; |
---|
| 1792 | var1=var1[1..size(var1)-1]; |
---|
| 1793 | hilf[1]=var1; |
---|
| 1794 | hilf[2]="lp"; |
---|
| 1795 | hilf[3]=di; |
---|
| 1796 | resu[n]=hilf; |
---|
| 1797 | } |
---|
| 1798 | else |
---|
| 1799 | { |
---|
| 1800 | resu[n]=varstr(basering),ordstr(basering),0; |
---|
| 1801 | } |
---|
| 1802 | } |
---|
| 1803 | return(resu); |
---|
[d6db1f2] | 1804 | } |
---|
| 1805 | example |
---|
| 1806 | { "EXAMPLE:"; echo = 2; |
---|
| 1807 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
---|
| 1808 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
---|
| 1809 | i=std(i); |
---|
| 1810 | list l=maxIndependSet(i); |
---|
| 1811 | l; |
---|
| 1812 | i=i,g; |
---|
| 1813 | l=maxIndependSet(i); |
---|
| 1814 | l; |
---|
| 1815 | |
---|
| 1816 | ring s=0,(x,y,z),lp; |
---|
| 1817 | ideal i=z,yx; |
---|
| 1818 | list l=maxIndependSet(i); |
---|
| 1819 | l; |
---|
| 1820 | |
---|
| 1821 | |
---|
| 1822 | } |
---|
| 1823 | |
---|
| 1824 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1825 | |
---|
[07c623] | 1826 | static proc prepareQuotientring (int nnp) |
---|
[d2b2a7] | 1827 | "USAGE: prepareQuotientring(nnp); nnp int |
---|
[d6db1f2] | 1828 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
---|
[18dd47] | 1829 | NOTE: |
---|
[d6db1f2] | 1830 | EXAMPLE: example independentSet; shows an example |
---|
[d2b2a7] | 1831 | " |
---|
[18dd47] | 1832 | { |
---|
[d6db1f2] | 1833 | ideal @ih,@jh; |
---|
| 1834 | int npar=npars(basering); |
---|
| 1835 | int @n; |
---|
[18dd47] | 1836 | |
---|
[d6db1f2] | 1837 | string quotring= "ring quring = ("+charstr(basering); |
---|
| 1838 | for(@n=nnp+1;@n<=nvars(basering);@n++) |
---|
| 1839 | { |
---|
[a36e78] | 1840 | quotring=quotring+",var("+string(@n)+")"; |
---|
| 1841 | @ih=@ih+var(@n); |
---|
[d6db1f2] | 1842 | } |
---|
[18dd47] | 1843 | |
---|
[d6db1f2] | 1844 | quotring=quotring+"),(var(1)"; |
---|
| 1845 | @jh=@jh+var(1); |
---|
| 1846 | for(@n=2;@n<=nnp;@n++) |
---|
| 1847 | { |
---|
| 1848 | quotring=quotring+",var("+string(@n)+")"; |
---|
| 1849 | @jh=@jh+var(@n); |
---|
| 1850 | } |
---|
[e801fe] | 1851 | quotring=quotring+"),(C,lp);"; |
---|
[18dd47] | 1852 | |
---|
[d6db1f2] | 1853 | return(quotring); |
---|
| 1854 | |
---|
| 1855 | } |
---|
| 1856 | example |
---|
| 1857 | { "EXAMPLE:"; echo = 2; |
---|
| 1858 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
---|
| 1859 | def @Q=basering; |
---|
| 1860 | list l= prepareQuotientring(3); |
---|
| 1861 | l; |
---|
[2d2cad9] | 1862 | execute(l[1]); |
---|
| 1863 | execute(l[2]); |
---|
[d6db1f2] | 1864 | basering; |
---|
| 1865 | phi; |
---|
| 1866 | setring @Q; |
---|
[18dd47] | 1867 | |
---|
[d6db1f2] | 1868 | } |
---|
| 1869 | |
---|
[091424] | 1870 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 1871 | static proc cleanPrimary(list l) |
---|
[d6db1f2] | 1872 | { |
---|
[a36e78] | 1873 | int i,j; |
---|
| 1874 | list lh; |
---|
| 1875 | for(i=1;i<=size(l)/2;i++) |
---|
| 1876 | { |
---|
| 1877 | if(deg(l[2*i-1][1])>0) |
---|
| 1878 | { |
---|
| 1879 | j++; |
---|
| 1880 | lh[j]=l[2*i-1]; |
---|
| 1881 | j++; |
---|
| 1882 | lh[j]=l[2*i]; |
---|
| 1883 | } |
---|
| 1884 | } |
---|
| 1885 | return(lh); |
---|
[d6db1f2] | 1886 | } |
---|
| 1887 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1888 | |
---|
[840745] | 1889 | |
---|
| 1890 | proc minAssPrimesold(ideal i, list #) |
---|
[d2b2a7] | 1891 | "USAGE: minAssPrimes(i); i ideal |
---|
[d6db1f2] | 1892 | minAssPrimes(i,1); i ideal (to use also the factorizing Groebner) |
---|
| 1893 | RETURN: list = the minimal associated prime ideals of i |
---|
| 1894 | EXAMPLE: example minAssPrimes; shows an example |
---|
[d2b2a7] | 1895 | " |
---|
[d6db1f2] | 1896 | { |
---|
[a36e78] | 1897 | def @P=basering; |
---|
| 1898 | if(size(i)==0){return(list(ideal(0)));} |
---|
| 1899 | list qr=simplifyIdeal(i); |
---|
| 1900 | map phi=@P,qr[2]; |
---|
| 1901 | i=qr[1]; |
---|
[3939bc] | 1902 | |
---|
[a36e78] | 1903 | execute ("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[2d2cad9] | 1904 | +ordstr(basering)+");"); |
---|
[67bd4c] | 1905 | |
---|
| 1906 | |
---|
[a36e78] | 1907 | ideal i=fetch(@P,i); |
---|
| 1908 | if(size(#)==0) |
---|
| 1909 | { |
---|
| 1910 | int @wr; |
---|
| 1911 | list tluser,@res; |
---|
| 1912 | list primary=decomp(i,2); |
---|
[d6db1f2] | 1913 | |
---|
[a36e78] | 1914 | @res[1]=primary; |
---|
[d6db1f2] | 1915 | |
---|
[a36e78] | 1916 | tluser=union(@res); |
---|
| 1917 | setring @P; |
---|
| 1918 | list @res=imap(gnir,tluser); |
---|
| 1919 | return(phi(@res)); |
---|
| 1920 | } |
---|
| 1921 | list @res,empty; |
---|
| 1922 | ideal ser; |
---|
| 1923 | option(redSB); |
---|
| 1924 | list @pr=facstd(i); |
---|
| 1925 | //if(size(@pr)==1) |
---|
[17407e] | 1926 | // { |
---|
| 1927 | // attrib(@pr[1],"isSB",1); |
---|
| 1928 | // if((dim(@pr[1])==0)&&(homog(@pr[1])==1)) |
---|
| 1929 | // { |
---|
| 1930 | // setring @P; |
---|
| 1931 | // list @res=maxideal(1); |
---|
| 1932 | // return(phi(@res)); |
---|
| 1933 | // } |
---|
| 1934 | // if(dim(@pr[1])>1) |
---|
| 1935 | // { |
---|
| 1936 | // setring @P; |
---|
| 1937 | // // kill gnir; |
---|
| 1938 | // execute ("ring gnir1 = ("+charstr(basering)+"), |
---|
| 1939 | // ("+varstr(basering)+"),(C,lp);"); |
---|
| 1940 | // ideal i=fetch(@P,i); |
---|
| 1941 | // list @pr=facstd(i); |
---|
| 1942 | // // ideal ser; |
---|
| 1943 | // setring gnir; |
---|
| 1944 | // @pr=fetch(gnir1,@pr); |
---|
| 1945 | // kill gnir1; |
---|
| 1946 | // } |
---|
| 1947 | // } |
---|
[a36e78] | 1948 | option(noredSB); |
---|
| 1949 | int j,k,odim,ndim,count; |
---|
| 1950 | attrib(@pr[1],"isSB",1); |
---|
| 1951 | if(#[1]==77) |
---|
| 1952 | { |
---|
| 1953 | odim=dim(@pr[1]); |
---|
| 1954 | count=1; |
---|
| 1955 | intvec pos; |
---|
| 1956 | pos[size(@pr)]=0; |
---|
| 1957 | for(j=2;j<=size(@pr);j++) |
---|
| 1958 | { |
---|
| 1959 | attrib(@pr[j],"isSB",1); |
---|
| 1960 | ndim=dim(@pr[j]); |
---|
| 1961 | if(ndim>odim) |
---|
[80b3cd] | 1962 | { |
---|
[a36e78] | 1963 | for(k=count;k<=j-1;k++) |
---|
| 1964 | { |
---|
| 1965 | pos[k]=1; |
---|
| 1966 | } |
---|
| 1967 | count=j; |
---|
| 1968 | odim=ndim; |
---|
[80b3cd] | 1969 | } |
---|
[a36e78] | 1970 | if(ndim<odim) |
---|
| 1971 | { |
---|
| 1972 | pos[j]=1; |
---|
| 1973 | } |
---|
| 1974 | } |
---|
| 1975 | for(j=1;j<=size(@pr);j++) |
---|
| 1976 | { |
---|
| 1977 | if(pos[j]!=1) |
---|
| 1978 | { |
---|
| 1979 | @res[j]=decomp(@pr[j],2); |
---|
| 1980 | } |
---|
| 1981 | else |
---|
| 1982 | { |
---|
| 1983 | @res[j]=empty; |
---|
| 1984 | } |
---|
| 1985 | } |
---|
| 1986 | } |
---|
| 1987 | else |
---|
| 1988 | { |
---|
| 1989 | ser=ideal(1); |
---|
| 1990 | for(j=1;j<=size(@pr);j++) |
---|
| 1991 | { |
---|
[e801fe] | 1992 | //@pr[j]; |
---|
[917fb5] | 1993 | //pause(); |
---|
[a36e78] | 1994 | @res[j]=decomp(@pr[j],2); |
---|
[e801fe] | 1995 | // @res[j]=decomp(@pr[j],2,@pr[j],ser); |
---|
| 1996 | // for(k=1;k<=size(@res[j]);k++) |
---|
| 1997 | // { |
---|
[d950c5] | 1998 | // ser=intersect(ser,@res[j][k]); |
---|
[e801fe] | 1999 | // } |
---|
[a36e78] | 2000 | } |
---|
| 2001 | } |
---|
[d6db1f2] | 2002 | |
---|
[a36e78] | 2003 | @res=union(@res); |
---|
| 2004 | setring @P; |
---|
| 2005 | list @res=imap(gnir,@res); |
---|
| 2006 | return(phi(@res)); |
---|
[d6db1f2] | 2007 | } |
---|
| 2008 | example |
---|
| 2009 | { "EXAMPLE:"; echo = 2; |
---|
| 2010 | ring r = 32003,(x,y,z),lp; |
---|
| 2011 | poly p = z2+1; |
---|
| 2012 | poly q = z4+2; |
---|
| 2013 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
[091424] | 2014 | list pr= minAssPrimes(i); pr; |
---|
| 2015 | |
---|
[9050ca] | 2016 | minAssPrimes(i,1); |
---|
[d6db1f2] | 2017 | } |
---|
| 2018 | |
---|
[24f458] | 2019 | static proc primT(ideal i) |
---|
| 2020 | { |
---|
[a36e78] | 2021 | //assumes that all generators of i are irreducible |
---|
| 2022 | //i is standard basis |
---|
[840745] | 2023 | |
---|
[a36e78] | 2024 | attrib(i,"isSB",1); |
---|
| 2025 | int j=size(i); |
---|
| 2026 | int k; |
---|
| 2027 | while(j>0) |
---|
| 2028 | { |
---|
| 2029 | if(deg(i[j])>1){break;} |
---|
| 2030 | j--; |
---|
| 2031 | } |
---|
| 2032 | if(j==0){return(1);} |
---|
| 2033 | if(deg(i[j])==vdim(i)){return(1);} |
---|
| 2034 | return(0); |
---|
[24f458] | 2035 | } |
---|
[840745] | 2036 | |
---|
| 2037 | static proc minAssPrimes(ideal i, list #) |
---|
| 2038 | "USAGE: minAssPrimes(i); i ideal |
---|
[808a9f3] | 2039 | Optional parameters in list #: (can be entered in any order) |
---|
| 2040 | 0, "facstd" -> uses facstd to first decompose the ideal |
---|
| 2041 | 1, "noFacstd" -> does not use facstd (default) |
---|
| 2042 | "SL" -> the new algorithm is used (default) |
---|
| 2043 | "GTZ" -> the old algorithm is used |
---|
[840745] | 2044 | RETURN: list = the minimal associated prime ideals of i |
---|
| 2045 | EXAMPLE: example minAssPrimes; shows an example |
---|
| 2046 | " |
---|
| 2047 | { |
---|
[70ab73] | 2048 | if(size(i) == 0){return(list(ideal(0)));} |
---|
| 2049 | string algorithm; // Algorithm to be used |
---|
| 2050 | string facstdOption; // To uses proc facstd |
---|
| 2051 | int j; // Counter |
---|
| 2052 | def P0 = basering; |
---|
| 2053 | list Pl=ringlist(P0); |
---|
| 2054 | intvec dp_w; |
---|
| 2055 | for(j=nvars(P0);j>0;j--) {dp_w[j]=1;} |
---|
| 2056 | Pl[3]=list(list("dp",dp_w),list("C",0)); |
---|
| 2057 | def P=ring(Pl); |
---|
| 2058 | setring P; |
---|
| 2059 | ideal i=imap(P0,i); |
---|
[24f458] | 2060 | |
---|
[70ab73] | 2061 | // Set input parameters |
---|
| 2062 | algorithm = "SL"; // Default: SL algorithm |
---|
[fc1526c] | 2063 | facstdOption = "Facstd"; // Default: facstd is not used |
---|
[70ab73] | 2064 | if(size(#) > 0) |
---|
| 2065 | { |
---|
| 2066 | int valid; |
---|
| 2067 | for(j = 1; j <= size(#); j++) |
---|
| 2068 | { |
---|
| 2069 | valid = 0; |
---|
| 2070 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
---|
[f54c83] | 2071 | { |
---|
[70ab73] | 2072 | if (#[j] == 0) {facstdOption = "noFacstd"; valid = 1;} // If #[j] == 0, facstd is not used. |
---|
| 2073 | if (#[j] == 1) {facstdOption = "facstd"; valid = 1;} // If #[j] == 1, facstd is used. |
---|
[f54c83] | 2074 | } |
---|
[70ab73] | 2075 | if(typeof(#[j]) == "string") |
---|
[f54c83] | 2076 | { |
---|
[70ab73] | 2077 | if(#[j] == "GTZ" || #[j] == "SL") |
---|
| 2078 | { |
---|
| 2079 | algorithm = #[j]; |
---|
| 2080 | valid = 1; |
---|
| 2081 | } |
---|
| 2082 | if(#[j] == "noFacstd" || #[j] == "facstd") |
---|
| 2083 | { |
---|
| 2084 | facstdOption = #[j]; |
---|
| 2085 | valid = 1; |
---|
| 2086 | } |
---|
[f54c83] | 2087 | } |
---|
[70ab73] | 2088 | if(valid == 0) |
---|
[24a90ca] | 2089 | { |
---|
[70ab73] | 2090 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
---|
[7f7c25e] | 2091 | } |
---|
[70ab73] | 2092 | } |
---|
| 2093 | } |
---|
[4d63da] | 2094 | |
---|
[70ab73] | 2095 | list q = simplifyIdeal(i); |
---|
| 2096 | list re = maxideal(1); |
---|
| 2097 | int a, k; |
---|
| 2098 | intvec op = option(get); |
---|
| 2099 | map phi = P,q[2]; |
---|
| 2100 | |
---|
| 2101 | list result; |
---|
| 2102 | |
---|
| 2103 | if(npars(P) == 0){option(redSB);} |
---|
| 2104 | |
---|
| 2105 | if(attrib(i,"isSB")!=1) |
---|
| 2106 | { |
---|
| 2107 | i=groebner(q[1]); |
---|
| 2108 | } |
---|
| 2109 | else |
---|
| 2110 | { |
---|
| 2111 | for(j=1;j<=nvars(basering);j++) |
---|
| 2112 | { |
---|
| 2113 | if(q[2][j]!=var(j)){k=1;break;} |
---|
| 2114 | } |
---|
| 2115 | if(k) |
---|
| 2116 | { |
---|
| 2117 | i=groebner(q[1]); |
---|
| 2118 | } |
---|
| 2119 | } |
---|
| 2120 | |
---|
| 2121 | if(dim(i) == -1){setring P0;return(ideal(1));} |
---|
| 2122 | if((dim(i) == 0) && (npars(P) == 0)) |
---|
| 2123 | { |
---|
| 2124 | int di = vdim(i); |
---|
| 2125 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
---|
| 2126 | ideal J = std(imap(P,i)); |
---|
| 2127 | attrib(J, "isSB", 1); |
---|
| 2128 | if(vdim(J) != di) |
---|
| 2129 | { |
---|
| 2130 | J = fglm(P, i); |
---|
| 2131 | } |
---|
[b0db25] | 2132 | // list pr = triangMH(J,2); HIER KOENNEN verschiedene Mengen zu gleichen |
---|
| 2133 | // asoziierten Primidealen fuehren |
---|
| 2134 | // Aenderung |
---|
[85e68dd] | 2135 | list pr = triangMH(J,2); |
---|
[70ab73] | 2136 | list qr, re; |
---|
| 2137 | for(k = 1; k <= size(pr); k++) |
---|
| 2138 | { |
---|
[fc1526c] | 2139 | if(primT(pr[k])&&(0)) |
---|
[840745] | 2140 | { |
---|
[70ab73] | 2141 | re[size(re) + 1] = pr[k]; |
---|
[840745] | 2142 | } |
---|
| 2143 | else |
---|
| 2144 | { |
---|
[70ab73] | 2145 | attrib(pr[k], "isSB", 1); |
---|
| 2146 | // Lines changed |
---|
| 2147 | if (algorithm == "GTZ") |
---|
| 2148 | { |
---|
| 2149 | qr = decomp(pr[k], 2); |
---|
| 2150 | } |
---|
| 2151 | else |
---|
| 2152 | { |
---|
| 2153 | qr = minAssSL(pr[k]); |
---|
| 2154 | } |
---|
| 2155 | for(j = 1; j <= size(qr) / 2; j++) |
---|
| 2156 | { |
---|
[fc1526c] | 2157 | re[size(re) + 1] = std(qr[2 * j]); |
---|
[70ab73] | 2158 | } |
---|
[840745] | 2159 | } |
---|
[70ab73] | 2160 | } |
---|
| 2161 | setring P; |
---|
| 2162 | re = imap(gnir, re); |
---|
| 2163 | re=phi(re); |
---|
| 2164 | option(set, op); |
---|
| 2165 | setring(P0); |
---|
| 2166 | list re=imap(P,re); |
---|
| 2167 | return(re); |
---|
| 2168 | } |
---|
| 2169 | |
---|
| 2170 | // Lines changed |
---|
| 2171 | if ((facstdOption == "noFacstd") || (dim(i) == 0)) |
---|
| 2172 | { |
---|
| 2173 | if (algorithm == "GTZ") |
---|
| 2174 | { |
---|
| 2175 | re[1] = decomp(i, 2); |
---|
| 2176 | } |
---|
| 2177 | else |
---|
| 2178 | { |
---|
| 2179 | re[1] = minAssSL(i); |
---|
| 2180 | } |
---|
| 2181 | re = union(re); |
---|
| 2182 | option(set, op); |
---|
| 2183 | re=phi(re); |
---|
| 2184 | setring(P0); |
---|
| 2185 | list re=imap(P,re); |
---|
| 2186 | return(re); |
---|
| 2187 | } |
---|
| 2188 | q = facstd(i); |
---|
| 2189 | |
---|
| 2190 | /* |
---|
| 2191 | if((size(q) == 1) && (dim(i) > 1)) |
---|
| 2192 | { |
---|
| 2193 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
---|
| 2194 | list p = facstd(fetch(P, i)); |
---|
| 2195 | if(size(p) > 1) |
---|
| 2196 | { |
---|
| 2197 | a = 1; |
---|
| 2198 | setring P; |
---|
| 2199 | q = fetch(gnir,p); |
---|
| 2200 | } |
---|
| 2201 | else |
---|
| 2202 | { |
---|
| 2203 | setring P; |
---|
| 2204 | } |
---|
| 2205 | kill gnir; |
---|
| 2206 | } |
---|
[f54c83] | 2207 | */ |
---|
[70ab73] | 2208 | option(set,op); |
---|
| 2209 | // Debug |
---|
| 2210 | dbprint(printlevel - voice, "Components returned by facstd", size(q), q); |
---|
| 2211 | for(j = 1; j <= size(q); j++) |
---|
| 2212 | { |
---|
| 2213 | if(a == 0){attrib(q[j], "isSB", 1);} |
---|
| 2214 | // Debug |
---|
| 2215 | dbprint(printlevel - voice, "We compute the decomp of component", j); |
---|
| 2216 | // Lines changed |
---|
| 2217 | if (algorithm == "GTZ") |
---|
| 2218 | { |
---|
| 2219 | re[j] = decomp(q[j], 2); |
---|
| 2220 | } |
---|
| 2221 | else |
---|
| 2222 | { |
---|
| 2223 | re[j] = minAssSL(q[j]); |
---|
| 2224 | } |
---|
| 2225 | // Debug |
---|
| 2226 | dbprint(printlevel - voice, "Number of components obtained for this component:", size(re[j]) / 2); |
---|
| 2227 | dbprint(printlevel - voice, "re[j]:", re[j]); |
---|
| 2228 | } |
---|
| 2229 | re = union(re); |
---|
| 2230 | re=phi(re); |
---|
| 2231 | setring(P0); |
---|
| 2232 | list re=imap(P,re); |
---|
| 2233 | return(re); |
---|
[840745] | 2234 | } |
---|
| 2235 | example |
---|
| 2236 | { "EXAMPLE:"; echo = 2; |
---|
| 2237 | ring r = 32003,(x,y,z),lp; |
---|
| 2238 | poly p = z2+1; |
---|
| 2239 | poly q = z4+2; |
---|
| 2240 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 2241 | list pr= minAssPrimes(i); pr; |
---|
| 2242 | |
---|
| 2243 | minAssPrimes(i,1); |
---|
| 2244 | } |
---|
| 2245 | |
---|
[07c623] | 2246 | static proc union(list li) |
---|
[d6db1f2] | 2247 | { |
---|
| 2248 | int i,j,k; |
---|
[67bd4c] | 2249 | |
---|
| 2250 | def P=basering; |
---|
| 2251 | |
---|
[2d2cad9] | 2252 | execute("ring ir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
[67bd4c] | 2253 | list l=fetch(P,li); |
---|
[d6db1f2] | 2254 | list @erg; |
---|
| 2255 | |
---|
| 2256 | for(k=1;k<=size(l);k++) |
---|
| 2257 | { |
---|
[a36e78] | 2258 | for(j=1;j<=size(l[k])/2;j++) |
---|
| 2259 | { |
---|
| 2260 | if(deg(l[k][2*j][1])!=0) |
---|
| 2261 | { |
---|
| 2262 | i++; |
---|
| 2263 | @erg[i]=l[k][2*j]; |
---|
| 2264 | } |
---|
| 2265 | } |
---|
[d6db1f2] | 2266 | } |
---|
| 2267 | |
---|
| 2268 | list @wos; |
---|
| 2269 | i=0; |
---|
| 2270 | ideal i1,i2; |
---|
| 2271 | while(i<size(@erg)-1) |
---|
| 2272 | { |
---|
[a36e78] | 2273 | i++; |
---|
| 2274 | k=i+1; |
---|
| 2275 | i1=lead(@erg[i]); |
---|
| 2276 | attrib(i1,"isSB",1); |
---|
| 2277 | attrib(@erg[i],"isSB",1); |
---|
[d6db1f2] | 2278 | |
---|
[a36e78] | 2279 | while(k<=size(@erg)) |
---|
| 2280 | { |
---|
| 2281 | if(deg(@erg[i][1])==0) |
---|
| 2282 | { |
---|
| 2283 | break; |
---|
| 2284 | } |
---|
| 2285 | i2=lead(@erg[k]); |
---|
| 2286 | attrib(@erg[k],"isSB",1); |
---|
| 2287 | attrib(i2,"isSB",1); |
---|
[d6db1f2] | 2288 | |
---|
[a36e78] | 2289 | if(size(reduce(i1,i2,1))==0) |
---|
[d6db1f2] | 2290 | { |
---|
[a36e78] | 2291 | if(size(reduce(@erg[i],@erg[k],1))==0) |
---|
| 2292 | { |
---|
| 2293 | @erg[k]=ideal(1); |
---|
| 2294 | i2=ideal(1); |
---|
| 2295 | } |
---|
[d6db1f2] | 2296 | } |
---|
[a36e78] | 2297 | if(size(reduce(i2,i1,1))==0) |
---|
[d6db1f2] | 2298 | { |
---|
[a36e78] | 2299 | if(size(reduce(@erg[k],@erg[i],1))==0) |
---|
| 2300 | { |
---|
| 2301 | break; |
---|
| 2302 | } |
---|
[d6db1f2] | 2303 | } |
---|
[a36e78] | 2304 | k++; |
---|
| 2305 | if(k>size(@erg)) |
---|
| 2306 | { |
---|
| 2307 | @wos[size(@wos)+1]=@erg[i]; |
---|
| 2308 | } |
---|
| 2309 | } |
---|
[d6db1f2] | 2310 | } |
---|
| 2311 | if(deg(@erg[size(@erg)][1])!=0) |
---|
| 2312 | { |
---|
[a36e78] | 2313 | @wos[size(@wos)+1]=@erg[size(@erg)]; |
---|
[d6db1f2] | 2314 | } |
---|
[67bd4c] | 2315 | setring P; |
---|
| 2316 | list @ser=fetch(ir,@wos); |
---|
| 2317 | return(@ser); |
---|
[d6db1f2] | 2318 | } |
---|
| 2319 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d8d3af] | 2320 | proc equidim(ideal i,list #) |
---|
[b9b906] | 2321 | "USAGE: equidim(i) or equidim(i,1) ; i ideal |
---|
[7b3971] | 2322 | RETURN: list of equidimensional ideals a[1],...,a[s] with: |
---|
[25c431] | 2323 | - a[s] the equidimensional locus of i, i.e. the intersection |
---|
| 2324 | of the primary ideals of dimension of i |
---|
[367e88] | 2325 | - a[1],...,a[s-1] the lower dimensional equidimensional loci. |
---|
| 2326 | NOTE: An embedded component q (primary ideal) of i can be replaced in the |
---|
[7b3971] | 2327 | decomposition by a primary ideal q1 with the same radical as q. @* |
---|
| 2328 | @code{equidim(i,1)} uses the algorithm of Eisenbud/Huneke/Vasconcelos. |
---|
| 2329 | |
---|
[07c623] | 2330 | EXAMPLE:example equidim; shows an example |
---|
[ba94539] | 2331 | " |
---|
| 2332 | { |
---|
[d88470] | 2333 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 2334 | { |
---|
[a36e78] | 2335 | ERROR( |
---|
[07c623] | 2336 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 2337 | ); |
---|
[07c623] | 2338 | } |
---|
[b9b906] | 2339 | intvec op ; |
---|
[ba94539] | 2340 | def P = basering; |
---|
| 2341 | list eq; |
---|
| 2342 | intvec w; |
---|
[4d68980] | 2343 | int n,m; |
---|
[6d6ed5b] | 2344 | int g=size(i); |
---|
[ba94539] | 2345 | int a=attrib(i,"isSB"); |
---|
| 2346 | int homo=homog(i); |
---|
[d8d3af] | 2347 | if(size(#)!=0) |
---|
| 2348 | { |
---|
[4d68980] | 2349 | m=1; |
---|
| 2350 | } |
---|
| 2351 | |
---|
[ba94539] | 2352 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
---|
| 2353 | &&(find(ordstr(basering),"s")==0)) |
---|
| 2354 | { |
---|
[a36e78] | 2355 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[2d2cad9] | 2356 | +ordstr(basering)+");"); |
---|
[a36e78] | 2357 | ideal i=imap(P,i); |
---|
| 2358 | ideal j=i; |
---|
| 2359 | if(a==1) |
---|
| 2360 | { |
---|
| 2361 | attrib(j,"isSB",1); |
---|
| 2362 | } |
---|
| 2363 | else |
---|
| 2364 | { |
---|
| 2365 | j=groebner(i); |
---|
| 2366 | } |
---|
[ba94539] | 2367 | } |
---|
| 2368 | else |
---|
| 2369 | { |
---|
[a36e78] | 2370 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
---|
| 2371 | ideal i=imap(P,i); |
---|
| 2372 | ideal j=groebner(i); |
---|
[b9b906] | 2373 | } |
---|
[ba94539] | 2374 | if(homo==1) |
---|
| 2375 | { |
---|
[a36e78] | 2376 | for(n=1;n<=nvars(basering);n++) |
---|
| 2377 | { |
---|
| 2378 | w[n]=ord(var(n)); |
---|
| 2379 | } |
---|
| 2380 | intvec hil=hilb(j,1,w); |
---|
[ba94539] | 2381 | } |
---|
[4d68980] | 2382 | |
---|
[6d6ed5b] | 2383 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
---|
| 2384 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
---|
[ba94539] | 2385 | { |
---|
| 2386 | setring P; |
---|
[6d6ed5b] | 2387 | eq[1]=i; |
---|
[ba94539] | 2388 | return(eq); |
---|
| 2389 | } |
---|
| 2390 | |
---|
[4d68980] | 2391 | if(m==0) |
---|
[ba94539] | 2392 | { |
---|
[a36e78] | 2393 | ideal k=equidimMax(j); |
---|
[ba94539] | 2394 | } |
---|
| 2395 | else |
---|
| 2396 | { |
---|
[a36e78] | 2397 | ideal k=equidimMaxEHV(j); |
---|
[ba94539] | 2398 | } |
---|
[6d6ed5b] | 2399 | if(size(reduce(k,j,1))==0) |
---|
| 2400 | { |
---|
| 2401 | setring P; |
---|
| 2402 | eq[1]=i; |
---|
| 2403 | kill gnir; |
---|
| 2404 | return(eq); |
---|
| 2405 | } |
---|
[466f80] | 2406 | op=option(get); |
---|
[b9b906] | 2407 | option(returnSB); |
---|
[651953] | 2408 | j=quotient(j,k); |
---|
[02335e] | 2409 | option(set,op); |
---|
[d8d3af] | 2410 | |
---|
[b9b906] | 2411 | list equi=equidim(j); |
---|
[4d68980] | 2412 | if(deg(equi[size(equi)][1])<=0) |
---|
[a9cf54] | 2413 | { |
---|
[a36e78] | 2414 | equi[size(equi)]=k; |
---|
[a9cf54] | 2415 | } |
---|
| 2416 | else |
---|
| 2417 | { |
---|
[4d68980] | 2418 | equi[size(equi)+1]=k; |
---|
[a9cf54] | 2419 | } |
---|
[ba94539] | 2420 | setring P; |
---|
[4d68980] | 2421 | eq=imap(gnir,equi); |
---|
[ba94539] | 2422 | kill gnir; |
---|
| 2423 | return(eq); |
---|
| 2424 | } |
---|
| 2425 | example |
---|
| 2426 | { "EXAMPLE:"; echo = 2; |
---|
| 2427 | ring r = 32003,(x,y,z),dp; |
---|
[7b3971] | 2428 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
[ba94539] | 2429 | equidim(i); |
---|
| 2430 | } |
---|
[6d6ed5b] | 2431 | |
---|
[03f29c] | 2432 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2433 | proc equidimMax(ideal i) |
---|
[b9b906] | 2434 | "USAGE: equidimMax(i); i ideal |
---|
[07c623] | 2435 | RETURN: ideal of equidimensional locus (of maximal dimension) of i. |
---|
| 2436 | EXAMPLE: example equidimMax; shows an example |
---|
[03f29c] | 2437 | " |
---|
| 2438 | { |
---|
[d88470] | 2439 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 2440 | { |
---|
[a36e78] | 2441 | ERROR( |
---|
[07c623] | 2442 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 2443 | ); |
---|
[07c623] | 2444 | } |
---|
[03f29c] | 2445 | def P = basering; |
---|
| 2446 | ideal eq; |
---|
| 2447 | intvec w; |
---|
| 2448 | int n; |
---|
[6d6ed5b] | 2449 | int g=size(i); |
---|
[03f29c] | 2450 | int a=attrib(i,"isSB"); |
---|
| 2451 | int homo=homog(i); |
---|
[b9b906] | 2452 | |
---|
[03f29c] | 2453 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
---|
| 2454 | &&(find(ordstr(basering),"s")==0)) |
---|
| 2455 | { |
---|
[a36e78] | 2456 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[2d2cad9] | 2457 | +ordstr(basering)+");"); |
---|
[a36e78] | 2458 | ideal i=imap(P,i); |
---|
| 2459 | ideal j=i; |
---|
| 2460 | if(a==1) |
---|
| 2461 | { |
---|
| 2462 | attrib(j,"isSB",1); |
---|
| 2463 | } |
---|
| 2464 | else |
---|
| 2465 | { |
---|
| 2466 | j=groebner(i); |
---|
| 2467 | } |
---|
[03f29c] | 2468 | } |
---|
| 2469 | else |
---|
| 2470 | { |
---|
[a36e78] | 2471 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
---|
| 2472 | ideal i=imap(P,i); |
---|
| 2473 | ideal j=groebner(i); |
---|
[03f29c] | 2474 | } |
---|
| 2475 | list indep; |
---|
| 2476 | ideal equ,equi; |
---|
| 2477 | if(homo==1) |
---|
| 2478 | { |
---|
[a36e78] | 2479 | for(n=1;n<=nvars(basering);n++) |
---|
| 2480 | { |
---|
| 2481 | w[n]=ord(var(n)); |
---|
| 2482 | } |
---|
| 2483 | intvec hil=hilb(j,1,w); |
---|
[03f29c] | 2484 | } |
---|
[6d6ed5b] | 2485 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
---|
| 2486 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
---|
[03f29c] | 2487 | { |
---|
| 2488 | setring P; |
---|
[a9cf54] | 2489 | return(i); |
---|
[03f29c] | 2490 | } |
---|
| 2491 | |
---|
| 2492 | indep=maxIndependSet(j); |
---|
[a9cf54] | 2493 | |
---|
[2d2cad9] | 2494 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[1][1]+"),(" |
---|
| 2495 | +indep[1][2]+");"); |
---|
[03f29c] | 2496 | if(homo==1) |
---|
| 2497 | { |
---|
[a36e78] | 2498 | ideal j=std(imap(gnir,j),hil,w); |
---|
[03f29c] | 2499 | } |
---|
| 2500 | else |
---|
| 2501 | { |
---|
[a36e78] | 2502 | ideal j=groebner(imap(gnir,j)); |
---|
[03f29c] | 2503 | } |
---|
| 2504 | string quotring=prepareQuotientring(nvars(basering)-indep[1][3]); |
---|
[2d2cad9] | 2505 | execute(quotring); |
---|
[03f29c] | 2506 | ideal j=imap(gnir1,j); |
---|
| 2507 | kill gnir1; |
---|
| 2508 | j=clearSB(j); |
---|
| 2509 | ideal h; |
---|
| 2510 | for(n=1;n<=size(j);n++) |
---|
| 2511 | { |
---|
[a36e78] | 2512 | h[n]=leadcoef(j[n]); |
---|
[03f29c] | 2513 | } |
---|
| 2514 | setring gnir; |
---|
| 2515 | ideal h=imap(quring,h); |
---|
| 2516 | kill quring; |
---|
[6d6ed5b] | 2517 | |
---|
[03f29c] | 2518 | list l=minSat(j,h); |
---|
[b9b906] | 2519 | |
---|
[b1d1e8c] | 2520 | if(deg(l[2])>0) |
---|
| 2521 | { |
---|
| 2522 | equ=l[1]; |
---|
| 2523 | attrib(equ,"isSB",1); |
---|
| 2524 | j=std(j,l[2]); |
---|
[6d6ed5b] | 2525 | |
---|
[b1d1e8c] | 2526 | if(dim(equ)==dim(j)) |
---|
| 2527 | { |
---|
| 2528 | equi=equidimMax(j); |
---|
| 2529 | equ=interred(intersect(equ,equi)); |
---|
| 2530 | } |
---|
| 2531 | } |
---|
| 2532 | else |
---|
[03f29c] | 2533 | { |
---|
[b1d1e8c] | 2534 | equ=i; |
---|
[03f29c] | 2535 | } |
---|
[b1d1e8c] | 2536 | |
---|
[03f29c] | 2537 | setring P; |
---|
| 2538 | eq=imap(gnir,equ); |
---|
| 2539 | kill gnir; |
---|
| 2540 | return(eq); |
---|
| 2541 | } |
---|
| 2542 | example |
---|
| 2543 | { "EXAMPLE:"; echo = 2; |
---|
| 2544 | ring r = 32003,(x,y,z),dp; |
---|
[7b3971] | 2545 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
[03f29c] | 2546 | equidimMax(i); |
---|
| 2547 | } |
---|
[24f458] | 2548 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2549 | static proc islp() |
---|
| 2550 | { |
---|
[a36e78] | 2551 | string s=ordstr(basering); |
---|
| 2552 | int n=find(s,"lp"); |
---|
| 2553 | if(!n){return(0);} |
---|
| 2554 | int k=find(s,","); |
---|
| 2555 | string t=s[k+1..size(s)]; |
---|
| 2556 | int l=find(t,","); |
---|
| 2557 | t=s[1..k-1]; |
---|
| 2558 | int m=find(t,","); |
---|
| 2559 | if(l+m){return(0);} |
---|
| 2560 | return(1); |
---|
[24f458] | 2561 | } |
---|
| 2562 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2563 | |
---|
| 2564 | proc algeDeco(ideal i, int w) |
---|
| 2565 | { |
---|
| 2566 | //reduces primery decomposition over algebraic extensions to |
---|
| 2567 | //the other cases |
---|
[a36e78] | 2568 | def R=basering; |
---|
| 2569 | int n=nvars(R); |
---|
[fc5095] | 2570 | |
---|
| 2571 | //---Anfang Provisorium |
---|
[a36e78] | 2572 | if((size(i)==2) && (w==2)) |
---|
| 2573 | { |
---|
| 2574 | option(redSB); |
---|
| 2575 | ideal J=std(i); |
---|
| 2576 | option(noredSB); |
---|
| 2577 | if((size(J)==2)&&(deg(J[1])==1)) |
---|
| 2578 | { |
---|
| 2579 | ideal keep; |
---|
| 2580 | poly f; |
---|
| 2581 | int j; |
---|
| 2582 | for(j=1;j<=nvars(basering);j++) |
---|
| 2583 | { |
---|
| 2584 | f=J[2]; |
---|
| 2585 | while((f/var(j))*var(j)-f==0) |
---|
| 2586 | { |
---|
| 2587 | f=f/var(j); |
---|
| 2588 | keep=keep,var(j); |
---|
| 2589 | } |
---|
| 2590 | J[2]=f; |
---|
| 2591 | } |
---|
| 2592 | ideal K=factorize(J[2],1); |
---|
| 2593 | if(deg(K[1])==0){K=0;} |
---|
| 2594 | K=K+std(keep); |
---|
| 2595 | ideal L; |
---|
| 2596 | list resu; |
---|
| 2597 | for(j=1;j<=size(K);j++) |
---|
| 2598 | { |
---|
| 2599 | L=J[1],K[j]; |
---|
| 2600 | resu[j]=L; |
---|
| 2601 | } |
---|
| 2602 | return(resu); |
---|
[70ab73] | 2603 | } |
---|
[a36e78] | 2604 | } |
---|
[fc5095] | 2605 | //---Ende Provisorium |
---|
[a36e78] | 2606 | string mp="poly p="+string(minpoly)+";"; |
---|
| 2607 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+","+string(par(1)) |
---|
[24f458] | 2608 | +"),dp;"; |
---|
[a36e78] | 2609 | execute(gnir); |
---|
| 2610 | execute(mp); |
---|
| 2611 | ideal i=imap(R,i); |
---|
| 2612 | ideal I=subst(i,var(nvars(basering)),0); |
---|
| 2613 | int j; |
---|
| 2614 | for(j=1;j<=ncols(i);j++) |
---|
| 2615 | { |
---|
| 2616 | if(i[j]!=I[j]){break;} |
---|
| 2617 | } |
---|
| 2618 | if((j>ncols(i))&&(deg(p)==1)) |
---|
| 2619 | { |
---|
| 2620 | setring R; |
---|
| 2621 | kill RH; |
---|
| 2622 | kill gnir; |
---|
| 2623 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+"),dp;"; |
---|
| 2624 | execute(gnir); |
---|
| 2625 | ideal i=imap(R,i); |
---|
| 2626 | ideal J; |
---|
| 2627 | } |
---|
| 2628 | else |
---|
| 2629 | { |
---|
| 2630 | i=i,p; |
---|
| 2631 | } |
---|
| 2632 | list pr; |
---|
[24f458] | 2633 | |
---|
[a36e78] | 2634 | if(w==0) |
---|
| 2635 | { |
---|
| 2636 | pr=decomp(i); |
---|
| 2637 | } |
---|
| 2638 | if(w==1) |
---|
| 2639 | { |
---|
| 2640 | pr=prim_dec(i,1); |
---|
| 2641 | pr=reconvList(pr); |
---|
| 2642 | } |
---|
| 2643 | if(w==2) |
---|
| 2644 | { |
---|
| 2645 | pr=minAssPrimes(i); |
---|
| 2646 | } |
---|
| 2647 | if(n<nvars(basering)) |
---|
| 2648 | { |
---|
| 2649 | gnir="ring RS="+string(char(R))+",("+varstr(RH) |
---|
[24f458] | 2650 | +"),(dp("+string(n)+"),lp);"; |
---|
[a36e78] | 2651 | execute(gnir); |
---|
| 2652 | list pr=imap(RH,pr); |
---|
| 2653 | ideal K; |
---|
| 2654 | for(j=1;j<=size(pr);j++) |
---|
| 2655 | { |
---|
| 2656 | K=groebner(pr[j]); |
---|
| 2657 | K=K[2..size(K)]; |
---|
| 2658 | pr[j]=K; |
---|
| 2659 | } |
---|
| 2660 | setring R; |
---|
| 2661 | list pr=imap(RS,pr); |
---|
| 2662 | } |
---|
| 2663 | else |
---|
| 2664 | { |
---|
| 2665 | setring R; |
---|
| 2666 | list pr=imap(RH,pr); |
---|
| 2667 | } |
---|
| 2668 | list re; |
---|
| 2669 | if(w==2) |
---|
| 2670 | { |
---|
| 2671 | re=pr; |
---|
| 2672 | } |
---|
| 2673 | else |
---|
| 2674 | { |
---|
| 2675 | re=convList(pr); |
---|
| 2676 | } |
---|
| 2677 | return(re); |
---|
[24f458] | 2678 | } |
---|
[ab8937] | 2679 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2680 | static proc prepare_absprimdec(list primary) |
---|
| 2681 | { |
---|
| 2682 | list resu,tempo; |
---|
| 2683 | string absotto; |
---|
| 2684 | resu[size(primary)/2]=list(); |
---|
| 2685 | for(int ab=1;ab<=size(primary)/2;ab++) |
---|
| 2686 | { |
---|
| 2687 | absotto= absFactorize(primary[2*ab][1],77); |
---|
| 2688 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 2689 | resu[ab]=tempo; |
---|
| 2690 | } |
---|
| 2691 | return(resu); |
---|
| 2692 | } |
---|
[67bd4c] | 2693 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 2694 | static proc decomp(ideal i,list #) |
---|
[7a7df90] | 2695 | "USAGE: decomp(i); i ideal (for primary decomposition) (resp. |
---|
| 2696 | decomp(i,1); (for the associated primes of dimension of i) ) |
---|
| 2697 | decomp(i,2); (for the minimal associated primes) ) |
---|
[6fa3af] | 2698 | decomp(i,3); (for the absolute primary decomposition) ) |
---|
[d6db1f2] | 2699 | RETURN: list = list of primary ideals and their associated primes |
---|
| 2700 | (at even positions in the list) |
---|
| 2701 | (resp. a list of the minimal associated primes) |
---|
[7b3971] | 2702 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
[d6db1f2] | 2703 | EXAMPLE: example decomp; shows an example |
---|
[d2b2a7] | 2704 | " |
---|
[d6db1f2] | 2705 | { |
---|
[7cd077] | 2706 | intvec op,@vv; |
---|
[d6db1f2] | 2707 | def @P = basering; |
---|
[67bd4c] | 2708 | list primary,indep,ltras; |
---|
[d36f7f] | 2709 | intvec @vh,isat,@w; |
---|
[6fa3af] | 2710 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi,abspri,ab,nn; |
---|
[d6db1f2] | 2711 | ideal peek=i; |
---|
| 2712 | ideal ser,tras; |
---|
[24f458] | 2713 | int isS=(attrib(i,"isSB")==1); |
---|
[18dd47] | 2714 | |
---|
[6fa3af] | 2715 | |
---|
[d6db1f2] | 2716 | if(size(#)>0) |
---|
| 2717 | { |
---|
[1d430ab] | 2718 | if((#[1]==1)||(#[1]==2)||(#[1]==3)) |
---|
| 2719 | { |
---|
| 2720 | @wr=#[1]; |
---|
| 2721 | if(@wr==3){abspri=1;@wr=0;} |
---|
| 2722 | if(size(#)>1) |
---|
[d6db1f2] | 2723 | { |
---|
[e801fe] | 2724 | seri=1; |
---|
[1d430ab] | 2725 | peek=#[2]; |
---|
| 2726 | ser=#[3]; |
---|
[d6db1f2] | 2727 | } |
---|
[1d430ab] | 2728 | } |
---|
| 2729 | else |
---|
| 2730 | { |
---|
| 2731 | seri=1; |
---|
| 2732 | peek=#[1]; |
---|
| 2733 | ser=#[2]; |
---|
| 2734 | } |
---|
[d6db1f2] | 2735 | } |
---|
[6fa3af] | 2736 | if(abspri) |
---|
| 2737 | { |
---|
[1d430ab] | 2738 | list absprimary,abskeep,absprimarytmp,abskeeptmp; |
---|
[6fa3af] | 2739 | } |
---|
[e801fe] | 2740 | homo=homog(i); |
---|
[d6db1f2] | 2741 | if(homo==1) |
---|
| 2742 | { |
---|
[e801fe] | 2743 | if(attrib(i,"isSB")!=1) |
---|
| 2744 | { |
---|
[17407e] | 2745 | //ltras=mstd(i); |
---|
| 2746 | tras=groebner(i); |
---|
| 2747 | ltras=tras,tras; |
---|
[e801fe] | 2748 | attrib(ltras[1],"isSB",1); |
---|
| 2749 | } |
---|
| 2750 | else |
---|
| 2751 | { |
---|
| 2752 | ltras=i,i; |
---|
[24f458] | 2753 | attrib(ltras[1],"isSB",1); |
---|
[e801fe] | 2754 | } |
---|
| 2755 | tras=ltras[1]; |
---|
[24f458] | 2756 | attrib(tras,"isSB",1); |
---|
[adde988] | 2757 | if((dim(tras)==0) && (!abspri)) |
---|
[e801fe] | 2758 | { |
---|
[1d430ab] | 2759 | primary[1]=ltras[2]; |
---|
| 2760 | primary[2]=maxideal(1); |
---|
| 2761 | if(@wr>0) |
---|
| 2762 | { |
---|
| 2763 | list l; |
---|
| 2764 | l[1]=maxideal(1); |
---|
| 2765 | l[2]=maxideal(1); |
---|
| 2766 | return(l); |
---|
| 2767 | } |
---|
| 2768 | return(primary); |
---|
| 2769 | } |
---|
| 2770 | for(@n=1;@n<=nvars(basering);@n++) |
---|
| 2771 | { |
---|
| 2772 | @w[@n]=ord(var(@n)); |
---|
| 2773 | } |
---|
| 2774 | intvec @hilb=hilb(tras,1,@w); |
---|
| 2775 | intvec keephilb=@hilb; |
---|
[a36e78] | 2776 | } |
---|
| 2777 | |
---|
| 2778 | //---------------------------------------------------------------- |
---|
[d6db1f2] | 2779 | //i is the zero-ideal |
---|
| 2780 | //---------------------------------------------------------------- |
---|
[18dd47] | 2781 | |
---|
[d6db1f2] | 2782 | if(size(i)==0) |
---|
| 2783 | { |
---|
[810a4af] | 2784 | primary=ideal(0),ideal(0); |
---|
[ab8937] | 2785 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[1d430ab] | 2786 | return(primary); |
---|
[d6db1f2] | 2787 | } |
---|
[18dd47] | 2788 | |
---|
[d6db1f2] | 2789 | //---------------------------------------------------------------- |
---|
| 2790 | //pass to the lexicographical ordering and compute a standardbasis |
---|
| 2791 | //---------------------------------------------------------------- |
---|
| 2792 | |
---|
[24f458] | 2793 | int lp=islp(); |
---|
| 2794 | |
---|
[2d2cad9] | 2795 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
[466f80] | 2796 | op=option(get); |
---|
[d6db1f2] | 2797 | option(redSB); |
---|
[e801fe] | 2798 | |
---|
[3939bc] | 2799 | ideal ser=fetch(@P,ser); |
---|
[18dd47] | 2800 | |
---|
[d6db1f2] | 2801 | if(homo==1) |
---|
| 2802 | { |
---|
[1d430ab] | 2803 | if(!lp) |
---|
| 2804 | { |
---|
| 2805 | ideal @j=std(fetch(@P,i),@hilb,@w); |
---|
| 2806 | } |
---|
| 2807 | else |
---|
| 2808 | { |
---|
| 2809 | ideal @j=fetch(@P,tras); |
---|
| 2810 | attrib(@j,"isSB",1); |
---|
| 2811 | } |
---|
[d6db1f2] | 2812 | } |
---|
| 2813 | else |
---|
| 2814 | { |
---|
[1d430ab] | 2815 | if(lp&&isS) |
---|
| 2816 | { |
---|
| 2817 | ideal @j=fetch(@P,i); |
---|
| 2818 | attrib(@j,"isSB",1); |
---|
| 2819 | } |
---|
| 2820 | else |
---|
| 2821 | { |
---|
| 2822 | ideal @j=groebner(fetch(@P,i)); |
---|
| 2823 | } |
---|
[d6db1f2] | 2824 | } |
---|
[02335e] | 2825 | option(set,op); |
---|
[e801fe] | 2826 | if(seri==1) |
---|
| 2827 | { |
---|
| 2828 | ideal peek=fetch(@P,peek); |
---|
| 2829 | attrib(peek,"isSB",1); |
---|
| 2830 | } |
---|
| 2831 | else |
---|
| 2832 | { |
---|
| 2833 | ideal peek=@j; |
---|
| 2834 | } |
---|
[6fa3af] | 2835 | if((size(ser)==0)&&(!abspri)) |
---|
[e801fe] | 2836 | { |
---|
| 2837 | ideal fried; |
---|
| 2838 | @n=size(@j); |
---|
| 2839 | for(@k=1;@k<=@n;@k++) |
---|
| 2840 | { |
---|
| 2841 | if(deg(lead(@j[@k]))==1) |
---|
| 2842 | { |
---|
| 2843 | fried[size(fried)+1]=@j[@k]; |
---|
| 2844 | @j[@k]=0; |
---|
| 2845 | } |
---|
| 2846 | } |
---|
[5674d5] | 2847 | if(size(fried)==nvars(basering)) |
---|
| 2848 | { |
---|
[1d430ab] | 2849 | setring @P; |
---|
| 2850 | primary[1]=i; |
---|
| 2851 | primary[2]=i; |
---|
[ab8937] | 2852 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[1d430ab] | 2853 | return(primary); |
---|
[5674d5] | 2854 | } |
---|
[e801fe] | 2855 | if(size(fried)>0) |
---|
| 2856 | { |
---|
[1d430ab] | 2857 | string newva; |
---|
| 2858 | string newma; |
---|
| 2859 | for(@k=1;@k<=nvars(basering);@k++) |
---|
| 2860 | { |
---|
| 2861 | @n1=0; |
---|
| 2862 | for(@n=1;@n<=size(fried);@n++) |
---|
| 2863 | { |
---|
| 2864 | if(leadmonom(fried[@n])==var(@k)) |
---|
[a36e78] | 2865 | { |
---|
[1d430ab] | 2866 | @n1=1; |
---|
| 2867 | break; |
---|
[a36e78] | 2868 | } |
---|
[1d430ab] | 2869 | } |
---|
| 2870 | if(@n1==0) |
---|
| 2871 | { |
---|
| 2872 | newva=newva+string(var(@k))+","; |
---|
| 2873 | newma=newma+string(var(@k))+","; |
---|
| 2874 | } |
---|
| 2875 | else |
---|
| 2876 | { |
---|
| 2877 | newma=newma+string(0)+","; |
---|
| 2878 | } |
---|
| 2879 | } |
---|
| 2880 | newva[size(newva)]=")"; |
---|
| 2881 | newma[size(newma)]=";"; |
---|
| 2882 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
---|
| 2883 | execute("map @kappa=gnir,"+newma); |
---|
| 2884 | ideal @j= @kappa(@j); |
---|
| 2885 | @j=simplify(@j,2); |
---|
| 2886 | attrib(@j,"isSB",1); |
---|
| 2887 | list pr=decomp(@j); |
---|
| 2888 | setring gnir; |
---|
| 2889 | list pr=imap(@deirf,pr); |
---|
| 2890 | for(@k=1;@k<=size(pr);@k++) |
---|
| 2891 | { |
---|
| 2892 | @j=pr[@k]+fried; |
---|
| 2893 | pr[@k]=@j; |
---|
| 2894 | } |
---|
| 2895 | setring @P; |
---|
[810a4af] | 2896 | primary=imap(gnir,pr); |
---|
| 2897 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
| 2898 | return(primary); |
---|
[e801fe] | 2899 | } |
---|
| 2900 | } |
---|
[d6db1f2] | 2901 | //---------------------------------------------------------------- |
---|
| 2902 | //j is the ring |
---|
| 2903 | //---------------------------------------------------------------- |
---|
| 2904 | |
---|
| 2905 | if (dim(@j)==-1) |
---|
| 2906 | { |
---|
[e801fe] | 2907 | setring @P; |
---|
[651953] | 2908 | primary=ideal(1),ideal(1); |
---|
[ab8937] | 2909 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[651953] | 2910 | return(primary); |
---|
[d6db1f2] | 2911 | } |
---|
[18dd47] | 2912 | |
---|
[d6db1f2] | 2913 | //---------------------------------------------------------------- |
---|
| 2914 | // the case of one variable |
---|
| 2915 | //---------------------------------------------------------------- |
---|
| 2916 | |
---|
| 2917 | if(nvars(basering)==1) |
---|
| 2918 | { |
---|
[1d430ab] | 2919 | list fac=factor(@j[1]); |
---|
| 2920 | list gprimary; |
---|
| 2921 | for(@k=1;@k<=size(fac[1]);@k++) |
---|
| 2922 | { |
---|
| 2923 | if(@wr==0) |
---|
| 2924 | { |
---|
| 2925 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
---|
| 2926 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 2927 | } |
---|
| 2928 | else |
---|
| 2929 | { |
---|
| 2930 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
---|
| 2931 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 2932 | } |
---|
| 2933 | } |
---|
| 2934 | setring @P; |
---|
| 2935 | primary=fetch(gnir,gprimary); |
---|
[d6db1f2] | 2936 | |
---|
[6fa3af] | 2937 | //HIER |
---|
[ab8937] | 2938 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[1d430ab] | 2939 | return(primary); |
---|
[d6db1f2] | 2940 | } |
---|
[3939bc] | 2941 | |
---|
[d6db1f2] | 2942 | //------------------------------------------------------------------ |
---|
| 2943 | //the zero-dimensional case |
---|
| 2944 | //------------------------------------------------------------------ |
---|
| 2945 | if (dim(@j)==0) |
---|
| 2946 | { |
---|
[466f80] | 2947 | op=option(get); |
---|
[e801fe] | 2948 | option(redSB); |
---|
| 2949 | list gprimary= zero_decomp(@j,ser,@wr); |
---|
[6fa3af] | 2950 | |
---|
[e801fe] | 2951 | setring @P; |
---|
| 2952 | primary=fetch(gnir,gprimary); |
---|
[6fa3af] | 2953 | |
---|
[e801fe] | 2954 | if(size(ser)>0) |
---|
| 2955 | { |
---|
| 2956 | primary=cleanPrimary(primary); |
---|
| 2957 | } |
---|
[6fa3af] | 2958 | //HIER |
---|
| 2959 | if(abspri) |
---|
| 2960 | { |
---|
[1d430ab] | 2961 | setring gnir; |
---|
| 2962 | list primary=imap(@P,primary); |
---|
| 2963 | list resu,tempo; |
---|
| 2964 | string absotto; |
---|
| 2965 | map sigma,invsigma; |
---|
| 2966 | ideal II,jmap; |
---|
| 2967 | nn=nvars(basering); |
---|
| 2968 | for(ab=1;ab<=size(primary)/2;ab++) |
---|
| 2969 | { |
---|
| 2970 | II=primary[2*ab]; |
---|
| 2971 | attrib(II,"isSB",1); |
---|
| 2972 | if(deg(II[1])==vdim(II)) |
---|
| 2973 | { |
---|
[a36e78] | 2974 | absotto= absFactorize(primary[2*ab][1],77); |
---|
[1d430ab] | 2975 | tempo= |
---|
| 2976 | primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 2977 | } |
---|
| 2978 | else |
---|
| 2979 | { |
---|
| 2980 | invsigma=basering,maxideal(1); |
---|
| 2981 | jmap=randomLast(50); |
---|
| 2982 | sigma=basering,jmap; |
---|
| 2983 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 2984 | invsigma=basering,jmap; |
---|
| 2985 | II=groebner(sigma(II)); |
---|
| 2986 | absotto = absFactorize(II[1],77); |
---|
| 2987 | II=var(nn); |
---|
| 2988 | tempo= primary[2*ab-1],primary[2*ab],absotto,string(invsigma(II)); |
---|
| 2989 | } |
---|
| 2990 | resu[ab]=tempo; |
---|
| 2991 | } |
---|
| 2992 | primary=resu; |
---|
| 2993 | setring @P; |
---|
| 2994 | primary=imap(gnir,primary); |
---|
[6fa3af] | 2995 | } |
---|
[e801fe] | 2996 | return(primary); |
---|
| 2997 | } |
---|
[d6db1f2] | 2998 | |
---|
| 2999 | poly @gs,@gh,@p; |
---|
| 3000 | string @va,quotring; |
---|
| 3001 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
---|
| 3002 | ideal @h; |
---|
| 3003 | int jdim=dim(@j); |
---|
| 3004 | list fett; |
---|
[e801fe] | 3005 | int lauf,di,newtest; |
---|
[67bd4c] | 3006 | //------------------------------------------------------------------ |
---|
| 3007 | //search for a maximal independent set indep,i.e. |
---|
| 3008 | //look for subring such that the intersection with the ideal is zero |
---|
| 3009 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
---|
[9050ca] | 3010 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
---|
[67bd4c] | 3011 | //------------------------------------------------------------------ |
---|
[d6db1f2] | 3012 | if(@wr!=1) |
---|
| 3013 | { |
---|
[1d430ab] | 3014 | allindep=independSet(@j); |
---|
| 3015 | for(@m=1;@m<=size(allindep);@m++) |
---|
| 3016 | { |
---|
| 3017 | if(allindep[@m][3]==jdim) |
---|
| 3018 | { |
---|
| 3019 | di++; |
---|
| 3020 | indep[di]=allindep[@m]; |
---|
| 3021 | } |
---|
| 3022 | else |
---|
| 3023 | { |
---|
| 3024 | lauf++; |
---|
| 3025 | restindep[lauf]=allindep[@m]; |
---|
| 3026 | } |
---|
| 3027 | } |
---|
| 3028 | } |
---|
| 3029 | else |
---|
| 3030 | { |
---|
| 3031 | indep=maxIndependSet(@j); |
---|
| 3032 | } |
---|
[3939bc] | 3033 | |
---|
[d6db1f2] | 3034 | ideal jkeep=@j; |
---|
| 3035 | if(ordstr(@P)[1]=="w") |
---|
| 3036 | { |
---|
[1d430ab] | 3037 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
---|
[d6db1f2] | 3038 | } |
---|
| 3039 | else |
---|
| 3040 | { |
---|
[1d430ab] | 3041 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
---|
[e801fe] | 3042 | } |
---|
| 3043 | |
---|
| 3044 | if(homo==1) |
---|
| 3045 | { |
---|
| 3046 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
---|
| 3047 | ||(ordstr(@P)[3]=="w")) |
---|
| 3048 | { |
---|
| 3049 | ideal jwork=imap(@P,tras); |
---|
| 3050 | attrib(jwork,"isSB",1); |
---|
| 3051 | } |
---|
| 3052 | else |
---|
| 3053 | { |
---|
[2d2c8be] | 3054 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
---|
[e801fe] | 3055 | } |
---|
| 3056 | } |
---|
| 3057 | else |
---|
| 3058 | { |
---|
[9a384e] | 3059 | ideal jwork=groebner(imap(gnir,@j)); |
---|
[d6db1f2] | 3060 | } |
---|
[e801fe] | 3061 | list hquprimary; |
---|
[d6db1f2] | 3062 | poly @p,@q; |
---|
[e801fe] | 3063 | ideal @h,fac,ser; |
---|
[5c7562] | 3064 | ideal @Ptest=1; |
---|
[d6db1f2] | 3065 | di=dim(jwork); |
---|
[e801fe] | 3066 | keepdi=di; |
---|
[3939bc] | 3067 | |
---|
[d6db1f2] | 3068 | setring gnir; |
---|
| 3069 | for(@m=1;@m<=size(indep);@m++) |
---|
| 3070 | { |
---|
[1d430ab] | 3071 | isat=0; |
---|
| 3072 | @n2=0; |
---|
| 3073 | if((indep[@m][1]==varstr(basering))&&(@m==1)) |
---|
| 3074 | //this is the good case, nothing to do, just to have the same notations |
---|
| 3075 | //change the ring |
---|
| 3076 | { |
---|
| 3077 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[a36e78] | 3078 | +ordstr(basering)+");"); |
---|
[1d430ab] | 3079 | ideal @j=fetch(gnir,@j); |
---|
| 3080 | attrib(@j,"isSB",1); |
---|
| 3081 | ideal ser=fetch(gnir,ser); |
---|
| 3082 | } |
---|
| 3083 | else |
---|
| 3084 | { |
---|
| 3085 | @va=string(maxideal(1)); |
---|
| 3086 | if(@m==1) |
---|
| 3087 | { |
---|
| 3088 | @j=fetch(@P,i); |
---|
| 3089 | } |
---|
| 3090 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[@m][1]+"),(" |
---|
[2d2cad9] | 3091 | +indep[@m][2]+");"); |
---|
[1d430ab] | 3092 | execute("map phi=gnir,"+@va+";"); |
---|
| 3093 | op=option(get); |
---|
| 3094 | option(redSB); |
---|
| 3095 | if(homo==1) |
---|
| 3096 | { |
---|
| 3097 | ideal @j=std(phi(@j),@hilb,@w); |
---|
| 3098 | } |
---|
| 3099 | else |
---|
| 3100 | { |
---|
| 3101 | ideal @j=groebner(phi(@j)); |
---|
| 3102 | } |
---|
| 3103 | ideal ser=phi(ser); |
---|
[3939bc] | 3104 | |
---|
[1d430ab] | 3105 | option(set,op); |
---|
| 3106 | } |
---|
| 3107 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
---|
| 3108 | { |
---|
| 3109 | setring gnir; |
---|
| 3110 | break; |
---|
| 3111 | } |
---|
| 3112 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 3113 | { |
---|
| 3114 | fett[lauf]=size(@j[lauf]); |
---|
| 3115 | } |
---|
| 3116 | //------------------------------------------------------------------------ |
---|
| 3117 | //we have now the following situation: |
---|
| 3118 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
| 3119 | //to this quotientring, j is their still a standardbasis, the |
---|
| 3120 | //leading coefficients of the polynomials there (polynomials in |
---|
| 3121 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 3122 | //we need their ggt, gh, because of the following: let |
---|
| 3123 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3124 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 3125 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 3126 | |
---|
| 3127 | //------------------------------------------------------------------------ |
---|
| 3128 | |
---|
| 3129 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
---|
| 3130 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
---|
| 3131 | //------------------------------------------------------------------------ |
---|
| 3132 | |
---|
| 3133 | quotring=prepareQuotientring(nvars(basering)-indep[@m][3]); |
---|
| 3134 | |
---|
| 3135 | //--------------------------------------------------------------------- |
---|
| 3136 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3137 | //--------------------------------------------------------------------- |
---|
| 3138 | |
---|
| 3139 | ideal @jj=lead(@j); //!! vorn vereinbaren |
---|
| 3140 | execute(quotring); |
---|
| 3141 | |
---|
| 3142 | ideal @jj=imap(gnir1,@jj); |
---|
| 3143 | @vv=clearSBNeu(@jj,fett); //!! vorn vereinbaren |
---|
| 3144 | setring gnir1; |
---|
| 3145 | @k=size(@j); |
---|
| 3146 | for (lauf=1;lauf<=@k;lauf++) |
---|
| 3147 | { |
---|
| 3148 | if(@vv[lauf]==1) |
---|
| 3149 | { |
---|
| 3150 | @j[lauf]=0; |
---|
| 3151 | } |
---|
| 3152 | } |
---|
| 3153 | @j=simplify(@j,2); |
---|
| 3154 | setring quring; |
---|
| 3155 | // @j considered in the quotientring |
---|
| 3156 | ideal @j=imap(gnir1,@j); |
---|
[70ab73] | 3157 | |
---|
[1d430ab] | 3158 | ideal ser=imap(gnir1,ser); |
---|
[70ab73] | 3159 | |
---|
[1d430ab] | 3160 | kill gnir1; |
---|
[70ab73] | 3161 | |
---|
[1d430ab] | 3162 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 3163 | //here it becomes minimal |
---|
[70ab73] | 3164 | |
---|
[1d430ab] | 3165 | attrib(@j,"isSB",1); |
---|
[70ab73] | 3166 | |
---|
[1d430ab] | 3167 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 3168 | ideal @h; |
---|
| 3169 | if(deg(@j[1])>0) |
---|
| 3170 | { |
---|
| 3171 | for(@n=1;@n<=size(@j);@n++) |
---|
| 3172 | { |
---|
| 3173 | @h[@n]=leadcoef(@j[@n]); |
---|
| 3174 | } |
---|
| 3175 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3176 | op=option(get); |
---|
| 3177 | option(redSB); |
---|
[70ab73] | 3178 | |
---|
[1d430ab] | 3179 | list uprimary= zero_decomp(@j,ser,@wr); |
---|
[a36e78] | 3180 | //HIER |
---|
[1d430ab] | 3181 | if(abspri) |
---|
| 3182 | { |
---|
| 3183 | ideal II; |
---|
| 3184 | ideal jmap; |
---|
| 3185 | map sigma; |
---|
| 3186 | nn=nvars(basering); |
---|
| 3187 | map invsigma=basering,maxideal(1); |
---|
| 3188 | for(ab=1;ab<=size(uprimary)/2;ab++) |
---|
[a36e78] | 3189 | { |
---|
[1d430ab] | 3190 | II=uprimary[2*ab]; |
---|
| 3191 | attrib(II,"isSB",1); |
---|
| 3192 | if(deg(II[1])!=vdim(II)) |
---|
| 3193 | { |
---|
| 3194 | jmap=randomLast(50); |
---|
| 3195 | sigma=basering,jmap; |
---|
| 3196 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 3197 | invsigma=basering,jmap; |
---|
| 3198 | II=groebner(sigma(II)); |
---|
| 3199 | } |
---|
| 3200 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 3201 | II=var(nn); |
---|
| 3202 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 3203 | invsigma=basering,maxideal(1); |
---|
[a36e78] | 3204 | } |
---|
[1d430ab] | 3205 | } |
---|
| 3206 | option(set,op); |
---|
| 3207 | } |
---|
| 3208 | else |
---|
| 3209 | { |
---|
| 3210 | list uprimary; |
---|
| 3211 | uprimary[1]=ideal(1); |
---|
| 3212 | uprimary[2]=ideal(1); |
---|
| 3213 | } |
---|
| 3214 | //we need the intersection of the ideals in the list quprimary with the |
---|
| 3215 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
| 3216 | //but fi polynomials, then the intersection of q with the polynomialring |
---|
| 3217 | //is the saturation of the ideal generated by f1,...,fr with respect to |
---|
| 3218 | //h which is the lcm of the leading coefficients of the fi considered in |
---|
| 3219 | //in the quotientring: this is coded in saturn |
---|
[f54c83] | 3220 | |
---|
[1d430ab] | 3221 | list saturn; |
---|
| 3222 | ideal hpl; |
---|
[d6db1f2] | 3223 | |
---|
[1d430ab] | 3224 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 3225 | { |
---|
| 3226 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
---|
| 3227 | hpl=0; |
---|
| 3228 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 3229 | { |
---|
| 3230 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 3231 | } |
---|
| 3232 | saturn[@n]=hpl; |
---|
| 3233 | } |
---|
[18dd47] | 3234 | |
---|
[1d430ab] | 3235 | //-------------------------------------------------------------------- |
---|
| 3236 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3237 | //back to the polynomialring |
---|
| 3238 | //--------------------------------------------------------------------- |
---|
| 3239 | setring gnir; |
---|
[d6db1f2] | 3240 | |
---|
[1d430ab] | 3241 | collectprimary=imap(quring,uprimary); |
---|
| 3242 | lsau=imap(quring,saturn); |
---|
| 3243 | @h=imap(quring,@h); |
---|
[d6db1f2] | 3244 | |
---|
[1d430ab] | 3245 | kill quring; |
---|
[7a7df90] | 3246 | |
---|
[1d430ab] | 3247 | @n2=size(quprimary); |
---|
| 3248 | @n3=@n2; |
---|
[a36e78] | 3249 | |
---|
[1d430ab] | 3250 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
---|
| 3251 | { |
---|
| 3252 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 3253 | { |
---|
| 3254 | @n2++; |
---|
| 3255 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 3256 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 3257 | @n2++; |
---|
| 3258 | lnew[@n2]=lsau[2*@n1]; |
---|
| 3259 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 3260 | if(abspri) |
---|
[d6db1f2] | 3261 | { |
---|
[1d430ab] | 3262 | absprimary[@n2/2]=absprimarytmp[@n1]; |
---|
| 3263 | abskeep[@n2/2]=abskeeptmp[@n1]; |
---|
[d6db1f2] | 3264 | } |
---|
[1d430ab] | 3265 | } |
---|
| 3266 | } |
---|
| 3267 | //here the intersection with the polynomialring |
---|
| 3268 | //mentioned above is really computed |
---|
| 3269 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
| 3270 | { |
---|
| 3271 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
| 3272 | { |
---|
| 3273 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 3274 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 3275 | } |
---|
| 3276 | else |
---|
| 3277 | { |
---|
| 3278 | if(@wr==0) |
---|
[d6db1f2] | 3279 | { |
---|
[1d430ab] | 3280 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
[d6db1f2] | 3281 | } |
---|
[1d430ab] | 3282 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
| 3283 | } |
---|
| 3284 | } |
---|
[3939bc] | 3285 | |
---|
[1d430ab] | 3286 | if(size(@h)>0) |
---|
| 3287 | { |
---|
| 3288 | //--------------------------------------------------------------- |
---|
| 3289 | //we change to @Phelp to have the ordering dp for saturation |
---|
| 3290 | //--------------------------------------------------------------- |
---|
| 3291 | setring @Phelp; |
---|
| 3292 | @h=imap(gnir,@h); |
---|
| 3293 | if(@wr!=1) |
---|
| 3294 | { |
---|
| 3295 | if(defined(@LL)){kill @LL;} |
---|
| 3296 | list @LL=minSat(jwork,@h); |
---|
| 3297 | @Ptest=intersect(@Ptest,@LL[1]); |
---|
| 3298 | @q=@LL[2]; |
---|
| 3299 | } |
---|
| 3300 | else |
---|
| 3301 | { |
---|
| 3302 | fac=ideal(0); |
---|
| 3303 | for(lauf=1;lauf<=ncols(@h);lauf++) |
---|
[a36e78] | 3304 | { |
---|
[1d430ab] | 3305 | if(deg(@h[lauf])>0) |
---|
| 3306 | { |
---|
| 3307 | fac=fac+factorize(@h[lauf],1); |
---|
| 3308 | } |
---|
[a36e78] | 3309 | } |
---|
[1d430ab] | 3310 | fac=simplify(fac,6); |
---|
| 3311 | @q=1; |
---|
| 3312 | for(lauf=1;lauf<=size(fac);lauf++) |
---|
[a36e78] | 3313 | { |
---|
[1d430ab] | 3314 | @q=@q*fac[lauf]; |
---|
[a36e78] | 3315 | } |
---|
[1d430ab] | 3316 | } |
---|
| 3317 | jwork=std(jwork,@q); |
---|
| 3318 | keepdi=dim(jwork); |
---|
| 3319 | if(keepdi<di) |
---|
| 3320 | { |
---|
[d6db1f2] | 3321 | setring gnir; |
---|
| 3322 | @j=imap(@Phelp,jwork); |
---|
[1d430ab] | 3323 | break; |
---|
| 3324 | } |
---|
| 3325 | if(homo==1) |
---|
| 3326 | { |
---|
| 3327 | @hilb=hilb(jwork,1,@w); |
---|
| 3328 | } |
---|
| 3329 | |
---|
| 3330 | setring gnir; |
---|
| 3331 | @j=imap(@Phelp,jwork); |
---|
| 3332 | } |
---|
[d6db1f2] | 3333 | } |
---|
[7a7df90] | 3334 | |
---|
| 3335 | if((size(quprimary)==0)&&(@wr==1)) |
---|
[d6db1f2] | 3336 | { |
---|
[1d430ab] | 3337 | @j=ideal(1); |
---|
| 3338 | quprimary[1]=ideal(1); |
---|
| 3339 | quprimary[2]=ideal(1); |
---|
[d6db1f2] | 3340 | } |
---|
[e801fe] | 3341 | if((size(quprimary)==0)) |
---|
| 3342 | { |
---|
| 3343 | keepdi=di-1; |
---|
[17407e] | 3344 | quprimary[1]=ideal(1); |
---|
| 3345 | quprimary[2]=ideal(1); |
---|
[3939bc] | 3346 | } |
---|
[d6db1f2] | 3347 | //--------------------------------------------------------------- |
---|
| 3348 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
---|
| 3349 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
---|
| 3350 | //--------------------------------------------------------------- |
---|
| 3351 | if((deg(@j[1])!=0)&&(@wr!=1)) |
---|
| 3352 | { |
---|
[1d430ab] | 3353 | if(size(quprimary)>0) |
---|
| 3354 | { |
---|
| 3355 | setring @Phelp; |
---|
| 3356 | ser=imap(gnir,ser); |
---|
| 3357 | hquprimary=imap(gnir,quprimary); |
---|
| 3358 | if(@wr==0) |
---|
| 3359 | { |
---|
| 3360 | //HIER STATT DURCHSCHNITT SATURIEREN! |
---|
| 3361 | ideal htest=@Ptest; |
---|
| 3362 | } |
---|
| 3363 | else |
---|
| 3364 | { |
---|
| 3365 | ideal htest=hquprimary[2]; |
---|
| 3366 | |
---|
| 3367 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
---|
[d6db1f2] | 3368 | { |
---|
[1d430ab] | 3369 | htest=intersect(htest,hquprimary[2*@n1]); |
---|
[d6db1f2] | 3370 | } |
---|
[1d430ab] | 3371 | } |
---|
[d6db1f2] | 3372 | |
---|
[1d430ab] | 3373 | if(size(ser)>0) |
---|
| 3374 | { |
---|
| 3375 | ser=intersect(htest,ser); |
---|
| 3376 | } |
---|
| 3377 | else |
---|
| 3378 | { |
---|
| 3379 | ser=htest; |
---|
| 3380 | } |
---|
| 3381 | setring gnir; |
---|
| 3382 | ser=imap(@Phelp,ser); |
---|
| 3383 | } |
---|
| 3384 | if(size(reduce(ser,peek,1))!=0) |
---|
| 3385 | { |
---|
| 3386 | for(@m=1;@m<=size(restindep);@m++) |
---|
| 3387 | { |
---|
| 3388 | // if(restindep[@m][3]>=keepdi) |
---|
| 3389 | // { |
---|
| 3390 | isat=0; |
---|
| 3391 | @n2=0; |
---|
[e801fe] | 3392 | |
---|
[1d430ab] | 3393 | if(restindep[@m][1]==varstr(basering)) |
---|
| 3394 | //the good case, nothing to do, just to have the same notations |
---|
| 3395 | //change the ring |
---|
[3939bc] | 3396 | { |
---|
[1d430ab] | 3397 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
| 3398 | varstr(basering)+"),("+ordstr(basering)+");"); |
---|
| 3399 | ideal @j=fetch(gnir,jkeep); |
---|
| 3400 | attrib(@j,"isSB",1); |
---|
[d6db1f2] | 3401 | } |
---|
[a36e78] | 3402 | else |
---|
[d6db1f2] | 3403 | { |
---|
[1d430ab] | 3404 | @va=string(maxideal(1)); |
---|
| 3405 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
[a36e78] | 3406 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
---|
[1d430ab] | 3407 | execute("map phi=gnir,"+@va+";"); |
---|
| 3408 | op=option(get); |
---|
| 3409 | option(redSB); |
---|
| 3410 | if(homo==1) |
---|
| 3411 | { |
---|
| 3412 | ideal @j=std(phi(jkeep),keephilb,@w); |
---|
| 3413 | } |
---|
| 3414 | else |
---|
| 3415 | { |
---|
| 3416 | ideal @j=groebner(phi(jkeep)); |
---|
| 3417 | } |
---|
| 3418 | ideal ser=phi(ser); |
---|
| 3419 | option(set,op); |
---|
| 3420 | } |
---|
[a36e78] | 3421 | |
---|
[1d430ab] | 3422 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 3423 | { |
---|
| 3424 | fett[lauf]=size(@j[lauf]); |
---|
| 3425 | } |
---|
| 3426 | //------------------------------------------------------------------ |
---|
| 3427 | //we have now the following situation: |
---|
| 3428 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
---|
| 3429 | //pass to this quotientring, j is their still a standardbasis, the |
---|
| 3430 | //leading coefficients of the polynomials there (polynomials in |
---|
| 3431 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 3432 | //we need their ggt, gh, because of the following: |
---|
| 3433 | //let (j:gh^n)=(j:gh^infinity) then |
---|
| 3434 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3435 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 3436 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 3437 | |
---|
| 3438 | //------------------------------------------------------------------ |
---|
| 3439 | |
---|
| 3440 | //the arrangement for the quotientring |
---|
| 3441 | // K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3442 | //and the map phi:K[var(1),...,var(nva)] ----> |
---|
| 3443 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
| 3444 | //------------------------------------------------------------------ |
---|
| 3445 | |
---|
| 3446 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
---|
| 3447 | |
---|
| 3448 | //------------------------------------------------------------------ |
---|
| 3449 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 3450 | //------------------------------------------------------------------ |
---|
| 3451 | |
---|
| 3452 | execute(quotring); |
---|
| 3453 | |
---|
| 3454 | // @j considered in the quotientring |
---|
| 3455 | ideal @j=imap(gnir1,@j); |
---|
| 3456 | ideal ser=imap(gnir1,ser); |
---|
| 3457 | |
---|
| 3458 | kill gnir1; |
---|
| 3459 | |
---|
| 3460 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 3461 | //here it becomes minimal |
---|
| 3462 | @j=clearSB(@j,fett); |
---|
| 3463 | attrib(@j,"isSB",1); |
---|
[a36e78] | 3464 | |
---|
[1d430ab] | 3465 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 3466 | ideal @h; |
---|
[a36e78] | 3467 | |
---|
[1d430ab] | 3468 | for(@n=1;@n<=size(@j);@n++) |
---|
| 3469 | { |
---|
| 3470 | @h[@n]=leadcoef(@j[@n]); |
---|
| 3471 | } |
---|
| 3472 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
[a36e78] | 3473 | |
---|
[1d430ab] | 3474 | op=option(get); |
---|
| 3475 | option(redSB); |
---|
| 3476 | list uprimary= zero_decomp(@j,ser,@wr); |
---|
[a36e78] | 3477 | //HIER |
---|
[1d430ab] | 3478 | if(abspri) |
---|
| 3479 | { |
---|
| 3480 | ideal II; |
---|
| 3481 | ideal jmap; |
---|
| 3482 | map sigma; |
---|
| 3483 | nn=nvars(basering); |
---|
| 3484 | map invsigma=basering,maxideal(1); |
---|
| 3485 | for(ab=1;ab<=size(uprimary)/2;ab++) |
---|
| 3486 | { |
---|
| 3487 | II=uprimary[2*ab]; |
---|
| 3488 | attrib(II,"isSB",1); |
---|
| 3489 | if(deg(II[1])!=vdim(II)) |
---|
| 3490 | { |
---|
| 3491 | jmap=randomLast(50); |
---|
| 3492 | sigma=basering,jmap; |
---|
| 3493 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 3494 | invsigma=basering,jmap; |
---|
| 3495 | II=groebner(sigma(II)); |
---|
| 3496 | } |
---|
| 3497 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 3498 | II=var(nn); |
---|
| 3499 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 3500 | invsigma=basering,maxideal(1); |
---|
| 3501 | } |
---|
| 3502 | } |
---|
| 3503 | option(set,op); |
---|
[a36e78] | 3504 | |
---|
[1d430ab] | 3505 | //we need the intersection of the ideals in the list quprimary with |
---|
| 3506 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
---|
| 3507 | //such an ideal but fi polynomials, then the intersection of q with |
---|
| 3508 | //the polynomialring is the saturation of the ideal generated by |
---|
| 3509 | //f1,...,fr with respect toh which is the lcm of the leading |
---|
| 3510 | //coefficients of the fi considered in the quotientring: |
---|
| 3511 | //this is coded in saturn |
---|
[a36e78] | 3512 | |
---|
[1d430ab] | 3513 | list saturn; |
---|
| 3514 | ideal hpl; |
---|
[a36e78] | 3515 | |
---|
[1d430ab] | 3516 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 3517 | { |
---|
| 3518 | hpl=0; |
---|
| 3519 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 3520 | { |
---|
| 3521 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 3522 | } |
---|
| 3523 | saturn[@n]=hpl; |
---|
| 3524 | } |
---|
| 3525 | //------------------------------------------------------------------ |
---|
| 3526 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 3527 | //back to the polynomialring |
---|
| 3528 | //------------------------------------------------------------------ |
---|
| 3529 | setring gnir; |
---|
| 3530 | collectprimary=imap(quring,uprimary); |
---|
| 3531 | lsau=imap(quring,saturn); |
---|
| 3532 | @h=imap(quring,@h); |
---|
[a36e78] | 3533 | |
---|
[1d430ab] | 3534 | kill quring; |
---|
[a36e78] | 3535 | |
---|
[1d430ab] | 3536 | @n2=size(quprimary); |
---|
| 3537 | @n3=@n2; |
---|
[a36e78] | 3538 | |
---|
[1d430ab] | 3539 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
---|
| 3540 | { |
---|
| 3541 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 3542 | { |
---|
| 3543 | @n2++; |
---|
| 3544 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 3545 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 3546 | @n2++; |
---|
| 3547 | lnew[@n2]=lsau[2*@n1]; |
---|
| 3548 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 3549 | if(abspri) |
---|
| 3550 | { |
---|
| 3551 | absprimary[@n2/2]=absprimarytmp[@n1]; |
---|
| 3552 | abskeep[@n2/2]=abskeeptmp[@n1]; |
---|
| 3553 | } |
---|
| 3554 | } |
---|
| 3555 | } |
---|
[a36e78] | 3556 | |
---|
| 3557 | |
---|
[1d430ab] | 3558 | //here the intersection with the polynomialring |
---|
| 3559 | //mentioned above is really computed |
---|
[70ab73] | 3560 | |
---|
[1d430ab] | 3561 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
[6fa3af] | 3562 | { |
---|
[1d430ab] | 3563 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
[6fa3af] | 3564 | { |
---|
[1d430ab] | 3565 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 3566 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 3567 | } |
---|
| 3568 | else |
---|
| 3569 | { |
---|
| 3570 | if(@wr==0) |
---|
| 3571 | { |
---|
| 3572 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 3573 | } |
---|
| 3574 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
[6fa3af] | 3575 | } |
---|
| 3576 | } |
---|
[1d430ab] | 3577 | if(@n2>=@n3+2) |
---|
[d6db1f2] | 3578 | { |
---|
[1d430ab] | 3579 | setring @Phelp; |
---|
| 3580 | ser=imap(gnir,ser); |
---|
| 3581 | hquprimary=imap(gnir,quprimary); |
---|
| 3582 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
| 3583 | { |
---|
| 3584 | if(@wr==0) |
---|
| 3585 | { |
---|
| 3586 | ser=intersect(ser,hquprimary[2*@n-1]); |
---|
| 3587 | } |
---|
| 3588 | else |
---|
| 3589 | { |
---|
| 3590 | ser=intersect(ser,hquprimary[2*@n]); |
---|
| 3591 | } |
---|
| 3592 | } |
---|
| 3593 | setring gnir; |
---|
| 3594 | ser=imap(@Phelp,ser); |
---|
[d6db1f2] | 3595 | } |
---|
[3939bc] | 3596 | |
---|
[1d430ab] | 3597 | // } |
---|
| 3598 | } |
---|
| 3599 | //HIER |
---|
[6fa3af] | 3600 | if(abspri) |
---|
| 3601 | { |
---|
| 3602 | list resu,tempo; |
---|
| 3603 | for(ab=1;ab<=size(quprimary)/2;ab++) |
---|
| 3604 | { |
---|
[1d430ab] | 3605 | if (deg(quprimary[2*ab][1])!=0) |
---|
| 3606 | { |
---|
| 3607 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 3608 | absprimary[ab],abskeep[ab]; |
---|
| 3609 | resu[ab]=tempo; |
---|
| 3610 | } |
---|
[6fa3af] | 3611 | } |
---|
| 3612 | quprimary=resu; |
---|
[1d430ab] | 3613 | @wr=3; |
---|
[70ab73] | 3614 | } |
---|
[1d430ab] | 3615 | if(size(reduce(ser,peek,1))!=0) |
---|
| 3616 | { |
---|
| 3617 | if(@wr>0) |
---|
| 3618 | { |
---|
| 3619 | htprimary=decomp(@j,@wr,peek,ser); |
---|
| 3620 | } |
---|
| 3621 | else |
---|
| 3622 | { |
---|
| 3623 | htprimary=decomp(@j,peek,ser); |
---|
| 3624 | } |
---|
| 3625 | // here we collect now both results primary(sat(j,gh)) |
---|
| 3626 | // and primary(j,gh^n) |
---|
| 3627 | @n=size(quprimary); |
---|
| 3628 | for (@k=1;@k<=size(htprimary);@k++) |
---|
| 3629 | { |
---|
| 3630 | quprimary[@n+@k]=htprimary[@k]; |
---|
| 3631 | } |
---|
| 3632 | } |
---|
| 3633 | } |
---|
| 3634 | } |
---|
| 3635 | else |
---|
| 3636 | { |
---|
| 3637 | if(abspri) |
---|
| 3638 | { |
---|
| 3639 | list resu,tempo; |
---|
| 3640 | for(ab=1;ab<=size(quprimary)/2;ab++) |
---|
| 3641 | { |
---|
| 3642 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 3643 | absprimary[ab],abskeep[ab]; |
---|
| 3644 | resu[ab]=tempo; |
---|
| 3645 | } |
---|
| 3646 | quprimary=resu; |
---|
| 3647 | } |
---|
| 3648 | } |
---|
[091424] | 3649 | //--------------------------------------------------------------------------- |
---|
[d6db1f2] | 3650 | //back to the ring we started with |
---|
| 3651 | //the final result: primary |
---|
[091424] | 3652 | //--------------------------------------------------------------------------- |
---|
[d6db1f2] | 3653 | setring @P; |
---|
| 3654 | primary=imap(gnir,quprimary); |
---|
[0ccdf4] | 3655 | if(!abspri) |
---|
| 3656 | { |
---|
[1d430ab] | 3657 | primary=cleanPrimary(primary); |
---|
[0ccdf4] | 3658 | } |
---|
[d92713] | 3659 | if (abspri && (typeof(primary[1][1])=="poly")) |
---|
| 3660 | { return(prepare_absprimdec(primary));} |
---|
[d6db1f2] | 3661 | return(primary); |
---|
| 3662 | } |
---|
[a36e78] | 3663 | |
---|
| 3664 | |
---|
[d6db1f2] | 3665 | example |
---|
| 3666 | { "EXAMPLE:"; echo = 2; |
---|
| 3667 | ring r = 32003,(x,y,z),lp; |
---|
| 3668 | poly p = z2+1; |
---|
| 3669 | poly q = z4+2; |
---|
| 3670 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 3671 | list pr= decomp(i); |
---|
| 3672 | pr; |
---|
[18dd47] | 3673 | testPrimary( pr, i); |
---|
[d6db1f2] | 3674 | } |
---|
[67bd4c] | 3675 | |
---|
| 3676 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 3677 | static proc powerCoeffs(poly f,int e) |
---|
[80654d] | 3678 | //computes a polynomial with the same monomials as f but coefficients |
---|
| 3679 | //the p^e th power of the coefficients of f |
---|
[67bd4c] | 3680 | { |
---|
[a36e78] | 3681 | int i; |
---|
| 3682 | poly g; |
---|
| 3683 | int ex=char(basering)^e; |
---|
| 3684 | for(i=1;i<=size(f);i++) |
---|
| 3685 | { |
---|
| 3686 | g=g+leadcoef(f[i])^ex*leadmonom(f[i]); |
---|
| 3687 | } |
---|
| 3688 | return(g); |
---|
[80654d] | 3689 | } |
---|
| 3690 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 3691 | |
---|
[fc5095] | 3692 | proc sep(poly f,int i, list #) |
---|
[80654d] | 3693 | "USAGE: input: a polynomial f depending on the i-th variable and optional |
---|
| 3694 | an integer k considering the polynomial f defined over Fp(t1,...,tm) |
---|
| 3695 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
| 3696 | RETURN: the separabel part of f as polynomial in Fp(t1,...,tm) |
---|
[b9b906] | 3697 | and an integer k to indicate that f should be considerd |
---|
[80654d] | 3698 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
| 3699 | EXAMPLE: example sep; shows an example |
---|
| 3700 | { |
---|
[a36e78] | 3701 | def R=basering; |
---|
| 3702 | int k; |
---|
| 3703 | if(size(#)>0){k=#[1];} |
---|
[fc5095] | 3704 | |
---|
[80654d] | 3705 | |
---|
[a36e78] | 3706 | poly h=gcd(f,diff(f,var(i))); |
---|
| 3707 | if((reduce(f,std(h))!=0)||(reduce(diff(f,var(i)),std(h))!=0)) |
---|
| 3708 | { |
---|
| 3709 | ERROR("FEHLER IN GCD"); |
---|
| 3710 | } |
---|
| 3711 | poly g1=lift(h,f)[1][1]; // f/h |
---|
| 3712 | poly h1; |
---|
| 3713 | |
---|
| 3714 | while(h!=h1) |
---|
| 3715 | { |
---|
| 3716 | h1=h; |
---|
| 3717 | h=gcd(h,diff(h,var(i))); |
---|
| 3718 | } |
---|
[80654d] | 3719 | |
---|
[a36e78] | 3720 | if(deg(h1)==0){return(list(g1,k));} //in characteristic 0 we return here |
---|
[80654d] | 3721 | |
---|
[a36e78] | 3722 | k++; |
---|
[80654d] | 3723 | |
---|
[a36e78] | 3724 | ideal ma=maxideal(1); |
---|
| 3725 | ma[i]=var(i)^char(R); |
---|
| 3726 | map phi=R,ma; |
---|
| 3727 | ideal hh=h; //this is technical because preimage works only for ideals |
---|
[80654d] | 3728 | |
---|
[a36e78] | 3729 | poly u=preimage(R,phi,hh)[1]; //h=u(x(i)^p) |
---|
[80654d] | 3730 | |
---|
[a36e78] | 3731 | list g2=sep(u,i,k); //we consider u(t(1)^(p^-1),...,t(m)^(p^-1)) |
---|
| 3732 | g1=powerCoeffs(g1,g2[2]-k+1); //to have g1 over the same field as g2[1] |
---|
[80654d] | 3733 | |
---|
[a36e78] | 3734 | list g3=sep(g1*g2[1],i,g2[2]); |
---|
| 3735 | return(g3); |
---|
[80654d] | 3736 | } |
---|
| 3737 | example |
---|
| 3738 | { "EXAMPLE:"; echo = 2; |
---|
| 3739 | ring R=(5,t,s),(x,y,z),dp; |
---|
| 3740 | poly f=(x^25-t*x^5+t)*(x^3+s); |
---|
| 3741 | sep(f,1); |
---|
| 3742 | } |
---|
| 3743 | |
---|
| 3744 | /////////////////////////////////////////////////////////////////////////////// |
---|
[24f458] | 3745 | proc zeroRad(ideal I,list #) |
---|
[80654d] | 3746 | "USAGE: zeroRad(I) , I a zero-dimensional ideal |
---|
| 3747 | RETURN: the radical of I |
---|
| 3748 | NOTE: Algorithm of Kemper |
---|
| 3749 | EXAMPLE: example zeroRad; shows an example |
---|
| 3750 | { |
---|
[a36e78] | 3751 | if(homog(I)==1){return(maxideal(1));} |
---|
| 3752 | //I needs to be a reduced standard basis |
---|
| 3753 | def R=basering; |
---|
| 3754 | int m=npars(R); |
---|
| 3755 | int n=nvars(R); |
---|
| 3756 | int p=char(R); |
---|
| 3757 | int d=vdim(I); |
---|
| 3758 | int i,k; |
---|
| 3759 | list l; |
---|
| 3760 | if(((p==0)||(p>d))&&(d==deg(I[1]))) |
---|
| 3761 | { |
---|
| 3762 | intvec e=leadexp(I[1]); |
---|
| 3763 | for(i=1;i<=nvars(basering);i++) |
---|
| 3764 | { |
---|
| 3765 | if(e[i]!=0) break; |
---|
| 3766 | } |
---|
| 3767 | I[1]=sep(I[1],i)[1]; |
---|
| 3768 | return(interred(I)); |
---|
| 3769 | } |
---|
| 3770 | intvec op=option(get); |
---|
[80654d] | 3771 | |
---|
[a36e78] | 3772 | option(redSB); |
---|
| 3773 | ideal F=finduni(I);//F[i] generates I intersected with K[var(i)] |
---|
[25c431] | 3774 | |
---|
[a36e78] | 3775 | option(set,op); |
---|
| 3776 | if(size(#)>0){I=#[1];} |
---|
[80654d] | 3777 | |
---|
[a36e78] | 3778 | for(i=1;i<=n;i++) |
---|
| 3779 | { |
---|
| 3780 | l[i]=sep(F[i],i); |
---|
| 3781 | F[i]=l[i][1]; |
---|
| 3782 | if(l[i][2]>k){k=l[i][2];} //computation of the maximal k |
---|
| 3783 | } |
---|
[80654d] | 3784 | |
---|
[a90eb0] | 3785 | if((k==0)||(m==0)) //the separable case |
---|
| 3786 | { |
---|
| 3787 | intvec save=option(get);option(redSB); |
---|
| 3788 | I=interred(I+F);option(set,save);return(I); |
---|
| 3789 | } |
---|
[80654d] | 3790 | |
---|
[a36e78] | 3791 | for(i=1;i<=n;i++) //consider all polynomials over |
---|
| 3792 | { //Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
| 3793 | F[i]=powerCoeffs(F[i],k-l[i][2]); |
---|
| 3794 | } |
---|
[24f458] | 3795 | |
---|
[a36e78] | 3796 | string cR="ring @R="+string(p)+",("+parstr(R)+","+varstr(R)+"),dp;"; |
---|
| 3797 | execute(cR); |
---|
| 3798 | ideal F=imap(R,F); |
---|
[24f458] | 3799 | |
---|
[a36e78] | 3800 | string nR="ring @S="+string(p)+",(y(1..m),"+varstr(R)+","+parstr(R)+"),dp;"; |
---|
| 3801 | execute(nR); |
---|
[80654d] | 3802 | |
---|
[a36e78] | 3803 | ideal G=fetch(@R,F); //G[i](t(1)^(p^-k),...,t(m)^(p^-k),x(i))=sep(F[i]) |
---|
[24f458] | 3804 | |
---|
[a36e78] | 3805 | ideal I=imap(R,I); |
---|
| 3806 | ideal J=I+G; |
---|
| 3807 | poly el=1; |
---|
| 3808 | k=p^k; |
---|
| 3809 | for(i=1;i<=m;i++) |
---|
| 3810 | { |
---|
| 3811 | J=J,var(i)^k-var(m+n+i); |
---|
| 3812 | el=el*y(i); |
---|
| 3813 | } |
---|
[80654d] | 3814 | |
---|
[a36e78] | 3815 | J=eliminate(J,el); |
---|
| 3816 | setring R; |
---|
| 3817 | ideal J=imap(@S,J); |
---|
| 3818 | return(J); |
---|
[80654d] | 3819 | } |
---|
| 3820 | example |
---|
| 3821 | { "EXAMPLE:"; echo = 2; |
---|
| 3822 | ring R=(5,t),(x,y),dp; |
---|
| 3823 | ideal I=x^5-t,y^5-t; |
---|
[24f458] | 3824 | zeroRad(I); |
---|
[80654d] | 3825 | } |
---|
| 3826 | |
---|
[ebecf83] | 3827 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3828 | |
---|
[07c623] | 3829 | proc radicalEHV(ideal i) |
---|
| 3830 | "USAGE: radicalEHV(i); i ideal. |
---|
| 3831 | RETURN: ideal, the radical of i. |
---|
[7a7df90] | 3832 | NOTE: Uses the algorithm of Eisenbud/Huneke/Vasconcelos, which |
---|
[50cbdc] | 3833 | reduces the computation to the complete intersection case, |
---|
[7a7df90] | 3834 | by taking, in the general case, a generic linear combination |
---|
| 3835 | of the input. |
---|
[07c623] | 3836 | Works only in characteristic 0 or p large. |
---|
| 3837 | EXAMPLE: example radicalEHV; shows an example |
---|
| 3838 | " |
---|
[67bd4c] | 3839 | { |
---|
[a36e78] | 3840 | if(attrib(basering,"global")!=1) |
---|
| 3841 | { |
---|
| 3842 | ERROR( |
---|
[07c623] | 3843 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 3844 | ); |
---|
| 3845 | } |
---|
| 3846 | if((char(basering)<100)&&(char(basering)!=0)) |
---|
| 3847 | { |
---|
| 3848 | "WARNING: The characteristic is too small, the result may be wrong"; |
---|
| 3849 | } |
---|
| 3850 | ideal J,I,I0,radI0,L,radI1,I2,radI2; |
---|
| 3851 | int l,n; |
---|
| 3852 | intvec op=option(get); |
---|
| 3853 | matrix M; |
---|
| 3854 | |
---|
| 3855 | option(redSB); |
---|
| 3856 | list m=mstd(i); |
---|
| 3857 | I=m[2]; |
---|
| 3858 | option(set,op); |
---|
| 3859 | |
---|
| 3860 | int cod=nvars(basering)-dim(m[1]); |
---|
| 3861 | //-------------------complete intersection case:---------------------- |
---|
| 3862 | if(cod==size(m[2])) |
---|
| 3863 | { |
---|
| 3864 | J=minor(jacob(I),cod); |
---|
| 3865 | return(quotient(I,J)); |
---|
| 3866 | } |
---|
| 3867 | //-----first codim elements of I are a complete intersection:--------- |
---|
| 3868 | for(l=1;l<=cod;l++) |
---|
| 3869 | { |
---|
| 3870 | I0[l]=I[l]; |
---|
| 3871 | } |
---|
| 3872 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3873 | //-----last codim elements of I are a complete intersection:---------- |
---|
| 3874 | if(n!=0) |
---|
| 3875 | { |
---|
| 3876 | for(l=1;l<=cod;l++) |
---|
| 3877 | { |
---|
| 3878 | I0[l]=I[size(I)-l+1]; |
---|
| 3879 | } |
---|
| 3880 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3881 | } |
---|
| 3882 | //-----taking a generic linear combination of the input:-------------- |
---|
| 3883 | if(n!=0) |
---|
| 3884 | { |
---|
| 3885 | M=transpose(sparsetriag(size(m[2]),cod,95,1)); |
---|
| 3886 | I0=ideal(M*transpose(I)); |
---|
| 3887 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3888 | } |
---|
| 3889 | //-----taking a more generic linear combination of the input:--------- |
---|
| 3890 | if(n!=0) |
---|
| 3891 | { |
---|
| 3892 | M=transpose(sparsetriag(size(m[2]),cod,0,100)); |
---|
| 3893 | I0=ideal(M*transpose(I)); |
---|
| 3894 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3895 | } |
---|
| 3896 | if(n==0) |
---|
| 3897 | { |
---|
| 3898 | J=minor(jacob(I0),cod); |
---|
| 3899 | radI0=quotient(I0,J); |
---|
| 3900 | L=quotient(radI0,I); |
---|
| 3901 | radI1=quotient(radI0,L); |
---|
[67bd4c] | 3902 | |
---|
[a36e78] | 3903 | if(size(reduce(radI1,m[1],1))==0) |
---|
| 3904 | { |
---|
| 3905 | return(I); |
---|
| 3906 | } |
---|
[70ab73] | 3907 | |
---|
[a36e78] | 3908 | I2=sat(I,radI1)[1]; |
---|
| 3909 | |
---|
| 3910 | if(deg(I2[1])<=0) |
---|
| 3911 | { |
---|
| 3912 | return(radI1); |
---|
| 3913 | } |
---|
| 3914 | return(intersect(radI1,radicalEHV(I2))); |
---|
| 3915 | } |
---|
| 3916 | //---------------------general case------------------------------------- |
---|
| 3917 | return(radical(I)); |
---|
[67bd4c] | 3918 | } |
---|
[07c623] | 3919 | example |
---|
| 3920 | { "EXAMPLE:"; echo = 2; |
---|
| 3921 | ring r = 0,(x,y,z),dp; |
---|
| 3922 | poly p = z2+1; |
---|
| 3923 | poly q = z3+2; |
---|
| 3924 | ideal i = p*q^2,y-z2; |
---|
| 3925 | ideal pr= radicalEHV(i); |
---|
| 3926 | pr; |
---|
| 3927 | } |
---|
| 3928 | |
---|
[ebecf83] | 3929 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3930 | |
---|
[24f458] | 3931 | proc Ann(module M) |
---|
[76aca2] | 3932 | "USAGE: Ann(M); M module |
---|
| 3933 | RETURN: ideal, the annihilator of coker(M) |
---|
| 3934 | NOTE: The output is the ideal of all elements a of the basering R such that |
---|
| 3935 | a * R^m is contained in M (m=number of rows of M). |
---|
| 3936 | EXAMPLE: example Ann; shows an example |
---|
| 3937 | " |
---|
[67bd4c] | 3938 | { |
---|
| 3939 | M=prune(M); //to obtain a small embedding |
---|
[d950c5] | 3940 | ideal ann=quotient1(M,freemodule(nrows(M))); |
---|
[e801fe] | 3941 | return(ann); |
---|
[67bd4c] | 3942 | } |
---|
[76aca2] | 3943 | example |
---|
| 3944 | { "EXAMPLE:"; echo = 2; |
---|
| 3945 | ring r = 0,(x,y,z),lp; |
---|
| 3946 | module M = x2-y2,z3; |
---|
| 3947 | Ann(M); |
---|
| 3948 | M = [1,x2],[y,x]; |
---|
| 3949 | Ann(M); |
---|
| 3950 | qring Q=std(xy-1); |
---|
| 3951 | module M=imap(r,M); |
---|
| 3952 | Ann(M); |
---|
| 3953 | } |
---|
| 3954 | |
---|
[ebecf83] | 3955 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3956 | |
---|
| 3957 | //computes the equidimensional part of the ideal i of codimension e |
---|
[07c623] | 3958 | static proc int_ass_primary_e(ideal i, int e) |
---|
[67bd4c] | 3959 | { |
---|
| 3960 | if(homog(i)!=1) |
---|
| 3961 | { |
---|
[a36e78] | 3962 | i=std(i); |
---|
[67bd4c] | 3963 | } |
---|
| 3964 | list re=sres(i,0); //the resolution |
---|
| 3965 | re=minres(re); //minimized resolution |
---|
| 3966 | ideal ann=AnnExt_R(e,re); |
---|
| 3967 | if(nvars(basering)-dim(std(ann))!=e) |
---|
| 3968 | { |
---|
| 3969 | return(ideal(1)); |
---|
| 3970 | } |
---|
| 3971 | return(ann); |
---|
[3939bc] | 3972 | } |
---|
| 3973 | |
---|
[ebecf83] | 3974 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3975 | |
---|
| 3976 | //computes the annihilator of Ext^n(R/i,R) with given resolution re |
---|
| 3977 | //n is not necessarily the number of variables |
---|
[07c623] | 3978 | static proc AnnExt_R(int n,list re) |
---|
[67bd4c] | 3979 | { |
---|
| 3980 | if(n<nvars(basering)) |
---|
| 3981 | { |
---|
[a36e78] | 3982 | matrix f=transpose(re[n+1]); //Hom(_,R) |
---|
| 3983 | module k=nres(f,2)[2]; //the kernel |
---|
| 3984 | matrix g=transpose(re[n]); //the image of Hom(_,R) |
---|
[d950c5] | 3985 | |
---|
[a36e78] | 3986 | ideal ann=quotient1(g,k); //the anihilator |
---|
[67bd4c] | 3987 | } |
---|
| 3988 | else |
---|
| 3989 | { |
---|
[a36e78] | 3990 | ideal ann=Ann(transpose(re[n])); |
---|
[67bd4c] | 3991 | } |
---|
[3939bc] | 3992 | return(ann); |
---|
[e801fe] | 3993 | } |
---|
[ebecf83] | 3994 | /////////////////////////////////////////////////////////////////////////////// |
---|
[e801fe] | 3995 | |
---|
[07c623] | 3996 | static proc analyze(list pr) |
---|
[3939bc] | 3997 | { |
---|
[a36e78] | 3998 | int ii,jj; |
---|
| 3999 | for(ii=1;ii<=size(pr)/2;ii++) |
---|
| 4000 | { |
---|
| 4001 | dim(std(pr[2*ii])); |
---|
| 4002 | idealsEqual(pr[2*ii-1],pr[2*ii]); |
---|
| 4003 | "==========================="; |
---|
| 4004 | } |
---|
[e801fe] | 4005 | |
---|
[a36e78] | 4006 | for(ii=size(pr)/2;ii>1;ii--) |
---|
| 4007 | { |
---|
| 4008 | for(jj=1;jj<ii;jj++) |
---|
[e801fe] | 4009 | { |
---|
[a36e78] | 4010 | if(size(reduce(pr[2*jj],std(pr[2*ii],1)))==0) |
---|
| 4011 | { |
---|
| 4012 | "eingebette Komponente"; |
---|
| 4013 | jj; |
---|
| 4014 | ii; |
---|
| 4015 | } |
---|
[e801fe] | 4016 | } |
---|
[a36e78] | 4017 | } |
---|
[e801fe] | 4018 | } |
---|
| 4019 | |
---|
[ebecf83] | 4020 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 4021 | // |
---|
| 4022 | // Shimoyama-Yokoyama |
---|
| 4023 | // |
---|
| 4024 | /////////////////////////////////////////////////////////////////////////////// |
---|
[e801fe] | 4025 | |
---|
[07c623] | 4026 | static proc simplifyIdeal(ideal i) |
---|
[e801fe] | 4027 | { |
---|
| 4028 | def r=basering; |
---|
[3939bc] | 4029 | |
---|
[e801fe] | 4030 | int j,k; |
---|
| 4031 | map phi; |
---|
| 4032 | poly p; |
---|
[3939bc] | 4033 | |
---|
[e801fe] | 4034 | ideal iwork=i; |
---|
| 4035 | ideal imap1=maxideal(1); |
---|
| 4036 | ideal imap2=maxideal(1); |
---|
[3939bc] | 4037 | |
---|
[e801fe] | 4038 | |
---|
| 4039 | for(j=1;j<=nvars(basering);j++) |
---|
| 4040 | { |
---|
[a36e78] | 4041 | for(k=1;k<=size(i);k++) |
---|
[e801fe] | 4042 | { |
---|
| 4043 | if(deg(iwork[k]/var(j))==0) |
---|
| 4044 | { |
---|
| 4045 | p=-1/leadcoef(iwork[k]/var(j))*iwork[k]; |
---|
| 4046 | imap1[j]=p+2*var(j); |
---|
| 4047 | phi=r,imap1; |
---|
| 4048 | iwork=phi(iwork); |
---|
| 4049 | iwork=subst(iwork,var(j),0); |
---|
| 4050 | iwork[k]=var(j); |
---|
| 4051 | imap1=maxideal(1); |
---|
[3939bc] | 4052 | imap2[j]=-p; |
---|
[e801fe] | 4053 | break; |
---|
| 4054 | } |
---|
| 4055 | } |
---|
| 4056 | } |
---|
| 4057 | return(iwork,imap2); |
---|
| 4058 | } |
---|
| 4059 | |
---|
[3939bc] | 4060 | |
---|
[e801fe] | 4061 | /////////////////////////////////////////////////////// |
---|
| 4062 | // ini_mod |
---|
| 4063 | // input: a polynomial p |
---|
| 4064 | // output: the initial term of p as needed |
---|
| 4065 | // in the context of characteristic sets |
---|
| 4066 | ////////////////////////////////////////////////////// |
---|
| 4067 | |
---|
[07c623] | 4068 | static proc ini_mod(poly p) |
---|
[e801fe] | 4069 | { |
---|
| 4070 | if (p==0) |
---|
| 4071 | { |
---|
| 4072 | return(0); |
---|
| 4073 | } |
---|
| 4074 | int n; matrix m; |
---|
[70ab73] | 4075 | for( n=nvars(basering); n>0; n--) |
---|
[e801fe] | 4076 | { |
---|
| 4077 | m=coef(p,var(n)); |
---|
| 4078 | if(m[1,1]!=1) |
---|
| 4079 | { |
---|
| 4080 | p=m[2,1]; |
---|
| 4081 | break; |
---|
| 4082 | } |
---|
| 4083 | } |
---|
| 4084 | if(deg(p)==0) |
---|
| 4085 | { |
---|
| 4086 | p=0; |
---|
| 4087 | } |
---|
| 4088 | return(p); |
---|
| 4089 | } |
---|
| 4090 | /////////////////////////////////////////////////////// |
---|
| 4091 | // min_ass_prim_charsets |
---|
| 4092 | // input: generators of an ideal PS and an integer cho |
---|
| 4093 | // If cho=0, the given ordering of the variables is used. |
---|
| 4094 | // Otherwise, the system tries to find an "optimal ordering", |
---|
| 4095 | // which in some cases may considerably speed up the algorithm |
---|
| 4096 | // output: the minimal associated primes of PS |
---|
| 4097 | // algorithm: via characteriostic sets |
---|
| 4098 | ////////////////////////////////////////////////////// |
---|
| 4099 | |
---|
| 4100 | |
---|
[07c623] | 4101 | static proc min_ass_prim_charsets (ideal PS, int cho) |
---|
[e801fe] | 4102 | { |
---|
| 4103 | if((cho<0) and (cho>1)) |
---|
| 4104 | { |
---|
[a36e78] | 4105 | ERROR("<int> must be 0 or 1"); |
---|
[e801fe] | 4106 | } |
---|
[70ab73] | 4107 | option(notWarnSB); |
---|
[e801fe] | 4108 | if(cho==0) |
---|
| 4109 | { |
---|
| 4110 | return(min_ass_prim_charsets0(PS)); |
---|
| 4111 | } |
---|
| 4112 | else |
---|
| 4113 | { |
---|
[a36e78] | 4114 | return(min_ass_prim_charsets1(PS)); |
---|
[e801fe] | 4115 | } |
---|
[67bd4c] | 4116 | } |
---|
[e801fe] | 4117 | /////////////////////////////////////////////////////// |
---|
| 4118 | // min_ass_prim_charsets0 |
---|
| 4119 | // input: generators of an ideal PS |
---|
| 4120 | // output: the minimal associated primes of PS |
---|
| 4121 | // algorithm: via characteristic sets |
---|
| 4122 | // the given ordering of the variables is used |
---|
| 4123 | ////////////////////////////////////////////////////// |
---|
[67bd4c] | 4124 | |
---|
[e801fe] | 4125 | |
---|
[07c623] | 4126 | static proc min_ass_prim_charsets0 (ideal PS) |
---|
[e801fe] | 4127 | { |
---|
[466f80] | 4128 | intvec op; |
---|
[e801fe] | 4129 | matrix m=char_series(PS); // We compute an irreducible |
---|
| 4130 | // characteristic series |
---|
| 4131 | int i,j,k; |
---|
| 4132 | list PSI; |
---|
| 4133 | list PHI; // the ideals given by the characteristic series |
---|
| 4134 | for(i=nrows(m);i>=1; i--) |
---|
| 4135 | { |
---|
[70ab73] | 4136 | PHI[i]=ideal(m[i,1..ncols(m)]); |
---|
[e801fe] | 4137 | } |
---|
| 4138 | // We compute the radical of each ideal in PHI |
---|
| 4139 | ideal I,JS,II; |
---|
| 4140 | int sizeJS, sizeII; |
---|
| 4141 | for(i=size(PHI);i>=1; i--) |
---|
| 4142 | { |
---|
[70ab73] | 4143 | I=0; |
---|
| 4144 | for(j=size(PHI[i]);j>0;j--) |
---|
| 4145 | { |
---|
| 4146 | I=I+ini_mod(PHI[i][j]); |
---|
| 4147 | } |
---|
| 4148 | JS=std(PHI[i]); |
---|
[a36e78] | 4149 | sizeJS=size(JS); |
---|
| 4150 | for(j=size(I);j>0;j--) |
---|
[70ab73] | 4151 | { |
---|
| 4152 | II=0; |
---|
| 4153 | sizeII=0; |
---|
| 4154 | k=0; |
---|
| 4155 | while(k<=sizeII) // successive saturation |
---|
| 4156 | { |
---|
| 4157 | op=option(get); |
---|
| 4158 | option(returnSB); |
---|
| 4159 | II=quotient(JS,I[j]); |
---|
| 4160 | option(set,op); |
---|
[a36e78] | 4161 | sizeII=size(II); |
---|
[70ab73] | 4162 | if(sizeII==sizeJS) |
---|
| 4163 | { |
---|
| 4164 | for(k=1;k<=sizeII;k++) |
---|
| 4165 | { |
---|
| 4166 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
---|
| 4167 | } |
---|
| 4168 | } |
---|
| 4169 | JS=II; |
---|
| 4170 | sizeJS=sizeII; |
---|
| 4171 | } |
---|
[e801fe] | 4172 | } |
---|
| 4173 | PSI=insert(PSI,JS); |
---|
| 4174 | } |
---|
| 4175 | int sizePSI=size(PSI); |
---|
| 4176 | // We eliminate redundant ideals |
---|
| 4177 | for(i=1;i<sizePSI;i++) |
---|
| 4178 | { |
---|
| 4179 | for(j=i+1;j<=sizePSI;j++) |
---|
| 4180 | { |
---|
| 4181 | if(size(PSI[i])!=0) |
---|
| 4182 | { |
---|
| 4183 | if(size(PSI[j])!=0) |
---|
| 4184 | { |
---|
| 4185 | if(size(NF(PSI[i],PSI[j],1))==0) |
---|
| 4186 | { |
---|
| 4187 | PSI[j]=ideal(0); |
---|
| 4188 | } |
---|
| 4189 | else |
---|
| 4190 | { |
---|
| 4191 | if(size(NF(PSI[j],PSI[i],1))==0) |
---|
| 4192 | { |
---|
| 4193 | PSI[i]=ideal(0); |
---|
| 4194 | } |
---|
| 4195 | } |
---|
| 4196 | } |
---|
| 4197 | } |
---|
| 4198 | } |
---|
| 4199 | } |
---|
| 4200 | for(i=sizePSI;i>=1;i--) |
---|
| 4201 | { |
---|
| 4202 | if(size(PSI[i])==0) |
---|
| 4203 | { |
---|
| 4204 | PSI=delete(PSI,i); |
---|
| 4205 | } |
---|
| 4206 | } |
---|
| 4207 | return (PSI); |
---|
| 4208 | } |
---|
| 4209 | |
---|
| 4210 | /////////////////////////////////////////////////////// |
---|
| 4211 | // min_ass_prim_charsets1 |
---|
| 4212 | // input: generators of an ideal PS |
---|
| 4213 | // output: the minimal associated primes of PS |
---|
| 4214 | // algorithm: via characteristic sets |
---|
| 4215 | // input: generators of an ideal PS and an integer i |
---|
| 4216 | // The system tries to find an "optimal ordering" of |
---|
| 4217 | // the variables |
---|
| 4218 | ////////////////////////////////////////////////////// |
---|
| 4219 | |
---|
| 4220 | |
---|
[07c623] | 4221 | static proc min_ass_prim_charsets1 (ideal PS) |
---|
[e801fe] | 4222 | { |
---|
[466f80] | 4223 | intvec op; |
---|
[e801fe] | 4224 | def oldring=basering; |
---|
| 4225 | string n=system("neworder",PS); |
---|
[2d2cad9] | 4226 | execute("ring r=("+charstr(oldring)+"),("+n+"),dp;"); |
---|
[e801fe] | 4227 | ideal PS=imap(oldring,PS); |
---|
| 4228 | matrix m=char_series(PS); // We compute an irreducible |
---|
| 4229 | // characteristic series |
---|
| 4230 | int i,j,k; |
---|
| 4231 | ideal I; |
---|
| 4232 | list PSI; |
---|
| 4233 | list PHI; // the ideals given by the characteristic series |
---|
| 4234 | list ITPHI; // their initial terms |
---|
| 4235 | for(i=nrows(m);i>=1; i--) |
---|
| 4236 | { |
---|
[70ab73] | 4237 | PHI[i]=ideal(m[i,1..ncols(m)]); |
---|
| 4238 | I=0; |
---|
| 4239 | for(j=size(PHI[i]);j>0;j=j-1) |
---|
| 4240 | { |
---|
| 4241 | I=I,ini_mod(PHI[i][j]); |
---|
| 4242 | } |
---|
| 4243 | I=I[2..ncols(I)]; |
---|
| 4244 | ITPHI[i]=I; |
---|
[e801fe] | 4245 | } |
---|
| 4246 | setring oldring; |
---|
| 4247 | matrix m=imap(r,m); |
---|
| 4248 | list PHI=imap(r,PHI); |
---|
| 4249 | list ITPHI=imap(r,ITPHI); |
---|
| 4250 | // We compute the radical of each ideal in PHI |
---|
| 4251 | ideal I,JS,II; |
---|
| 4252 | int sizeJS, sizeII; |
---|
| 4253 | for(i=size(PHI);i>=1; i--) |
---|
| 4254 | { |
---|
[70ab73] | 4255 | I=0; |
---|
| 4256 | for(j=size(PHI[i]);j>0;j--) |
---|
| 4257 | { |
---|
| 4258 | I=I+ITPHI[i][j]; |
---|
| 4259 | } |
---|
| 4260 | JS=std(PHI[i]); |
---|
| 4261 | sizeJS=size(JS); |
---|
[a36e78] | 4262 | for(j=size(I);j>0;j--) |
---|
[70ab73] | 4263 | { |
---|
| 4264 | II=0; |
---|
| 4265 | sizeII=0; |
---|
| 4266 | k=0; |
---|
| 4267 | while(k<=sizeII) // successive iteration |
---|
| 4268 | { |
---|
| 4269 | op=option(get); |
---|
| 4270 | option(returnSB); |
---|
| 4271 | II=quotient(JS,I[j]); |
---|
| 4272 | option(set,op); |
---|
[e801fe] | 4273 | //std |
---|
[a36e78] | 4274 | // II=std(II); |
---|
| 4275 | sizeII=size(II); |
---|
[70ab73] | 4276 | if(sizeII==sizeJS) |
---|
| 4277 | { |
---|
| 4278 | for(k=1;k<=sizeII;k++) |
---|
| 4279 | { |
---|
| 4280 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
---|
| 4281 | } |
---|
| 4282 | } |
---|
| 4283 | JS=II; |
---|
| 4284 | sizeJS=sizeII; |
---|
| 4285 | } |
---|
[e801fe] | 4286 | } |
---|
| 4287 | PSI=insert(PSI,JS); |
---|
| 4288 | } |
---|
| 4289 | int sizePSI=size(PSI); |
---|
| 4290 | // We eliminate redundant ideals |
---|
| 4291 | for(i=1;i<sizePSI;i++) |
---|
| 4292 | { |
---|
| 4293 | for(j=i+1;j<=sizePSI;j++) |
---|
| 4294 | { |
---|
| 4295 | if(size(PSI[i])!=0) |
---|
| 4296 | { |
---|
| 4297 | if(size(PSI[j])!=0) |
---|
| 4298 | { |
---|
| 4299 | if(size(NF(PSI[i],PSI[j],1))==0) |
---|
| 4300 | { |
---|
| 4301 | PSI[j]=ideal(0); |
---|
| 4302 | } |
---|
| 4303 | else |
---|
| 4304 | { |
---|
| 4305 | if(size(NF(PSI[j],PSI[i],1))==0) |
---|
| 4306 | { |
---|
| 4307 | PSI[i]=ideal(0); |
---|
| 4308 | } |
---|
| 4309 | } |
---|
| 4310 | } |
---|
| 4311 | } |
---|
| 4312 | } |
---|
| 4313 | } |
---|
| 4314 | for(i=sizePSI;i>=1;i--) |
---|
| 4315 | { |
---|
| 4316 | if(size(PSI[i])==0) |
---|
| 4317 | { |
---|
| 4318 | PSI=delete(PSI,i); |
---|
| 4319 | } |
---|
| 4320 | } |
---|
| 4321 | return (PSI); |
---|
| 4322 | } |
---|
| 4323 | |
---|
| 4324 | |
---|
| 4325 | ///////////////////////////////////////////////////// |
---|
| 4326 | // proc prim_dec |
---|
| 4327 | // input: generators of an ideal I and an integer choose |
---|
| 4328 | // If choose=0, min_ass_prim_charsets with the given |
---|
| 4329 | // ordering of the variables is used. |
---|
| 4330 | // If choose=1, min_ass_prim_charsets with the "optimized" |
---|
| 4331 | // ordering of the variables is used. |
---|
| 4332 | // If choose=2, minAssPrimes from primdec.lib is used |
---|
| 4333 | // If choose=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
| 4334 | // output: a primary decomposition of I, i.e., a list |
---|
| 4335 | // of pairs consisting of a standard basis of a primary component |
---|
| 4336 | // of I and a standard basis of the corresponding associated prime. |
---|
| 4337 | // To compute the minimal associated primes of a given ideal |
---|
| 4338 | // min_ass_prim_l is called, i.e., the minimal associated primes |
---|
| 4339 | // are computed via characteristic sets. |
---|
| 4340 | // In the homogeneous case, the performance of the procedure |
---|
| 4341 | // will be improved if I is already given by a minimal set of |
---|
| 4342 | // generators. Apply minbase if necessary. |
---|
| 4343 | ////////////////////////////////////////////////////////// |
---|
| 4344 | |
---|
| 4345 | |
---|
[07c623] | 4346 | static proc prim_dec(ideal I, int choose) |
---|
[e801fe] | 4347 | { |
---|
[70ab73] | 4348 | if((choose<0) or (choose>3)) |
---|
| 4349 | { |
---|
| 4350 | ERROR("ERROR: <int> must be 0 or 1 or 2 or 3"); |
---|
| 4351 | } |
---|
| 4352 | option(notWarnSB); |
---|
[e801fe] | 4353 | ideal H=1; // The intersection of the primary components |
---|
| 4354 | list U; // the leaves of the decomposition tree, i.e., |
---|
| 4355 | // pairs consisting of a primary component of I |
---|
| 4356 | // and the corresponding associated prime |
---|
| 4357 | list W; // the non-leaf vertices in the decomposition tree. |
---|
| 4358 | // every entry has 6 components: |
---|
| 4359 | // 1- the vertex itself , i.e., a standard bais of the |
---|
| 4360 | // given ideal I (type 1), or a standard basis of a |
---|
| 4361 | // pseudo-primary component arising from |
---|
| 4362 | // pseudo-primary decomposition (type 2), or a |
---|
| 4363 | // standard basis of a remaining component arising from |
---|
| 4364 | // pseudo-primary decomposition or extraction (type 3) |
---|
| 4365 | // 2- the type of the vertex as indicated above |
---|
| 4366 | // 3- the weighted_tree_depth of the vertex |
---|
| 4367 | // 4- the tester of the vertex |
---|
| 4368 | // 5- a standard basis of the associated prime |
---|
| 4369 | // of a vertex of type 2, or 0 otherwise |
---|
| 4370 | // 6- a list of pairs consisting of a standard |
---|
| 4371 | // basis of a minimal associated prime ideal |
---|
| 4372 | // of the father of the vertex and the |
---|
| 4373 | // irreducible factors of the "minimal |
---|
| 4374 | // divisor" of the seperator or extractor |
---|
| 4375 | // corresponding to the prime ideal |
---|
| 4376 | // as computed by the procedure minsat, |
---|
| 4377 | // if the vertex is of type 3, or |
---|
| 4378 | // the empty list otherwise |
---|
| 4379 | ideal SI=std(I); |
---|
[333b889] | 4380 | if(SI[1]==1) // primdecSY(ideal(1)) |
---|
| 4381 | { |
---|
| 4382 | return(list()); |
---|
| 4383 | } |
---|
[e801fe] | 4384 | int ncolsSI=ncols(SI); |
---|
| 4385 | int ncolsH=1; |
---|
| 4386 | W[1]=list(I,1,0,poly(1),ideal(0),list()); // The root of the tree |
---|
| 4387 | int weighted_tree_depth; |
---|
| 4388 | int i,j; |
---|
| 4389 | int check; |
---|
| 4390 | list V; // current vertex |
---|
| 4391 | list VV; // new vertex |
---|
| 4392 | list QQ; |
---|
| 4393 | list WI; |
---|
| 4394 | ideal Qi,SQ,SRest,fac; |
---|
| 4395 | poly tester; |
---|
| 4396 | |
---|
| 4397 | while(1) |
---|
| 4398 | { |
---|
| 4399 | i=1; |
---|
| 4400 | while(1) |
---|
| 4401 | { |
---|
| 4402 | while(i<=size(W)) // find vertex V of smallest weighted tree-depth |
---|
| 4403 | { |
---|
| 4404 | if (W[i][3]<=weighted_tree_depth) break; |
---|
| 4405 | i++; |
---|
| 4406 | } |
---|
| 4407 | if (i<=size(W)) break; |
---|
| 4408 | i=1; |
---|
| 4409 | weighted_tree_depth++; |
---|
| 4410 | } |
---|
| 4411 | V=W[i]; |
---|
| 4412 | W=delete(W,i); // delete V from W |
---|
| 4413 | |
---|
| 4414 | // now proceed by type of vertex V |
---|
| 4415 | |
---|
| 4416 | if (V[2]==2) // extraction needed |
---|
| 4417 | { |
---|
| 4418 | SQ,SRest,fac=extraction(V[1],V[5]); |
---|
| 4419 | // standard basis of primary component, |
---|
| 4420 | // standard basis of remaining component, |
---|
| 4421 | // irreducible factors of |
---|
| 4422 | // the "minimal divisor" of the extractor |
---|
| 4423 | // as computed by the procedure minsat, |
---|
| 4424 | check=0; |
---|
| 4425 | for(j=1;j<=ncolsH;j++) |
---|
| 4426 | { |
---|
| 4427 | if (NF(H[j],SQ,1)!=0) // Q is not redundant |
---|
| 4428 | { |
---|
| 4429 | check=1; |
---|
| 4430 | break; |
---|
| 4431 | } |
---|
| 4432 | } |
---|
| 4433 | if(check==1) // Q is not redundant |
---|
| 4434 | { |
---|
| 4435 | QQ=list(); |
---|
| 4436 | QQ[1]=list(SQ,V[5]); // primary component, associated prime, |
---|
| 4437 | // i.e., standard bases thereof |
---|
| 4438 | U=U+QQ; |
---|
[d950c5] | 4439 | H=intersect(H,SQ); |
---|
[e801fe] | 4440 | H=std(H); |
---|
| 4441 | ncolsH=ncols(H); |
---|
| 4442 | check=0; |
---|
| 4443 | if(ncolsH==ncolsSI) |
---|
| 4444 | { |
---|
| 4445 | for(j=1;j<=ncolsSI;j++) |
---|
| 4446 | { |
---|
| 4447 | if(leadexp(H[j])!=leadexp(SI[j])) |
---|
| 4448 | { |
---|
| 4449 | check=1; |
---|
| 4450 | break; |
---|
| 4451 | } |
---|
| 4452 | } |
---|
| 4453 | } |
---|
| 4454 | else |
---|
| 4455 | { |
---|
| 4456 | check=1; |
---|
| 4457 | } |
---|
| 4458 | if(check==0) // H==I => U is a primary decomposition |
---|
| 4459 | { |
---|
| 4460 | return(U); |
---|
| 4461 | } |
---|
| 4462 | } |
---|
| 4463 | if (SRest[1]!=1) // the remaining component is not |
---|
| 4464 | // the whole ring |
---|
| 4465 | { |
---|
| 4466 | if (rad_con(V[4],SRest)==0) // the new vertex is not the |
---|
| 4467 | // root of a redundant subtree |
---|
| 4468 | { |
---|
| 4469 | VV[1]=SRest; // remaining component |
---|
| 4470 | VV[2]=3; // pseudoprimdec_special |
---|
| 4471 | VV[3]=V[3]+1; // weighted depth |
---|
| 4472 | VV[4]=V[4]; // the tester did not change |
---|
| 4473 | VV[5]=ideal(0); |
---|
| 4474 | VV[6]=list(list(V[5],fac)); |
---|
| 4475 | W=insert(W,VV,size(W)); |
---|
| 4476 | } |
---|
| 4477 | } |
---|
| 4478 | } |
---|
| 4479 | else |
---|
| 4480 | { |
---|
| 4481 | if (V[2]==3) // pseudo_prim_dec_special is needed |
---|
| 4482 | { |
---|
| 4483 | QQ,SRest=pseudo_prim_dec_special_charsets(V[1],V[6],choose); |
---|
| 4484 | // QQ = quadruples: |
---|
| 4485 | // standard basis of pseudo-primary component, |
---|
| 4486 | // standard basis of corresponding prime, |
---|
| 4487 | // seperator, irreducible factors of |
---|
| 4488 | // the "minimal divisor" of the seperator |
---|
| 4489 | // as computed by the procedure minsat, |
---|
| 4490 | // SRest=standard basis of remaining component |
---|
| 4491 | } |
---|
| 4492 | else // V is the root, pseudo_prim_dec is needed |
---|
| 4493 | { |
---|
| 4494 | QQ,SRest=pseudo_prim_dec_charsets(I,SI,choose); |
---|
| 4495 | // QQ = quadruples: |
---|
| 4496 | // standard basis of pseudo-primary component, |
---|
| 4497 | // standard basis of corresponding prime, |
---|
| 4498 | // seperator, irreducible factors of |
---|
| 4499 | // the "minimal divisor" of the seperator |
---|
| 4500 | // as computed by the procedure minsat, |
---|
| 4501 | // SRest=standard basis of remaining component |
---|
| 4502 | |
---|
| 4503 | } |
---|
[091424] | 4504 | //check |
---|
[e801fe] | 4505 | for(i=size(QQ);i>=1;i--) |
---|
| 4506 | //for(i=1;i<=size(QQ);i++) |
---|
| 4507 | { |
---|
| 4508 | tester=QQ[i][3]*V[4]; |
---|
| 4509 | Qi=QQ[i][2]; |
---|
| 4510 | if(NF(tester,Qi,1)!=0) // the new vertex is not the |
---|
| 4511 | // root of a redundant subtree |
---|
| 4512 | { |
---|
| 4513 | VV[1]=QQ[i][1]; |
---|
| 4514 | VV[2]=2; |
---|
| 4515 | VV[3]=V[3]+1; |
---|
| 4516 | VV[4]=tester; // the new tester as computed above |
---|
| 4517 | VV[5]=Qi; // QQ[i][2]; |
---|
| 4518 | VV[6]=list(); |
---|
| 4519 | W=insert(W,VV,size(W)); |
---|
| 4520 | } |
---|
| 4521 | } |
---|
| 4522 | if (SRest[1]!=1) // the remaining component is not |
---|
| 4523 | // the whole ring |
---|
| 4524 | { |
---|
| 4525 | if (rad_con(V[4],SRest)==0) // the vertex is not the root |
---|
| 4526 | // of a redundant subtree |
---|
| 4527 | { |
---|
| 4528 | VV[1]=SRest; |
---|
| 4529 | VV[2]=3; |
---|
| 4530 | VV[3]=V[3]+2; |
---|
| 4531 | VV[4]=V[4]; // the tester did not change |
---|
| 4532 | VV[5]=ideal(0); |
---|
| 4533 | WI=list(); |
---|
| 4534 | for(i=1;i<=size(QQ);i++) |
---|
| 4535 | { |
---|
| 4536 | WI=insert(WI,list(QQ[i][2],QQ[i][4])); |
---|
| 4537 | } |
---|
| 4538 | VV[6]=WI; |
---|
| 4539 | W=insert(W,VV,size(W)); |
---|
| 4540 | } |
---|
| 4541 | } |
---|
| 4542 | } |
---|
| 4543 | } |
---|
| 4544 | } |
---|
| 4545 | |
---|
| 4546 | ////////////////////////////////////////////////////////////////////////// |
---|
| 4547 | // proc pseudo_prim_dec_charsets |
---|
| 4548 | // input: Generators of an arbitrary ideal I, a standard basis SI of I, |
---|
| 4549 | // and an integer choo |
---|
| 4550 | // If choo=0, min_ass_prim_charsets with the given |
---|
| 4551 | // ordering of the variables is used. |
---|
| 4552 | // If choo=1, min_ass_prim_charsets with the "optimized" |
---|
| 4553 | // ordering of the variables is used. |
---|
| 4554 | // If choo=2, minAssPrimes from primdec.lib is used |
---|
| 4555 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
| 4556 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
| 4557 | // of pseudo primary components together with a standard basis of the |
---|
| 4558 | // remaining component. Each pseudo primary component is |
---|
| 4559 | // represented by a quadrupel: A standard basis of the component, |
---|
| 4560 | // a standard basis of the corresponding associated prime, the |
---|
| 4561 | // seperator of the component, and the irreducible factors of the |
---|
| 4562 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
| 4563 | // calls proc pseudo_prim_dec_i |
---|
| 4564 | ////////////////////////////////////////////////////////////////////////// |
---|
| 4565 | |
---|
| 4566 | |
---|
[07c623] | 4567 | static proc pseudo_prim_dec_charsets (ideal I, ideal SI, int choo) |
---|
[e801fe] | 4568 | { |
---|
| 4569 | list L; // The list of minimal associated primes, |
---|
| 4570 | // each one given by a standard basis |
---|
| 4571 | if((choo==0) or (choo==1)) |
---|
[70ab73] | 4572 | { |
---|
| 4573 | L=min_ass_prim_charsets(I,choo); |
---|
| 4574 | } |
---|
| 4575 | else |
---|
| 4576 | { |
---|
| 4577 | if(choo==2) |
---|
[e801fe] | 4578 | { |
---|
[70ab73] | 4579 | L=minAssPrimes(I); |
---|
[e801fe] | 4580 | } |
---|
[70ab73] | 4581 | else |
---|
[e801fe] | 4582 | { |
---|
[70ab73] | 4583 | L=minAssPrimes(I,1); |
---|
[e801fe] | 4584 | } |
---|
[70ab73] | 4585 | for(int i=size(L);i>=1;i--) |
---|
| 4586 | { |
---|
| 4587 | L[i]=std(L[i]); |
---|
| 4588 | } |
---|
| 4589 | } |
---|
[e801fe] | 4590 | return (pseudo_prim_dec_i(SI,L)); |
---|
| 4591 | } |
---|
| 4592 | |
---|
| 4593 | //////////////////////////////////////////////////////////////// |
---|
| 4594 | // proc pseudo_prim_dec_special_charsets |
---|
| 4595 | // input: a standard basis of an ideal I whose radical is the |
---|
| 4596 | // intersection of the radicals of ideals generated by one prime ideal |
---|
| 4597 | // P_i together with one polynomial f_i, the list V6 must be the list of |
---|
| 4598 | // pairs (standard basis of P_i, irreducible factors of f_i), |
---|
| 4599 | // and an integer choo |
---|
| 4600 | // If choo=0, min_ass_prim_charsets with the given |
---|
| 4601 | // ordering of the variables is used. |
---|
| 4602 | // If choo=1, min_ass_prim_charsets with the "optimized" |
---|
| 4603 | // ordering of the variables is used. |
---|
| 4604 | // If choo=2, minAssPrimes from primdec.lib is used |
---|
| 4605 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
| 4606 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
| 4607 | // of pseudo primary components together with a standard basis of the |
---|
| 4608 | // remaining component. Each pseudo primary component is |
---|
| 4609 | // represented by a quadrupel: A standard basis of the component, |
---|
| 4610 | // a standard basis of the corresponding associated prime, the |
---|
| 4611 | // seperator of the component, and the irreducible factors of the |
---|
| 4612 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
| 4613 | // calls proc pseudo_prim_dec_i |
---|
| 4614 | //////////////////////////////////////////////////////////////// |
---|
| 4615 | |
---|
| 4616 | |
---|
[07c623] | 4617 | static proc pseudo_prim_dec_special_charsets (ideal SI,list V6, int choo) |
---|
[e801fe] | 4618 | { |
---|
| 4619 | int i,j,l; |
---|
| 4620 | list m; |
---|
| 4621 | list L; |
---|
| 4622 | int sizeL; |
---|
| 4623 | ideal P,SP; ideal fac; |
---|
| 4624 | int dimSP; |
---|
| 4625 | for(l=size(V6);l>=1;l--) // creates a list of associated primes |
---|
| 4626 | // of I, possibly redundant |
---|
| 4627 | { |
---|
| 4628 | P=V6[l][1]; |
---|
| 4629 | fac=V6[l][2]; |
---|
| 4630 | for(i=ncols(fac);i>=1;i--) |
---|
| 4631 | { |
---|
| 4632 | SP=P+fac[i]; |
---|
| 4633 | SP=std(SP); |
---|
| 4634 | if(SP[1]!=1) |
---|
| 4635 | { |
---|
| 4636 | if((choo==0) or (choo==1)) |
---|
| 4637 | { |
---|
| 4638 | m=min_ass_prim_charsets(SP,choo); // a list of SB |
---|
| 4639 | } |
---|
| 4640 | else |
---|
| 4641 | { |
---|
| 4642 | if(choo==2) |
---|
| 4643 | { |
---|
| 4644 | m=minAssPrimes(SP); |
---|
| 4645 | } |
---|
| 4646 | else |
---|
| 4647 | { |
---|
| 4648 | m=minAssPrimes(SP,1); |
---|
| 4649 | } |
---|
| 4650 | for(j=size(m);j>=1;j=j-1) |
---|
[a36e78] | 4651 | { |
---|
| 4652 | m[j]=std(m[j]); |
---|
| 4653 | } |
---|
[e801fe] | 4654 | } |
---|
[3939bc] | 4655 | dimSP=dim(SP); |
---|
[e801fe] | 4656 | for(j=size(m);j>=1; j--) |
---|
| 4657 | { |
---|
| 4658 | if(dim(m[j])==dimSP) |
---|
| 4659 | { |
---|
| 4660 | L=insert(L,m[j],size(L)); |
---|
| 4661 | } |
---|
| 4662 | } |
---|
| 4663 | } |
---|
| 4664 | } |
---|
| 4665 | } |
---|
| 4666 | sizeL=size(L); |
---|
| 4667 | for(i=1;i<sizeL;i++) // get rid of redundant primes |
---|
| 4668 | { |
---|
| 4669 | for(j=i+1;j<=sizeL;j++) |
---|
| 4670 | { |
---|
| 4671 | if(size(L[i])!=0) |
---|
| 4672 | { |
---|
| 4673 | if(size(L[j])!=0) |
---|
| 4674 | { |
---|
| 4675 | if(size(NF(L[i],L[j],1))==0) |
---|
| 4676 | { |
---|
| 4677 | L[j]=ideal(0); |
---|
| 4678 | } |
---|
| 4679 | else |
---|
| 4680 | { |
---|
| 4681 | if(size(NF(L[j],L[i],1))==0) |
---|
| 4682 | { |
---|
| 4683 | L[i]=ideal(0); |
---|
| 4684 | } |
---|
| 4685 | } |
---|
| 4686 | } |
---|
| 4687 | } |
---|
| 4688 | } |
---|
| 4689 | } |
---|
| 4690 | for(i=sizeL;i>=1;i--) |
---|
| 4691 | { |
---|
| 4692 | if(size(L[i])==0) |
---|
| 4693 | { |
---|
| 4694 | L=delete(L,i); |
---|
| 4695 | } |
---|
| 4696 | } |
---|
| 4697 | return (pseudo_prim_dec_i(SI,L)); |
---|
| 4698 | } |
---|
| 4699 | |
---|
| 4700 | |
---|
| 4701 | //////////////////////////////////////////////////////////////// |
---|
| 4702 | // proc pseudo_prim_dec_i |
---|
| 4703 | // input: A standard basis of an arbitrary ideal I, and standard bases |
---|
| 4704 | // of the minimal associated primes of I |
---|
| 4705 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
| 4706 | // of pseudo primary components together with a standard basis of the |
---|
| 4707 | // remaining component. Each pseudo primary component is |
---|
| 4708 | // represented by a quadrupel: A standard basis of the component Q_i, |
---|
| 4709 | // a standard basis of the corresponding associated prime P_i, the |
---|
| 4710 | // seperator of the component, and the irreducible factors of the |
---|
| 4711 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
| 4712 | //////////////////////////////////////////////////////////////// |
---|
| 4713 | |
---|
| 4714 | |
---|
[07c623] | 4715 | static proc pseudo_prim_dec_i (ideal SI, list L) |
---|
[e801fe] | 4716 | { |
---|
| 4717 | list Q; |
---|
| 4718 | if (size(L)==1) // one minimal associated prime only |
---|
| 4719 | // the ideal is already pseudo primary |
---|
| 4720 | { |
---|
| 4721 | Q=SI,L[1],1; |
---|
| 4722 | list QQ; |
---|
| 4723 | QQ[1]=Q; |
---|
| 4724 | return (QQ,ideal(1)); |
---|
| 4725 | } |
---|
| 4726 | |
---|
| 4727 | poly f0,f,g; |
---|
| 4728 | ideal fac; |
---|
| 4729 | int i,j,k,l; |
---|
| 4730 | ideal SQi; |
---|
| 4731 | ideal I'=SI; |
---|
| 4732 | list QP; |
---|
| 4733 | int sizeL=size(L); |
---|
| 4734 | for(i=1;i<=sizeL;i++) |
---|
| 4735 | { |
---|
| 4736 | fac=0; |
---|
| 4737 | for(j=1;j<=sizeL;j++) // compute the seperator sep_i |
---|
| 4738 | // of the i-th component |
---|
| 4739 | { |
---|
| 4740 | if (i!=j) // search g not in L[i], but L[j] |
---|
| 4741 | { |
---|
| 4742 | for(k=1;k<=ncols(L[j]);k++) |
---|
| 4743 | { |
---|
| 4744 | if(NF(L[j][k],L[i],1)!=0) |
---|
| 4745 | { |
---|
| 4746 | break; |
---|
| 4747 | } |
---|
| 4748 | } |
---|
| 4749 | fac=fac+L[j][k]; |
---|
| 4750 | } |
---|
| 4751 | } |
---|
| 4752 | // delete superfluous polynomials |
---|
[7f38f4] | 4753 | fac=simplify(fac,8+2); |
---|
[e801fe] | 4754 | // saturation |
---|
| 4755 | SQi,f0,f,fac=minsat_ppd(SI,fac); |
---|
| 4756 | I'=I',f; |
---|
| 4757 | QP=SQi,L[i],f0,fac; |
---|
| 4758 | // the quadrupel: |
---|
| 4759 | // a standard basis of Q_i, |
---|
| 4760 | // a standard basis of P_i, |
---|
| 4761 | // sep_i, |
---|
| 4762 | // irreducible factors of |
---|
| 4763 | // the "minimal divisor" of the seperator |
---|
| 4764 | // as computed by the procedure minsat, |
---|
| 4765 | Q[i]=QP; |
---|
| 4766 | } |
---|
| 4767 | I'=std(I'); |
---|
| 4768 | return (Q, I'); |
---|
| 4769 | // I' = remaining component |
---|
| 4770 | } |
---|
| 4771 | |
---|
| 4772 | |
---|
| 4773 | //////////////////////////////////////////////////////////////// |
---|
| 4774 | // proc extraction |
---|
| 4775 | // input: A standard basis of a pseudo primary ideal I, and a standard |
---|
| 4776 | // basis of the unique minimal associated prime P of I |
---|
| 4777 | // output: an extraction of I, i.e., a standard basis of the primary |
---|
| 4778 | // component Q of I with associated prime P, a standard basis of the |
---|
| 4779 | // remaining component, and the irreducible factors of the |
---|
| 4780 | // "minimal divisor" of the extractor as computed by the procedure minsat |
---|
| 4781 | //////////////////////////////////////////////////////////////// |
---|
| 4782 | |
---|
| 4783 | |
---|
[07c623] | 4784 | static proc extraction (ideal SI, ideal SP) |
---|
[e801fe] | 4785 | { |
---|
[aa3811c] | 4786 | list indsets=indepSet(SP,0); |
---|
[e801fe] | 4787 | poly f; |
---|
| 4788 | if(size(indsets)!=0) //check, whether dim P != 0 |
---|
| 4789 | { |
---|
| 4790 | intvec v; // a maximal independent set of variables |
---|
| 4791 | // modulo P |
---|
| 4792 | string U; // the independent variables |
---|
| 4793 | string A; // the dependent variables |
---|
| 4794 | int j,k; |
---|
| 4795 | int a; // the size of A |
---|
| 4796 | int degf; |
---|
| 4797 | ideal g; |
---|
| 4798 | list polys; |
---|
| 4799 | int sizepolys; |
---|
| 4800 | list newpoly; |
---|
| 4801 | def R=basering; |
---|
| 4802 | //intvec hv=hilb(SI,1); |
---|
| 4803 | for (k=1;k<=size(indsets);k++) |
---|
| 4804 | { |
---|
| 4805 | v=indsets[k]; |
---|
| 4806 | for (j=1;j<=nvars(R);j++) |
---|
| 4807 | { |
---|
| 4808 | if (v[j]==1) |
---|
| 4809 | { |
---|
| 4810 | U=U+varstr(j)+","; |
---|
| 4811 | } |
---|
| 4812 | else |
---|
| 4813 | { |
---|
| 4814 | A=A+varstr(j)+","; |
---|
| 4815 | a++; |
---|
| 4816 | } |
---|
| 4817 | } |
---|
| 4818 | |
---|
| 4819 | U[size(U)]=")"; // we compute the extractor of I (w.r.t. U) |
---|
[24f458] | 4820 | execute("ring RAU=("+charstr(basering)+"),("+A+U+",(dp("+string(a)+"),dp);"); |
---|
[e801fe] | 4821 | ideal I=imap(R,SI); |
---|
| 4822 | //I=std(I,hv); // the standard basis in (R[U])[A] |
---|
| 4823 | I=std(I); // the standard basis in (R[U])[A] |
---|
| 4824 | A[size(A)]=")"; |
---|
[2d2cad9] | 4825 | execute("ring Rloc=("+charstr(basering)+","+U+",("+A+",dp;"); |
---|
[e801fe] | 4826 | ideal I=imap(RAU,I); |
---|
| 4827 | //"std in lokalisierung:"+newline,I; |
---|
| 4828 | ideal h; |
---|
| 4829 | for(j=ncols(I);j>=1;j--) |
---|
| 4830 | { |
---|
| 4831 | h[j]=leadcoef(I[j]); // consider I in (R(U))[A] |
---|
| 4832 | } |
---|
| 4833 | setring R; |
---|
| 4834 | g=imap(Rloc,h); |
---|
| 4835 | kill RAU,Rloc; |
---|
| 4836 | U=""; |
---|
| 4837 | A=""; |
---|
| 4838 | a=0; |
---|
| 4839 | f=lcm(g); |
---|
| 4840 | newpoly[1]=f; |
---|
| 4841 | polys=polys+newpoly; |
---|
| 4842 | newpoly=list(); |
---|
| 4843 | } |
---|
| 4844 | f=polys[1]; |
---|
| 4845 | degf=deg(f); |
---|
| 4846 | sizepolys=size(polys); |
---|
| 4847 | for (k=2;k<=sizepolys;k++) |
---|
| 4848 | { |
---|
| 4849 | if (deg(polys[k])<degf) |
---|
| 4850 | { |
---|
| 4851 | f=polys[k]; |
---|
[3939bc] | 4852 | degf=deg(f); |
---|
[e801fe] | 4853 | } |
---|
| 4854 | } |
---|
| 4855 | } |
---|
| 4856 | else |
---|
| 4857 | { |
---|
| 4858 | f=1; |
---|
| 4859 | } |
---|
| 4860 | poly f0,h0; ideal SQ; ideal fac; |
---|
| 4861 | if(f!=1) |
---|
| 4862 | { |
---|
| 4863 | SQ,f0,h0,fac=minsat(SI,f); |
---|
| 4864 | return(SQ,std(SI+h0),fac); |
---|
| 4865 | // the tripel |
---|
| 4866 | // a standard basis of Q, |
---|
| 4867 | // a standard basis of remaining component, |
---|
| 4868 | // irreducible factors of |
---|
| 4869 | // the "minimal divisor" of the extractor |
---|
| 4870 | // as computed by the procedure minsat |
---|
| 4871 | } |
---|
| 4872 | else |
---|
| 4873 | { |
---|
| 4874 | return(SI,ideal(1),ideal(1)); |
---|
| 4875 | } |
---|
| 4876 | } |
---|
| 4877 | |
---|
| 4878 | ///////////////////////////////////////////////////// |
---|
| 4879 | // proc minsat |
---|
| 4880 | // input: a standard basis of an ideal I and a polynomial p |
---|
| 4881 | // output: a standard basis IS of the saturation of I w.r. to p, |
---|
| 4882 | // the maximal squarefree factor f0 of p, |
---|
| 4883 | // the "minimal divisor" f of f0 such that the saturation of |
---|
| 4884 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
---|
| 4885 | // the irreducible factors of f |
---|
| 4886 | ////////////////////////////////////////////////////////// |
---|
| 4887 | |
---|
| 4888 | |
---|
[07c623] | 4889 | static proc minsat(ideal SI, poly p) |
---|
[e801fe] | 4890 | { |
---|
| 4891 | ideal fac=factorize(p,1); //the irreducible factors of p |
---|
| 4892 | fac=sort(fac)[1]; |
---|
| 4893 | int i,k; |
---|
| 4894 | poly f0=1; |
---|
| 4895 | for(i=ncols(fac);i>=1;i--) |
---|
| 4896 | { |
---|
| 4897 | f0=f0*fac[i]; |
---|
| 4898 | } |
---|
| 4899 | poly f=1; |
---|
| 4900 | ideal iold; |
---|
| 4901 | list quotM; |
---|
| 4902 | quotM[1]=SI; |
---|
| 4903 | quotM[2]=fac; |
---|
| 4904 | quotM[3]=f0; |
---|
| 4905 | // we deal seperately with the first quotient; |
---|
| 4906 | // factors, which do not contribute to this one, |
---|
| 4907 | // are omitted |
---|
| 4908 | iold=quotM[1]; |
---|
| 4909 | quotM=minquot(quotM); |
---|
| 4910 | fac=quotM[2]; |
---|
| 4911 | if(quotM[3]==1) |
---|
[a36e78] | 4912 | { |
---|
| 4913 | return(quotM[1],f0,f,fac); |
---|
| 4914 | } |
---|
[e801fe] | 4915 | while(special_ideals_equal(iold,quotM[1])==0) |
---|
[a36e78] | 4916 | { |
---|
| 4917 | f=f*quotM[3]; |
---|
| 4918 | iold=quotM[1]; |
---|
| 4919 | quotM=minquot(quotM); |
---|
| 4920 | } |
---|
[e801fe] | 4921 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
---|
| 4922 | } |
---|
| 4923 | |
---|
| 4924 | ///////////////////////////////////////////////////// |
---|
| 4925 | // proc minsat_ppd |
---|
| 4926 | // input: a standard basis of an ideal I and a polynomial p |
---|
| 4927 | // output: a standard basis IS of the saturation of I w.r. to p, |
---|
| 4928 | // the maximal squarefree factor f0 of p, |
---|
| 4929 | // the "minimal divisor" f of f0 such that the saturation of |
---|
| 4930 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
---|
| 4931 | // the irreducible factors of f |
---|
| 4932 | ////////////////////////////////////////////////////////// |
---|
| 4933 | |
---|
| 4934 | |
---|
[07c623] | 4935 | static proc minsat_ppd(ideal SI, ideal fac) |
---|
[e801fe] | 4936 | { |
---|
| 4937 | fac=sort(fac)[1]; |
---|
| 4938 | int i,k; |
---|
| 4939 | poly f0=1; |
---|
| 4940 | for(i=ncols(fac);i>=1;i--) |
---|
| 4941 | { |
---|
| 4942 | f0=f0*fac[i]; |
---|
| 4943 | } |
---|
| 4944 | poly f=1; |
---|
| 4945 | ideal iold; |
---|
| 4946 | list quotM; |
---|
| 4947 | quotM[1]=SI; |
---|
| 4948 | quotM[2]=fac; |
---|
| 4949 | quotM[3]=f0; |
---|
| 4950 | // we deal seperately with the first quotient; |
---|
| 4951 | // factors, which do not contribute to this one, |
---|
| 4952 | // are omitted |
---|
| 4953 | iold=quotM[1]; |
---|
| 4954 | quotM=minquot(quotM); |
---|
| 4955 | fac=quotM[2]; |
---|
| 4956 | if(quotM[3]==1) |
---|
[a36e78] | 4957 | { |
---|
| 4958 | return(quotM[1],f0,f,fac); |
---|
| 4959 | } |
---|
[e801fe] | 4960 | while(special_ideals_equal(iold,quotM[1])==0) |
---|
| 4961 | { |
---|
| 4962 | f=f*quotM[3]; |
---|
| 4963 | iold=quotM[1]; |
---|
| 4964 | quotM=minquot(quotM); |
---|
| 4965 | k++; |
---|
| 4966 | } |
---|
[a36e78] | 4967 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
---|
[e801fe] | 4968 | } |
---|
| 4969 | ///////////////////////////////////////////////////////////////// |
---|
| 4970 | // proc minquot |
---|
| 4971 | // input: a list with 3 components: a standard basis |
---|
| 4972 | // of an ideal I, a set of irreducible polynomials, and |
---|
| 4973 | // there product f0 |
---|
| 4974 | // output: a standard basis of the ideal (I:f0), the irreducible |
---|
| 4975 | // factors of the "minimal divisor" f of f0 with (I:f0) = (I:f), |
---|
| 4976 | // the "minimal divisor" f |
---|
| 4977 | ///////////////////////////////////////////////////////////////// |
---|
| 4978 | |
---|
[07c623] | 4979 | static proc minquot(list tsil) |
---|
[e801fe] | 4980 | { |
---|
[a36e78] | 4981 | intvec op; |
---|
| 4982 | int i,j,k,action; |
---|
| 4983 | ideal verg; |
---|
| 4984 | list l; |
---|
| 4985 | poly g; |
---|
| 4986 | ideal laedi=tsil[1]; |
---|
| 4987 | ideal fac=tsil[2]; |
---|
| 4988 | poly f=tsil[3]; |
---|
[e801fe] | 4989 | |
---|
| 4990 | //std |
---|
| 4991 | // ideal star=quotient(laedi,f); |
---|
| 4992 | // star=std(star); |
---|
[a36e78] | 4993 | op=option(get); |
---|
| 4994 | option(returnSB); |
---|
| 4995 | ideal star=quotient(laedi,f); |
---|
| 4996 | option(set,op); |
---|
| 4997 | if(special_ideals_equal(laedi,star)==1) |
---|
| 4998 | { |
---|
| 4999 | return(laedi,ideal(1),1); |
---|
| 5000 | } |
---|
| 5001 | action=1; |
---|
| 5002 | while(action==1) |
---|
| 5003 | { |
---|
| 5004 | if(size(fac)==1) |
---|
[e801fe] | 5005 | { |
---|
[a36e78] | 5006 | action=0; |
---|
| 5007 | break; |
---|
[e801fe] | 5008 | } |
---|
[a36e78] | 5009 | for(i=1;i<=size(fac);i++) |
---|
| 5010 | { |
---|
| 5011 | g=1; |
---|
| 5012 | for(j=1;j<=size(fac);j++) |
---|
| 5013 | { |
---|
| 5014 | if(i!=j) |
---|
| 5015 | { |
---|
| 5016 | g=g*fac[j]; |
---|
| 5017 | } |
---|
| 5018 | } |
---|
[e801fe] | 5019 | //std |
---|
| 5020 | // verg=quotient(laedi,g); |
---|
| 5021 | // verg=std(verg); |
---|
[a36e78] | 5022 | op=option(get); |
---|
| 5023 | option(returnSB); |
---|
| 5024 | verg=quotient(laedi,g); |
---|
| 5025 | option(set,op); |
---|
| 5026 | if(special_ideals_equal(verg,star)==1) |
---|
| 5027 | { |
---|
| 5028 | f=g; |
---|
| 5029 | fac[i]=0; |
---|
| 5030 | fac=simplify(fac,2); |
---|
| 5031 | break; |
---|
| 5032 | } |
---|
| 5033 | if(i==size(fac)) |
---|
| 5034 | { |
---|
| 5035 | action=0; |
---|
| 5036 | } |
---|
[70ab73] | 5037 | } |
---|
[a36e78] | 5038 | } |
---|
| 5039 | l=star,fac,f; |
---|
| 5040 | return(l); |
---|
[e801fe] | 5041 | } |
---|
| 5042 | ///////////////////////////////////////////////// |
---|
| 5043 | // proc special_ideals_equal |
---|
| 5044 | // input: standard bases of ideal k1 and k2 such that |
---|
| 5045 | // k1 is contained in k2, or k2 is contained ink1 |
---|
| 5046 | // output: 1, if k1 equals k2, 0 otherwise |
---|
| 5047 | ////////////////////////////////////////////////// |
---|
| 5048 | |
---|
[07c623] | 5049 | static proc special_ideals_equal( ideal k1, ideal k2) |
---|
[e801fe] | 5050 | { |
---|
[a36e78] | 5051 | int j; |
---|
| 5052 | if(size(k1)==size(k2)) |
---|
| 5053 | { |
---|
| 5054 | for(j=1;j<=size(k1);j++) |
---|
[e801fe] | 5055 | { |
---|
[a36e78] | 5056 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
| 5057 | { |
---|
| 5058 | return(0); |
---|
| 5059 | } |
---|
[70ab73] | 5060 | } |
---|
[a36e78] | 5061 | return(1); |
---|
| 5062 | } |
---|
| 5063 | return(0); |
---|
[e801fe] | 5064 | } |
---|
[3939bc] | 5065 | |
---|
| 5066 | |
---|
[ebecf83] | 5067 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5068 | |
---|
[07c623] | 5069 | static proc convList(list l) |
---|
[ebecf83] | 5070 | { |
---|
[a36e78] | 5071 | int i; |
---|
| 5072 | list re,he; |
---|
| 5073 | for(i=1;i<=size(l)/2;i++) |
---|
| 5074 | { |
---|
| 5075 | he=l[2*i-1],l[2*i]; |
---|
| 5076 | re[i]=he; |
---|
| 5077 | } |
---|
| 5078 | return(re); |
---|
[ebecf83] | 5079 | } |
---|
| 5080 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5081 | |
---|
[07c623] | 5082 | static proc reconvList(list l) |
---|
[ebecf83] | 5083 | { |
---|
[a36e78] | 5084 | int i; |
---|
| 5085 | list re; |
---|
| 5086 | for(i=1;i<=size(l);i++) |
---|
| 5087 | { |
---|
| 5088 | re[2*i-1]=l[i][1]; |
---|
| 5089 | re[2*i]=l[i][2]; |
---|
| 5090 | } |
---|
| 5091 | return(re); |
---|
[ebecf83] | 5092 | } |
---|
| 5093 | |
---|
| 5094 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5095 | // |
---|
| 5096 | // The main procedures |
---|
| 5097 | // |
---|
| 5098 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5099 | |
---|
| 5100 | proc primdecGTZ(ideal i) |
---|
[091424] | 5101 | "USAGE: primdecGTZ(i); i ideal |
---|
[07c623] | 5102 | RETURN: a list pr of primary ideals and their associated primes: |
---|
[367e88] | 5103 | @format |
---|
[7b3971] | 5104 | pr[i][1] the i-th primary component, |
---|
| 5105 | pr[i][2] the i-th prime component. |
---|
| 5106 | @end format |
---|
| 5107 | NOTE: Algorithm of Gianni/Trager/Zacharias. |
---|
[b9b906] | 5108 | Designed for characteristic 0, works also in char k > 0, if it |
---|
[091424] | 5109 | terminates (may result in an infinite loop in small characteristic!) |
---|
[ebecf83] | 5110 | EXAMPLE: example primdecGTZ; shows an example |
---|
| 5111 | " |
---|
| 5112 | { |
---|
[a36e78] | 5113 | if(attrib(basering,"global")!=1) |
---|
| 5114 | { |
---|
| 5115 | ERROR( |
---|
[07c623] | 5116 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 5117 | ); |
---|
| 5118 | } |
---|
| 5119 | if(minpoly!=0) |
---|
| 5120 | { |
---|
| 5121 | return(algeDeco(i,0)); |
---|
| 5122 | ERROR( |
---|
[24f458] | 5123 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
---|
[a36e78] | 5124 | ); |
---|
| 5125 | } |
---|
[24f458] | 5126 | return(convList(decomp(i))); |
---|
[ebecf83] | 5127 | } |
---|
| 5128 | example |
---|
| 5129 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5130 | ring r = 0,(x,y,z),lp; |
---|
[ebecf83] | 5131 | poly p = z2+1; |
---|
[07c623] | 5132 | poly q = z3+2; |
---|
| 5133 | ideal i = p*q^2,y-z2; |
---|
[091424] | 5134 | list pr = primdecGTZ(i); |
---|
[ebecf83] | 5135 | pr; |
---|
| 5136 | } |
---|
| 5137 | /////////////////////////////////////////////////////////////////////////////// |
---|
[6fa3af] | 5138 | proc absPrimdecGTZ(ideal I) |
---|
| 5139 | "USAGE: absPrimdecGTZ(I); I ideal |
---|
| 5140 | ASSUME: Ground field has characteristic 0. |
---|
| 5141 | RETURN: a ring containing two lists: @code{absolute_primes} (the absolute |
---|
| 5142 | prime components of I) and @code{primary_decomp} (the output of |
---|
| 5143 | @code{primdecGTZ(I)}). |
---|
| 5144 | The list absolute_primes has to be interpreted as follows: |
---|
| 5145 | each entry describes a class of conjugated absolute primes, |
---|
| 5146 | @format |
---|
[326dba] | 5147 | absolute_primes[i][1] the absolute prime component, |
---|
[6fa3af] | 5148 | absolute_primes[i][2] the number of conjugates. |
---|
| 5149 | @end format |
---|
| 5150 | The first entry of @code{absolute_primes[i][1]} is the minimal |
---|
| 5151 | polynomial of a minimal finite field extension over which the |
---|
| 5152 | absolute prime component is defined. |
---|
| 5153 | NOTE: Algorithm of Gianni/Trager/Zacharias combined with the |
---|
| 5154 | @code{absFactorize} command. |
---|
| 5155 | SEE ALSO: primdecGTZ; absFactorize |
---|
| 5156 | EXAMPLE: example absPrimdecGTZ; shows an example |
---|
| 5157 | " |
---|
| 5158 | { |
---|
[70ab73] | 5159 | if (char(basering) != 0) |
---|
| 5160 | { |
---|
[6fa3af] | 5161 | ERROR("primdec.lib::absPrimdecGTZ is only implemented for "+ |
---|
| 5162 | +"characteristic 0"); |
---|
| 5163 | } |
---|
| 5164 | |
---|
[70ab73] | 5165 | if(attrib(basering,"global")!=1) |
---|
| 5166 | { |
---|
| 5167 | ERROR( |
---|
[6fa3af] | 5168 | "// Not implemented for this ordering, please change to global ordering." |
---|
[70ab73] | 5169 | ); |
---|
| 5170 | } |
---|
| 5171 | if(minpoly!=0) |
---|
| 5172 | { |
---|
| 5173 | //return(algeDeco(i,0)); |
---|
| 5174 | ERROR( |
---|
[6fa3af] | 5175 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
---|
[70ab73] | 5176 | ); |
---|
| 5177 | } |
---|
[6fa3af] | 5178 | def R=basering; |
---|
| 5179 | int n=nvars(R); |
---|
| 5180 | list L=decomp(I,3); |
---|
[4719f0] | 5181 | string newvar=L[1][3]; |
---|
[6fa3af] | 5182 | int k=find(newvar,",",find(newvar,",")+1); |
---|
| 5183 | newvar=newvar[k+1..size(newvar)]; |
---|
| 5184 | list lR=ringlist(R); |
---|
[1d430ab] | 5185 | int i,de,ii; |
---|
| 5186 | intvec vv=1:n; |
---|
| 5187 | //for(i=1;i<=n;i++){vv[i]=1;} |
---|
[6fa3af] | 5188 | |
---|
| 5189 | list orst; |
---|
| 5190 | orst[1]=list("dp",vv); |
---|
| 5191 | orst[2]=list("dp",intvec(1)); |
---|
| 5192 | orst[3]=list("C",0); |
---|
| 5193 | lR[3]=orst; |
---|
| 5194 | lR[2][n+1] = newvar; |
---|
| 5195 | def Rz = ring(lR); |
---|
| 5196 | setring Rz; |
---|
| 5197 | list L=imap(R,L); |
---|
| 5198 | list absolute_primes,primary_decomp; |
---|
| 5199 | ideal I,M,N,K; |
---|
| 5200 | M=maxideal(1); |
---|
| 5201 | N=maxideal(1); |
---|
| 5202 | poly p,q,f,g; |
---|
| 5203 | map phi,psi; |
---|
[1d430ab] | 5204 | string tvar; |
---|
[6fa3af] | 5205 | for(i=1;i<=size(L);i++) |
---|
| 5206 | { |
---|
[1d430ab] | 5207 | tvar=L[i][4]; |
---|
| 5208 | ii=find(tvar,"+"); |
---|
| 5209 | while(ii) |
---|
| 5210 | { |
---|
| 5211 | tvar=tvar[ii+1..size(tvar)]; |
---|
| 5212 | ii=find(tvar,"+"); |
---|
| 5213 | } |
---|
| 5214 | for(ii=1;ii<=nvars(basering);ii++) |
---|
| 5215 | { |
---|
| 5216 | if(tvar==string(var(ii))) break; |
---|
| 5217 | } |
---|
[6fa3af] | 5218 | I=L[i][2]; |
---|
| 5219 | execute("K="+L[i][3]+";"); |
---|
| 5220 | p=K[1]; |
---|
| 5221 | q=K[2]; |
---|
| 5222 | execute("f="+L[i][4]+";"); |
---|
[1d430ab] | 5223 | g=2*var(ii)-f; |
---|
| 5224 | M[ii]=f; |
---|
| 5225 | N[ii]=g; |
---|
[9d7c01] | 5226 | de=deg(p); |
---|
[1d430ab] | 5227 | psi=Rz,M; |
---|
| 5228 | phi=Rz,N; |
---|
[6fa3af] | 5229 | I=phi(I),p,q; |
---|
| 5230 | I=std(I); |
---|
[9d7c01] | 5231 | absolute_primes[i]=list(psi(I),de); |
---|
[6fa3af] | 5232 | primary_decomp[i]=list(L[i][1],L[i][2]); |
---|
| 5233 | } |
---|
| 5234 | export(primary_decomp); |
---|
| 5235 | export(absolute_primes); |
---|
| 5236 | setring R; |
---|
| 5237 | dbprint( printlevel-voice+3," |
---|
| 5238 | // 'absPrimdecGTZ' created a ring, in which two lists absolute_primes (the |
---|
| 5239 | // absolute prime components) and primary_decomp (the primary and prime |
---|
| 5240 | // components over the current basering) are stored. |
---|
| 5241 | // To access the list of absolute prime components, type (if the name S was |
---|
| 5242 | // assigned to the return value): |
---|
| 5243 | setring S; absolute_primes; "); |
---|
| 5244 | |
---|
| 5245 | return(Rz); |
---|
| 5246 | } |
---|
| 5247 | example |
---|
| 5248 | { "EXAMPLE:"; echo = 2; |
---|
| 5249 | ring r = 0,(x,y,z),lp; |
---|
| 5250 | poly p = z2+1; |
---|
| 5251 | poly q = z3+2; |
---|
| 5252 | ideal i = p*q^2,y-z2; |
---|
| 5253 | def S = absPrimdecGTZ(i); |
---|
| 5254 | setring S; |
---|
| 5255 | absolute_primes; |
---|
| 5256 | } |
---|
[1d430ab] | 5257 | |
---|
[6fa3af] | 5258 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5259 | |
---|
[7b3971] | 5260 | proc primdecSY(ideal i, list #) |
---|
[7f7c25e] | 5261 | "USAGE: primdecSY(I, c); I ideal, c int (optional) |
---|
[07c623] | 5262 | RETURN: a list pr of primary ideals and their associated primes: |
---|
[367e88] | 5263 | @format |
---|
[7b3971] | 5264 | pr[i][1] the i-th primary component, |
---|
| 5265 | pr[i][2] the i-th prime component. |
---|
| 5266 | @end format |
---|
| 5267 | NOTE: Algorithm of Shimoyama/Yokoyama. |
---|
| 5268 | @format |
---|
| 5269 | if c=0, the given ordering of the variables is used, |
---|
[7f7c25e] | 5270 | if c=1, minAssChar tries to use an optimal ordering (default), |
---|
[7b3971] | 5271 | if c=2, minAssGTZ is used, |
---|
| 5272 | if c=3, minAssGTZ and facstd are used. |
---|
| 5273 | @end format |
---|
[ebecf83] | 5274 | EXAMPLE: example primdecSY; shows an example |
---|
| 5275 | " |
---|
| 5276 | { |
---|
[a36e78] | 5277 | if(attrib(basering,"global")!=1) |
---|
| 5278 | { |
---|
| 5279 | ERROR( |
---|
[07c623] | 5280 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 5281 | ); |
---|
| 5282 | } |
---|
| 5283 | i=simplify(i,2); |
---|
| 5284 | if ((i[1]==0)||(i[1]==1)) |
---|
| 5285 | { |
---|
| 5286 | list L=list(ideal(i[1]),ideal(i[1])); |
---|
| 5287 | return(list(L)); |
---|
| 5288 | } |
---|
| 5289 | if(minpoly!=0) |
---|
| 5290 | { |
---|
| 5291 | return(algeDeco(i,1)); |
---|
| 5292 | } |
---|
| 5293 | if (size(#)==1) |
---|
| 5294 | { return(prim_dec(i,#[1])); } |
---|
| 5295 | else |
---|
| 5296 | { return(prim_dec(i,1)); } |
---|
[ebecf83] | 5297 | } |
---|
| 5298 | example |
---|
| 5299 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5300 | ring r = 0,(x,y,z),lp; |
---|
[ebecf83] | 5301 | poly p = z2+1; |
---|
[07c623] | 5302 | poly q = z3+2; |
---|
| 5303 | ideal i = p*q^2,y-z2; |
---|
[091424] | 5304 | list pr = primdecSY(i); |
---|
[ebecf83] | 5305 | pr; |
---|
| 5306 | } |
---|
| 5307 | /////////////////////////////////////////////////////////////////////////////// |
---|
[25c431] | 5308 | proc minAssGTZ(ideal i,list #) |
---|
[7f7c25e] | 5309 | "USAGE: minAssGTZ(I[, l]); I ideal, l list (optional) |
---|
| 5310 | @* Optional parameters in list l (can be entered in any order): |
---|
| 5311 | @* 0, \"facstd\" -> uses facstd to first decompose the ideal (default) |
---|
| 5312 | @* 1, \"noFacstd\" -> does not use facstd |
---|
| 5313 | @* \"GTZ\" -> the original algorithm by Gianni, Trager and Zacharias is used |
---|
| 5314 | @* \"SL\" -> GTZ algorithm with modificiations by Laplagne is used (default) |
---|
| 5315 | |
---|
| 5316 | RETURN: a list, the minimal associated prime ideals of I. |
---|
[24f458] | 5317 | NOTE: Designed for characteristic 0, works also in char k > 0 based |
---|
| 5318 | on an algorithm of Yokoyama |
---|
[ebecf83] | 5319 | EXAMPLE: example minAssGTZ; shows an example |
---|
| 5320 | " |
---|
| 5321 | { |
---|
[70ab73] | 5322 | int j; |
---|
| 5323 | string algorithm; |
---|
| 5324 | string facstdOption; |
---|
| 5325 | int useFac; |
---|
[808a9f3] | 5326 | |
---|
[70ab73] | 5327 | // Set input parameters |
---|
| 5328 | algorithm = "SL"; // Default: SL algorithm |
---|
| 5329 | facstdOption = "facstd"; |
---|
| 5330 | if(size(#) > 0) |
---|
| 5331 | { |
---|
| 5332 | int valid; |
---|
| 5333 | for(j = 1; j <= size(#); j++) |
---|
| 5334 | { |
---|
| 5335 | valid = 0; |
---|
| 5336 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
---|
| 5337 | { |
---|
| 5338 | if (#[j] == 1) {facstdOption = "noFacstd"; valid = 1;} // If #[j] == 1, facstd is not used. |
---|
| 5339 | if (#[j] == 0) {facstdOption = "facstd"; valid = 1;} // If #[j] == 0, facstd is used. |
---|
| 5340 | } |
---|
| 5341 | if(typeof(#[j]) == "string") |
---|
| 5342 | { |
---|
| 5343 | if((#[j] == "GTZ") || (#[j] == "SL")) |
---|
| 5344 | { |
---|
| 5345 | algorithm = #[j]; |
---|
| 5346 | valid = 1; |
---|
| 5347 | } |
---|
| 5348 | if((#[j] == "noFacstd") || (#[j] == "facstd")) |
---|
| 5349 | { |
---|
| 5350 | facstdOption = #[j]; |
---|
| 5351 | valid = 1; |
---|
| 5352 | } |
---|
| 5353 | } |
---|
| 5354 | if(valid == 0) |
---|
| 5355 | { |
---|
| 5356 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
---|
| 5357 | } |
---|
| 5358 | } |
---|
| 5359 | } |
---|
| 5360 | |
---|
| 5361 | if(attrib(basering,"global")!=1) |
---|
| 5362 | { |
---|
| 5363 | ERROR( |
---|
[07c623] | 5364 | "// Not implemented for this ordering, please change to global ordering." |
---|
[70ab73] | 5365 | ); |
---|
| 5366 | } |
---|
| 5367 | if(minpoly!=0) |
---|
| 5368 | { |
---|
| 5369 | return(algeDeco(i,2)); |
---|
| 5370 | } |
---|
[808a9f3] | 5371 | |
---|
[70ab73] | 5372 | list result = minAssPrimes(i, facstdOption, algorithm); |
---|
| 5373 | return(result); |
---|
[ebecf83] | 5374 | } |
---|
| 5375 | example |
---|
| 5376 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5377 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 5378 | poly p = z2+1; |
---|
[07c623] | 5379 | poly q = z3+2; |
---|
| 5380 | ideal i = p*q^2,y-z2; |
---|
| 5381 | list pr = minAssGTZ(i); |
---|
[ebecf83] | 5382 | pr; |
---|
| 5383 | } |
---|
| 5384 | |
---|
| 5385 | /////////////////////////////////////////////////////////////////////////////// |
---|
[2d3c9b] | 5386 | proc minAssChar(ideal i, list #) |
---|
[7f7c25e] | 5387 | "USAGE: minAssChar(I[,c]); i ideal, c int (optional). |
---|
[7b3971] | 5388 | RETURN: list, the minimal associated prime ideals of i. |
---|
| 5389 | NOTE: If c=0, the given ordering of the variables is used. @* |
---|
[2d3c9b] | 5390 | Otherwise, the system tries to find an optimal ordering, |
---|
[7b3971] | 5391 | which in some cases may considerably speed up the algorithm. @* |
---|
| 5392 | Due to a bug in the factorization, the result may be not completely |
---|
[07c623] | 5393 | decomposed in small characteristic. |
---|
[9050ca] | 5394 | EXAMPLE: example minAssChar; shows an example |
---|
[22c0fc9] | 5395 | " |
---|
| 5396 | { |
---|
[a36e78] | 5397 | if(attrib(basering,"global")!=1) |
---|
| 5398 | { |
---|
| 5399 | ERROR( |
---|
[07c623] | 5400 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 5401 | ); |
---|
| 5402 | } |
---|
| 5403 | if (size(#)==1) |
---|
| 5404 | { return(min_ass_prim_charsets(i,#[1])); } |
---|
| 5405 | else |
---|
| 5406 | { return(min_ass_prim_charsets(i,1)); } |
---|
[22c0fc9] | 5407 | } |
---|
| 5408 | example |
---|
| 5409 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5410 | ring r = 0,(x,y,z),dp; |
---|
[22c0fc9] | 5411 | poly p = z2+1; |
---|
[07c623] | 5412 | poly q = z3+2; |
---|
| 5413 | ideal i = p*q^2,y-z2; |
---|
| 5414 | list pr = minAssChar(i); |
---|
[22c0fc9] | 5415 | pr; |
---|
| 5416 | } |
---|
[ebecf83] | 5417 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5418 | proc equiRadical(ideal i) |
---|
[7f7c25e] | 5419 | "USAGE: equiRadical(I); I ideal |
---|
| 5420 | RETURN: ideal, intersection of associated primes of I of maximal dimension. |
---|
| 5421 | NOTE: A combination of the algorithms of Krick/Logar (with modifications by Laplagne) and Kemper is used. |
---|
[07c623] | 5422 | Works also in positive characteristic (Kempers algorithm). |
---|
[ebecf83] | 5423 | EXAMPLE: example equiRadical; shows an example |
---|
| 5424 | " |
---|
| 5425 | { |
---|
[d88470] | 5426 | if(attrib(basering,"global")!=1) |
---|
| 5427 | { |
---|
[a36e78] | 5428 | ERROR( |
---|
[d88470] | 5429 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 5430 | ); |
---|
[d88470] | 5431 | } |
---|
| 5432 | return(radical(i, 1)); |
---|
[ebecf83] | 5433 | } |
---|
| 5434 | example |
---|
| 5435 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5436 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 5437 | poly p = z2+1; |
---|
[07c623] | 5438 | poly q = z3+2; |
---|
| 5439 | ideal i = p*q^2,y-z2; |
---|
[ebecf83] | 5440 | ideal pr= equiRadical(i); |
---|
| 5441 | pr; |
---|
| 5442 | } |
---|
[fc5095] | 5443 | |
---|
[ebecf83] | 5444 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f0daaa2] | 5445 | proc radical(ideal i, list #) |
---|
[7f7c25e] | 5446 | "USAGE: radical(I[, l]); I ideal, l list (optional) |
---|
| 5447 | @* Optional parameters in list l (can be entered in any order): |
---|
| 5448 | @* 0, \"fullRad\" -> full radical is computed (default) |
---|
| 5449 | @* 1, \"equiRad\" -> equiRadical is computed |
---|
| 5450 | @* \"KL\" -> Krick/Logar algorithm is used |
---|
| 5451 | @* \"SL\" -> modifications by Laplagne are used (default) |
---|
| 5452 | @* \"facstd\" -> uses facstd to first decompose the ideal (default for non homogeneous ideals) |
---|
| 5453 | @* \"noFacstd\" -> does not use facstd (default for homogeneous ideals) |
---|
| 5454 | RETURN: ideal, the radical of I (or the equiradical if required in the input parameters) |
---|
| 5455 | NOTE: A combination of the algorithms of Krick/Logar (with modifications by Laplagne) and Kemper is used. |
---|
[07c623] | 5456 | Works also in positive characteristic (Kempers algorithm). |
---|
[ebecf83] | 5457 | EXAMPLE: example radical; shows an example |
---|
| 5458 | " |
---|
| 5459 | { |
---|
[d88470] | 5460 | dbprint(printlevel - voice, "Radical, version 2006.05.08"); |
---|
| 5461 | if(attrib(basering,"global")!=1) |
---|
| 5462 | { |
---|
[70ab73] | 5463 | ERROR( |
---|
[d88470] | 5464 | "// Not implemented for this ordering, please change to global ordering." |
---|
[70ab73] | 5465 | ); |
---|
[d88470] | 5466 | } |
---|
| 5467 | if(size(i) == 0){return(ideal(0));} |
---|
| 5468 | int j; |
---|
| 5469 | def P0 = basering; |
---|
| 5470 | list Pl=ringlist(P0); |
---|
| 5471 | intvec dp_w; |
---|
| 5472 | for(j=nvars(P0);j>0;j--) {dp_w[j]=1;} |
---|
| 5473 | Pl[3]=list(list("dp",dp_w),list("C",0)); |
---|
| 5474 | def @P=ring(Pl); |
---|
| 5475 | setring @P; |
---|
| 5476 | ideal i=imap(P0,i); |
---|
| 5477 | |
---|
| 5478 | int il; |
---|
| 5479 | string algorithm; |
---|
| 5480 | int useFac; |
---|
| 5481 | |
---|
| 5482 | // Set input parameters |
---|
| 5483 | algorithm = "SL"; // Default: SL algorithm |
---|
| 5484 | il = 0; // Default: Full radical (not only equiRadical) |
---|
| 5485 | if (homog(i) == 1) |
---|
| 5486 | { // Default: facStd is used, except if the ideal is homogeneous. |
---|
[70ab73] | 5487 | useFac = 0; |
---|
| 5488 | } |
---|
| 5489 | else |
---|
| 5490 | { |
---|
| 5491 | useFac = 1; |
---|
[d88470] | 5492 | } |
---|
[70ab73] | 5493 | if(size(#) > 0) |
---|
| 5494 | { |
---|
[d88470] | 5495 | int valid; |
---|
| 5496 | for(j = 1; j <= size(#); j++) |
---|
| 5497 | { |
---|
| 5498 | valid = 0; |
---|
| 5499 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
---|
| 5500 | { |
---|
| 5501 | il = #[j]; // If il == 1, equiRadical is computed |
---|
| 5502 | valid = 1; |
---|
[7f7c25e] | 5503 | } |
---|
[70ab73] | 5504 | if(typeof(#[j]) == "string") |
---|
| 5505 | { |
---|
| 5506 | if(#[j] == "KL") |
---|
| 5507 | { |
---|
[d88470] | 5508 | algorithm = "KL"; |
---|
| 5509 | valid = 1; |
---|
| 5510 | } |
---|
[70ab73] | 5511 | if(#[j] == "SL") |
---|
| 5512 | { |
---|
[d88470] | 5513 | algorithm = "SL"; |
---|
| 5514 | valid = 1; |
---|
| 5515 | } |
---|
[70ab73] | 5516 | if(#[j] == "noFacstd") |
---|
| 5517 | { |
---|
[d88470] | 5518 | useFac = 0; |
---|
[70ab73] | 5519 | valid = 1; |
---|
| 5520 | } |
---|
| 5521 | if(#[j] == "facstd") |
---|
| 5522 | { |
---|
[d88470] | 5523 | useFac = 1; |
---|
[70ab73] | 5524 | valid = 1; |
---|
| 5525 | } |
---|
| 5526 | if(#[j] == "equiRad") |
---|
| 5527 | { |
---|
[d88470] | 5528 | il = 1; |
---|
[70ab73] | 5529 | valid = 1; |
---|
| 5530 | } |
---|
| 5531 | if(#[j] == "fullRad") |
---|
| 5532 | { |
---|
[d88470] | 5533 | il = 0; |
---|
[70ab73] | 5534 | valid = 1; |
---|
| 5535 | } |
---|
[d88470] | 5536 | } |
---|
| 5537 | if(valid == 0) |
---|
| 5538 | { |
---|
| 5539 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
---|
| 5540 | } |
---|
| 5541 | } |
---|
| 5542 | } |
---|
[f0daaa2] | 5543 | |
---|
[d88470] | 5544 | ideal rad = 1; |
---|
| 5545 | intvec op = option(get); |
---|
| 5546 | list qr = simplifyIdeal(i); |
---|
| 5547 | map phi = @P, qr[2]; |
---|
[0c33fb] | 5548 | |
---|
[d88470] | 5549 | option(redSB); |
---|
| 5550 | i = groebner(qr[1]); |
---|
| 5551 | option(set, op); |
---|
| 5552 | int di = dim(i); |
---|
[0c33fb] | 5553 | |
---|
[d88470] | 5554 | if(di == 0) |
---|
| 5555 | { |
---|
| 5556 | i = zeroRad(i, qr[1]); |
---|
[a90eb0] | 5557 | option(redSB); |
---|
[d88470] | 5558 | i=interred(phi(i)); |
---|
[a90eb0] | 5559 | option(set, op); |
---|
[d88470] | 5560 | setring(P0); |
---|
| 5561 | i=imap(@P,i); |
---|
| 5562 | return(i); |
---|
| 5563 | } |
---|
[0c33fb] | 5564 | |
---|
[d88470] | 5565 | option(redSB); |
---|
| 5566 | list pr; |
---|
| 5567 | if(useFac == 1) |
---|
| 5568 | { |
---|
| 5569 | pr = facstd(i); |
---|
[70ab73] | 5570 | } |
---|
| 5571 | else |
---|
| 5572 | { |
---|
[d88470] | 5573 | pr = i; |
---|
| 5574 | } |
---|
| 5575 | option(set, op); |
---|
| 5576 | int s = size(pr); |
---|
[70ab73] | 5577 | if(useFac == 1) |
---|
| 5578 | { |
---|
[d88470] | 5579 | dbprint(printlevel - voice, "Number of components returned by facstd: ", s); |
---|
| 5580 | } |
---|
| 5581 | for(j = 1; j <= s; j++) |
---|
| 5582 | { |
---|
| 5583 | attrib(pr[s + 1 - j], "isSB", 1); |
---|
| 5584 | if((size(reduce(rad, pr[s + 1 - j], 1)) != 0) && ((dim(pr[s + 1 - j]) == di) || !il)) |
---|
| 5585 | { |
---|
| 5586 | // SL Debug messages |
---|
| 5587 | dbprint(printlevel-voice, "We shall compute the radical of ", pr[s + 1 - j]); |
---|
| 5588 | dbprint(printlevel-voice, "The dimension is: ", dim(pr[s+1-j])); |
---|
[f0daaa2] | 5589 | |
---|
[d88470] | 5590 | if(algorithm == "KL") |
---|
[0266ac] | 5591 | { |
---|
[d88470] | 5592 | rad = intersect(rad, radicalKL(pr[s + 1 - j], rad, il)); |
---|
[7f7c25e] | 5593 | } |
---|
[70ab73] | 5594 | if(algorithm == "SL") |
---|
| 5595 | { |
---|
[d88470] | 5596 | rad = intersect(rad, radicalSL(pr[s + 1 - j], il)); |
---|
| 5597 | } |
---|
| 5598 | } |
---|
| 5599 | else |
---|
| 5600 | { |
---|
| 5601 | // SL Debug |
---|
| 5602 | dbprint(printlevel-voice, "The radical of this component is not needed."); |
---|
| 5603 | dbprint(printlevel-voice, "size(reduce(rad, pr[s + 1 - j], 1))", |
---|
| 5604 | size(reduce(rad, pr[s + 1 - j], 1))); |
---|
| 5605 | dbprint(printlevel-voice, "dim(pr[s + 1 - j])", dim(pr[s + 1 - j])); |
---|
| 5606 | dbprint(printlevel-voice, "il", il); |
---|
| 5607 | } |
---|
| 5608 | } |
---|
| 5609 | rad=interred(phi(rad)); |
---|
| 5610 | setring(P0); |
---|
| 5611 | i=imap(@P,rad); |
---|
| 5612 | return(i); |
---|
[1918008] | 5613 | } |
---|
[ebecf83] | 5614 | example |
---|
| 5615 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5616 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 5617 | poly p = z2+1; |
---|
[07c623] | 5618 | poly q = z3+2; |
---|
| 5619 | ideal i = p*q^2,y-z2; |
---|
[f0daaa2] | 5620 | ideal pr = radical(i); |
---|
[ebecf83] | 5621 | pr; |
---|
| 5622 | } |
---|
[f0daaa2] | 5623 | |
---|
| 5624 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5625 | // |
---|
| 5626 | // Computes the radical of I using KL algorithm. |
---|
[7d56875] | 5627 | // The only difference with the previous implementation of KL algorithm is |
---|
[f0daaa2] | 5628 | // that now it uses block dp instead of lp ordering for the reduction to the |
---|
| 5629 | // zerodimensional case. |
---|
[f995aa] | 5630 | // The reduction step has been moved to the new routine radicalReduction, so that it can be |
---|
| 5631 | // used also by radicalSL procedure. |
---|
[f0daaa2] | 5632 | // |
---|
[70ab73] | 5633 | static proc radicalKL(ideal I, ideal ser, list #) |
---|
| 5634 | { |
---|
[f3c6e5] | 5635 | // ideal I The ideal for which the radical is computed |
---|
| 5636 | // ideal ser Used to reduce components already obtained |
---|
| 5637 | // list # If #[1] = 1, equiradical is computed. |
---|
[f0daaa2] | 5638 | |
---|
[70ab73] | 5639 | // I needs to be a Groebner basis. |
---|
| 5640 | if (attrib(I, "isSB") != 1) |
---|
| 5641 | { |
---|
| 5642 | I = groebner(I); |
---|
| 5643 | } |
---|
[f0daaa2] | 5644 | |
---|
[70ab73] | 5645 | ideal rad; // The radical |
---|
| 5646 | int allIndep = 1; // All max independent sets are used |
---|
[0266ac] | 5647 | |
---|
[70ab73] | 5648 | list result = radicalReduction(I, ser, allIndep, #); |
---|
| 5649 | int done = result[3]; |
---|
| 5650 | rad = result[1]; |
---|
| 5651 | if (done == 0) |
---|
| 5652 | { |
---|
| 5653 | rad = intersect(rad, radicalKL(result[2], ideal(1), #)); |
---|
| 5654 | } |
---|
| 5655 | return(rad); |
---|
| 5656 | } |
---|
[f0daaa2] | 5657 | |
---|
| 5658 | |
---|
| 5659 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5660 | // |
---|
[f995aa] | 5661 | // Computes the radical of I via Laplagne algorithm, using zerodimensional radical in |
---|
[f0daaa2] | 5662 | // the zero dimensional case. |
---|
| 5663 | // For the reduction to the zerodimensional case, it uses the procedure |
---|
[f995aa] | 5664 | // radical, with some modifications to avoid the recursion. |
---|
[f0daaa2] | 5665 | // |
---|
[f995aa] | 5666 | static proc radicalSL(ideal I, list #) |
---|
[f0daaa2] | 5667 | // Input = I, ideal |
---|
| 5668 | // #, list. If #[1] = 1, then computes only the equiradical. |
---|
| 5669 | // Output = (P, primaryDec) where P = rad(I) and primaryDec is the list of the radicals |
---|
| 5670 | // obtained in intermediate steps. |
---|
| 5671 | { |
---|
[70ab73] | 5672 | ideal rad = 1; |
---|
| 5673 | ideal equiRad = 1; |
---|
| 5674 | list primes; |
---|
| 5675 | int k; // Counter |
---|
| 5676 | int il; // If il = 1, only the equiradical is required. |
---|
| 5677 | int iDim; // The dimension of I |
---|
| 5678 | int stop = 0; // Checks if the radical has been obtained |
---|
| 5679 | |
---|
| 5680 | if (attrib(I, "isSB") != 1) |
---|
| 5681 | { |
---|
| 5682 | I = groebner(I); |
---|
| 5683 | } |
---|
| 5684 | iDim = dim(I); |
---|
| 5685 | |
---|
| 5686 | // Checks if only equiradical is required |
---|
| 5687 | if (size(#) > 0) |
---|
| 5688 | { |
---|
| 5689 | il = #[1]; |
---|
| 5690 | } |
---|
| 5691 | |
---|
| 5692 | while(stop == 0) |
---|
| 5693 | { |
---|
| 5694 | dbprint (printlevel-voice, "// We call radLoopR to find new prime ideals."); |
---|
| 5695 | primes = radicalSLIteration(I, rad); // A list of primes or intersections of primes, not included in P |
---|
| 5696 | dbprint (printlevel - voice, "// Output of Iteration Step:"); |
---|
| 5697 | dbprint (printlevel - voice, primes); |
---|
| 5698 | if (size(primes) > 0) |
---|
| 5699 | { |
---|
| 5700 | dbprint (printlevel - voice, "// We intersect P with the ideal just obtained."); |
---|
| 5701 | for(k = 1; k <= size(primes); k++) |
---|
| 5702 | { |
---|
| 5703 | rad = intersect(rad, primes[k]); |
---|
| 5704 | if (il == 1) |
---|
| 5705 | { |
---|
| 5706 | if (attrib(primes[k], "isSB") != 1) |
---|
| 5707 | { |
---|
| 5708 | primes[k] = groebner(primes[k]); |
---|
| 5709 | } |
---|
| 5710 | if (iDim == dim(primes[k])) |
---|
| 5711 | { |
---|
| 5712 | equiRad = intersect(equiRad, primes[k]); |
---|
| 5713 | } |
---|
[7f7c25e] | 5714 | } |
---|
[70ab73] | 5715 | } |
---|
[7f7c25e] | 5716 | } |
---|
[70ab73] | 5717 | else |
---|
| 5718 | { |
---|
| 5719 | stop = 1; |
---|
[7f7c25e] | 5720 | } |
---|
[70ab73] | 5721 | } |
---|
| 5722 | if (il == 0) |
---|
| 5723 | { |
---|
| 5724 | return(rad); |
---|
| 5725 | } |
---|
| 5726 | else |
---|
| 5727 | { |
---|
| 5728 | return(equiRad); |
---|
| 5729 | } |
---|
| 5730 | } |
---|
[f0daaa2] | 5731 | |
---|
| 5732 | ////////////////////////////////////////////////////////////////////////// |
---|
| 5733 | // Based on radicalKL. |
---|
[f995aa] | 5734 | // It contains all of old version of proc radicalKL except the recursion call. |
---|
[a36e78] | 5735 | // |
---|
[f0daaa2] | 5736 | // Output: |
---|
| 5737 | // #1 -> output ideal, the part of the radical that has been computed |
---|
| 5738 | // #2 -> complementary ideal, the part of the ideal I whose radical remains to be computed |
---|
| 5739 | // = (I, h) in KL algorithm |
---|
| 5740 | // This is not used in the new algorithm. It is part of KL algorithm |
---|
| 5741 | // #3 -> done, 1: output = radical, there is no need to continue |
---|
| 5742 | // 0: radical = output \cap \sqrt{complementary ideal} |
---|
| 5743 | // This is not used in the new algorithm. It is part of KL algorithm |
---|
| 5744 | |
---|
[70ab73] | 5745 | static proc radicalReduction(ideal I, ideal ser, int allIndep, list #) |
---|
| 5746 | { |
---|
[6fd3a2] | 5747 | // allMaximal 1 -> Indicates that the reduction to the zerodim case |
---|
| 5748 | // must be done for all indep set of the leading terms ideal |
---|
| 5749 | // 0 -> Otherwise |
---|
| 5750 | // ideal ser Only for radicalKL. (Same as in radicalKL) |
---|
[7d56875] | 5751 | // list # Only for radicalKL (If #[1] = 1, |
---|
[6fd3a2] | 5752 | // only equiradical is required. |
---|
| 5753 | // It is used to set the value of done.) |
---|
[f0daaa2] | 5754 | |
---|
[70ab73] | 5755 | attrib(I, "isSB", 1); // I needs to be a reduced standard basis |
---|
| 5756 | list indep, fett; |
---|
| 5757 | intvec @w, @hilb, op; |
---|
| 5758 | int @wr, @n, @m, lauf, di; |
---|
| 5759 | ideal fac, @h, collectrad, lsau; |
---|
| 5760 | poly @q; |
---|
| 5761 | string @va, quotring; |
---|
| 5762 | |
---|
| 5763 | def @P = basering; |
---|
| 5764 | int jdim = dim(I); // Computes the dimension of I |
---|
| 5765 | int homo = homog(I); // Finds out if I is homogeneous |
---|
| 5766 | ideal rad = ideal(1); // The unit ideal |
---|
| 5767 | ideal te = ser; |
---|
| 5768 | if(size(#) > 0) |
---|
| 5769 | { |
---|
| 5770 | @wr = #[1]; |
---|
| 5771 | } |
---|
| 5772 | if(homo == 1) |
---|
| 5773 | { |
---|
| 5774 | for(@n = 1; @n <= nvars(basering); @n++) |
---|
| 5775 | { |
---|
| 5776 | @w[@n] = ord(var(@n)); |
---|
| 5777 | } |
---|
| 5778 | @hilb = hilb(I, 1, @w); |
---|
| 5779 | } |
---|
[f0daaa2] | 5780 | |
---|
[70ab73] | 5781 | // SL 2006.04.11 1 Debug messages |
---|
| 5782 | dbprint(printlevel-voice, "//Computes the radical of the ideal:", I); |
---|
| 5783 | // SL 2006.04.11 2 Debug messages |
---|
[f0daaa2] | 5784 | |
---|
| 5785 | //--------------------------------------------------------------------------- |
---|
| 5786 | //j is the ring |
---|
| 5787 | //--------------------------------------------------------------------------- |
---|
| 5788 | |
---|
[70ab73] | 5789 | if (jdim==-1) |
---|
| 5790 | { |
---|
| 5791 | return(ideal(1), ideal(1), 1); |
---|
| 5792 | } |
---|
[f0daaa2] | 5793 | |
---|
| 5794 | //--------------------------------------------------------------------------- |
---|
| 5795 | //the zero-dimensional case |
---|
| 5796 | //--------------------------------------------------------------------------- |
---|
| 5797 | |
---|
[70ab73] | 5798 | if (jdim==0) |
---|
| 5799 | { |
---|
| 5800 | return(zeroRad(I), ideal(1), 1); |
---|
| 5801 | } |
---|
[f0daaa2] | 5802 | |
---|
[70ab73] | 5803 | //------------------------------------------------------------------------- |
---|
| 5804 | //search for a maximal independent set indep,i.e. |
---|
| 5805 | //look for subring such that the intersection with the ideal is zero |
---|
| 5806 | //j intersected with K[var(indep[3]+1),...,var(nvar)] is zero, |
---|
| 5807 | //indep[1] is the new varstring, indep[2] the string for the block-ordering |
---|
| 5808 | //------------------------------------------------------------------------- |
---|
| 5809 | |
---|
[6fd3a2] | 5810 | // SL 2006-04-24 1 If allIndep = 0, then it only computes one maximal |
---|
| 5811 | // independent set. |
---|
| 5812 | // This looks better for the new algorithm but not for KL |
---|
| 5813 | // algorithm |
---|
[70ab73] | 5814 | list parameters = allIndep; |
---|
| 5815 | indep = newMaxIndependSetDp(I, parameters); |
---|
| 5816 | // SL 2006-04-24 2 |
---|
| 5817 | |
---|
| 5818 | for(@m = 1; @m <= size(indep); @m++) |
---|
| 5819 | { |
---|
| 5820 | if((indep[@m][1] == varstr(basering)) && (@m == 1)) |
---|
| 5821 | //this is the good case, nothing to do, just to have the same notations |
---|
| 5822 | //change the ring |
---|
| 5823 | { |
---|
| 5824 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[f0daaa2] | 5825 | +ordstr(basering)+");"); |
---|
[70ab73] | 5826 | ideal @j = fetch(@P, I); |
---|
| 5827 | attrib(@j, "isSB", 1); |
---|
| 5828 | } |
---|
| 5829 | else |
---|
| 5830 | { |
---|
| 5831 | @va = string(maxideal(1)); |
---|
[f0daaa2] | 5832 | |
---|
[70ab73] | 5833 | execute("ring gnir1 = (" + charstr(basering) + "), (" + indep[@m][1] + "),(" |
---|
[f0daaa2] | 5834 | + indep[@m][2] + ");"); |
---|
[70ab73] | 5835 | execute("map phi = @P," + @va + ";"); |
---|
| 5836 | if(homo == 1) |
---|
| 5837 | { |
---|
| 5838 | ideal @j = std(phi(I), @hilb, @w); |
---|
[0266ac] | 5839 | } |
---|
| 5840 | else |
---|
| 5841 | { |
---|
[70ab73] | 5842 | ideal @j = groebner(phi(I)); |
---|
| 5843 | } |
---|
| 5844 | } |
---|
| 5845 | if((deg(@j[1]) == 0) || (dim(@j) < jdim)) |
---|
| 5846 | { |
---|
| 5847 | setring @P; |
---|
| 5848 | break; |
---|
| 5849 | } |
---|
| 5850 | for (lauf = 1; lauf <= size(@j); lauf++) |
---|
| 5851 | { |
---|
| 5852 | fett[lauf] = size(@j[lauf]); |
---|
| 5853 | } |
---|
| 5854 | //------------------------------------------------------------------------ |
---|
| 5855 | // We have now the following situation: |
---|
| 5856 | // j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
| 5857 | // to this quotientring, j is there still a standardbasis, the |
---|
| 5858 | // leading coefficients of the polynomials there (polynomials in |
---|
| 5859 | // K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 5860 | // we need their LCM, gh, because of the following: |
---|
| 5861 | // let (j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 5862 | // intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 5863 | // on the other hand j = ((j, gh^n) intersected with (j : gh^n)) |
---|
| 5864 | |
---|
| 5865 | //------------------------------------------------------------------------ |
---|
| 5866 | // The arrangement for the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 5867 | // and the map phi:K[var(1),...,var(nva)] -----> |
---|
| 5868 | // K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
| 5869 | //------------------------------------------------------------------------ |
---|
| 5870 | quotring = prepareQuotientRingDp(nvars(basering) - indep[@m][3]); |
---|
| 5871 | //------------------------------------------------------------------------ |
---|
| 5872 | // We pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 5873 | //------------------------------------------------------------------------ |
---|
| 5874 | |
---|
| 5875 | execute(quotring); |
---|
| 5876 | |
---|
| 5877 | // @j considered in the quotientring |
---|
| 5878 | ideal @j = imap(gnir1, @j); |
---|
| 5879 | |
---|
| 5880 | kill gnir1; |
---|
| 5881 | |
---|
| 5882 | // j is a standardbasis in the quotientring but usually not minimal |
---|
| 5883 | // here it becomes minimal |
---|
| 5884 | |
---|
| 5885 | @j = clearSB(@j, fett); |
---|
| 5886 | |
---|
| 5887 | // We need later LCM(h[1],...) = gh for saturation |
---|
| 5888 | ideal @h; |
---|
| 5889 | if(deg(@j[1]) > 0) |
---|
| 5890 | { |
---|
| 5891 | for(@n = 1; @n <= size(@j); @n++) |
---|
| 5892 | { |
---|
| 5893 | @h[@n] = leadcoef(@j[@n]); |
---|
[0266ac] | 5894 | } |
---|
[70ab73] | 5895 | op = option(get); |
---|
| 5896 | option(redSB); |
---|
[6fd3a2] | 5897 | @j = std(@j); //to obtain a reduced standardbasis |
---|
[70ab73] | 5898 | option(set, op); |
---|
| 5899 | |
---|
| 5900 | // SL 1 Debug messages |
---|
| 5901 | dbprint(printlevel - voice, "zero_rad", basering, @j, dim(groebner(@j))); |
---|
| 5902 | ideal zero_rad = zeroRad(@j); |
---|
| 5903 | dbprint(printlevel - voice, "zero_rad passed"); |
---|
| 5904 | // SL 2 |
---|
| 5905 | } |
---|
| 5906 | else |
---|
| 5907 | { |
---|
| 5908 | ideal zero_rad = ideal(1); |
---|
| 5909 | } |
---|
[f0daaa2] | 5910 | |
---|
[70ab73] | 5911 | // We need the intersection of the ideals in the list quprimary with the |
---|
| 5912 | // polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
| 5913 | // but fi polynomials, then the intersection of q with the polynomialring |
---|
| 5914 | // is the saturation of the ideal generated by f1,...,fr with respect to |
---|
| 5915 | // h which is the lcm of the leading coefficients of the fi considered in |
---|
| 5916 | // the quotientring: this is coded in saturn |
---|
[f0daaa2] | 5917 | |
---|
[70ab73] | 5918 | zero_rad = std(zero_rad); |
---|
[f0daaa2] | 5919 | |
---|
[70ab73] | 5920 | ideal hpl; |
---|
[f0daaa2] | 5921 | |
---|
[70ab73] | 5922 | for(@n = 1; @n <= size(zero_rad); @n++) |
---|
| 5923 | { |
---|
| 5924 | hpl = hpl, leadcoef(zero_rad[@n]); |
---|
| 5925 | } |
---|
[f0daaa2] | 5926 | |
---|
[70ab73] | 5927 | //------------------------------------------------------------------------ |
---|
| 5928 | // We leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 5929 | // back to the polynomialring |
---|
| 5930 | //------------------------------------------------------------------------ |
---|
| 5931 | setring @P; |
---|
[f0daaa2] | 5932 | |
---|
[70ab73] | 5933 | collectrad = imap(quring, zero_rad); |
---|
| 5934 | lsau = simplify(imap(quring, hpl), 2); |
---|
| 5935 | @h = imap(quring, @h); |
---|
[f0daaa2] | 5936 | |
---|
[70ab73] | 5937 | kill quring; |
---|
[f0daaa2] | 5938 | |
---|
[70ab73] | 5939 | // Here the intersection with the polynomialring |
---|
| 5940 | // mentioned above is really computed |
---|
[f0daaa2] | 5941 | |
---|
[70ab73] | 5942 | collectrad = sat2(collectrad, lsau)[1]; |
---|
| 5943 | if(deg(@h[1])>=0) |
---|
| 5944 | { |
---|
| 5945 | fac = ideal(0); |
---|
| 5946 | for(lauf = 1; lauf <= ncols(@h); lauf++) |
---|
[f0daaa2] | 5947 | { |
---|
[70ab73] | 5948 | if(deg(@h[lauf]) > 0) |
---|
| 5949 | { |
---|
| 5950 | fac = fac + factorize(@h[lauf], 1); |
---|
| 5951 | } |
---|
[f0daaa2] | 5952 | } |
---|
[70ab73] | 5953 | fac = simplify(fac, 6); |
---|
| 5954 | @q = 1; |
---|
| 5955 | for(lauf = 1; lauf <= size(fac); lauf++) |
---|
[f0daaa2] | 5956 | { |
---|
[70ab73] | 5957 | @q = @q * fac[lauf]; |
---|
[f0daaa2] | 5958 | } |
---|
[70ab73] | 5959 | op = option(get); |
---|
| 5960 | option(returnSB); |
---|
| 5961 | option(redSB); |
---|
| 5962 | I = quotient(I + ideal(@q), rad); |
---|
| 5963 | attrib(I, "isSB", 1); |
---|
| 5964 | option(set, op); |
---|
| 5965 | } |
---|
| 5966 | if((deg(rad[1]) > 0) && (deg(collectrad[1]) > 0)) |
---|
| 5967 | { |
---|
| 5968 | rad = intersect(rad, collectrad); |
---|
| 5969 | te = intersect(te, collectrad); |
---|
| 5970 | te = simplify(reduce(te, I, 1), 2); |
---|
| 5971 | } |
---|
| 5972 | else |
---|
| 5973 | { |
---|
| 5974 | if(deg(collectrad[1]) > 0) |
---|
[f0daaa2] | 5975 | { |
---|
[70ab73] | 5976 | rad = collectrad; |
---|
| 5977 | te = intersect(te, collectrad); |
---|
| 5978 | te = simplify(reduce(te, I, 1), 2); |
---|
[f0daaa2] | 5979 | } |
---|
[70ab73] | 5980 | } |
---|
[f0daaa2] | 5981 | |
---|
[70ab73] | 5982 | if((dim(I) < jdim)||(size(te) == 0)) |
---|
| 5983 | { |
---|
| 5984 | break; |
---|
| 5985 | } |
---|
| 5986 | if(homo==1) |
---|
| 5987 | { |
---|
| 5988 | @hilb = hilb(I, 1, @w); |
---|
| 5989 | } |
---|
| 5990 | } |
---|
[f0daaa2] | 5991 | |
---|
[70ab73] | 5992 | // SL 2006.04.11 1 Debug messages |
---|
| 5993 | dbprint (printlevel-voice, "// Part of the Radical already computed:", rad); |
---|
| 5994 | dbprint (printlevel-voice, "// Dimension:", dim(groebner(rad))); |
---|
| 5995 | // SL 2006.04.11 2 Debug messages |
---|
[f0daaa2] | 5996 | |
---|
[6fd3a2] | 5997 | // SL 2006.04.21 1 New variable "done". |
---|
| 5998 | // It tells if the radical is already computed or |
---|
| 5999 | // if it still has to be computed the radical of the new ideal I |
---|
[70ab73] | 6000 | int done; |
---|
| 6001 | if(((@wr == 1) && (dim(I)<jdim)) || (deg(I[1])==0) || (size(te) == 0)) |
---|
| 6002 | { |
---|
| 6003 | done = 1; |
---|
| 6004 | } |
---|
| 6005 | else |
---|
| 6006 | { |
---|
| 6007 | done = 0; |
---|
| 6008 | } |
---|
| 6009 | // SL 2006.04.21 2 |
---|
[f0daaa2] | 6010 | |
---|
[6fd3a2] | 6011 | // SL 2006.04.21 1 See details of the output at the beginning of this proc. |
---|
[70ab73] | 6012 | list result = rad, I, done; |
---|
| 6013 | return(result); |
---|
| 6014 | // SL 2006.04.21 2 |
---|
| 6015 | } |
---|
[f0daaa2] | 6016 | |
---|
| 6017 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6018 | // Given an ideal I and an ideal P (intersection of some minimal prime ideals |
---|
| 6019 | // associated to I), it calculates the intersection of new minimal prime ideals |
---|
| 6020 | // associated to I which where not used to calculate P. |
---|
| 6021 | // This version uses ZD Radical in the zerodimensional case. |
---|
[f995aa] | 6022 | static proc radicalSLIteration (ideal I, ideal P); |
---|
[f0daaa2] | 6023 | // Input: I, ideal. The ideal from which new prime components will be obtained. |
---|
| 6024 | // P, ideal. Intersection of some prime ideals of I. |
---|
| 6025 | // Output: ideal. Intersection of some primes of I different from the ones in P. |
---|
| 6026 | { |
---|
[6fd3a2] | 6027 | int k = 1; // Counter |
---|
| 6028 | int good = 0; // Checks if an element of P is in rad(I) |
---|
[70ab73] | 6029 | |
---|
| 6030 | dbprint (printlevel-voice, "// We search for an element in P - sqrt(I)."); |
---|
| 6031 | while ((k <= size(P)) and (good == 0)) |
---|
| 6032 | { |
---|
| 6033 | dbprint (printlevel-voice, "// We try with:", P[k]); |
---|
| 6034 | good = 1 - rad_con(P[k], I); |
---|
| 6035 | k++; |
---|
| 6036 | } |
---|
| 6037 | k--; |
---|
| 6038 | if (good == 0) |
---|
| 6039 | { |
---|
| 6040 | dbprint (printlevel-voice, "// No element was found, P = sqrt(I)."); |
---|
| 6041 | list emptyList = list(); |
---|
| 6042 | return (emptyList); |
---|
| 6043 | } |
---|
| 6044 | dbprint(printlevel - voice, "// That one was good!"); |
---|
| 6045 | dbprint(printlevel - voice, "// We saturate I with respect to this element."); |
---|
| 6046 | if (P[k] != 1) |
---|
| 6047 | { |
---|
[6fd3a2] | 6048 | intvec oo=option(get); |
---|
| 6049 | option(redSB); |
---|
[70ab73] | 6050 | ideal J = sat(I, P[k])[1]; |
---|
[6fd3a2] | 6051 | option(set,oo); |
---|
| 6052 | |
---|
[a36e78] | 6053 | } |
---|
| 6054 | else |
---|
| 6055 | { |
---|
| 6056 | dbprint(printlevel - voice, "// The polynomial is 1, the saturation in not actually computed."); |
---|
| 6057 | ideal J = I; |
---|
| 6058 | } |
---|
[7f7c25e] | 6059 | |
---|
[a36e78] | 6060 | // We now call proc radicalNew; |
---|
| 6061 | dbprint(printlevel - voice, "// We do the reduction to the zerodimensional case, via radical."); |
---|
| 6062 | dbprint(printlevel - voice, "// The ideal is ", J); |
---|
| 6063 | dbprint(printlevel - voice, "// The dimension is ", dim(groebner(J))); |
---|
[7f7c25e] | 6064 | |
---|
[6fd3a2] | 6065 | int allMaximal = 0; // Compute the zerodim reduction for only one indep set. |
---|
| 6066 | ideal re = 1; // No reduction is need, |
---|
| 6067 | // there are not redundant components. |
---|
| 6068 | list emptyList = list(); // Look for primes of any dimension, |
---|
| 6069 | // not only of max dimension. |
---|
[a36e78] | 6070 | list result = radicalReduction(J, re, allMaximal, emptyList); |
---|
[f0daaa2] | 6071 | |
---|
[a36e78] | 6072 | return(result[1]); |
---|
[70ab73] | 6073 | } |
---|
[f0daaa2] | 6074 | |
---|
| 6075 | /////////////////////////////////////////////////////////////////////////////////// |
---|
| 6076 | // Based on maxIndependSet |
---|
| 6077 | // Added list # as parameter |
---|
| 6078 | // If the first element of # is 0, the output is only 1 max indep set. |
---|
| 6079 | // If no list is specified or #[1] = 1, the output is all the max indep set of the |
---|
| 6080 | // leading terms ideal. This is the original output of maxIndependSet |
---|
| 6081 | |
---|
| 6082 | // The ordering given in the output has been changed to block dp instead of lp. |
---|
| 6083 | |
---|
[f995aa] | 6084 | proc newMaxIndependSetDp(ideal j, list #) |
---|
[7f7c25e] | 6085 | "USAGE: newMaxIndependentSetDp(I); I ideal (returns all maximal independent sets of the corresponding leading terms ideal) |
---|
| 6086 | newMaxIndependentSetDp(I, 0); I ideal (returns only one maximal independent set) |
---|
[f0daaa2] | 6087 | RETURN: list = #1. new varstring with the maximal independent set at the end, |
---|
[f995aa] | 6088 | #2. ordstring with the corresponding dp block ordering, |
---|
[f0daaa2] | 6089 | #3. the number of independent variables |
---|
| 6090 | NOTE: |
---|
[f995aa] | 6091 | EXAMPLE: example newMaxIndependentSetDp; shows an example |
---|
[f0daaa2] | 6092 | " |
---|
| 6093 | { |
---|
[70ab73] | 6094 | int n, k, di; |
---|
| 6095 | list resu, hilf; |
---|
| 6096 | string var1, var2; |
---|
| 6097 | list v = indepSet(j, 0); |
---|
| 6098 | |
---|
| 6099 | // SL 2006.04.21 1 Lines modified to use only one independent Set |
---|
| 6100 | int allMaximal; |
---|
| 6101 | if (size(#) > 0) |
---|
| 6102 | { |
---|
| 6103 | allMaximal = #[1]; |
---|
[a36e78] | 6104 | } |
---|
| 6105 | else |
---|
| 6106 | { |
---|
| 6107 | allMaximal = 1; |
---|
| 6108 | } |
---|
[f0daaa2] | 6109 | |
---|
[a36e78] | 6110 | int nMax; |
---|
| 6111 | if (allMaximal == 1) |
---|
| 6112 | { |
---|
| 6113 | nMax = size(v); |
---|
| 6114 | } |
---|
| 6115 | else |
---|
| 6116 | { |
---|
| 6117 | nMax = 1; |
---|
| 6118 | } |
---|
[f0daaa2] | 6119 | |
---|
[a36e78] | 6120 | for(n = 1; n <= nMax; n++) |
---|
| 6121 | // SL 2006.04.21 2 |
---|
| 6122 | { |
---|
| 6123 | di = 0; |
---|
| 6124 | var1 = ""; |
---|
| 6125 | var2 = ""; |
---|
| 6126 | for(k = 1; k <= size(v[n]); k++) |
---|
| 6127 | { |
---|
| 6128 | if(v[n][k] != 0) |
---|
| 6129 | { |
---|
| 6130 | di++; |
---|
| 6131 | var2 = var2 + "var(" + string(k) + "), "; |
---|
| 6132 | } |
---|
| 6133 | else |
---|
| 6134 | { |
---|
| 6135 | var1 = var1 + "var(" + string(k) + "), "; |
---|
| 6136 | } |
---|
| 6137 | } |
---|
| 6138 | if(di > 0) |
---|
| 6139 | { |
---|
| 6140 | var1 = var1 + var2; |
---|
| 6141 | var1 = var1[1..size(var1) - 2]; // The "- 2" removes the trailer comma |
---|
| 6142 | hilf[1] = var1; |
---|
| 6143 | // SL 2006.21.04 1 The order is now block dp instead of lp |
---|
| 6144 | hilf[2] = "dp(" + string(nvars(basering) - di) + "), dp(" + string(di) + ")"; |
---|
| 6145 | // SL 2006.21.04 2 |
---|
| 6146 | hilf[3] = di; |
---|
| 6147 | resu[n] = hilf; |
---|
| 6148 | } |
---|
| 6149 | else |
---|
| 6150 | { |
---|
| 6151 | resu[n] = varstr(basering), ordstr(basering), 0; |
---|
| 6152 | } |
---|
| 6153 | } |
---|
| 6154 | return(resu); |
---|
[f0daaa2] | 6155 | } |
---|
| 6156 | example |
---|
| 6157 | { "EXAMPLE:"; echo = 2; |
---|
| 6158 | ring s1 = (0, x, y), (a, b, c, d, e, f, g), lp; |
---|
| 6159 | ideal i = ea - fbg, fa + be, ec - fdg, fc + de; |
---|
| 6160 | i = std(i); |
---|
[f995aa] | 6161 | list l = newMaxIndependSetDp(i); |
---|
[f0daaa2] | 6162 | l; |
---|
| 6163 | i = i, g; |
---|
[f995aa] | 6164 | l = newMaxIndependSetDp(i); |
---|
[f0daaa2] | 6165 | l; |
---|
| 6166 | |
---|
| 6167 | ring s = 0, (x, y, z), lp; |
---|
| 6168 | ideal i = z, yx; |
---|
[f995aa] | 6169 | list l = newMaxIndependSetDp(i); |
---|
[f0daaa2] | 6170 | l; |
---|
| 6171 | } |
---|
| 6172 | |
---|
| 6173 | |
---|
| 6174 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6175 | // based on prepareQuotientring |
---|
| 6176 | // The order returned is now (C, dp) instead of (C, lp) |
---|
| 6177 | |
---|
[f995aa] | 6178 | static proc prepareQuotientRingDp (int nnp) |
---|
| 6179 | "USAGE: prepareQuotientRingDp(nnp); nnp int |
---|
[f0daaa2] | 6180 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
---|
| 6181 | NOTE: |
---|
[f995aa] | 6182 | EXAMPLE: example prepareQuotientRingDp; shows an example |
---|
[f0daaa2] | 6183 | " |
---|
| 6184 | { |
---|
| 6185 | ideal @ih,@jh; |
---|
| 6186 | int npar=npars(basering); |
---|
| 6187 | int @n; |
---|
| 6188 | |
---|
| 6189 | string quotring= "ring quring = ("+charstr(basering); |
---|
| 6190 | for(@n = nnp + 1; @n <= nvars(basering); @n++) |
---|
| 6191 | { |
---|
[a36e78] | 6192 | quotring = quotring + ", var(" + string(@n) + ")"; |
---|
| 6193 | @ih = @ih + var(@n); |
---|
[f0daaa2] | 6194 | } |
---|
| 6195 | |
---|
| 6196 | quotring = quotring+"),(var(1)"; |
---|
| 6197 | @jh = @jh + var(1); |
---|
| 6198 | for(@n = 2; @n <= nnp; @n++) |
---|
| 6199 | { |
---|
| 6200 | quotring = quotring + ", var(" + string(@n) + ")"; |
---|
| 6201 | @jh = @jh + var(@n); |
---|
| 6202 | } |
---|
| 6203 | // SL 2006-04-21 1 The order returned is now (C, dp) instead of (C, lp) |
---|
| 6204 | quotring = quotring + "), (C, dp);"; |
---|
| 6205 | // SL 2006-04-21 2 |
---|
| 6206 | |
---|
| 6207 | return(quotring); |
---|
| 6208 | } |
---|
| 6209 | example |
---|
| 6210 | { "EXAMPLE:"; echo = 2; |
---|
| 6211 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
---|
| 6212 | def @Q=basering; |
---|
[f995aa] | 6213 | list l= prepareQuotientRingDp(3); |
---|
[f0daaa2] | 6214 | l; |
---|
| 6215 | execute(l[1]); |
---|
| 6216 | execute(l[2]); |
---|
| 6217 | basering; |
---|
| 6218 | phi; |
---|
| 6219 | setring @Q; |
---|
[a36e78] | 6220 | |
---|
[f0daaa2] | 6221 | } |
---|
| 6222 | |
---|
[ebecf83] | 6223 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6224 | proc prepareAss(ideal i) |
---|
[7f7c25e] | 6225 | "USAGE: prepareAss(I); I ideal |
---|
| 6226 | RETURN: list, the radicals of the maximal dimensional components of I. |
---|
[7b3971] | 6227 | NOTE: Uses algorithm of Eisenbud/Huneke/Vasconcelos. |
---|
[ebecf83] | 6228 | EXAMPLE: example prepareAss; shows an example |
---|
| 6229 | " |
---|
| 6230 | { |
---|
[d88470] | 6231 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 6232 | { |
---|
[a36e78] | 6233 | ERROR( |
---|
[07c623] | 6234 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 6235 | ); |
---|
[07c623] | 6236 | } |
---|
[ebecf83] | 6237 | ideal j=std(i); |
---|
[d950c5] | 6238 | int cod=nvars(basering)-dim(j); |
---|
[ebecf83] | 6239 | int e; |
---|
| 6240 | list er; |
---|
| 6241 | ideal ann; |
---|
| 6242 | if(homog(i)==1) |
---|
| 6243 | { |
---|
[a36e78] | 6244 | list re=sres(j,0); //the resolution |
---|
| 6245 | re=minres(re); //minimized resolution |
---|
[ebecf83] | 6246 | } |
---|
| 6247 | else |
---|
| 6248 | { |
---|
[3939bc] | 6249 | list re=mres(i,0); |
---|
| 6250 | } |
---|
[ebecf83] | 6251 | for(e=cod;e<=nvars(basering);e++) |
---|
| 6252 | { |
---|
[a36e78] | 6253 | ann=AnnExt_R(e,re); |
---|
[d950c5] | 6254 | |
---|
[a36e78] | 6255 | if(nvars(basering)-dim(std(ann))==e) |
---|
| 6256 | { |
---|
| 6257 | er[size(er)+1]=equiRadical(ann); |
---|
| 6258 | } |
---|
[ebecf83] | 6259 | } |
---|
| 6260 | return(er); |
---|
[3939bc] | 6261 | } |
---|
[ebecf83] | 6262 | example |
---|
| 6263 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 6264 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 6265 | poly p = z2+1; |
---|
[07c623] | 6266 | poly q = z3+2; |
---|
| 6267 | ideal i = p*q^2,y-z2; |
---|
| 6268 | list pr = prepareAss(i); |
---|
[ebecf83] | 6269 | pr; |
---|
| 6270 | } |
---|
[03f29c] | 6271 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6272 | proc equidimMaxEHV(ideal i) |
---|
[7f7c25e] | 6273 | "USAGE: equidimMaxEHV(I); I ideal |
---|
| 6274 | RETURN: ideal, the equidimensional component (of maximal dimension) of I. |
---|
[07c623] | 6275 | NOTE: Uses algorithm of Eisenbud, Huneke and Vasconcelos. |
---|
[03f29c] | 6276 | EXAMPLE: example equidimMaxEHV; shows an example |
---|
| 6277 | " |
---|
| 6278 | { |
---|
[d88470] | 6279 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 6280 | { |
---|
[a36e78] | 6281 | ERROR( |
---|
[07c623] | 6282 | "// Not implemented for this ordering, please change to global ordering." |
---|
[a36e78] | 6283 | ); |
---|
[07c623] | 6284 | } |
---|
[0ad359] | 6285 | ideal j=groebner(i); |
---|
[03f29c] | 6286 | int cod=nvars(basering)-dim(j); |
---|
| 6287 | int e; |
---|
| 6288 | ideal ann; |
---|
| 6289 | if(homog(i)==1) |
---|
| 6290 | { |
---|
[a36e78] | 6291 | list re=sres(j,0); //the resolution |
---|
| 6292 | re=minres(re); //minimized resolution |
---|
[03f29c] | 6293 | } |
---|
| 6294 | else |
---|
| 6295 | { |
---|
| 6296 | list re=mres(i,0); |
---|
| 6297 | } |
---|
| 6298 | ann=AnnExt_R(cod,re); |
---|
| 6299 | return(ann); |
---|
| 6300 | } |
---|
| 6301 | example |
---|
| 6302 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 6303 | ring r = 0,(x,y,z),dp; |
---|
[03f29c] | 6304 | ideal i=intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
| 6305 | equidimMaxEHV(i); |
---|
| 6306 | } |
---|
[ebecf83] | 6307 | |
---|
[838d37] | 6308 | proc testPrimary(list pr, ideal k) |
---|
[7b3971] | 6309 | "USAGE: testPrimary(pr,k); pr a list, k an ideal. |
---|
| 6310 | ASSUME: pr is the result of primdecGTZ(k) or primdecSY(k). |
---|
| 6311 | RETURN: int, 1 if the intersection of the ideals in pr is k, 0 if not |
---|
[091424] | 6312 | EXAMPLE: example testPrimary; shows an example |
---|
[ebecf83] | 6313 | " |
---|
| 6314 | { |
---|
[a36e78] | 6315 | int i; |
---|
| 6316 | pr=reconvList(pr); |
---|
| 6317 | ideal j=pr[1]; |
---|
| 6318 | for (i=2;i<=size(pr)/2;i++) |
---|
| 6319 | { |
---|
| 6320 | j=intersect(j,pr[2*i-1]); |
---|
| 6321 | } |
---|
| 6322 | return(idealsEqual(j,k)); |
---|
[ebecf83] | 6323 | } |
---|
| 6324 | example |
---|
| 6325 | { "EXAMPLE:"; echo = 2; |
---|
| 6326 | ring r = 32003,(x,y,z),dp; |
---|
| 6327 | poly p = z2+1; |
---|
| 6328 | poly q = z4+2; |
---|
| 6329 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
[091424] | 6330 | list pr = primdecGTZ(i); |
---|
[ebecf83] | 6331 | testPrimary(pr,i); |
---|
| 6332 | } |
---|
| 6333 | |
---|
[55fcff] | 6334 | /////////////////////////////////////////////////////////////////////////////// |
---|
[7f24dd7] | 6335 | proc zerodec(ideal I) |
---|
| 6336 | "USAGE: zerodec(I); I ideal |
---|
[7b3971] | 6337 | ASSUME: I is zero-dimensional, the characteristic of the ground field is 0 |
---|
| 6338 | RETURN: list of primary ideals, the zero-dimensional decomposition of I |
---|
| 6339 | NOTE: The algorithm (of Monico), works well only for a small total number |
---|
[367e88] | 6340 | of solutions (@code{vdim(std(I))} should be < 100) and without |
---|
| 6341 | parameters. In practice, it works also in large characteristic p>0 |
---|
[7b3971] | 6342 | but may fail for small p. |
---|
| 6343 | @* If printlevel > 0 (default = 0) additional information is displayed. |
---|
[55fcff] | 6344 | EXAMPLE: example zerodec; shows an example |
---|
[7f24dd7] | 6345 | " |
---|
| 6346 | { |
---|
[d88470] | 6347 | if(attrib(basering,"global")!=1) |
---|
| 6348 | { |
---|
| 6349 | ERROR( |
---|
| 6350 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 6351 | ); |
---|
| 6352 | } |
---|
| 6353 | def R=basering; |
---|
| 6354 | poly q; |
---|
| 6355 | int j,time; |
---|
| 6356 | matrix m; |
---|
| 6357 | list re; |
---|
| 6358 | poly va=var(1); |
---|
| 6359 | ideal J=groebner(I); |
---|
| 6360 | ideal ba=kbase(J); |
---|
| 6361 | int d=vdim(J); |
---|
| 6362 | dbprint(printlevel-voice+2,"// multiplicity of ideal : "+ string(d)); |
---|
[55fcff] | 6363 | //------ compute matrix of multiplication on R/I with generic element p ----- |
---|
[d88470] | 6364 | int e=nvars(basering); |
---|
| 6365 | poly p=randomLast(100)[e]+random(-50,50); //the generic element |
---|
| 6366 | matrix n[d][d]; |
---|
| 6367 | time = timer; |
---|
| 6368 | for(j=2;j<=e;j++) |
---|
| 6369 | { |
---|
| 6370 | va=va*var(j); |
---|
| 6371 | } |
---|
| 6372 | for(j=1;j<=d;j++) |
---|
| 6373 | { |
---|
| 6374 | q=reduce(p*ba[j],J); |
---|
| 6375 | m=coeffs(q,ba,va); |
---|
| 6376 | n[j,1..d]=m[1..d,1]; |
---|
| 6377 | } |
---|
| 6378 | dbprint(printlevel-voice+2, |
---|
| 6379 | "// time for computing multiplication matrix (with generic element) : "+ |
---|
| 6380 | string(timer-time)); |
---|
[55fcff] | 6381 | //---------------- compute characteristic polynomial of matrix -------------- |
---|
[d88470] | 6382 | execute("ring P1=("+charstr(R)+"),T,dp;"); |
---|
| 6383 | matrix n=imap(R,n); |
---|
| 6384 | time = timer; |
---|
| 6385 | poly charpol=det(n-T*freemodule(d)); |
---|
| 6386 | dbprint(printlevel-voice+2,"// time for computing char poly: "+ |
---|
| 6387 | string(timer-time)); |
---|
[55fcff] | 6388 | //------------------- factorize characteristic polynomial ------------------- |
---|
[b9b906] | 6389 | //check first if constant term of charpoly is != 0 (which is true for |
---|
[55fcff] | 6390 | //sufficiently generic element) |
---|
[d88470] | 6391 | if(charpol[size(charpol)]!=0) |
---|
| 6392 | { |
---|
| 6393 | time = timer; |
---|
| 6394 | list fac=factor(charpol); |
---|
| 6395 | testFactor(fac,charpol); |
---|
| 6396 | dbprint(printlevel-voice+2,"// time for factorizing char poly: "+ |
---|
| 6397 | string(timer-time)); |
---|
| 6398 | int f=size(fac[1]); |
---|
[55fcff] | 6399 | //--------------------------- the irreducible case -------------------------- |
---|
[d88470] | 6400 | if(f==1) |
---|
| 6401 | { |
---|
| 6402 | setring R; |
---|
| 6403 | re=I; |
---|
| 6404 | return(re); |
---|
| 6405 | } |
---|
[55fcff] | 6406 | //---------------------------- the reducible case --------------------------- |
---|
[b9b906] | 6407 | //if f_i are the irreducible factors of charpoly, mult=ri, then <I,g_i^ri> |
---|
| 6408 | //are the primary components where g_i = f_i(p). However, substituting p in |
---|
| 6409 | //f_i may result in a huge object although the final result may be small. |
---|
| 6410 | //Hence it is better to simultaneously reduce with I. For this we need a new |
---|
[55fcff] | 6411 | //ring. |
---|
[d88470] | 6412 | execute("ring P=("+charstr(R)+"),(T,"+varstr(R)+"),(dp(1),dp);"); |
---|
| 6413 | list rfac=imap(P1,fac); |
---|
| 6414 | intvec ov=option(get);; |
---|
| 6415 | option(redSB); |
---|
| 6416 | list re1; |
---|
| 6417 | ideal new = T-imap(R,p),imap(R,J); |
---|
| 6418 | attrib(new, "isSB",1); //we know that new is a standard basis |
---|
| 6419 | for(j=1;j<=f;j++) |
---|
| 6420 | { |
---|
| 6421 | re1[j]=reduce(rfac[1][j]^rfac[2][j],new); |
---|
| 6422 | } |
---|
| 6423 | setring R; |
---|
| 6424 | re = imap(P,re1); |
---|
| 6425 | for(j=1;j<=f;j++) |
---|
| 6426 | { |
---|
| 6427 | J=I,re[j]; |
---|
| 6428 | re[j]=interred(J); |
---|
| 6429 | } |
---|
| 6430 | option(set,ov); |
---|
| 6431 | return(re); |
---|
[7f24dd7] | 6432 | } |
---|
| 6433 | else |
---|
[55fcff] | 6434 | //------------------- choice of generic element failed ------------------- |
---|
[7f24dd7] | 6435 | { |
---|
[d88470] | 6436 | dbprint(printlevel-voice+2,"// try new generic element!"); |
---|
| 6437 | setring R; |
---|
| 6438 | return(zerodec(I)); |
---|
[7f24dd7] | 6439 | } |
---|
| 6440 | } |
---|
| 6441 | example |
---|
| 6442 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 6443 | ring r = 0,(x,y),dp; |
---|
| 6444 | ideal i = x2-2,y2-2; |
---|
| 6445 | list pr = zerodec(i); |
---|
[7f24dd7] | 6446 | pr; |
---|
| 6447 | } |
---|
[808a9f3] | 6448 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6449 | static proc newDecompStep(ideal i, list #) |
---|
| 6450 | "USAGE: newDecompStep(i); i ideal (for primary decomposition) |
---|
| 6451 | newDecompStep(i,1); (for the associated primes of dimension of i) |
---|
| 6452 | newDecompStep(i,2); (for the minimal associated primes) |
---|
[f995aa] | 6453 | newDecompStep(i,3); (for the absolute primary decomposition (not tested!)) |
---|
[808a9f3] | 6454 | "oneIndep"; (for using only one max indep set) |
---|
| 6455 | "intersect"; (returns alse the intersection of the components founded) |
---|
| 6456 | |
---|
| 6457 | RETURN: list = list of primary ideals and their associated primes |
---|
| 6458 | (at even positions in the list) |
---|
| 6459 | (resp. a list of the minimal associated primes) |
---|
| 6460 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
| 6461 | EXAMPLE: example newDecompStep; shows an example |
---|
| 6462 | " |
---|
| 6463 | { |
---|
| 6464 | intvec op,@vv; |
---|
| 6465 | def @P = basering; |
---|
| 6466 | list primary,indep,ltras; |
---|
| 6467 | intvec @vh,isat,@w; |
---|
| 6468 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi,abspri,ab,nn; |
---|
| 6469 | ideal peek=i; |
---|
| 6470 | ideal ser,tras; |
---|
| 6471 | list data; |
---|
| 6472 | list result; |
---|
| 6473 | intvec @hilb; |
---|
| 6474 | int isS=(attrib(i,"isSB")==1); |
---|
| 6475 | |
---|
| 6476 | // Debug |
---|
| 6477 | dbprint(printlevel - voice, "newDecompStep, v2.0"); |
---|
| 6478 | |
---|
| 6479 | string indepOption = "allIndep"; |
---|
| 6480 | string intersectOption = "noIntersect"; |
---|
| 6481 | |
---|
| 6482 | if(size(#)>0) |
---|
| 6483 | { |
---|
[70ab73] | 6484 | int count = 1; |
---|
| 6485 | if(typeof(#[count]) == "string") |
---|
| 6486 | { |
---|
| 6487 | if ((#[count] == "oneIndep") or (#[count] == "allIndep")) |
---|
| 6488 | { |
---|
| 6489 | indepOption = #[count]; |
---|
| 6490 | count++; |
---|
[7f7c25e] | 6491 | } |
---|
[70ab73] | 6492 | } |
---|
| 6493 | if(typeof(#[count]) == "string") |
---|
| 6494 | { |
---|
| 6495 | if ((#[count] == "intersect") or (#[count] == "noIntersect")) |
---|
[7f7c25e] | 6496 | { |
---|
[70ab73] | 6497 | intersectOption = #[count]; |
---|
| 6498 | count++; |
---|
[7f7c25e] | 6499 | } |
---|
[70ab73] | 6500 | } |
---|
| 6501 | if((typeof(#[count]) == "int") or (typeof(#[count]) == "number")) |
---|
| 6502 | { |
---|
| 6503 | if ((#[count]==1)||(#[count]==2)||(#[count]==3)) |
---|
[7f7c25e] | 6504 | { |
---|
[70ab73] | 6505 | @wr=#[count]; |
---|
| 6506 | if(@wr==3){abspri = 1; @wr = 0;} |
---|
| 6507 | count++; |
---|
[7f7c25e] | 6508 | } |
---|
[70ab73] | 6509 | } |
---|
| 6510 | if(size(#)>count) |
---|
| 6511 | { |
---|
| 6512 | seri=1; |
---|
| 6513 | peek=#[count + 1]; |
---|
| 6514 | ser=#[count + 2]; |
---|
| 6515 | } |
---|
| 6516 | } |
---|
| 6517 | if(abspri) |
---|
| 6518 | { |
---|
| 6519 | list absprimary,abskeep,absprimarytmp,abskeeptmp; |
---|
| 6520 | } |
---|
| 6521 | homo=homog(i); |
---|
| 6522 | if(homo==1) |
---|
| 6523 | { |
---|
| 6524 | if(attrib(i,"isSB")!=1) |
---|
| 6525 | { |
---|
| 6526 | //ltras=mstd(i); |
---|
| 6527 | tras=groebner(i); |
---|
| 6528 | ltras=tras,tras; |
---|
| 6529 | attrib(ltras[1],"isSB",1); |
---|
| 6530 | } |
---|
| 6531 | else |
---|
| 6532 | { |
---|
| 6533 | ltras=i,i; |
---|
| 6534 | attrib(ltras[1],"isSB",1); |
---|
| 6535 | } |
---|
| 6536 | tras = ltras[1]; |
---|
| 6537 | attrib(tras,"isSB",1); |
---|
| 6538 | if(dim(tras)==0) |
---|
| 6539 | { |
---|
| 6540 | primary[1]=ltras[2]; |
---|
| 6541 | primary[2]=maxideal(1); |
---|
| 6542 | if(@wr>0) |
---|
| 6543 | { |
---|
| 6544 | list l; |
---|
| 6545 | l[2]=maxideal(1); |
---|
| 6546 | l[1]=maxideal(1); |
---|
| 6547 | if (intersectOption == "intersect") |
---|
[808a9f3] | 6548 | { |
---|
[70ab73] | 6549 | return(list(l, maxideal(1))); |
---|
[808a9f3] | 6550 | } |
---|
[70ab73] | 6551 | else |
---|
| 6552 | { |
---|
| 6553 | return(l); |
---|
[7f7c25e] | 6554 | } |
---|
[70ab73] | 6555 | } |
---|
| 6556 | if (intersectOption == "intersect") |
---|
| 6557 | { |
---|
| 6558 | return(list(primary, primary[1])); |
---|
| 6559 | } |
---|
| 6560 | else |
---|
| 6561 | { |
---|
| 6562 | return(primary); |
---|
| 6563 | } |
---|
| 6564 | } |
---|
| 6565 | for(@n=1;@n<=nvars(basering);@n++) |
---|
| 6566 | { |
---|
| 6567 | @w[@n]=ord(var(@n)); |
---|
| 6568 | } |
---|
| 6569 | @hilb=hilb(tras,1,@w); |
---|
| 6570 | intvec keephilb=@hilb; |
---|
[808a9f3] | 6571 | } |
---|
| 6572 | |
---|
| 6573 | //---------------------------------------------------------------- |
---|
| 6574 | //i is the zero-ideal |
---|
| 6575 | //---------------------------------------------------------------- |
---|
| 6576 | |
---|
| 6577 | if(size(i)==0) |
---|
| 6578 | { |
---|
[7f7c25e] | 6579 | primary=i,i; |
---|
[70ab73] | 6580 | if (intersectOption == "intersect") |
---|
| 6581 | { |
---|
| 6582 | return(list(primary, i)); |
---|
| 6583 | } |
---|
| 6584 | else |
---|
| 6585 | { |
---|
| 6586 | return(primary); |
---|
[7f7c25e] | 6587 | } |
---|
[808a9f3] | 6588 | } |
---|
| 6589 | |
---|
| 6590 | //---------------------------------------------------------------- |
---|
| 6591 | //pass to the lexicographical ordering and compute a standardbasis |
---|
| 6592 | //---------------------------------------------------------------- |
---|
| 6593 | |
---|
| 6594 | int lp=islp(); |
---|
| 6595 | |
---|
| 6596 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
| 6597 | op=option(get); |
---|
| 6598 | option(redSB); |
---|
| 6599 | |
---|
| 6600 | ideal ser=fetch(@P,ser); |
---|
| 6601 | if(homo==1) |
---|
| 6602 | { |
---|
[70ab73] | 6603 | if(!lp) |
---|
| 6604 | { |
---|
| 6605 | ideal @j=std(fetch(@P,i),@hilb,@w); |
---|
| 6606 | } |
---|
| 6607 | else |
---|
| 6608 | { |
---|
| 6609 | ideal @j=fetch(@P,tras); |
---|
| 6610 | attrib(@j,"isSB",1); |
---|
| 6611 | } |
---|
[808a9f3] | 6612 | } |
---|
| 6613 | else |
---|
| 6614 | { |
---|
[70ab73] | 6615 | if(lp&&isS) |
---|
| 6616 | { |
---|
| 6617 | ideal @j=fetch(@P,i); |
---|
| 6618 | attrib(@j,"isSB",1); |
---|
| 6619 | } |
---|
| 6620 | else |
---|
| 6621 | { |
---|
| 6622 | ideal @j=groebner(fetch(@P,i)); |
---|
| 6623 | } |
---|
[808a9f3] | 6624 | } |
---|
| 6625 | option(set,op); |
---|
| 6626 | if(seri==1) |
---|
| 6627 | { |
---|
| 6628 | ideal peek=fetch(@P,peek); |
---|
| 6629 | attrib(peek,"isSB",1); |
---|
| 6630 | } |
---|
| 6631 | else |
---|
| 6632 | { |
---|
| 6633 | ideal peek=@j; |
---|
| 6634 | } |
---|
| 6635 | if((size(ser)==0)&&(!abspri)) |
---|
| 6636 | { |
---|
| 6637 | ideal fried; |
---|
| 6638 | @n=size(@j); |
---|
| 6639 | for(@k=1;@k<=@n;@k++) |
---|
| 6640 | { |
---|
| 6641 | if(deg(lead(@j[@k]))==1) |
---|
| 6642 | { |
---|
| 6643 | fried[size(fried)+1]=@j[@k]; |
---|
| 6644 | @j[@k]=0; |
---|
| 6645 | } |
---|
| 6646 | } |
---|
| 6647 | if(size(fried)==nvars(basering)) |
---|
| 6648 | { |
---|
[70ab73] | 6649 | setring @P; |
---|
| 6650 | primary[1]=i; |
---|
| 6651 | primary[2]=i; |
---|
| 6652 | if (intersectOption == "intersect") |
---|
| 6653 | { |
---|
| 6654 | return(list(primary, i)); |
---|
| 6655 | } |
---|
| 6656 | else |
---|
| 6657 | { |
---|
| 6658 | return(primary); |
---|
| 6659 | } |
---|
[808a9f3] | 6660 | } |
---|
| 6661 | if(size(fried)>0) |
---|
| 6662 | { |
---|
[70ab73] | 6663 | string newva; |
---|
| 6664 | string newma; |
---|
| 6665 | for(@k=1;@k<=nvars(basering);@k++) |
---|
| 6666 | { |
---|
| 6667 | @n1=0; |
---|
| 6668 | for(@n=1;@n<=size(fried);@n++) |
---|
| 6669 | { |
---|
| 6670 | if(leadmonom(fried[@n])==var(@k)) |
---|
[808a9f3] | 6671 | { |
---|
[70ab73] | 6672 | @n1=1; |
---|
| 6673 | break; |
---|
[808a9f3] | 6674 | } |
---|
[70ab73] | 6675 | } |
---|
| 6676 | if(@n1==0) |
---|
| 6677 | { |
---|
| 6678 | newva=newva+string(var(@k))+","; |
---|
| 6679 | newma=newma+string(var(@k))+","; |
---|
| 6680 | } |
---|
| 6681 | else |
---|
| 6682 | { |
---|
| 6683 | newma=newma+string(0)+","; |
---|
| 6684 | } |
---|
| 6685 | } |
---|
| 6686 | newva[size(newva)]=")"; |
---|
| 6687 | newma[size(newma)]=";"; |
---|
| 6688 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
---|
| 6689 | execute("map @kappa=gnir,"+newma); |
---|
| 6690 | ideal @j= @kappa(@j); |
---|
| 6691 | @j=simplify(@j, 2); |
---|
| 6692 | attrib(@j,"isSB",1); |
---|
| 6693 | result = newDecompStep(@j, indepOption, intersectOption, @wr); |
---|
| 6694 | if (intersectOption == "intersect") |
---|
| 6695 | { |
---|
[a36e78] | 6696 | list pr = result[1]; |
---|
| 6697 | ideal intersection = result[2]; |
---|
[70ab73] | 6698 | } |
---|
| 6699 | else |
---|
| 6700 | { |
---|
| 6701 | list pr = result; |
---|
| 6702 | } |
---|
[808a9f3] | 6703 | |
---|
[70ab73] | 6704 | setring gnir; |
---|
| 6705 | list pr=imap(@deirf,pr); |
---|
| 6706 | for(@k=1;@k<=size(pr);@k++) |
---|
| 6707 | { |
---|
| 6708 | @j=pr[@k]+fried; |
---|
| 6709 | pr[@k]=@j; |
---|
| 6710 | } |
---|
| 6711 | if (intersectOption == "intersect") |
---|
| 6712 | { |
---|
| 6713 | ideal intersection = imap(@deirf, intersection); |
---|
| 6714 | @j = intersection + fried; |
---|
| 6715 | intersection = @j; |
---|
| 6716 | } |
---|
| 6717 | setring @P; |
---|
| 6718 | if (intersectOption == "intersect") |
---|
| 6719 | { |
---|
| 6720 | return(list(imap(gnir,pr), imap(gnir,intersection))); |
---|
| 6721 | } |
---|
| 6722 | else |
---|
| 6723 | { |
---|
| 6724 | return(imap(gnir,pr)); |
---|
| 6725 | } |
---|
[808a9f3] | 6726 | } |
---|
| 6727 | } |
---|
| 6728 | //---------------------------------------------------------------- |
---|
| 6729 | //j is the ring |
---|
| 6730 | //---------------------------------------------------------------- |
---|
| 6731 | |
---|
| 6732 | if (dim(@j)==-1) |
---|
| 6733 | { |
---|
| 6734 | setring @P; |
---|
| 6735 | primary=ideal(1),ideal(1); |
---|
[70ab73] | 6736 | if (intersectOption == "intersect") |
---|
| 6737 | { |
---|
[7f7c25e] | 6738 | return(list(primary, ideal(1))); |
---|
[70ab73] | 6739 | } |
---|
| 6740 | else |
---|
| 6741 | { |
---|
[7f7c25e] | 6742 | return(primary); |
---|
| 6743 | } |
---|
[808a9f3] | 6744 | } |
---|
| 6745 | |
---|
| 6746 | //---------------------------------------------------------------- |
---|
| 6747 | // the case of one variable |
---|
| 6748 | //---------------------------------------------------------------- |
---|
| 6749 | |
---|
| 6750 | if(nvars(basering)==1) |
---|
| 6751 | { |
---|
[70ab73] | 6752 | list fac=factor(@j[1]); |
---|
| 6753 | list gprimary; |
---|
| 6754 | poly generator; |
---|
| 6755 | ideal gIntersection; |
---|
| 6756 | for(@k=1;@k<=size(fac[1]);@k++) |
---|
| 6757 | { |
---|
| 6758 | if(@wr==0) |
---|
| 6759 | { |
---|
| 6760 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
---|
| 6761 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 6762 | } |
---|
| 6763 | else |
---|
| 6764 | { |
---|
| 6765 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
---|
| 6766 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 6767 | } |
---|
| 6768 | if (intersectOption == "intersect") |
---|
| 6769 | { |
---|
| 6770 | generator = generator * fac[1][@k]; |
---|
| 6771 | } |
---|
| 6772 | } |
---|
| 6773 | if (intersectOption == "intersect") |
---|
| 6774 | { |
---|
| 6775 | gIntersection = generator; |
---|
| 6776 | } |
---|
| 6777 | setring @P; |
---|
| 6778 | primary=fetch(gnir,gprimary); |
---|
| 6779 | if (intersectOption == "intersect") |
---|
| 6780 | { |
---|
| 6781 | ideal intersection = fetch(gnir,gIntersection); |
---|
| 6782 | } |
---|
[808a9f3] | 6783 | |
---|
| 6784 | //HIER |
---|
| 6785 | if(abspri) |
---|
| 6786 | { |
---|
[70ab73] | 6787 | list resu,tempo; |
---|
| 6788 | string absotto; |
---|
| 6789 | for(ab=1;ab<=size(primary)/2;ab++) |
---|
| 6790 | { |
---|
| 6791 | absotto= absFactorize(primary[2*ab][1],77); |
---|
| 6792 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 6793 | resu[ab]=tempo; |
---|
| 6794 | } |
---|
| 6795 | primary=resu; |
---|
| 6796 | intersection = 1; |
---|
| 6797 | for(ab=1;ab<=size(primary);ab++) |
---|
| 6798 | { |
---|
| 6799 | intersection = intersect(intersection, primary[ab][2]); |
---|
| 6800 | } |
---|
| 6801 | } |
---|
| 6802 | if (intersectOption == "intersect") |
---|
| 6803 | { |
---|
| 6804 | return(list(primary, intersection)); |
---|
| 6805 | } |
---|
| 6806 | else |
---|
| 6807 | { |
---|
| 6808 | return(primary); |
---|
| 6809 | } |
---|
[808a9f3] | 6810 | } |
---|
| 6811 | |
---|
| 6812 | //------------------------------------------------------------------ |
---|
| 6813 | //the zero-dimensional case |
---|
| 6814 | //------------------------------------------------------------------ |
---|
| 6815 | if (dim(@j)==0) |
---|
| 6816 | { |
---|
| 6817 | op=option(get); |
---|
| 6818 | option(redSB); |
---|
| 6819 | list gprimary= newZero_decomp(@j,ser,@wr); |
---|
| 6820 | |
---|
| 6821 | setring @P; |
---|
| 6822 | primary=fetch(gnir,gprimary); |
---|
| 6823 | |
---|
| 6824 | if(size(ser)>0) |
---|
| 6825 | { |
---|
| 6826 | primary=cleanPrimary(primary); |
---|
| 6827 | } |
---|
| 6828 | //HIER |
---|
| 6829 | if(abspri) |
---|
| 6830 | { |
---|
[70ab73] | 6831 | list resu,tempo; |
---|
| 6832 | string absotto; |
---|
| 6833 | for(ab=1;ab<=size(primary)/2;ab++) |
---|
| 6834 | { |
---|
| 6835 | absotto= absFactorize(primary[2*ab][1],77); |
---|
| 6836 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 6837 | resu[ab]=tempo; |
---|
| 6838 | } |
---|
| 6839 | primary=resu; |
---|
[808a9f3] | 6840 | } |
---|
[70ab73] | 6841 | if (intersectOption == "intersect") |
---|
| 6842 | { |
---|
[7f7c25e] | 6843 | return(list(primary, fetch(gnir,@j))); |
---|
[70ab73] | 6844 | } |
---|
| 6845 | else |
---|
| 6846 | { |
---|
[7f7c25e] | 6847 | return(primary); |
---|
| 6848 | } |
---|
[808a9f3] | 6849 | } |
---|
| 6850 | |
---|
| 6851 | poly @gs,@gh,@p; |
---|
| 6852 | string @va,quotring; |
---|
| 6853 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
---|
| 6854 | ideal @h; |
---|
| 6855 | int jdim=dim(@j); |
---|
| 6856 | list fett; |
---|
| 6857 | int lauf,di,newtest; |
---|
| 6858 | //------------------------------------------------------------------ |
---|
| 6859 | //search for a maximal independent set indep,i.e. |
---|
| 6860 | //look for subring such that the intersection with the ideal is zero |
---|
| 6861 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
---|
| 6862 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
---|
| 6863 | //------------------------------------------------------------------ |
---|
| 6864 | if(@wr!=1) |
---|
| 6865 | { |
---|
[70ab73] | 6866 | allindep = newMaxIndependSetLp(@j, indepOption); |
---|
| 6867 | for(@m=1;@m<=size(allindep);@m++) |
---|
| 6868 | { |
---|
| 6869 | if(allindep[@m][3]==jdim) |
---|
| 6870 | { |
---|
| 6871 | di++; |
---|
| 6872 | indep[di]=allindep[@m]; |
---|
| 6873 | } |
---|
| 6874 | else |
---|
| 6875 | { |
---|
| 6876 | lauf++; |
---|
| 6877 | restindep[lauf]=allindep[@m]; |
---|
| 6878 | } |
---|
| 6879 | } |
---|
| 6880 | } |
---|
| 6881 | else |
---|
| 6882 | { |
---|
| 6883 | indep = newMaxIndependSetLp(@j, indepOption); |
---|
| 6884 | } |
---|
[808a9f3] | 6885 | |
---|
| 6886 | ideal jkeep=@j; |
---|
| 6887 | if(ordstr(@P)[1]=="w") |
---|
| 6888 | { |
---|
[70ab73] | 6889 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
---|
[808a9f3] | 6890 | } |
---|
| 6891 | else |
---|
| 6892 | { |
---|
[70ab73] | 6893 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
---|
[808a9f3] | 6894 | } |
---|
| 6895 | |
---|
| 6896 | if(homo==1) |
---|
| 6897 | { |
---|
| 6898 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
---|
| 6899 | ||(ordstr(@P)[3]=="w")) |
---|
| 6900 | { |
---|
| 6901 | ideal jwork=imap(@P,tras); |
---|
| 6902 | attrib(jwork,"isSB",1); |
---|
| 6903 | } |
---|
| 6904 | else |
---|
| 6905 | { |
---|
| 6906 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
---|
| 6907 | } |
---|
| 6908 | } |
---|
| 6909 | else |
---|
| 6910 | { |
---|
| 6911 | ideal jwork=groebner(imap(gnir,@j)); |
---|
| 6912 | } |
---|
| 6913 | list hquprimary; |
---|
| 6914 | poly @p,@q; |
---|
| 6915 | ideal @h,fac,ser; |
---|
| 6916 | //Aenderung================ |
---|
| 6917 | ideal @Ptest=1; |
---|
| 6918 | //========================= |
---|
| 6919 | di=dim(jwork); |
---|
| 6920 | keepdi=di; |
---|
| 6921 | |
---|
| 6922 | ser = 1; |
---|
| 6923 | |
---|
| 6924 | setring gnir; |
---|
| 6925 | for(@m=1; @m<=size(indep); @m++) |
---|
| 6926 | { |
---|
| 6927 | data[1] = indep[@m]; |
---|
| 6928 | result = newReduction(@j, ser, @hilb, @w, jdim, abspri, @wr, data); |
---|
| 6929 | quprimary = quprimary + result[1]; |
---|
[70ab73] | 6930 | if(abspri) |
---|
| 6931 | { |
---|
[808a9f3] | 6932 | absprimary = absprimary + result[2]; |
---|
| 6933 | abskeep = abskeep + result[3]; |
---|
[7f7c25e] | 6934 | } |
---|
[808a9f3] | 6935 | @h = result[5]; |
---|
| 6936 | ser = result[4]; |
---|
[7f7c25e] | 6937 | if(size(@h)>0) |
---|
| 6938 | { |
---|
[70ab73] | 6939 | //--------------------------------------------------------------- |
---|
| 6940 | //we change to @Phelp to have the ordering dp for saturation |
---|
| 6941 | //--------------------------------------------------------------- |
---|
[808a9f3] | 6942 | |
---|
[70ab73] | 6943 | setring @Phelp; |
---|
| 6944 | @h=imap(gnir,@h); |
---|
[808a9f3] | 6945 | //Aenderung================================== |
---|
[70ab73] | 6946 | if(defined(@LL)){kill @LL;} |
---|
| 6947 | list @LL=minSat(jwork,@h); |
---|
| 6948 | @Ptest=intersect(@Ptest,@LL[1]); |
---|
| 6949 | ser = intersect(ser, @LL[1]); |
---|
[808a9f3] | 6950 | //=========================================== |
---|
| 6951 | |
---|
[70ab73] | 6952 | if(@wr!=1) |
---|
| 6953 | { |
---|
[808a9f3] | 6954 | //Aenderung================================== |
---|
[70ab73] | 6955 | @q=@LL[2]; |
---|
[808a9f3] | 6956 | //=========================================== |
---|
[70ab73] | 6957 | //@q=minSat(jwork,@h)[2]; |
---|
| 6958 | } |
---|
| 6959 | else |
---|
| 6960 | { |
---|
| 6961 | fac=ideal(0); |
---|
| 6962 | for(lauf=1;lauf<=ncols(@h);lauf++) |
---|
[808a9f3] | 6963 | { |
---|
[70ab73] | 6964 | if(deg(@h[lauf])>0) |
---|
| 6965 | { |
---|
| 6966 | fac=fac+factorize(@h[lauf],1); |
---|
| 6967 | } |
---|
[808a9f3] | 6968 | } |
---|
[70ab73] | 6969 | fac=simplify(fac,6); |
---|
| 6970 | @q=1; |
---|
| 6971 | for(lauf=1;lauf<=size(fac);lauf++) |
---|
[808a9f3] | 6972 | { |
---|
[70ab73] | 6973 | @q=@q*fac[lauf]; |
---|
[808a9f3] | 6974 | } |
---|
[70ab73] | 6975 | } |
---|
| 6976 | jwork = std(jwork,@q); |
---|
| 6977 | keepdi = dim(jwork); |
---|
| 6978 | if(keepdi < di) |
---|
| 6979 | { |
---|
[808a9f3] | 6980 | setring gnir; |
---|
| 6981 | @j = imap(@Phelp, jwork); |
---|
[70ab73] | 6982 | ser = imap(@Phelp, ser); |
---|
| 6983 | break; |
---|
| 6984 | } |
---|
| 6985 | if(homo == 1) |
---|
| 6986 | { |
---|
| 6987 | @hilb = hilb(jwork, 1, @w); |
---|
| 6988 | } |
---|
| 6989 | |
---|
| 6990 | setring gnir; |
---|
| 6991 | ser = imap(@Phelp, ser); |
---|
| 6992 | @j = imap(@Phelp, jwork); |
---|
| 6993 | } |
---|
[808a9f3] | 6994 | } |
---|
| 6995 | |
---|
| 6996 | if((size(quprimary)==0)&&(@wr==1)) |
---|
| 6997 | { |
---|
[a36e78] | 6998 | @j=ideal(1); |
---|
| 6999 | quprimary[1]=ideal(1); |
---|
| 7000 | quprimary[2]=ideal(1); |
---|
[808a9f3] | 7001 | } |
---|
| 7002 | if((size(quprimary)==0)) |
---|
| 7003 | { |
---|
| 7004 | keepdi = di - 1; |
---|
| 7005 | quprimary[1]=ideal(1); |
---|
| 7006 | quprimary[2]=ideal(1); |
---|
| 7007 | } |
---|
| 7008 | //--------------------------------------------------------------- |
---|
| 7009 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
---|
| 7010 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
---|
| 7011 | //--------------------------------------------------------------- |
---|
| 7012 | if((deg(@j[1])!=0)&&(@wr!=1)) |
---|
| 7013 | { |
---|
[a36e78] | 7014 | if(size(quprimary)>0) |
---|
| 7015 | { |
---|
| 7016 | setring @Phelp; |
---|
| 7017 | ser=imap(gnir,ser); |
---|
[808a9f3] | 7018 | |
---|
[a36e78] | 7019 | hquprimary=imap(gnir,quprimary); |
---|
| 7020 | if(@wr==0) |
---|
| 7021 | { |
---|
[808a9f3] | 7022 | //Aenderung==================================================== |
---|
| 7023 | //HIER STATT DURCHSCHNITT SATURIEREN! |
---|
[a36e78] | 7024 | ideal htest=@Ptest; |
---|
[808a9f3] | 7025 | /* |
---|
[a36e78] | 7026 | ideal htest=hquprimary[1]; |
---|
| 7027 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
---|
| 7028 | { |
---|
| 7029 | htest=intersect(htest,hquprimary[2*@n1-1]); |
---|
| 7030 | } |
---|
[808a9f3] | 7031 | */ |
---|
| 7032 | //============================================================= |
---|
| 7033 | } |
---|
| 7034 | else |
---|
| 7035 | { |
---|
[a36e78] | 7036 | ideal htest=hquprimary[2]; |
---|
[808a9f3] | 7037 | |
---|
[a36e78] | 7038 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
---|
| 7039 | { |
---|
| 7040 | htest=intersect(htest,hquprimary[2*@n1]); |
---|
| 7041 | } |
---|
[808a9f3] | 7042 | } |
---|
[70ab73] | 7043 | |
---|
[a36e78] | 7044 | if(size(ser)>0) |
---|
[808a9f3] | 7045 | { |
---|
[a36e78] | 7046 | ser=intersect(htest,ser); |
---|
[808a9f3] | 7047 | } |
---|
[a36e78] | 7048 | else |
---|
[808a9f3] | 7049 | { |
---|
[a36e78] | 7050 | ser=htest; |
---|
[70ab73] | 7051 | } |
---|
| 7052 | setring gnir; |
---|
[a36e78] | 7053 | ser=imap(@Phelp,ser); |
---|
| 7054 | } |
---|
| 7055 | if(size(reduce(ser,peek,1))!=0) |
---|
| 7056 | { |
---|
| 7057 | for(@m=1;@m<=size(restindep);@m++) |
---|
| 7058 | { |
---|
| 7059 | // if(restindep[@m][3]>=keepdi) |
---|
| 7060 | // { |
---|
| 7061 | isat=0; |
---|
| 7062 | @n2=0; |
---|
| 7063 | |
---|
| 7064 | if(restindep[@m][1]==varstr(basering)) |
---|
| 7065 | //the good case, nothing to do, just to have the same notations |
---|
| 7066 | //change the ring |
---|
| 7067 | { |
---|
| 7068 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
| 7069 | varstr(basering)+"),("+ordstr(basering)+");"); |
---|
| 7070 | ideal @j=fetch(gnir,jkeep); |
---|
| 7071 | attrib(@j,"isSB",1); |
---|
| 7072 | } |
---|
| 7073 | else |
---|
| 7074 | { |
---|
| 7075 | @va=string(maxideal(1)); |
---|
| 7076 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
| 7077 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
---|
| 7078 | execute("map phi=gnir,"+@va+";"); |
---|
| 7079 | op=option(get); |
---|
| 7080 | option(redSB); |
---|
| 7081 | if(homo==1) |
---|
| 7082 | { |
---|
| 7083 | ideal @j=std(phi(jkeep),keephilb,@w); |
---|
| 7084 | } |
---|
| 7085 | else |
---|
| 7086 | { |
---|
| 7087 | ideal @j=groebner(phi(jkeep)); |
---|
| 7088 | } |
---|
| 7089 | ideal ser=phi(ser); |
---|
| 7090 | option(set,op); |
---|
| 7091 | } |
---|
| 7092 | |
---|
| 7093 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 7094 | { |
---|
| 7095 | fett[lauf]=size(@j[lauf]); |
---|
| 7096 | } |
---|
| 7097 | //------------------------------------------------------------------ |
---|
| 7098 | //we have now the following situation: |
---|
| 7099 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
---|
| 7100 | //pass to this quotientring, j is their still a standardbasis, the |
---|
| 7101 | //leading coefficients of the polynomials there (polynomials in |
---|
| 7102 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 7103 | //we need their ggt, gh, because of the following: |
---|
| 7104 | //let (j:gh^n)=(j:gh^infinity) then |
---|
| 7105 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7106 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 7107 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 7108 | |
---|
| 7109 | //------------------------------------------------------------------ |
---|
| 7110 | |
---|
| 7111 | //the arrangement for the quotientring |
---|
| 7112 | // K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7113 | //and the map phi:K[var(1),...,var(nva)] ----> |
---|
| 7114 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
| 7115 | //------------------------------------------------------------------ |
---|
| 7116 | |
---|
| 7117 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
---|
| 7118 | |
---|
| 7119 | //------------------------------------------------------------------ |
---|
| 7120 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 7121 | //------------------------------------------------------------------ |
---|
| 7122 | |
---|
| 7123 | execute(quotring); |
---|
| 7124 | |
---|
| 7125 | // @j considered in the quotientring |
---|
| 7126 | ideal @j=imap(gnir1,@j); |
---|
| 7127 | ideal ser=imap(gnir1,ser); |
---|
| 7128 | |
---|
| 7129 | kill gnir1; |
---|
| 7130 | |
---|
| 7131 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 7132 | //here it becomes minimal |
---|
| 7133 | @j=clearSB(@j,fett); |
---|
| 7134 | attrib(@j,"isSB",1); |
---|
| 7135 | |
---|
| 7136 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 7137 | ideal @h; |
---|
| 7138 | |
---|
| 7139 | for(@n=1;@n<=size(@j);@n++) |
---|
| 7140 | { |
---|
| 7141 | @h[@n]=leadcoef(@j[@n]); |
---|
| 7142 | } |
---|
| 7143 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 7144 | |
---|
| 7145 | op=option(get); |
---|
| 7146 | option(redSB); |
---|
| 7147 | list uprimary= newZero_decomp(@j,ser,@wr); |
---|
| 7148 | //HIER |
---|
| 7149 | if(abspri) |
---|
| 7150 | { |
---|
| 7151 | ideal II; |
---|
| 7152 | ideal jmap; |
---|
| 7153 | map sigma; |
---|
| 7154 | nn=nvars(basering); |
---|
| 7155 | map invsigma=basering,maxideal(1); |
---|
| 7156 | for(ab=1;ab<=size(uprimary)/2;ab++) |
---|
| 7157 | { |
---|
| 7158 | II=uprimary[2*ab]; |
---|
| 7159 | attrib(II,"isSB",1); |
---|
| 7160 | if(deg(II[1])!=vdim(II)) |
---|
| 7161 | { |
---|
| 7162 | jmap=randomLast(50); |
---|
| 7163 | sigma=basering,jmap; |
---|
| 7164 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 7165 | invsigma=basering,jmap; |
---|
| 7166 | II=groebner(sigma(II)); |
---|
| 7167 | } |
---|
| 7168 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 7169 | II=var(nn); |
---|
| 7170 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 7171 | invsigma=basering,maxideal(1); |
---|
| 7172 | } |
---|
| 7173 | } |
---|
| 7174 | option(set,op); |
---|
| 7175 | |
---|
| 7176 | //we need the intersection of the ideals in the list quprimary with |
---|
| 7177 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
---|
| 7178 | //such an ideal but fi polynomials, then the intersection of q with |
---|
| 7179 | //the polynomialring is the saturation of the ideal generated by |
---|
| 7180 | //f1,...,fr with respect toh which is the lcm of the leading |
---|
| 7181 | //coefficients of the fi considered in the quotientring: |
---|
| 7182 | //this is coded in saturn |
---|
| 7183 | |
---|
| 7184 | list saturn; |
---|
| 7185 | ideal hpl; |
---|
| 7186 | |
---|
| 7187 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 7188 | { |
---|
| 7189 | hpl=0; |
---|
| 7190 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 7191 | { |
---|
| 7192 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 7193 | } |
---|
| 7194 | saturn[@n]=hpl; |
---|
| 7195 | } |
---|
| 7196 | //------------------------------------------------------------------ |
---|
| 7197 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 7198 | //back to the polynomialring |
---|
| 7199 | //------------------------------------------------------------------ |
---|
| 7200 | setring gnir; |
---|
| 7201 | collectprimary=imap(quring,uprimary); |
---|
| 7202 | lsau=imap(quring,saturn); |
---|
| 7203 | @h=imap(quring,@h); |
---|
| 7204 | |
---|
| 7205 | kill quring; |
---|
| 7206 | |
---|
| 7207 | |
---|
| 7208 | @n2=size(quprimary); |
---|
[808a9f3] | 7209 | //================NEU========================================= |
---|
[a36e78] | 7210 | if(deg(quprimary[1][1])<=0){ @n2=0; } |
---|
[808a9f3] | 7211 | //============================================================ |
---|
| 7212 | |
---|
[a36e78] | 7213 | @n3=@n2; |
---|
| 7214 | |
---|
| 7215 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
---|
| 7216 | { |
---|
| 7217 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 7218 | { |
---|
| 7219 | @n2++; |
---|
| 7220 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 7221 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 7222 | @n2++; |
---|
| 7223 | lnew[@n2]=lsau[2*@n1]; |
---|
| 7224 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 7225 | if(abspri) |
---|
| 7226 | { |
---|
| 7227 | absprimary[@n2/2]=absprimarytmp[@n1]; |
---|
| 7228 | abskeep[@n2/2]=abskeeptmp[@n1]; |
---|
| 7229 | } |
---|
| 7230 | } |
---|
| 7231 | } |
---|
| 7232 | |
---|
| 7233 | |
---|
| 7234 | //here the intersection with the polynomialring |
---|
| 7235 | //mentioned above is really computed |
---|
| 7236 | |
---|
| 7237 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
| 7238 | { |
---|
| 7239 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
| 7240 | { |
---|
| 7241 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7242 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 7243 | } |
---|
| 7244 | else |
---|
| 7245 | { |
---|
| 7246 | if(@wr==0) |
---|
| 7247 | { |
---|
| 7248 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7249 | } |
---|
| 7250 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
| 7251 | } |
---|
| 7252 | } |
---|
| 7253 | if(@n2>=@n3+2) |
---|
| 7254 | { |
---|
| 7255 | setring @Phelp; |
---|
| 7256 | ser=imap(gnir,ser); |
---|
| 7257 | hquprimary=imap(gnir,quprimary); |
---|
| 7258 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
| 7259 | { |
---|
| 7260 | if(@wr==0) |
---|
| 7261 | { |
---|
| 7262 | ser=intersect(ser,hquprimary[2*@n-1]); |
---|
| 7263 | } |
---|
| 7264 | else |
---|
| 7265 | { |
---|
| 7266 | ser=intersect(ser,hquprimary[2*@n]); |
---|
| 7267 | } |
---|
| 7268 | } |
---|
| 7269 | setring gnir; |
---|
| 7270 | ser=imap(@Phelp,ser); |
---|
| 7271 | } |
---|
[808a9f3] | 7272 | |
---|
[a36e78] | 7273 | // } |
---|
[808a9f3] | 7274 | } |
---|
[a36e78] | 7275 | //HIER |
---|
| 7276 | if(abspri) |
---|
[808a9f3] | 7277 | { |
---|
[a36e78] | 7278 | list resu,tempo; |
---|
| 7279 | for(ab=1;ab<=size(quprimary)/2;ab++) |
---|
[70ab73] | 7280 | { |
---|
[a36e78] | 7281 | if (deg(quprimary[2*ab][1])!=0) |
---|
| 7282 | { |
---|
| 7283 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 7284 | absprimary[ab],abskeep[ab]; |
---|
| 7285 | resu[ab]=tempo; |
---|
| 7286 | } |
---|
[808a9f3] | 7287 | } |
---|
[a36e78] | 7288 | quprimary=resu; |
---|
| 7289 | @wr=3; |
---|
[808a9f3] | 7290 | } |
---|
[a36e78] | 7291 | if(size(reduce(ser,peek,1))!=0) |
---|
[808a9f3] | 7292 | { |
---|
[a36e78] | 7293 | if(@wr>0) |
---|
| 7294 | { |
---|
| 7295 | // The following line was dropped to avoid the recursion step: |
---|
| 7296 | //htprimary=newDecompStep(@j,@wr,peek,ser); |
---|
| 7297 | htprimary = list(); |
---|
| 7298 | } |
---|
| 7299 | else |
---|
| 7300 | { |
---|
| 7301 | // The following line was dropped to avoid the recursion step: |
---|
| 7302 | //htprimary=newDecompStep(@j,peek,ser); |
---|
| 7303 | htprimary = list(); |
---|
| 7304 | } |
---|
| 7305 | // here we collect now both results primary(sat(j,gh)) |
---|
| 7306 | // and primary(j,gh^n) |
---|
| 7307 | @n=size(quprimary); |
---|
| 7308 | if (deg(quprimary[1][1])<=0) { @n=0; } |
---|
| 7309 | for (@k=1;@k<=size(htprimary);@k++) |
---|
| 7310 | { |
---|
| 7311 | quprimary[@n+@k]=htprimary[@k]; |
---|
| 7312 | } |
---|
[808a9f3] | 7313 | } |
---|
[a36e78] | 7314 | } |
---|
| 7315 | } |
---|
| 7316 | else |
---|
| 7317 | { |
---|
[808a9f3] | 7318 | if(abspri) |
---|
| 7319 | { |
---|
| 7320 | list resu,tempo; |
---|
| 7321 | for(ab=1;ab<=size(quprimary)/2;ab++) |
---|
| 7322 | { |
---|
[a36e78] | 7323 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 7324 | absprimary[ab],abskeep[ab]; |
---|
| 7325 | resu[ab]=tempo; |
---|
[808a9f3] | 7326 | } |
---|
| 7327 | quprimary=resu; |
---|
[70ab73] | 7328 | } |
---|
[a36e78] | 7329 | } |
---|
[808a9f3] | 7330 | //--------------------------------------------------------------------------- |
---|
| 7331 | //back to the ring we started with |
---|
| 7332 | //the final result: primary |
---|
| 7333 | //--------------------------------------------------------------------------- |
---|
| 7334 | |
---|
| 7335 | setring @P; |
---|
| 7336 | primary=imap(gnir,quprimary); |
---|
| 7337 | |
---|
[70ab73] | 7338 | if (intersectOption == "intersect") |
---|
| 7339 | { |
---|
[a36e78] | 7340 | return(list(primary, imap(gnir, ser))); |
---|
[70ab73] | 7341 | } |
---|
| 7342 | else |
---|
| 7343 | { |
---|
| 7344 | return(primary); |
---|
| 7345 | } |
---|
[808a9f3] | 7346 | } |
---|
| 7347 | example |
---|
| 7348 | { "EXAMPLE:"; echo = 2; |
---|
| 7349 | ring r = 32003,(x,y,z),lp; |
---|
| 7350 | poly p = z2+1; |
---|
| 7351 | poly q = z4+2; |
---|
| 7352 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 7353 | list pr= newDecompStep(i); |
---|
| 7354 | pr; |
---|
| 7355 | testPrimary( pr, i); |
---|
| 7356 | } |
---|
| 7357 | |
---|
[7f7c25e] | 7358 | // This was part of proc decomp. |
---|
| 7359 | // In proc newDecompStep, used for the computation of the minimal associated primes, |
---|
| 7360 | // this part was separated as a soubrutine to make the code more clear. |
---|
| 7361 | // Also, since the reduction is performed twice in proc newDecompStep, it should use both times this routine. |
---|
| 7362 | // This is not yet implemented, since the reduction is not exactly the same and some changes should be made. |
---|
| 7363 | static proc newReduction(ideal @j, ideal ser, intvec @hilb, intvec @w, int jdim, int abspri, int @wr, list data) |
---|
[808a9f3] | 7364 | { |
---|
[a36e78] | 7365 | string @va; |
---|
| 7366 | string quotring; |
---|
| 7367 | intvec op; |
---|
| 7368 | intvec @vv; |
---|
| 7369 | def gnir = basering; |
---|
| 7370 | ideal isat=0; |
---|
| 7371 | int @n; |
---|
| 7372 | int @n1 = 0; |
---|
| 7373 | int @n2 = 0; |
---|
| 7374 | int @n3 = 0; |
---|
| 7375 | int homo = homog(@j); |
---|
| 7376 | int lauf; |
---|
| 7377 | int @k; |
---|
| 7378 | list fett; |
---|
| 7379 | int keepdi; |
---|
| 7380 | list collectprimary; |
---|
| 7381 | list lsau; |
---|
| 7382 | list lnew; |
---|
| 7383 | ideal @h; |
---|
| 7384 | |
---|
| 7385 | list indepInfo = data[1]; |
---|
| 7386 | list quprimary = list(); |
---|
| 7387 | |
---|
| 7388 | //if(abspri) |
---|
| 7389 | //{ |
---|
[808a9f3] | 7390 | int ab; |
---|
| 7391 | list absprimarytmp,abskeeptmp; |
---|
| 7392 | list absprimary, abskeep; |
---|
[a36e78] | 7393 | //} |
---|
| 7394 | // Debug |
---|
| 7395 | dbprint(printlevel - voice, "newReduction, v2.0"); |
---|
[808a9f3] | 7396 | |
---|
[a36e78] | 7397 | if((indepInfo[1]==varstr(basering))) // &&(@m==1) |
---|
| 7398 | //this is the good case, nothing to do, just to have the same notations |
---|
| 7399 | //change the ring |
---|
| 7400 | { |
---|
| 7401 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[808a9f3] | 7402 | +ordstr(basering)+");"); |
---|
[a36e78] | 7403 | ideal @j = fetch(gnir, @j); |
---|
| 7404 | attrib(@j,"isSB",1); |
---|
| 7405 | ideal ser = fetch(gnir, ser); |
---|
| 7406 | } |
---|
| 7407 | else |
---|
| 7408 | { |
---|
| 7409 | @va=string(maxideal(1)); |
---|
[808a9f3] | 7410 | //Aenderung============== |
---|
[a36e78] | 7411 | //if(@m==1) |
---|
| 7412 | //{ |
---|
| 7413 | // @j=fetch(@P,i); |
---|
| 7414 | //} |
---|
[808a9f3] | 7415 | //======================= |
---|
[a36e78] | 7416 | execute("ring gnir1 = ("+charstr(basering)+"),("+indepInfo[1]+"),(" |
---|
[808a9f3] | 7417 | +indepInfo[2]+");"); |
---|
[a36e78] | 7418 | execute("map phi=gnir,"+@va+";"); |
---|
| 7419 | op=option(get); |
---|
| 7420 | option(redSB); |
---|
| 7421 | if(homo==1) |
---|
| 7422 | { |
---|
| 7423 | ideal @j=std(phi(@j),@hilb,@w); |
---|
| 7424 | } |
---|
| 7425 | else |
---|
| 7426 | { |
---|
| 7427 | ideal @j=groebner(phi(@j)); |
---|
| 7428 | } |
---|
| 7429 | ideal ser=phi(ser); |
---|
[808a9f3] | 7430 | |
---|
[a36e78] | 7431 | option(set,op); |
---|
| 7432 | } |
---|
| 7433 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
---|
| 7434 | { |
---|
| 7435 | setring gnir; |
---|
| 7436 | break; |
---|
| 7437 | } |
---|
| 7438 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 7439 | { |
---|
| 7440 | fett[lauf]=size(@j[lauf]); |
---|
| 7441 | } |
---|
| 7442 | //------------------------------------------------------------------------ |
---|
| 7443 | //we have now the following situation: |
---|
| 7444 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
| 7445 | //to this quotientring, j is their still a standardbasis, the |
---|
| 7446 | //leading coefficients of the polynomials there (polynomials in |
---|
| 7447 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 7448 | //we need their ggt, gh, because of the following: let |
---|
| 7449 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7450 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 7451 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 7452 | |
---|
| 7453 | //------------------------------------------------------------------------ |
---|
| 7454 | |
---|
| 7455 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
---|
| 7456 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
---|
| 7457 | //------------------------------------------------------------------------ |
---|
| 7458 | |
---|
| 7459 | quotring=prepareQuotientring(nvars(basering)-indepInfo[3]); |
---|
| 7460 | |
---|
| 7461 | //--------------------------------------------------------------------- |
---|
| 7462 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7463 | //--------------------------------------------------------------------- |
---|
| 7464 | |
---|
| 7465 | ideal @jj=lead(@j); //!! vorn vereinbaren |
---|
| 7466 | execute(quotring); |
---|
| 7467 | |
---|
| 7468 | ideal @jj=imap(gnir1,@jj); |
---|
| 7469 | @vv=clearSBNeu(@jj,fett); //!! vorn vereinbaren |
---|
| 7470 | setring gnir1; |
---|
| 7471 | @k=size(@j); |
---|
| 7472 | for (lauf=1;lauf<=@k;lauf++) |
---|
| 7473 | { |
---|
| 7474 | if(@vv[lauf]==1) |
---|
| 7475 | { |
---|
| 7476 | @j[lauf]=0; |
---|
| 7477 | } |
---|
| 7478 | } |
---|
| 7479 | @j=simplify(@j,2); |
---|
| 7480 | setring quring; |
---|
| 7481 | // @j considered in the quotientring |
---|
| 7482 | ideal @j=imap(gnir1,@j); |
---|
[808a9f3] | 7483 | |
---|
[a36e78] | 7484 | ideal ser=imap(gnir1,ser); |
---|
[808a9f3] | 7485 | |
---|
[a36e78] | 7486 | kill gnir1; |
---|
[808a9f3] | 7487 | |
---|
[a36e78] | 7488 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 7489 | //here it becomes minimal |
---|
[808a9f3] | 7490 | |
---|
[a36e78] | 7491 | attrib(@j,"isSB",1); |
---|
[808a9f3] | 7492 | |
---|
[a36e78] | 7493 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 7494 | ideal @h; |
---|
| 7495 | if(deg(@j[1])>0) |
---|
| 7496 | { |
---|
| 7497 | for(@n=1;@n<=size(@j);@n++) |
---|
| 7498 | { |
---|
| 7499 | @h[@n]=leadcoef(@j[@n]); |
---|
| 7500 | } |
---|
| 7501 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7502 | op=option(get); |
---|
| 7503 | option(redSB); |
---|
[808a9f3] | 7504 | |
---|
[a36e78] | 7505 | int zeroMinAss = @wr; |
---|
| 7506 | if (@wr == 2) {zeroMinAss = 1;} |
---|
| 7507 | list uprimary= newZero_decomp(@j, ser, zeroMinAss); |
---|
[808a9f3] | 7508 | |
---|
[a36e78] | 7509 | //HIER |
---|
| 7510 | if(abspri) |
---|
| 7511 | { |
---|
| 7512 | ideal II; |
---|
| 7513 | ideal jmap; |
---|
| 7514 | map sigma; |
---|
| 7515 | nn=nvars(basering); |
---|
| 7516 | map invsigma=basering,maxideal(1); |
---|
| 7517 | for(ab=1;ab<=size(uprimary)/2;ab++) |
---|
| 7518 | { |
---|
| 7519 | II=uprimary[2*ab]; |
---|
| 7520 | attrib(II,"isSB",1); |
---|
| 7521 | if(deg(II[1])!=vdim(II)) |
---|
| 7522 | { |
---|
| 7523 | jmap=randomLast(50); |
---|
| 7524 | sigma=basering,jmap; |
---|
| 7525 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 7526 | invsigma=basering,jmap; |
---|
| 7527 | II=groebner(sigma(II)); |
---|
| 7528 | } |
---|
| 7529 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 7530 | II=var(nn); |
---|
| 7531 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 7532 | invsigma=basering,maxideal(1); |
---|
| 7533 | } |
---|
| 7534 | } |
---|
| 7535 | option(set,op); |
---|
| 7536 | } |
---|
| 7537 | else |
---|
| 7538 | { |
---|
| 7539 | list uprimary; |
---|
| 7540 | uprimary[1]=ideal(1); |
---|
| 7541 | uprimary[2]=ideal(1); |
---|
| 7542 | } |
---|
| 7543 | //we need the intersection of the ideals in the list quprimary with the |
---|
| 7544 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
| 7545 | //but fi polynomials, then the intersection of q with the polynomialring |
---|
| 7546 | //is the saturation of the ideal generated by f1,...,fr with respect to |
---|
| 7547 | //h which is the lcm of the leading coefficients of the fi considered in |
---|
| 7548 | //in the quotientring: this is coded in saturn |
---|
[70ab73] | 7549 | |
---|
[a36e78] | 7550 | list saturn; |
---|
| 7551 | ideal hpl; |
---|
[70ab73] | 7552 | |
---|
[a36e78] | 7553 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 7554 | { |
---|
| 7555 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
---|
| 7556 | hpl=0; |
---|
| 7557 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 7558 | { |
---|
| 7559 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 7560 | } |
---|
| 7561 | saturn[@n]=hpl; |
---|
| 7562 | } |
---|
[808a9f3] | 7563 | |
---|
[a36e78] | 7564 | //-------------------------------------------------------------------- |
---|
| 7565 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7566 | //back to the polynomialring |
---|
| 7567 | //--------------------------------------------------------------------- |
---|
| 7568 | setring gnir; |
---|
[808a9f3] | 7569 | |
---|
[a36e78] | 7570 | collectprimary=imap(quring,uprimary); |
---|
| 7571 | lsau=imap(quring,saturn); |
---|
| 7572 | @h=imap(quring,@h); |
---|
[808a9f3] | 7573 | |
---|
[a36e78] | 7574 | kill quring; |
---|
[808a9f3] | 7575 | |
---|
[a36e78] | 7576 | @n2=size(quprimary); |
---|
| 7577 | @n3=@n2; |
---|
[808a9f3] | 7578 | |
---|
[a36e78] | 7579 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
---|
| 7580 | { |
---|
| 7581 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 7582 | { |
---|
| 7583 | @n2++; |
---|
| 7584 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 7585 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 7586 | @n2++; |
---|
| 7587 | lnew[@n2]=lsau[2*@n1]; |
---|
| 7588 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 7589 | if(abspri) |
---|
| 7590 | { |
---|
| 7591 | absprimary[@n2/2]=absprimarytmp[@n1]; |
---|
| 7592 | abskeep[@n2/2]=abskeeptmp[@n1]; |
---|
| 7593 | } |
---|
| 7594 | } |
---|
| 7595 | } |
---|
[808a9f3] | 7596 | |
---|
[a36e78] | 7597 | //here the intersection with the polynomialring |
---|
| 7598 | //mentioned above is really computed |
---|
| 7599 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
| 7600 | { |
---|
| 7601 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
| 7602 | { |
---|
| 7603 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7604 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 7605 | } |
---|
| 7606 | else |
---|
| 7607 | { |
---|
| 7608 | if(@wr==0) |
---|
| 7609 | { |
---|
| 7610 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7611 | } |
---|
| 7612 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
| 7613 | } |
---|
| 7614 | } |
---|
[808a9f3] | 7615 | |
---|
[a36e78] | 7616 | return(quprimary, absprimary, abskeep, ser, @h); |
---|
| 7617 | } |
---|
[808a9f3] | 7618 | |
---|
| 7619 | |
---|
[a36e78] | 7620 | //////////////////////////////////////////////////////////////////////////// |
---|
[808a9f3] | 7621 | |
---|
| 7622 | |
---|
| 7623 | |
---|
| 7624 | |
---|
| 7625 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7626 | // Based on minAssGTZ |
---|
| 7627 | |
---|
[f995aa] | 7628 | proc minAss(ideal i,list #) |
---|
[7f7c25e] | 7629 | "USAGE: minAss(I[, l]); i ideal, l list (optional) of parameters, same as minAssGTZ |
---|
| 7630 | RETURN: a list, the minimal associated prime ideals of I. |
---|
[808a9f3] | 7631 | NOTE: Designed for characteristic 0, works also in char k > 0 based |
---|
| 7632 | on an algorithm of Yokoyama |
---|
[f995aa] | 7633 | EXAMPLE: example minAss; shows an example |
---|
[808a9f3] | 7634 | " |
---|
| 7635 | { |
---|
[70ab73] | 7636 | return(minAssGTZ(i,#)); |
---|
[808a9f3] | 7637 | } |
---|
| 7638 | example |
---|
| 7639 | { "EXAMPLE:"; echo = 2; |
---|
| 7640 | ring r = 0, (x, y, z), dp; |
---|
| 7641 | poly p = z2 + 1; |
---|
| 7642 | poly q = z3 + 2; |
---|
| 7643 | ideal i = p * q^2, y - z2; |
---|
[f995aa] | 7644 | list pr = minAss(i); |
---|
[808a9f3] | 7645 | pr; |
---|
| 7646 | } |
---|
| 7647 | |
---|
| 7648 | |
---|
| 7649 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7650 | // |
---|
[f995aa] | 7651 | // Computes the minimal associated primes of I via Laplagne algorithm, |
---|
[808a9f3] | 7652 | // using primary decomposition in the zero dimensional case. |
---|
| 7653 | // For reduction to the zerodimensional case, it uses the procedure |
---|
[f995aa] | 7654 | // decomp, with some modifications to avoid the recursion. |
---|
[808a9f3] | 7655 | // |
---|
| 7656 | |
---|
[f995aa] | 7657 | static proc minAssSL(ideal I) |
---|
[808a9f3] | 7658 | // Input = I, ideal |
---|
| 7659 | // Output = primaryDec where primaryDec is the list of the minimal |
---|
| 7660 | // associated primes and the primary components corresponding to these primes. |
---|
| 7661 | { |
---|
| 7662 | ideal P = 1; |
---|
| 7663 | list pd = list(); |
---|
| 7664 | int k; |
---|
| 7665 | int stop = 0; |
---|
| 7666 | list primaryDec = list(); |
---|
| 7667 | |
---|
[70ab73] | 7668 | while (stop == 0) |
---|
| 7669 | { |
---|
[808a9f3] | 7670 | // Debug |
---|
[f995aa] | 7671 | dbprint(printlevel - voice, "// We call minAssSLIteration to find new prime ideals!"); |
---|
| 7672 | pd = minAssSLIteration(I, P); |
---|
[808a9f3] | 7673 | // Debug |
---|
[f995aa] | 7674 | dbprint(printlevel - voice, "// Output of minAssSLIteration:"); |
---|
[808a9f3] | 7675 | dbprint(printlevel - voice, pd); |
---|
[70ab73] | 7676 | if (size(pd[1]) > 0) |
---|
| 7677 | { |
---|
[808a9f3] | 7678 | primaryDec = primaryDec + pd[1]; |
---|
| 7679 | // Debug |
---|
| 7680 | dbprint(printlevel - voice, "// We intersect the prime ideals obtained."); |
---|
| 7681 | P = intersect(P, pd[2]); |
---|
| 7682 | // Debug |
---|
| 7683 | dbprint(printlevel - voice, "// Intersection finished."); |
---|
[70ab73] | 7684 | } |
---|
| 7685 | else |
---|
| 7686 | { |
---|
[f3c6e5] | 7687 | stop = 1; |
---|
[7f7c25e] | 7688 | } |
---|
| 7689 | } |
---|
[f3c6e5] | 7690 | |
---|
[808a9f3] | 7691 | // Returns only the primary components, not the radical. |
---|
| 7692 | return(primaryDec); |
---|
[f3c6e5] | 7693 | } |
---|
[808a9f3] | 7694 | |
---|
| 7695 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7696 | // Given an ideal I and an ideal P (intersection of some minimal prime ideals |
---|
| 7697 | // associated to I), it calculates new minimal prime ideals associated to I |
---|
| 7698 | // which were not used to calculate P. |
---|
| 7699 | // This version uses Primary Decomposition in the zerodimensional case. |
---|
[f995aa] | 7700 | static proc minAssSLIteration(ideal I, ideal P); |
---|
[808a9f3] | 7701 | { |
---|
| 7702 | int k = 1; |
---|
| 7703 | int good = 0; |
---|
| 7704 | list primaryDec = list(); |
---|
| 7705 | // Debug |
---|
| 7706 | dbprint (printlevel-voice, "// We search for an element in P - sqrt(I)."); |
---|
[70ab73] | 7707 | while ((k <= size(P)) and (good == 0)) |
---|
| 7708 | { |
---|
[808a9f3] | 7709 | good = 1 - rad_con(P[k], I); |
---|
| 7710 | k++; |
---|
[7f7c25e] | 7711 | } |
---|
[808a9f3] | 7712 | k--; |
---|
[70ab73] | 7713 | if (good == 0) |
---|
| 7714 | { |
---|
[808a9f3] | 7715 | // Debug |
---|
| 7716 | dbprint (printlevel - voice, "// No element was found, P = sqrt(I)."); |
---|
| 7717 | return (list(primaryDec, ideal(0))); |
---|
[7f7c25e] | 7718 | } |
---|
[808a9f3] | 7719 | // Debug |
---|
| 7720 | dbprint (printlevel - voice, "// We found h = ", P[k]); |
---|
| 7721 | dbprint (printlevel - voice, "// We calculate the saturation of I with respect to the element just founded."); |
---|
| 7722 | ideal J = sat(I, P[k])[1]; |
---|
| 7723 | |
---|
| 7724 | // Uses decomp from primdec, modified to avoid the recursion. |
---|
| 7725 | // Debug |
---|
| 7726 | dbprint(printlevel - voice, "// We do the reduction to the zerodimensional case, via decomp."); |
---|
| 7727 | |
---|
| 7728 | primaryDec = newDecompStep(J, "oneIndep", "intersect", 2); |
---|
| 7729 | // Debug |
---|
| 7730 | dbprint(printlevel - voice, "// Proc decomp has found", size(primaryDec) / 2, "new primary components."); |
---|
| 7731 | |
---|
| 7732 | return(primaryDec); |
---|
| 7733 | } |
---|
| 7734 | |
---|
| 7735 | |
---|
| 7736 | |
---|
| 7737 | /////////////////////////////////////////////////////////////////////////////////// |
---|
| 7738 | // Based on maxIndependSet |
---|
| 7739 | // Added list # as parameter |
---|
| 7740 | // If the first element of # is 0, the output is only 1 max indep set. |
---|
| 7741 | // If no list is specified or #[1] = 1, the output is all the max indep set of the |
---|
| 7742 | // leading terms ideal. This is the original output of maxIndependSet |
---|
| 7743 | |
---|
| 7744 | proc newMaxIndependSetLp(ideal j, list #) |
---|
[f995aa] | 7745 | "USAGE: newMaxIndependentSetLp(i); i ideal (returns all maximal independent sets of the corresponding leading terms ideal) |
---|
| 7746 | newMaxIndependentSetLp(i, 0); i ideal (returns only one maximal independent set) |
---|
[808a9f3] | 7747 | RETURN: list = #1. new varstring with the maximal independent set at the end, |
---|
[f995aa] | 7748 | #2. ordstring with the lp ordering, |
---|
[808a9f3] | 7749 | #3. the number of independent variables |
---|
| 7750 | NOTE: |
---|
[f995aa] | 7751 | EXAMPLE: example newMaxIndependentSetLp; shows an example |
---|
[808a9f3] | 7752 | " |
---|
| 7753 | { |
---|
[70ab73] | 7754 | int n, k, di; |
---|
| 7755 | list resu, hilf; |
---|
| 7756 | string var1, var2; |
---|
| 7757 | list v = indepSet(j, 0); |
---|
[808a9f3] | 7758 | |
---|
[70ab73] | 7759 | // SL 2006.04.21 1 Lines modified to use only one independent Set |
---|
| 7760 | string indepOption; |
---|
| 7761 | if (size(#) > 0) |
---|
| 7762 | { |
---|
| 7763 | indepOption = #[1]; |
---|
| 7764 | } |
---|
| 7765 | else |
---|
| 7766 | { |
---|
| 7767 | indepOption = "allIndep"; |
---|
| 7768 | } |
---|
[808a9f3] | 7769 | |
---|
[70ab73] | 7770 | int nMax; |
---|
| 7771 | if (indepOption == "allIndep") |
---|
| 7772 | { |
---|
| 7773 | nMax = size(v); |
---|
| 7774 | } |
---|
| 7775 | else |
---|
| 7776 | { |
---|
| 7777 | nMax = 1; |
---|
| 7778 | } |
---|
| 7779 | |
---|
| 7780 | for(n = 1; n <= nMax; n++) |
---|
| 7781 | // SL 2006.04.21 2 |
---|
| 7782 | { |
---|
| 7783 | di = 0; |
---|
| 7784 | var1 = ""; |
---|
| 7785 | var2 = ""; |
---|
| 7786 | for(k = 1; k <= size(v[n]); k++) |
---|
| 7787 | { |
---|
| 7788 | if(v[n][k] != 0) |
---|
| 7789 | { |
---|
| 7790 | di++; |
---|
| 7791 | var2 = var2 + "var(" + string(k) + "), "; |
---|
[808a9f3] | 7792 | } |
---|
| 7793 | else |
---|
| 7794 | { |
---|
[70ab73] | 7795 | var1 = var1 + "var(" + string(k) + "), "; |
---|
[808a9f3] | 7796 | } |
---|
[70ab73] | 7797 | } |
---|
| 7798 | if(di > 0) |
---|
| 7799 | { |
---|
| 7800 | var1 = var1 + var2; |
---|
| 7801 | var1 = var1[1..size(var1) - 2]; // The "- 2" removes the trailer comma |
---|
| 7802 | hilf[1] = var1; |
---|
| 7803 | // SL 2006.21.04 1 The order is now block dp instead of lp |
---|
| 7804 | //hilf[2] = "dp(" + string(nvars(basering) - di) + "), dp(" + string(di) + ")"; |
---|
| 7805 | // SL 2006.21.04 2 |
---|
| 7806 | // For decomp, lp ordering is needed. Nothing is changed. |
---|
| 7807 | hilf[2] = "lp"; |
---|
| 7808 | hilf[3] = di; |
---|
| 7809 | resu[n] = hilf; |
---|
| 7810 | } |
---|
| 7811 | else |
---|
| 7812 | { |
---|
| 7813 | resu[n] = varstr(basering), ordstr(basering), 0; |
---|
| 7814 | } |
---|
| 7815 | } |
---|
| 7816 | return(resu); |
---|
[808a9f3] | 7817 | } |
---|
| 7818 | example |
---|
| 7819 | { "EXAMPLE:"; echo = 2; |
---|
| 7820 | ring s1 = (0, x, y), (a, b, c, d, e, f, g), lp; |
---|
| 7821 | ideal i = ea - fbg, fa + be, ec - fdg, fc + de; |
---|
| 7822 | i = std(i); |
---|
| 7823 | list l = newMaxIndependSetLp(i); |
---|
| 7824 | l; |
---|
| 7825 | i = i, g; |
---|
| 7826 | l = newMaxIndependSetLp(i); |
---|
| 7827 | l; |
---|
| 7828 | |
---|
| 7829 | ring s = 0, (x, y, z), lp; |
---|
| 7830 | ideal i = z, yx; |
---|
| 7831 | list l = newMaxIndependSetLp(i); |
---|
| 7832 | l; |
---|
| 7833 | } |
---|
| 7834 | |
---|
| 7835 | |
---|
| 7836 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7837 | |
---|
| 7838 | proc newZero_decomp (ideal j, ideal ser, int @wr, list #) |
---|
| 7839 | "USAGE: newZero_decomp(j,ser,@wr); j,ser ideals, @wr=0 or 1 |
---|
[7f7c25e] | 7840 | (@wr=0 for primary decomposition, @wr=1 for computation of associated |
---|
[808a9f3] | 7841 | primes) |
---|
| 7842 | if #[1] = "nest", then #[2] indicates the nest level (number of recursive calls) |
---|
| 7843 | When the nest level is high it indicates that the computation is difficult, |
---|
| 7844 | and different methods are applied. |
---|
| 7845 | RETURN: list = list of primary ideals and their radicals (at even positions |
---|
| 7846 | in the list) if the input is zero-dimensional and a standardbases |
---|
| 7847 | with respect to lex-ordering |
---|
| 7848 | If ser!=(0) and ser is contained in j or if j is not zero-dimen- |
---|
| 7849 | sional then ideal(1),ideal(1) is returned |
---|
| 7850 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
| 7851 | EXAMPLE: example newZero_decomp; shows an example |
---|
| 7852 | " |
---|
| 7853 | { |
---|
| 7854 | def @P = basering; |
---|
| 7855 | int uytrewq; |
---|
| 7856 | int nva = nvars(basering); |
---|
| 7857 | int @k,@s,@n,@k1,zz; |
---|
| 7858 | list primary,lres0,lres1,act,@lh,@wh; |
---|
| 7859 | map phi,psi,phi1,psi1; |
---|
| 7860 | ideal jmap,jmap1,jmap2,helpprim,@qh,@qht,ser1; |
---|
| 7861 | intvec @vh,@hilb; |
---|
| 7862 | string @ri; |
---|
| 7863 | poly @f; |
---|
| 7864 | |
---|
| 7865 | // Debug |
---|
| 7866 | dbprint(printlevel - voice, "proc newZero_decomp"); |
---|
| 7867 | |
---|
| 7868 | if (dim(j)>0) |
---|
| 7869 | { |
---|
[70ab73] | 7870 | primary[1]=ideal(1); |
---|
| 7871 | primary[2]=ideal(1); |
---|
| 7872 | return(primary); |
---|
[808a9f3] | 7873 | } |
---|
| 7874 | j=interred(j); |
---|
| 7875 | |
---|
| 7876 | attrib(j,"isSB",1); |
---|
| 7877 | |
---|
| 7878 | int nestLevel = 0; |
---|
[70ab73] | 7879 | if (size(#) > 0) |
---|
| 7880 | { |
---|
| 7881 | if (typeof(#[1]) == "string") |
---|
| 7882 | { |
---|
| 7883 | if (#[1] == "nest") |
---|
| 7884 | { |
---|
[808a9f3] | 7885 | nestLevel = #[2]; |
---|
[7f7c25e] | 7886 | } |
---|
[808a9f3] | 7887 | # = list(); |
---|
[7f7c25e] | 7888 | } |
---|
| 7889 | } |
---|
[808a9f3] | 7890 | |
---|
| 7891 | if(vdim(j)==deg(j[1])) |
---|
| 7892 | { |
---|
[70ab73] | 7893 | act=factor(j[1]); |
---|
| 7894 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 7895 | { |
---|
| 7896 | @qh=j; |
---|
| 7897 | if(@wr==0) |
---|
| 7898 | { |
---|
| 7899 | @qh[1]=act[1][@k]^act[2][@k]; |
---|
| 7900 | } |
---|
| 7901 | else |
---|
| 7902 | { |
---|
| 7903 | @qh[1]=act[1][@k]; |
---|
| 7904 | } |
---|
| 7905 | primary[2*@k-1]=interred(@qh); |
---|
| 7906 | @qh=j; |
---|
| 7907 | @qh[1]=act[1][@k]; |
---|
| 7908 | primary[2*@k]=interred(@qh); |
---|
| 7909 | attrib( primary[2*@k-1],"isSB",1); |
---|
[808a9f3] | 7910 | |
---|
[70ab73] | 7911 | if((size(ser)>0)&&(size(reduce(ser,primary[2*@k-1],1))==0)) |
---|
| 7912 | { |
---|
| 7913 | primary[2*@k-1]=ideal(1); |
---|
| 7914 | primary[2*@k]=ideal(1); |
---|
| 7915 | } |
---|
| 7916 | } |
---|
| 7917 | return(primary); |
---|
[808a9f3] | 7918 | } |
---|
| 7919 | |
---|
| 7920 | if(homog(j)==1) |
---|
| 7921 | { |
---|
[70ab73] | 7922 | primary[1]=j; |
---|
| 7923 | if((size(ser)>0)&&(size(reduce(ser,j,1))==0)) |
---|
| 7924 | { |
---|
| 7925 | primary[1]=ideal(1); |
---|
| 7926 | primary[2]=ideal(1); |
---|
| 7927 | return(primary); |
---|
| 7928 | } |
---|
| 7929 | if(dim(j)==-1) |
---|
| 7930 | { |
---|
| 7931 | primary[1]=ideal(1); |
---|
| 7932 | primary[2]=ideal(1); |
---|
| 7933 | } |
---|
| 7934 | else |
---|
| 7935 | { |
---|
| 7936 | primary[2]=maxideal(1); |
---|
| 7937 | } |
---|
| 7938 | return(primary); |
---|
[808a9f3] | 7939 | } |
---|
| 7940 | |
---|
| 7941 | //the first element in the standardbase is factorized |
---|
| 7942 | if(deg(j[1])>0) |
---|
| 7943 | { |
---|
| 7944 | act=factor(j[1]); |
---|
| 7945 | testFactor(act,j[1]); |
---|
| 7946 | } |
---|
| 7947 | else |
---|
| 7948 | { |
---|
[70ab73] | 7949 | primary[1]=ideal(1); |
---|
| 7950 | primary[2]=ideal(1); |
---|
| 7951 | return(primary); |
---|
[808a9f3] | 7952 | } |
---|
| 7953 | |
---|
| 7954 | //with the factors new ideals (hopefully the primary decomposition) |
---|
| 7955 | //are created |
---|
| 7956 | if(size(act[1])>1) |
---|
| 7957 | { |
---|
[70ab73] | 7958 | if(size(#)>1) |
---|
| 7959 | { |
---|
| 7960 | primary[1]=ideal(1); |
---|
| 7961 | primary[2]=ideal(1); |
---|
| 7962 | primary[3]=ideal(1); |
---|
| 7963 | primary[4]=ideal(1); |
---|
| 7964 | return(primary); |
---|
| 7965 | } |
---|
| 7966 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 7967 | { |
---|
| 7968 | if(@wr==0) |
---|
| 7969 | { |
---|
| 7970 | primary[2*@k-1]=std(j,act[1][@k]^act[2][@k]); |
---|
| 7971 | } |
---|
| 7972 | else |
---|
| 7973 | { |
---|
| 7974 | primary[2*@k-1]=std(j,act[1][@k]); |
---|
| 7975 | } |
---|
| 7976 | if((act[2][@k]==1)&&(vdim(primary[2*@k-1])==deg(act[1][@k]))) |
---|
| 7977 | { |
---|
| 7978 | primary[2*@k] = primary[2*@k-1]; |
---|
| 7979 | } |
---|
| 7980 | else |
---|
| 7981 | { |
---|
| 7982 | primary[2*@k] = primaryTest(primary[2*@k-1],act[1][@k]); |
---|
| 7983 | } |
---|
| 7984 | } |
---|
[808a9f3] | 7985 | } |
---|
| 7986 | else |
---|
| 7987 | { |
---|
[70ab73] | 7988 | primary[1]=j; |
---|
| 7989 | if((size(#)>0)&&(act[2][1]>1)) |
---|
| 7990 | { |
---|
| 7991 | act[2]=1; |
---|
| 7992 | primary[1]=std(primary[1],act[1][1]); |
---|
| 7993 | } |
---|
| 7994 | if(@wr!=0) |
---|
| 7995 | { |
---|
| 7996 | primary[1]=std(j,act[1][1]); |
---|
| 7997 | } |
---|
| 7998 | if((act[2][1]==1)&&(vdim(primary[1])==deg(act[1][1]))) |
---|
| 7999 | { |
---|
| 8000 | primary[2]=primary[1]; |
---|
| 8001 | } |
---|
| 8002 | else |
---|
| 8003 | { |
---|
| 8004 | primary[2]=primaryTest(primary[1],act[1][1]); |
---|
| 8005 | } |
---|
[808a9f3] | 8006 | } |
---|
| 8007 | |
---|
| 8008 | if(size(#)==0) |
---|
| 8009 | { |
---|
[70ab73] | 8010 | primary=splitPrimary(primary,ser,@wr,act); |
---|
[808a9f3] | 8011 | } |
---|
| 8012 | |
---|
| 8013 | if((voice>=6)&&(char(basering)<=181)) |
---|
| 8014 | { |
---|
[70ab73] | 8015 | primary=splitCharp(primary); |
---|
[808a9f3] | 8016 | } |
---|
| 8017 | |
---|
| 8018 | if((@wr==2)&&(npars(basering)>0)&&(voice>=6)&&(char(basering)>0)) |
---|
| 8019 | { |
---|
| 8020 | //the prime decomposition of Yokoyama in characteristic p |
---|
[70ab73] | 8021 | list ke,ek; |
---|
| 8022 | @k=0; |
---|
| 8023 | while(@k<size(primary)/2) |
---|
| 8024 | { |
---|
| 8025 | @k++; |
---|
| 8026 | if(size(primary[2*@k])==0) |
---|
| 8027 | { |
---|
| 8028 | ek=insepDecomp(primary[2*@k-1]); |
---|
| 8029 | primary=delete(primary,2*@k); |
---|
| 8030 | primary=delete(primary,2*@k-1); |
---|
| 8031 | @k--; |
---|
| 8032 | } |
---|
| 8033 | ke=ke+ek; |
---|
| 8034 | } |
---|
| 8035 | for(@k=1;@k<=size(ke);@k++) |
---|
| 8036 | { |
---|
| 8037 | primary[size(primary)+1]=ke[@k]; |
---|
| 8038 | primary[size(primary)+1]=ke[@k]; |
---|
| 8039 | } |
---|
[808a9f3] | 8040 | } |
---|
| 8041 | |
---|
[7f7c25e] | 8042 | if(nestLevel > 1){primary=extF(primary);} |
---|
[808a9f3] | 8043 | |
---|
| 8044 | //test whether all ideals in the decomposition are primary and |
---|
| 8045 | //in general position |
---|
| 8046 | //if not after a random coordinate transformation of the last |
---|
| 8047 | //variable the corresponding ideal is decomposed again. |
---|
| 8048 | if((npars(basering)>0)&&(nestLevel > 1)) |
---|
| 8049 | { |
---|
[70ab73] | 8050 | poly randp; |
---|
| 8051 | for(zz=1;zz<nvars(basering);zz++) |
---|
| 8052 | { |
---|
| 8053 | randp=randp |
---|
[808a9f3] | 8054 | +(random(0,5)*par(1)^2+random(0,5)*par(1)+random(0,5))*var(zz); |
---|
[70ab73] | 8055 | } |
---|
| 8056 | randp=randp+var(nvars(basering)); |
---|
[808a9f3] | 8057 | } |
---|
| 8058 | @k=0; |
---|
| 8059 | while(@k<(size(primary)/2)) |
---|
| 8060 | { |
---|
| 8061 | @k++; |
---|
| 8062 | if (size(primary[2*@k])==0) |
---|
| 8063 | { |
---|
[70ab73] | 8064 | for(zz=1;zz<size(primary[2*@k-1])-1;zz++) |
---|
| 8065 | { |
---|
| 8066 | attrib(primary[2*@k-1],"isSB",1); |
---|
| 8067 | if(vdim(primary[2*@k-1])==deg(primary[2*@k-1][zz])) |
---|
| 8068 | { |
---|
| 8069 | primary[2*@k]=primary[2*@k-1]; |
---|
| 8070 | } |
---|
| 8071 | } |
---|
[808a9f3] | 8072 | } |
---|
| 8073 | } |
---|
| 8074 | |
---|
| 8075 | @k=0; |
---|
| 8076 | ideal keep; |
---|
| 8077 | while(@k<(size(primary)/2)) |
---|
| 8078 | { |
---|
| 8079 | @k++; |
---|
| 8080 | if (size(primary[2*@k])==0) |
---|
| 8081 | { |
---|
[70ab73] | 8082 | jmap=randomLast(100); |
---|
| 8083 | jmap1=maxideal(1); |
---|
| 8084 | jmap2=maxideal(1); |
---|
| 8085 | @qht=primary[2*@k-1]; |
---|
| 8086 | if((npars(basering)>0)&&(nestLevel > 1)) |
---|
| 8087 | { |
---|
| 8088 | jmap[size(jmap)]=randp; |
---|
| 8089 | } |
---|
[808a9f3] | 8090 | |
---|
[70ab73] | 8091 | for(@n=2;@n<=size(primary[2*@k-1]);@n++) |
---|
| 8092 | { |
---|
| 8093 | if(deg(lead(primary[2*@k-1][@n]))==1) |
---|
| 8094 | { |
---|
| 8095 | for(zz=1;zz<=nva;zz++) |
---|
[808a9f3] | 8096 | { |
---|
[70ab73] | 8097 | if(lead(primary[2*@k-1][@n])/var(zz)!=0) |
---|
| 8098 | { |
---|
| 8099 | jmap1[zz]=-1/leadcoef(primary[2*@k-1][@n])*primary[2*@k-1][@n] |
---|
[a36e78] | 8100 | +2/leadcoef(primary[2*@k-1][@n])*lead(primary[2*@k-1][@n]); |
---|
[70ab73] | 8101 | jmap2[zz]=primary[2*@k-1][@n]; |
---|
| 8102 | @qht[@n]=var(zz); |
---|
| 8103 | } |
---|
[808a9f3] | 8104 | } |
---|
[70ab73] | 8105 | jmap[nva]=subst(jmap[nva],lead(primary[2*@k-1][@n]),0); |
---|
| 8106 | } |
---|
| 8107 | } |
---|
| 8108 | if(size(subst(jmap[nva],var(1),0)-var(nva))!=0) |
---|
| 8109 | { |
---|
| 8110 | // jmap[nva]=subst(jmap[nva],var(1),0); |
---|
| 8111 | //hier geaendert +untersuchen!!!!!!!!!!!!!! |
---|
| 8112 | } |
---|
| 8113 | phi1=@P,jmap1; |
---|
| 8114 | phi=@P,jmap; |
---|
| 8115 | for(@n=1;@n<=nva;@n++) |
---|
| 8116 | { |
---|
| 8117 | jmap[@n]=-(jmap[@n]-2*var(@n)); |
---|
| 8118 | } |
---|
| 8119 | psi=@P,jmap; |
---|
| 8120 | psi1=@P,jmap2; |
---|
| 8121 | @qh=phi(@qht); |
---|
[808a9f3] | 8122 | |
---|
| 8123 | //=================== the new part ============================ |
---|
| 8124 | |
---|
[8992ed] | 8125 | if (npars(basering)>1) { @qh=groebner(@qh,"par2var"); } |
---|
| 8126 | else { @qh=groebner(@qh); } |
---|
[808a9f3] | 8127 | |
---|
| 8128 | //============================================================= |
---|
| 8129 | // if(npars(@P)>0) |
---|
| 8130 | // { |
---|
| 8131 | // @ri= "ring @Phelp =" |
---|
| 8132 | // +string(char(@P))+", |
---|
| 8133 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 8134 | // } |
---|
| 8135 | // else |
---|
| 8136 | // { |
---|
| 8137 | // @ri= "ring @Phelp =" |
---|
| 8138 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 8139 | // } |
---|
| 8140 | // execute(@ri); |
---|
| 8141 | // ideal @qh=homog(imap(@P,@qht),@t); |
---|
| 8142 | // |
---|
| 8143 | // ideal @qh1=std(@qh); |
---|
| 8144 | // @hilb=hilb(@qh1,1); |
---|
| 8145 | // @ri= "ring @Phelp1 =" |
---|
| 8146 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 8147 | // execute(@ri); |
---|
| 8148 | // ideal @qh=homog(imap(@P,@qh),@t); |
---|
| 8149 | // kill @Phelp; |
---|
| 8150 | // @qh=std(@qh,@hilb); |
---|
| 8151 | // @qh=subst(@qh,@t,1); |
---|
| 8152 | // setring @P; |
---|
| 8153 | // @qh=imap(@Phelp1,@qh); |
---|
| 8154 | // kill @Phelp1; |
---|
| 8155 | // @qh=clearSB(@qh); |
---|
| 8156 | // attrib(@qh,"isSB",1); |
---|
| 8157 | //============================================================= |
---|
| 8158 | |
---|
[70ab73] | 8159 | ser1=phi1(ser); |
---|
| 8160 | @lh=newZero_decomp (@qh,phi(ser1),@wr, list("nest", nestLevel + 1)); |
---|
[808a9f3] | 8161 | |
---|
[70ab73] | 8162 | kill lres0; |
---|
| 8163 | list lres0; |
---|
| 8164 | if(size(@lh)==2) |
---|
| 8165 | { |
---|
| 8166 | helpprim=@lh[2]; |
---|
| 8167 | lres0[1]=primary[2*@k-1]; |
---|
| 8168 | ser1=psi(helpprim); |
---|
| 8169 | lres0[2]=psi1(ser1); |
---|
| 8170 | if(size(reduce(lres0[2],lres0[1],1))==0) |
---|
| 8171 | { |
---|
| 8172 | primary[2*@k]=primary[2*@k-1]; |
---|
| 8173 | continue; |
---|
| 8174 | } |
---|
| 8175 | } |
---|
| 8176 | else |
---|
| 8177 | { |
---|
| 8178 | lres1=psi(@lh); |
---|
| 8179 | lres0=psi1(lres1); |
---|
| 8180 | } |
---|
[808a9f3] | 8181 | |
---|
| 8182 | //=================== the new part ============================ |
---|
| 8183 | |
---|
[70ab73] | 8184 | primary=delete(primary,2*@k-1); |
---|
| 8185 | primary=delete(primary,2*@k-1); |
---|
| 8186 | @k--; |
---|
| 8187 | if(size(lres0)==2) |
---|
| 8188 | { |
---|
[8992ed] | 8189 | if (npars(basering)>1) { lres0[2]=groebner(lres0[2],"par2var"); } |
---|
| 8190 | else { lres0[2]=groebner(lres0[2]); } |
---|
[70ab73] | 8191 | } |
---|
| 8192 | else |
---|
| 8193 | { |
---|
| 8194 | for(@n=1;@n<=size(lres0)/2;@n++) |
---|
| 8195 | { |
---|
| 8196 | if(specialIdealsEqual(lres0[2*@n-1],lres0[2*@n])==1) |
---|
[808a9f3] | 8197 | { |
---|
[a36e78] | 8198 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
[70ab73] | 8199 | lres0[2*@n]=lres0[2*@n-1]; |
---|
| 8200 | attrib(lres0[2*@n],"isSB",1); |
---|
[808a9f3] | 8201 | } |
---|
[70ab73] | 8202 | else |
---|
| 8203 | { |
---|
[a36e78] | 8204 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
| 8205 | lres0[2*@n]=groebner(lres0[2*@n]); |
---|
[70ab73] | 8206 | } |
---|
| 8207 | } |
---|
| 8208 | } |
---|
| 8209 | primary=primary+lres0; |
---|
[808a9f3] | 8210 | |
---|
| 8211 | //============================================================= |
---|
| 8212 | // if(npars(@P)>0) |
---|
| 8213 | // { |
---|
| 8214 | // @ri= "ring @Phelp =" |
---|
| 8215 | // +string(char(@P))+", |
---|
| 8216 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 8217 | // } |
---|
| 8218 | // else |
---|
| 8219 | // { |
---|
| 8220 | // @ri= "ring @Phelp =" |
---|
| 8221 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 8222 | // } |
---|
| 8223 | // execute(@ri); |
---|
| 8224 | // list @lvec; |
---|
| 8225 | // list @lr=imap(@P,lres0); |
---|
| 8226 | // ideal @lr1; |
---|
| 8227 | // |
---|
| 8228 | // if(size(@lr)==2) |
---|
| 8229 | // { |
---|
| 8230 | // @lr[2]=homog(@lr[2],@t); |
---|
| 8231 | // @lr1=std(@lr[2]); |
---|
| 8232 | // @lvec[2]=hilb(@lr1,1); |
---|
| 8233 | // } |
---|
| 8234 | // else |
---|
| 8235 | // { |
---|
| 8236 | // for(@n=1;@n<=size(@lr)/2;@n++) |
---|
| 8237 | // { |
---|
| 8238 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 8239 | // { |
---|
| 8240 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 8241 | // @lr1=std(@lr[2*@n-1]); |
---|
| 8242 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 8243 | // @lvec[2*@n]=@lvec[2*@n-1]; |
---|
| 8244 | // } |
---|
| 8245 | // else |
---|
| 8246 | // { |
---|
| 8247 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 8248 | // @lr1=std(@lr[2*@n-1]); |
---|
| 8249 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 8250 | // @lr[2*@n]=homog(@lr[2*@n],@t); |
---|
| 8251 | // @lr1=std(@lr[2*@n]); |
---|
| 8252 | // @lvec[2*@n]=hilb(@lr1,1); |
---|
| 8253 | // |
---|
| 8254 | // } |
---|
| 8255 | // } |
---|
| 8256 | // } |
---|
| 8257 | // @ri= "ring @Phelp1 =" |
---|
| 8258 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 8259 | // execute(@ri); |
---|
| 8260 | // list @lr=imap(@Phelp,@lr); |
---|
| 8261 | // |
---|
| 8262 | // kill @Phelp; |
---|
| 8263 | // if(size(@lr)==2) |
---|
| 8264 | // { |
---|
| 8265 | // @lr[2]=std(@lr[2],@lvec[2]); |
---|
| 8266 | // @lr[2]=subst(@lr[2],@t,1); |
---|
| 8267 | // |
---|
| 8268 | // } |
---|
| 8269 | // else |
---|
| 8270 | // { |
---|
| 8271 | // for(@n=1;@n<=size(@lr)/2;@n++) |
---|
| 8272 | // { |
---|
| 8273 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 8274 | // { |
---|
| 8275 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 8276 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 8277 | // @lr[2*@n]=@lr[2*@n-1]; |
---|
| 8278 | // attrib(@lr[2*@n],"isSB",1); |
---|
| 8279 | // } |
---|
| 8280 | // else |
---|
| 8281 | // { |
---|
| 8282 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 8283 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 8284 | // @lr[2*@n]=std(@lr[2*@n],@lvec[2*@n]); |
---|
| 8285 | // @lr[2*@n]=subst(@lr[2*@n],@t,1); |
---|
| 8286 | // } |
---|
| 8287 | // } |
---|
| 8288 | // } |
---|
| 8289 | // kill @lvec; |
---|
| 8290 | // setring @P; |
---|
| 8291 | // lres0=imap(@Phelp1,@lr); |
---|
| 8292 | // kill @Phelp1; |
---|
| 8293 | // for(@n=1;@n<=size(lres0);@n++) |
---|
| 8294 | // { |
---|
| 8295 | // lres0[@n]=clearSB(lres0[@n]); |
---|
| 8296 | // attrib(lres0[@n],"isSB",1); |
---|
| 8297 | // } |
---|
| 8298 | // |
---|
| 8299 | // primary[2*@k-1]=lres0[1]; |
---|
| 8300 | // primary[2*@k]=lres0[2]; |
---|
| 8301 | // @s=size(primary)/2; |
---|
| 8302 | // for(@n=1;@n<=size(lres0)/2-1;@n++) |
---|
| 8303 | // { |
---|
| 8304 | // primary[2*@s+2*@n-1]=lres0[2*@n+1]; |
---|
| 8305 | // primary[2*@s+2*@n]=lres0[2*@n+2]; |
---|
| 8306 | // } |
---|
| 8307 | // @k--; |
---|
| 8308 | //============================================================= |
---|
[70ab73] | 8309 | } |
---|
[808a9f3] | 8310 | } |
---|
| 8311 | return(primary); |
---|
| 8312 | } |
---|
| 8313 | example |
---|
| 8314 | { "EXAMPLE:"; echo = 2; |
---|
| 8315 | ring r = 0,(x,y,z),lp; |
---|
| 8316 | poly p = z2+1; |
---|
| 8317 | poly q = z4+2; |
---|
| 8318 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 8319 | i=std(i); |
---|
| 8320 | list pr= newZero_decomp(i,ideal(0),0); |
---|
| 8321 | pr; |
---|
| 8322 | } |
---|
| 8323 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 8324 | |
---|
[55fcff] | 8325 | //////////////////////////////////////////////////////////////////////////// |
---|
| 8326 | /* |
---|
| 8327 | //Beispiele Wenk-Dipl (in ~/Texfiles/Diplom/Wenk/Examples/) |
---|
| 8328 | //Zeiten: Singular/Singular/Singular -r123456789 -v :wilde13 (PentiumPro200) |
---|
| 8329 | //Singular for HPUX-9 version 1-3-8 (2000060214) Jun 2 2000 15:31:26 |
---|
| 8330 | //(wilde13) |
---|
| 8331 | |
---|
| 8332 | //1. vdim=20, 3 Komponenten |
---|
| 8333 | //zerodec-time:2(1) (matrix:1 charpoly:0 factor:1) |
---|
| 8334 | //primdecGTZ-time: 1(0) |
---|
| 8335 | ring rs= 0,(a,b,c),dp; |
---|
| 8336 | poly f1= a^2*b*c + a*b^2*c + a*b*c^2 + a*b*c + a*b + a*c + b*c; |
---|
| 8337 | poly f2= a^2*b^2*c + a*b^2*c^2 + a^2*b*c + a*b*c + b*c + a + c; |
---|
| 8338 | poly f3= a^2*b^2*c^2 + a^2*b^2*c + a*b^2*c + a*b*c + a*c + c + 1; |
---|
| 8339 | ideal gls=f1,f2,f3; |
---|
| 8340 | int time=timer; |
---|
| 8341 | printlevel =1; |
---|
| 8342 | time=timer; list pr1=zerodec(gls); timer-time;size(pr1); |
---|
| 8343 | time=timer; list pr =primdecGTZ(gls); timer-time;size(pr); |
---|
[07c623] | 8344 | time=timer; ideal ra =radical(gls); timer-time;size(pr); |
---|
[55fcff] | 8345 | |
---|
| 8346 | //2.cyclic5 vdim=70, 20 Komponenten |
---|
| 8347 | //zerodec-time:36(28) (matrix:1(0) charpoly:18(19) factor:17(9) |
---|
| 8348 | //primdecGTZ-time: 28(5) |
---|
[b9b906] | 8349 | //radical : 0 |
---|
[55fcff] | 8350 | ring rs= 0,(a,b,c,d,e),dp; |
---|
| 8351 | poly f0= a + b + c + d + e + 1; |
---|
| 8352 | poly f1= a + b + c + d + e; |
---|
| 8353 | poly f2= a*b + b*c + c*d + a*e + d*e; |
---|
| 8354 | poly f3= a*b*c + b*c*d + a*b*e + a*d*e + c*d*e; |
---|
| 8355 | poly f4= a*b*c*d + a*b*c*e + a*b*d*e + a*c*d*e + b*c*d*e; |
---|
| 8356 | poly f5= a*b*c*d*e - 1; |
---|
| 8357 | ideal gls= f1,f2,f3,f4,f5; |
---|
| 8358 | |
---|
| 8359 | //3. random vdim=40, 1 Komponente |
---|
| 8360 | //zerodec-time:126(304) (matrix:1 charpoly:115(298) factor:10(5)) |
---|
[b9b906] | 8361 | //primdecGTZ-time:17 (11) |
---|
[55fcff] | 8362 | ring rs=0,(x,y,z),dp; |
---|
| 8363 | poly f1=2*x^2 + 4*x + 3*y^2 + 7*x*z + 9*y*z + 5*z^2; |
---|
| 8364 | poly f2=7*x^3 + 8*x*y + 12*y^2 + 18*x*z + 3*y^4*z + 10*z^3 + 12; |
---|
| 8365 | poly f3=3*x^4 + 1*x*y*z + 6*y^3 + 3*x*z^2 + 2*y*z^2 + 4*z^2 + 5; |
---|
| 8366 | ideal gls=f1,f2,f3; |
---|
| 8367 | |
---|
| 8368 | //4. introduction into resultants, sturmfels, vdim=28, 1 Komponente |
---|
| 8369 | //zerodec-time:4 (matrix:0 charpoly:0 factor:4) |
---|
[b9b906] | 8370 | //primdecGTZ-time:1 |
---|
[55fcff] | 8371 | ring rs=0,(x,y),dp; |
---|
| 8372 | poly f1= x4+y4-1; |
---|
| 8373 | poly f2= x5y2-4x3y3+x2y5-1; |
---|
| 8374 | ideal gls=f1,f2; |
---|
| 8375 | |
---|
| 8376 | //5. 3 quadratic equations with random coeffs, vdim=8, 1 Komponente |
---|
| 8377 | //zerodec-time:0(0) (matrix:0 charpoly:0 factor:0) |
---|
[b9b906] | 8378 | //primdecGTZ-time:1(0) |
---|
[55fcff] | 8379 | ring rs=0,(x,y,z),dp; |
---|
| 8380 | poly f1=2*x^2 + 4*x*y + 3*y^2 + 7*x*z + 9*y*z + 5*z^2 + 2; |
---|
| 8381 | poly f2=7*x^2 + 8*x*y + 12*y^2 + 18*x*z + 3*y*z + 10*z^2 + 12; |
---|
| 8382 | poly f3=3*x^2 + 1*x*y + 6*y^2 + 3*x*z + 2*y*z + 4*z^2 + 5; |
---|
| 8383 | ideal gls=f1,f2,f3; |
---|
| 8384 | |
---|
| 8385 | //6. 3 polys vdim=24, 1 Komponente |
---|
| 8386 | // run("ex14",2); |
---|
| 8387 | //zerodec-time:5(4) (matrix:0 charpoly:3(3) factor:2(1)) |
---|
| 8388 | //primdecGTZ-time:4 (2) |
---|
| 8389 | ring rs=0,(x1,x2,x3,x4),dp; |
---|
| 8390 | poly f1=16*x1^2 + 3*x2^2 + 5*x3^4 - 1 - 4*x4 + x4^3; |
---|
| 8391 | poly f2=5*x1^3 + 3*x2^2 + 4*x3^2*x4 + 2*x1*x4 - 1 + x4 + 4*x1 + x2 + x3 + x4; |
---|
| 8392 | poly f3=-4*x1^2 + x2^2 + x3^2 - 3 + x4^2 + 4*x1 + x2 + x3 + x4; |
---|
| 8393 | poly f4=-4*x1 + x2 + x3 + x4; |
---|
| 8394 | ideal gls=f1,f2,f3,f4; |
---|
| 8395 | |
---|
| 8396 | //7. ex43, PoSSo, caprasse vdim=56, 16 Komponenten |
---|
[b9b906] | 8397 | //zerodec-time:23(15) (matrix:0 charpoly:16(13) factor:3(2)) |
---|
[55fcff] | 8398 | //primdecGTZ-time:3 (2) |
---|
| 8399 | ring rs= 0,(y,z,x,t),dp; |
---|
| 8400 | ideal gls=y^2*z+2*y*x*t-z-2*x, |
---|
| 8401 | 4*y^2*z*x-z*x^3+2*y^3*t+4*y*x^2*t-10*y^2+4*z*x+4*x^2-10*y*t+2, |
---|
| 8402 | 2*y*z*t+x*t^2-2*z-x, |
---|
| 8403 | -z^3*x+4*y*z^2*t+4*z*x*t^2+2*y*t^3+4*z^2+4*z*x-10*y*t-10*t^2+2; |
---|
| 8404 | |
---|
| 8405 | //8. Arnborg-System, n=6 (II), vdim=156, 90 Komponenten |
---|
| 8406 | //zerodec-time (char32003):127(45)(matrix:2(0) charpoly:106(37) factor:16(7)) |
---|
| 8407 | //primdecGTZ-time(char32003) :81 (18) |
---|
| 8408 | //ring rs= 0,(a,b,c,d,x,f),dp; |
---|
| 8409 | ring rs= 32003,(a,b,c,d,x,f),dp; |
---|
[b9b906] | 8410 | ideal gls=a+b+c+d+x+f, ab+bc+cd+dx+xf+af, abc+bcd+cdx+d*xf+axf+abf, |
---|
| 8411 | abcd+bcdx+cd*xf+ad*xf+abxf+abcf, abcdx+bcd*xf+acd*xf+abd*xf+abcxf+abcdf, |
---|
[55fcff] | 8412 | abcd*xf-1; |
---|
| 8413 | |
---|
| 8414 | //9. ex42, PoSSo, Methan6_1, vdim=27, 2 Komponenten |
---|
| 8415 | //zerodec-time:610 (matrix:10 charpoly:557 factor:26) |
---|
| 8416 | //primdecGTZ-time: 118 |
---|
| 8417 | //zerodec-time(char32003):2 |
---|
| 8418 | //primdecGTZ-time(char32003):4 |
---|
| 8419 | //ring rs= 0,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10),dp; |
---|
| 8420 | ring rs= 32003,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10),dp; |
---|
| 8421 | ideal gls=64*x2*x7-10*x1*x8+10*x7*x9+11*x7*x10-320000*x1, |
---|
| 8422 | -32*x2*x7-5*x2*x8-5*x2*x10+160000*x1-5000*x2, |
---|
| 8423 | -x3*x8+x6*x8+x9*x10+210*x6+1300000, |
---|
| 8424 | -x4*x8+700000, |
---|
| 8425 | x10^2-2*x5, |
---|
| 8426 | -x6*x8+x7*x9-210*x6, |
---|
| 8427 | -64*x2*x7-10*x7*x9-11*x7*x10+320000*x1-16*x7+7000000, |
---|
| 8428 | -10*x1*x8-10*x2*x8-10*x3*x8-10*x4*x8-10*x6*x8+10*x2*x10+11*x7*x10 |
---|
| 8429 | +20000*x2+14*x5, |
---|
| 8430 | x4*x8-x7*x9-x9*x10-410*x9, |
---|
| 8431 | 10*x2*x8+10*x3*x8+10*x6*x8+10*x7*x9-10*x2*x10-11*x7*x10-10*x9*x10 |
---|
| 8432 | -10*x10^2+1400*x6-4200*x10; |
---|
| 8433 | |
---|
| 8434 | //10. ex33, walk-s7, Diplomarbeit von Tim, vdim=114 |
---|
| 8435 | //zerfaellt in unterschiedlich viele Komponenten in versch. Charkteristiken: |
---|
| 8436 | //char32003:30, char0:3(2xdeg1,1xdeg112!), char181:4(2xdeg1,1xdeg28,1xdeg84) |
---|
| 8437 | //char 0: zerodec-time:10075 (ca 3h) (matrix:3 charpoly:9367, factor:680 |
---|
| 8438 | // + 24 sec fuer Normalform (anstatt einsetzen), total [29623k]) |
---|
| 8439 | // primdecGTZ-time: 214 |
---|
| 8440 | //char 32003:zerodec-time:197(68) (matrix:2(1) charpoly:173(60) factor:15(6)) |
---|
| 8441 | // primdecGTZ-time:14 (5) |
---|
| 8442 | //char 181:zerodec-time:(87) (matrix:(1) charpoly:(58) factor:(25)) |
---|
| 8443 | // primdecGTZ-time:(2) |
---|
| 8444 | //in char181 stimmen Ergebnisse von zerodec und primdecGTZ ueberein (gecheckt) |
---|
| 8445 | |
---|
| 8446 | //ring rs= 0,(a,b,c,d,e,f,g),dp; |
---|
| 8447 | ring rs= 32003,(a,b,c,d,e,f,g),dp; |
---|
| 8448 | poly f1= 2gb + 2fc + 2ed + a2 + a; |
---|
| 8449 | poly f2= 2gc + 2fd + e2 + 2ba + b; |
---|
| 8450 | poly f3= 2gd + 2fe + 2ca + c + b2; |
---|
| 8451 | poly f4= 2ge + f2 + 2da + d + 2cb; |
---|
| 8452 | poly f5= 2fg + 2ea + e + 2db + c2; |
---|
| 8453 | poly f6= g2 + 2fa + f + 2eb + 2dc; |
---|
| 8454 | poly f7= 2ga + g + 2fb + 2ec + d2; |
---|
| 8455 | ideal gls= f1,f2,f3,f4,f5,f6,f7; |
---|
| 8456 | |
---|
| 8457 | ~/Singular/Singular/Singular -r123456789 -v |
---|
| 8458 | LIB"./primdec.lib"; |
---|
| 8459 | timer=1; |
---|
| 8460 | int time=timer; |
---|
| 8461 | printlevel =1; |
---|
| 8462 | option(prot,mem); |
---|
| 8463 | time=timer; list pr1=zerodec(gls); timer-time; |
---|
| 8464 | |
---|
| 8465 | time=timer; list pr =primdecGTZ(gls); timer-time; |
---|
| 8466 | time=timer; list pr =primdecSY(gls); timer-time; |
---|
[07c623] | 8467 | time=timer; ideal ra =radical(gls); timer-time;size(pr); |
---|
[24f458] | 8468 | LIB"all.lib"; |
---|
| 8469 | |
---|
| 8470 | ring R=0,(a,b,c,d,e,f),dp; |
---|
| 8471 | ideal I=cyclic(6); |
---|
| 8472 | minAssGTZ(I); |
---|
| 8473 | |
---|
| 8474 | |
---|
| 8475 | ring S=(2,a,b),(x,y),lp; |
---|
| 8476 | ideal I=x8-b,y4+a; |
---|
| 8477 | minAssGTZ(I); |
---|
| 8478 | |
---|
| 8479 | ring S1=2,(x,y,a,b),lp; |
---|
| 8480 | ideal I=x8-b,y4+a; |
---|
| 8481 | minAssGTZ(I); |
---|
| 8482 | |
---|
| 8483 | |
---|
| 8484 | ring S2=(2,z),(x,y),dp; |
---|
| 8485 | minpoly=z2+z+1; |
---|
| 8486 | ideal I=y3+y+1,x4+x+1; |
---|
| 8487 | primdecGTZ(I); |
---|
| 8488 | minAssGTZ(I); |
---|
| 8489 | |
---|
| 8490 | ring S3=2,(x,y,z),dp; |
---|
| 8491 | ideal I=y3+y+1,x4+x+1,z2+z+1; |
---|
| 8492 | primdecGTZ(I); |
---|
| 8493 | minAssGTZ(I); |
---|
| 8494 | |
---|
| 8495 | |
---|
| 8496 | ring R1=2,(x,y,z),lp; |
---|
| 8497 | ideal I=y6+y5+y3+y2+1,x4+x+1,z2+z+1; |
---|
| 8498 | primdecGTZ(I); |
---|
| 8499 | minAssGTZ(I); |
---|
| 8500 | |
---|
| 8501 | |
---|
| 8502 | ring R2=(2,z),(x,y),lp; |
---|
| 8503 | minpoly=z3+z+1; |
---|
| 8504 | ideal I=y2+y+(z2+z+1),x4+x+1; |
---|
| 8505 | primdecGTZ(I); |
---|
| 8506 | minAssGTZ(I); |
---|
| 8507 | |
---|
[55fcff] | 8508 | */ |
---|