[e27e47] | 1 | ////////////////////////////////////////////////////////////////////////////// |
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[341696] | 2 | version="$Id$"; |
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[e27e47] | 3 | category="Commutative Algebra"; |
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| 4 | info=" |
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[906458] | 5 | LIBRARY: sagbi.lib Compute subalgebra bases analogous to Groebner bases for ideals |
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[e27e47] | 6 | AUTHORS: Gerhard Pfister, pfister@mathematik.uni-kl.de, |
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| 7 | @* Anen Lakhal, alakhal@mathematik.uni-kl.de |
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[a073dd] | 8 | |
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[e27e47] | 9 | PROCEDURES: |
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[804d68] | 10 | sagbiRreduction(p,I); perform one step subalgebra reducton (for short S-reduction) of p w.r.t I |
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[3da61f] | 11 | sagbiSPoly(I); compute the S-polynomials of the Subalgebra defined by the genartors of I |
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| 12 | sagbiNF(id,I); perform iterated S-reductions in order to compute Subalgebras normal forms |
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| 13 | sagbi(I); construct SAGBI basis for the Subalgebra defined by I |
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| 14 | sagbiPart(I); construct partial SAGBI basis for the Subalgebra defined by I |
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[e27e47] | 15 | "; |
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| 16 | |
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[7a68965] | 17 | LIB "algebra.lib"; |
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| 18 | LIB "elim.lib"; |
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[e27e47] | 19 | |
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| 20 | /////////////////////////////////////////////////////////////////////////////// |
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[a073dd] | 21 | proc sagbiSPoly(id ,list #) |
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| 22 | "USAGE: sagbiSPoly(id [,n]); id ideal, n positive integer. |
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[e27e47] | 23 | RETURN: an ideal |
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| 24 | @format |
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| 25 | - If (n=0 or default) an ideal, whose generators are the S-polynomials. |
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[906458] | 26 | - If (n=1) a list of size 2: |
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[e27e47] | 27 | the first element of this list is the ideal of S-polynomials. |
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[906458] | 28 | the second element of this list is the ring in which the ideal of algebraic |
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| 29 | relations is defined. |
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[e27e47] | 30 | @end format |
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[a073dd] | 31 | EXAMPLE: example sagbiSPoly; show an example " |
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[e27e47] | 32 | { |
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| 33 | if(size(#)==0) |
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[7a68965] | 34 | { |
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| 35 | #[1]=0; |
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| 36 | } |
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[e27e47] | 37 | degBound=0; |
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| 38 | def bsr=basering; |
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| 39 | ideal vars=maxideal(1); |
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| 40 | ideal B=ideal(bsr);//when the basering is quotient ring this "type casting" |
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| 41 | //gives th quotient ideal. |
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| 42 | int b=size(B); |
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| 43 | |
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| 44 | //In quotient rings,SINGULAR does not reduce polynomials w.r.t the |
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| 45 | //quotient ideal,therefore we should first 'reduce';if it is necessary for |
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| 46 | //computations to have a uniquely determined representant for each equivalent |
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| 47 | //class,which is the case of this procedure. |
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| 48 | |
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| 49 | if(b!=0) |
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[7a68965] | 50 | { |
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| 51 | id =reduce(id,groebner(0)); |
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| 52 | } |
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[e27e47] | 53 | int n,m=nvars(bsr),ncols(id); |
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| 54 | int z; |
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| 55 | string mp=string(minpoly); |
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| 56 | ideal P; |
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| 57 | list L; |
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| 58 | |
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| 59 | if(id==0) |
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[7a68965] | 60 | { |
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| 61 | if(#[1]==0) |
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[e27e47] | 62 | { |
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[7a68965] | 63 | return(P); |
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[e27e47] | 64 | } |
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[7a68965] | 65 | else |
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[e27e47] | 66 | { |
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[7a68965] | 67 | return(L); |
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| 68 | } |
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| 69 | } |
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| 70 | else |
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| 71 | { |
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[e27e47] | 72 | //==================create anew ring with extra variables================ |
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| 73 | |
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[3da61f] | 74 | execute("ring R1=("+charstr(bsr)+"),("+varstr(bsr)+",@y(1..m)),(dp(n),dp(m));"); |
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[7a68965] | 75 | execute("minpoly=number("+mp+");"); |
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| 76 | ideal id=imap(bsr,id); |
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| 77 | ideal A; |
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[e27e47] | 78 | |
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[7a68965] | 79 | for(z=1;z<=m;z++) |
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| 80 | { |
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| 81 | A[z]=lead(id[z])-@y(z); |
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| 82 | } |
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[e27e47] | 83 | |
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[7a68965] | 84 | A=groebner(A); |
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[c99fd4] | 85 | ideal kern=nselect(A,1..n);// "kern" is the kernel of the ring map: |
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[7a68965] | 86 | // R1----->bsr ;y(z)----> lead(id[z]). |
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| 87 | //"kern" is the ideal of algebraic relations between |
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| 88 | // lead(id[z]). |
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| 89 | export kern,A;// we export: |
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| 90 | // * the ideal A to avoid useless computations |
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| 91 | // between 2 steps in sagbi procedure. |
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| 92 | // * the ideal kern : some times we can get intresting |
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| 93 | // informations from this ideal. |
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| 94 | setring bsr; |
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| 95 | map phi=R1,vars,id; |
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[e27e47] | 96 | |
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[7a68965] | 97 | // the sagbiSPolynomials are the image by phi of the generators of kern |
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[e27e47] | 98 | |
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[7a68965] | 99 | P=simplify(phi(kern),1); |
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| 100 | if(#[1]==0) |
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| 101 | { |
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| 102 | return(P); |
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| 103 | } |
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| 104 | else |
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| 105 | { |
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| 106 | L=P,R1; |
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| 107 | kill phi,vars; |
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[e27e47] | 108 | |
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[7a68965] | 109 | dbprint(printlevel-voice+3," |
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[a073dd] | 110 | // 'sagbiSPoly' created a ring as 2nd element of the list. |
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[e27e47] | 111 | // The ring contains the ideal 'kern' of algebraic relations between the |
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| 112 | //leading terms of the generators of I. |
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| 113 | // To access to this ring and see 'kern' you should give the ring a name, |
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| 114 | // e.g.: |
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| 115 | def S = L[2]; setring S; kern; |
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[7a68965] | 116 | "); |
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| 117 | return(L); |
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[e27e47] | 118 | } |
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[7a68965] | 119 | } |
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[e27e47] | 120 | } |
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| 121 | example |
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| 122 | { "EXAMPLE:"; echo = 2; |
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| 123 | ring r=0, (x,y),dp; |
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| 124 | poly f1,f2,f3,f4=x2,y2,xy+y,2xy2; |
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| 125 | ideal I=f1,f2,f3,f4; |
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[a073dd] | 126 | sagbiSPoly(I); |
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| 127 | list L=sagbiSPoly(I,1); |
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[e27e47] | 128 | L[1]; |
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| 129 | def S= L[2]; setring S; kern; |
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| 130 | } |
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| 131 | |
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| 132 | /////////////////////////////////////////////////////////////////////////////// |
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| 133 | static proc std1(ideal J,ideal I,list #) |
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| 134 | // I is contained in J, and it is assumed to be a standard bases! |
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| 135 | //This procedure computes a Standard basis for J from I one's |
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| 136 | //This procedure is essential for Spoly1 procedure. |
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| 137 | { |
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| 138 | def br=basering; |
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| 139 | int tt; |
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| 140 | ideal Res,@result; |
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| 141 | |
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[7a68965] | 142 | if(size(#)>0) {tt=#[1];} |
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[e27e47] | 143 | |
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[7a68965] | 144 | if(size(I)==0) {@result=groebner(J);} |
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[e27e47] | 145 | |
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| 146 | if((size(I)!=0) && (size(J)-size(I)>=1)) |
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[7a68965] | 147 | { |
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| 148 | qring Q=I; |
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| 149 | ideal J=fetch(br,J); |
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| 150 | J=groebner(J); |
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| 151 | setring br; |
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| 152 | Res=fetch(Q,J);// Res contains the generators that we add to I |
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| 153 | // to get the generators of std(J); |
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| 154 | @result=Res+I; |
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| 155 | } |
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[e27e47] | 156 | |
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[7a68965] | 157 | if(tt==0) { return(@result);} |
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| 158 | else { return(Res);} |
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[e27e47] | 159 | } |
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| 160 | |
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| 161 | /////////////////////////////////////////////////////////////////////////////// |
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| 162 | |
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| 163 | static proc Spoly1(list l,ideal I,ideal J,int a) |
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| 164 | //an implementation of SAGBI construction Algorithm using Spoly |
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| 165 | //procedure leads to useless computations and affect the efficiency |
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| 166 | //of SAGBI bases computations. This procedure is a variant of Spoly |
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| 167 | //in order to avoid these useless compuations. |
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| 168 | { |
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| 169 | degBound=0; |
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| 170 | def br=basering; |
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| 171 | ideal vars=maxideal(1); |
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| 172 | ideal B=ideal(br); |
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| 173 | int b=size(B); |
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| 174 | |
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| 175 | if(b!=0) |
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[7a68965] | 176 | { |
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| 177 | I=reduce(I,groebner(0)); |
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| 178 | J=reduce(J,groebner(0)); |
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| 179 | } |
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[e27e47] | 180 | int n,ii,jj=nvars(br),ncols(I),ncols(J); |
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| 181 | int z; |
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| 182 | list @L; |
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| 183 | string mp =string(minpoly); |
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| 184 | |
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| 185 | if(size(J)==0) |
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[7a68965] | 186 | { |
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| 187 | @L =sagbiSPoly(I,1); |
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| 188 | } |
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| 189 | else |
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| 190 | { |
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| 191 | ideal @sum=I+J; |
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| 192 | ideal P1; |
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| 193 | ideal P=l[1];//P is the ideal of spolynomials of I; |
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| 194 | def R=l[2];setring R;int kk=nvars(R); |
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| 195 | ideal J=fetch(br,J); |
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| 196 | |
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| 197 | //================create a new ring with extra variables============== |
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[3da61f] | 198 | execute("ring R1=("+charstr(R)+"),("+varstr(R)+",@y((ii+1)..(ii+jj))),(dp(n),dp(kk+jj-n));"); |
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| 199 | // *levandov: would it not be easier and better to use |
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| 200 | // ring @Y = char(R),(@y((ii+1)..(ii+jj))),dp; |
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| 201 | // def R1 = R + @Y; |
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| 202 | // setring R1; |
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[a2c2031] | 203 | // -> thus |
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[7a68965] | 204 | ideal kern1; |
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| 205 | ideal A=fetch(R,A); |
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| 206 | attrib(A,"isSB",1); |
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| 207 | ideal J=fetch(R,J); |
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| 208 | ideal kern=fetch(R,kern); |
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| 209 | ideal A1; |
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| 210 | for(z=1;z<=jj;z++) |
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[e27e47] | 211 | { |
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[7a68965] | 212 | A1[z]=lead(J[z])-var(z+kk); |
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[e27e47] | 213 | } |
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[7a68965] | 214 | A1=A+A1; |
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| 215 | ideal @Res=std1(A1,A,1);// the generators of @Res are whose we have to add |
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| 216 | // to A to get std(A1). |
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| 217 | A=A+@Res; |
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[c99fd4] | 218 | kern1=nselect(@Res,1..n); |
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[7a68965] | 219 | kern=kern+kern1; |
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| 220 | export kern,kern1,A; |
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| 221 | setring br; |
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| 222 | map phi=R1,vars,@sum; |
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| 223 | P1=simplify(phi(kern1),1);//P1 is th ideal we add to P to get the ideal |
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| 224 | //of Spolynomials of @sum. |
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| 225 | P=P+P1; |
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| 226 | |
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| 227 | if (a==1) |
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[e27e47] | 228 | { |
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[7a68965] | 229 | @L=P,R1; |
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| 230 | kill phi,vars; |
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| 231 | dbprint(printlevel-voice+3," |
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[e27e47] | 232 | // 'Spoly1' created a ring as 2nd element of the list. |
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| 233 | // The ring contains the ideal 'kern' of algebraic relations between the |
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| 234 | //generators of I+J. |
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| 235 | // To access to this ring and see 'kern' you should give the ring a name, |
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| 236 | // e.g.: |
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| 237 | def @ring = L[2]; setring @ring ; kern; |
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[7a68965] | 238 | "); |
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[e27e47] | 239 | } |
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[7a68965] | 240 | if(a==2) |
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| 241 | { |
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| 242 | @L=P1,R1; |
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| 243 | kill phi,vars; |
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| 244 | } |
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| 245 | } |
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[e27e47] | 246 | return(@L); |
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| 247 | } |
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| 248 | /////////////////////////////////////////////////////////////////////////////// |
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| 249 | |
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[043cba] | 250 | proc sagbiReduction(poly p,ideal dom,list #) |
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[804d68] | 251 | "USAGE: sagbiReduction(p,dom[,n]); p poly , dom ideal |
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[043cba] | 252 | RETURN: a polynomial, after one step subalgebra reduction |
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[e27e47] | 253 | @format |
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[83f218] | 254 | Three algorithm variants are used to perform subalgebra reduction. |
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[043cba] | 255 | The positive interger n determines which variant should be used. |
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| 256 | n may take the values 0 (default), 1 or 2. |
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[e27e47] | 257 | @end format |
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[043cba] | 258 | EXAMPLE: sagbiReduction; shows an example" |
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[e27e47] | 259 | { |
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| 260 | def bsr=basering; |
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| 261 | ideal B=ideal(bsr);//When the basering is quotient ring this type casting |
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| 262 | // gives the quotient ideal. |
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| 263 | int b=size(B); |
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| 264 | int n=nvars(bsr); |
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| 265 | |
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| 266 | //In quotient rings, SINGULAR, usually does not reduce polynomials w.r.t the |
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| 267 | //quotient ideal,therefore we should first reduce ,when it is necessary for computations, |
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| 268 | // to have a uniquely determined representant for each equivalent |
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| 269 | //class,which is the case of this algorithm. |
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| 270 | |
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| 271 | if(b !=0) //means that the basering is a quotient ring |
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[7a68965] | 272 | { |
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[3da61f] | 273 | p=reduce(p,std(0)); |
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| 274 | dom=reduce(dom,std(0)); |
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[7a68965] | 275 | } |
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[e27e47] | 276 | |
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| 277 | int i,choose; |
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| 278 | int z=ncols(dom); |
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| 279 | |
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| 280 | if((size(#)>0) && (typeof(#[1])=="int")) |
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[7a68965] | 281 | { |
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| 282 | choose = #[1]; |
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| 283 | } |
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[e27e47] | 284 | if (size(#)>1) |
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[7a68965] | 285 | { |
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| 286 | choose =#[2]; |
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| 287 | } |
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[e27e47] | 288 | |
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| 289 | //=======================first algorithm(default)========================= |
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[7a68965] | 290 | if ( choose == 0 ) |
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[e27e47] | 291 | { |
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[7a68965] | 292 | list L = algebra_containment(lead(p),lead(dom),1); |
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| 293 | if( L[1]==1 ) |
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| 294 | { |
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| 295 | // the ring L[2] = char(bsr),(x(1..nvars(bsr)),y(1..z)),(dp(n),dp(m)), |
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| 296 | // contains poly check s.t. LT(p) is of the form check(LT(f1),...,LT(fr)) |
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| 297 | def s1 = L[2]; |
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| 298 | map psi = s1,maxideal(1),dom; |
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| 299 | poly re = p - psi(check); |
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| 300 | // divide by the maximal power of #[1] |
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| 301 | if ( (size(#)>0) && (typeof(#[1])=="poly") ) |
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[e27e47] | 302 | { |
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[7a68965] | 303 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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| 304 | { |
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| 305 | re=re/#[1]; |
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[e27e47] | 306 | } |
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[7a68965] | 307 | } |
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| 308 | return(re); |
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| 309 | } |
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| 310 | return(p); |
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[e27e47] | 311 | } |
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[7a68965] | 312 | //======================2end variant of algorithm========================= |
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| 313 | //It uses two different commands for elimaination. |
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| 314 | //if(choose==1):"elimainate"command. |
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| 315 | //if (choose==2):"nselect" command. |
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| 316 | else |
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| 317 | { |
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| 318 | poly v=product(maxideal(1)); |
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| 319 | |
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| 320 | //------------- change the basering bsr to bsr[@(0),...,@(z)] ---------- |
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[3da61f] | 321 | execute("ring s=("+charstr(basering)+"),("+varstr(basering)+",@(0..z)),dp;"); |
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[7a68965] | 322 | // Ev hier die Reihenfolge der Vars aendern. Dazu muss unten aber entsprechend |
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| 323 | // geaendert werden: |
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| 324 | // execute("ring s="+charstr(basering)+",(@(0..z),"+varstr(basering)+"),dp;"); |
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| 325 | |
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| 326 | //constructs the leading ideal of dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
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| 327 | ideal dom=imap(bsr,dom); |
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| 328 | for (i=1;i<=z;i++) |
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| 329 | { |
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| 330 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
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| 331 | } |
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| 332 | dom=lead(imap(bsr,p))-@(0),dom; |
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[e27e47] | 333 | |
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[7a68965] | 334 | //---------- eliminate the variables of the basering bsr -------------- |
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| 335 | //i.e. computes dom intersected with K[@(0),...,@(z)]. |
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| 336 | |
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| 337 | if(choose==1) |
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| 338 | { |
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| 339 | ideal kern=eliminate(dom,imap(bsr,v));//eliminate does not need a |
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| 340 | //standard basis as input. |
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| 341 | } |
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| 342 | if(choose==2) |
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| 343 | { |
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[c99fd4] | 344 | ideal kern= nselect(groebner(dom),1..n);//"nselect" is combinatorial command |
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[7a68965] | 345 | //which uses the internal command |
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| 346 | // "simplify" |
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| 347 | } |
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[e27e47] | 348 | |
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[7a68965] | 349 | //--------- test wether @(0)-h(@(1),...,@(z)) is in ker --------------- |
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| 350 | // for some poly h and divide by maximal power of q=#[1] |
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| 351 | poly h; |
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| 352 | z=size(kern); |
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| 353 | for (i=1;i<=z;i++) |
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| 354 | { |
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| 355 | h=kern[i]/@(0); |
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| 356 | if (deg(h)==0) |
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| 357 | { |
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| 358 | h=(1/h)*kern[i]; |
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| 359 | // define the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
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| 360 | setring bsr; |
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| 361 | map psi=s,maxideal(1),p,dom; |
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| 362 | poly re=psi(h); |
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| 363 | // divide by the maximal power of #[1] |
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| 364 | if ((size(#)>0) && (typeof(#[1])== "poly") ) |
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| 365 | { |
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| 366 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
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| 367 | { |
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| 368 | re=re/#[1]; |
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| 369 | } |
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| 370 | } |
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| 371 | return(re); |
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| 372 | } |
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| 373 | } |
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| 374 | setring bsr; |
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| 375 | return(p); |
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| 376 | } |
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[e27e47] | 377 | } |
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| 378 | example |
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| 379 | {"EXAMPLE:"; echo = 2; |
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| 380 | ring r= 0,(x,y),dp; |
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| 381 | ideal dom =x2,y2,xy-y; |
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| 382 | poly p=x4+x3y+xy2-y2; |
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[804d68] | 383 | sagbiReduction(p,dom); |
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| 384 | sagbiReduction(p,dom,1); |
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| 385 | sagbiReduction(p,dom,2); |
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[e27e47] | 386 | } |
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| 387 | |
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| 388 | /////////////////////////////////////////////////////////////////////////////// |
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| 389 | static proc completeReduction(poly p,ideal dom,list#)//reduction |
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| 390 | { |
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| 391 | poly p1=p; |
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[804d68] | 392 | poly p2=sagbiReduction(p,dom,#); |
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[e27e47] | 393 | while (p1!=p2) |
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| 394 | { |
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| 395 | p1=p2; |
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[804d68] | 396 | p2=sagbiReduction(p1,dom,#); |
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[e27e47] | 397 | } |
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| 398 | return(p2); |
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| 399 | } |
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| 400 | /////////////////////////////////////////////////////////////////////////////// |
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| 401 | |
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[83f218] | 402 | static proc completeReduction1(poly p,ideal dom,list #) //tail reduction |
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[e27e47] | 403 | { |
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| 404 | poly p1,p2,re; |
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| 405 | p1=p; |
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| 406 | while(p1!=0) |
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[7a68965] | 407 | { |
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[804d68] | 408 | p2=sagbiReduction(p1,dom,#); |
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[7a68965] | 409 | if(p2!=p1) |
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[e27e47] | 410 | { |
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[7a68965] | 411 | p1=p2; |
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[e27e47] | 412 | } |
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[7a68965] | 413 | else |
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| 414 | { |
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| 415 | re=re+lead(p2); |
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| 416 | p1=p2-lead(p2); |
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| 417 | } |
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| 418 | } |
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[e27e47] | 419 | return(re); |
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| 420 | } |
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| 421 | |
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| 422 | |
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| 423 | |
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| 424 | /////////////////////////////////////////////////////////////////////////////// |
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| 425 | |
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[a073dd] | 426 | proc sagbiNF(id,ideal dom,int k,list#) |
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| 427 | "USAGE: sagbiNF(id,dom,k[,n]); id either poly or ideal,dom ideal, k and n positive intergers. |
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[906458] | 428 | RETURN: same as type of id; ideal or polynomial. |
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[e27e47] | 429 | @format |
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[3da61f] | 430 | The integer k determines what kind of s-reduction is performed: |
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| 431 | - if (k=0) no tail s-reduction is performed. |
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| 432 | - if (k=1) tail s-reduction is performed. |
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[906458] | 433 | Three Algorithm variants are used to perform subalgebra reduction. |
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| 434 | The positive integer n determines which variant should be used. |
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[e27e47] | 435 | n may take the values (0 or default),1 or 2. |
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| 436 | @end format |
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[906458] | 437 | NOTE: computation of subalgebra normal forms may be performed in polynomial rings or quotients |
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| 438 | thereof |
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[a073dd] | 439 | EXAMPLE: example sagbiNF; show example " |
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[e27e47] | 440 | { |
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| 441 | int z; |
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| 442 | ideal Red; |
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| 443 | poly re; |
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| 444 | if(typeof(id)=="ideal") |
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[7a68965] | 445 | { |
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| 446 | int i=ncols(id); |
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| 447 | for(z=1;z<=i;z++) |
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[e27e47] | 448 | { |
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| 449 | if(k==0) |
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[7a68965] | 450 | { |
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| 451 | id[z]=completeReduction(id[z],dom,#); |
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| 452 | } |
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[e27e47] | 453 | else |
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[7a68965] | 454 | { |
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| 455 | id[z]=completeReduction1(id[z],dom,#);//tail reduction. |
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| 456 | } |
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[e27e47] | 457 | } |
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[7a68965] | 458 | Red=simplify(id,7); |
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| 459 | return(Red); |
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| 460 | } |
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[e27e47] | 461 | if(typeof(id)=="poly") |
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[7a68965] | 462 | { |
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| 463 | if(k==0) |
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[e27e47] | 464 | { |
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[7a68965] | 465 | re=completeReduction(id,dom,#); |
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[e27e47] | 466 | } |
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[7a68965] | 467 | else |
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| 468 | { |
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| 469 | re=completeReduction1(id,dom,#); |
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| 470 | } |
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| 471 | return(re); |
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| 472 | } |
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[e27e47] | 473 | } |
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| 474 | example |
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| 475 | {"EXAMPLE:"; echo = 2; |
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| 476 | ring r=0,(x,y),dp; |
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[a073dd] | 477 | ideal I= x2-xy; |
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[e27e47] | 478 | qring Q=std(I); |
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| 479 | ideal dom =x2,x2y+y,x3y2; |
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| 480 | poly p=x4+x2y+y; |
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[a073dd] | 481 | sagbiNF(p,dom,0); |
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[3da61f] | 482 | sagbiNF(p,dom,1);// tail subalgebra reduction is performed |
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[e27e47] | 483 | } |
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| 484 | |
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| 485 | |
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| 486 | /////////////////////////////////////////////////////////////////////////////// |
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| 487 | |
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| 488 | static proc intRed(id,int k, list #) |
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| 489 | { |
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| 490 | int i,z; |
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| 491 | ideal Rest,intRed; |
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| 492 | z=ncols(id); |
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| 493 | for(i=1;i<=z;i++) |
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[7a68965] | 494 | { |
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| 495 | Rest=id; |
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| 496 | Rest[i]=0; |
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| 497 | Rest=simplify(Rest,2); |
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| 498 | if(k==0) |
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[e27e47] | 499 | { |
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[7a68965] | 500 | intRed[i]=completeReduction(id[i],Rest,#); |
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[e27e47] | 501 | } |
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[7a68965] | 502 | else |
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| 503 | { |
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| 504 | intRed[i]=completeReduction1(id[i],Rest,#); |
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| 505 | } |
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| 506 | } |
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[e27e47] | 507 | intRed=simplify(intRed,7);//1+2+4 in simplify command |
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| 508 | return(intRed); |
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| 509 | } |
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| 510 | ////////////////////////////////////////////////////////////////////////////// |
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| 511 | |
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[63ba62] | 512 | proc sagbi(id,int k,list#) |
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[3da61f] | 513 | "USAGE: sagbi(id,k[,n]); id ideal, k and n positive integers. |
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[e27e47] | 514 | RETURN: A SAGBI basis for the subalgebra defined by the generators of id. |
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| 515 | @format |
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[906458] | 516 | k determines what kind of s-reduction is performed: |
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[3da61f] | 517 | - if (k=0) no tail s-reduction is performed. |
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| 518 | - if (k=1) tail s-reduction is performed, and S-interreduced SAGBI basis |
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[e27e47] | 519 | is returned. |
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[906458] | 520 | Three algorithm variants are used to perform subalgebra reduction. |
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[e27e47] | 521 | The positive interger n determine which variant should be used. |
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| 522 | n may take the values (0 or default),1 or 2. |
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| 523 | @end format |
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[906458] | 524 | NOTE: SAGBI bases computations may be performed in polynomial rings or quotients |
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| 525 | thereof. |
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[63ba62] | 526 | EXAMPLE: example sagbi; show example " |
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[e27e47] | 527 | { |
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| 528 | degBound=0; |
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| 529 | ideal S,oldS,Red; |
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| 530 | list L; |
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| 531 | S=intRed(id,k,#); |
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| 532 | while(size(S)!=size(oldS)) |
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[7a68965] | 533 | { |
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| 534 | L=Spoly1(L,S,Red,2); |
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| 535 | Red=L[1]; |
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| 536 | Red=sagbiNF(Red,S,k,#); |
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| 537 | oldS=S; |
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| 538 | S=S+Red; |
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| 539 | } |
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[b54a393] | 540 | return(interreduced(S)); |
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[e27e47] | 541 | } |
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| 542 | example |
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| 543 | { "EXAMPLE:"; echo = 2; |
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| 544 | ring r= 0,(x,y),dp; |
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| 545 | ideal I=x2,y2,xy+y; |
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[63ba62] | 546 | sagbi(I,1,1); |
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[e27e47] | 547 | } |
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[b54a393] | 548 | |
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| 549 | proc interreduced(ideal I) |
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| 550 | { |
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| 551 | ideal J,B; |
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| 552 | int i,j,k; |
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| 553 | poly f; |
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| 554 | for(k=1;k<=ncols(I);k++) |
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| 555 | { |
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| 556 | f=I[k]; |
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| 557 | I[k]=0; |
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| 558 | f=sagbiNF(f,I,1); |
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| 559 | I[k]=f; |
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| 560 | } |
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| 561 | I=simplify(I,2); |
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| 562 | return(I); |
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| 563 | } |
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| 564 | |
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[e27e47] | 565 | /////////////////////////////////////////////////////////////////////////////// |
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[a073dd] | 566 | proc sagbiPart(id,int k,int c,list #) |
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[906458] | 567 | "USAGE: sagbiPart(id,k,c[,n]); id ideal, k, c and n positive integers |
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[3da61f] | 568 | RETURN: A partial SAGBI basis for the subalgebra defined by the generators of id. |
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[e27e47] | 569 | @format |
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[906458] | 570 | k determines what kind of s-reduction is performed: |
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[3da61f] | 571 | - if (k=0) no tail s-reduction is performed. |
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| 572 | - if (k=1) tail s-reduction is performed, and S-intereduced SAGBI basis |
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[e27e47] | 573 | is returned. |
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[906458] | 574 | c determines, after how many loops the Sagbi basis computation should stop. |
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| 575 | Three algorithm variants are used to perform subalgebra reduction. |
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[3da61f] | 576 | The positive integer n determines which variant should be used. |
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[e27e47] | 577 | n may take the values (0 or default),1 or 2. |
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| 578 | @end format |
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[906458] | 579 | NOTE:- SAGBI bases computations may be performed in polynomial rings or quotients |
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| 580 | thereof. |
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| 581 | - This version of sagbi is interesting in the case of subalgebras with infinte |
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| 582 | SAGBI basis. In this case, it may be used to check, if the elements of this |
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| 583 | basis have a particular form. |
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[a073dd] | 584 | EXAMPLE: example sagbiPart; show example " |
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[e27e47] | 585 | { |
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| 586 | degBound=0; |
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| 587 | ideal S,oldS,Red; |
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| 588 | int counter; |
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| 589 | list L; |
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| 590 | S=intRed(id,k,#); |
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| 591 | while((size(S)!=size(oldS))&&(counter<=c)) |
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[7a68965] | 592 | { |
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| 593 | L=Spoly1(L,S,Red,2); |
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| 594 | Red=L[1]; |
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| 595 | Red=sagbiNF(Red,S,k,#); |
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| 596 | oldS=S; |
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| 597 | S=S+Red; |
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| 598 | counter=counter+1; |
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| 599 | } |
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[e27e47] | 600 | return(S); |
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| 601 | } |
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| 602 | example |
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| 603 | { "EXAMPLE:"; echo = 2; |
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| 604 | ring r= 0,(x,y),dp; |
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| 605 | ideal I=x,xy-y2,xy2;//the corresponding Subalgebra has an infinte SAGBI basis |
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[a073dd] | 606 | sagbiPart(I,1,3);// computations should stop after 3 turns. |
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[e27e47] | 607 | } |
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| 608 | ////////////////////////////////////////////////////////////////////////////// |
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