[731e67e] | 1 | // $Id: |
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[373856] | 2 | //GMG, last modified 28.9.01 |
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| 3 | /////////////////////////////////////////////////////////////////////////////// |
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[2c957af] | 4 | version="$Id: teachstd.lib,v 1.6 2006-07-25 12:02:44 Singular Exp $"; |
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[373856] | 5 | category="Teaching"; |
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| 6 | info=" |
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| 7 | LIBRARY: teachstd.lib Procedures for teaching standard bases |
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| 8 | AUTHOR: G.-M. Greuel, greuel@mathematik.uni-kl.de |
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| 9 | |
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[2c957af] | 10 | NOTE: The library is intended to be used for teaching purposes, but not |
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[731e67e] | 11 | for serious computations. Sufficiently high printlevel allows to |
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[9ece67] | 12 | control each step, thus illustrating the algorithms at work. |
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[2c957af] | 13 | The procedures are implemented exactly as described in the book |
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[9ece67] | 14 | 'A SINGULAR Introduction to Commutative Algebra' by G.-M. Greuel and |
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| 15 | G. Pfister (Springer 2002). |
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| 16 | |
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[c7c5ef] | 17 | PROCEDURES: |
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[373856] | 18 | ecart(f); ecart of f |
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| 19 | tail(f); tail of f |
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| 20 | sameComponent(f,g); test for same module component of lead(f) and lead(g) |
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| 21 | leadmonomial(f); leading monomial as poly (also for vectors) |
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| 22 | monomialLcm(m,n); lcm of monomials m and n as poly (also for vectors) |
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| 23 | spoly(f[,1]); s-polynomial of f [symmetric form] |
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| 24 | minEcart(T,h); element g from T of minimal ecart s.t. LM(g)|LM(h) |
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| 25 | NFMora(i); normal form of i w.r.t Mora algorithm |
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| 26 | prodcrit(f,g); test for product criterion |
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| 27 | chaincrit(f,g,h); test for chain criterion |
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| 28 | pairset(G); pairs form G neither satifying prodcrit nor chaincrit |
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[731e67e] | 29 | updatePairs(P,S,h); pairset P enlarded by not useless pairs (h,f), f in S |
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[373856] | 30 | standard(id); standard basis of ideal/module |
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| 31 | localstd(id); local standard basis of id using Lazard's method |
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| 32 | |
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[9ece67] | 33 | [parameters in square brackets are optional] |
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[373856] | 34 | "; |
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| 35 | |
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| 36 | LIB "poly.lib"; |
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| 37 | |
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| 38 | /////////////////////////////////////////////////////////////////////////////// |
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| 39 | proc ecart(f) |
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| 40 | "USAGE: ecart(f); f poly or vector |
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| 41 | RETURN: the ecart e of f of type int |
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| 42 | EXAMPLE: example ecart; shows an example |
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| 43 | " |
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| 44 | { |
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| 45 | int e = maxdeg1(f)-maxdeg1(lead(f)); |
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| 46 | return(e); |
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[731e67e] | 47 | } |
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[373856] | 48 | example |
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| 49 | { "EXAMPLE:"; echo = 2; |
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| 50 | ring r=0,(x,y,z),ls; |
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| 51 | ecart((y+z+x+xyz)**2); |
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| 52 | ring s=0,(x,y,z),dp; |
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[731e67e] | 53 | ecart((y+z+x+xyz)**2); |
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| 54 | } |
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[373856] | 55 | /////////////////////////////////////////////////////////////////////////////// |
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| 56 | proc leadmonomial(f) |
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| 57 | "USAGE: leadmonomial(f); f poly or vector |
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| 58 | RETURN: the leading monomial of f of type poly |
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| 59 | NOTE: if f is of type poly, leadmonomial(f)=leadmonom(f), if f is of type |
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| 60 | vector and if leadmonom(f)=m*gen(i) then leadmonomial(f)=m |
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| 61 | EXAMPLE: example leadmonomial; shows an example |
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| 62 | " |
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| 63 | { |
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| 64 | int e; |
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| 65 | poly m; |
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| 66 | if(typeof(f) == "vector") |
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| 67 | { |
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| 68 | e=leadexp(f)[nvars(basering)+1]; |
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| 69 | m=leadmonom(f)[e,1]; |
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| 70 | } |
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| 71 | if(typeof(f) == "poly") |
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| 72 | { |
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| 73 | m=leadmonom(f); |
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[731e67e] | 74 | } |
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[373856] | 75 | return(m); |
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[731e67e] | 76 | } |
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[373856] | 77 | example |
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| 78 | { "EXAMPLE:"; echo = 2; |
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| 79 | ring s=0,(x,y,z),(c,dp); |
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[731e67e] | 80 | leadmonomial((y+z+x+xyz)**2); |
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[373856] | 81 | leadmonomial([(y+z+x+xyz)**2,xyz5]); |
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[731e67e] | 82 | } |
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[373856] | 83 | /////////////////////////////////////////////////////////////////////////////// |
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| 84 | proc tail(f) |
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| 85 | "USAGE: tail(f); f poly or vector |
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| 86 | RETURN: f-lead(f), the tail of f of type poly |
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| 87 | EXAMPLE: example tail; shows an example |
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| 88 | " |
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| 89 | { |
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| 90 | def t = f-lead(f); |
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| 91 | return(t); |
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[731e67e] | 92 | } |
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[373856] | 93 | example |
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| 94 | { "EXAMPLE:"; echo = 2; |
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| 95 | ring r=0,(x,y,z),ls; |
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| 96 | tail((y+z+x+xyz)**2); |
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| 97 | ring s=0,(x,y,z),dp; |
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[731e67e] | 98 | tail((y+z+x+xyz)**2); |
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| 99 | } |
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[373856] | 100 | /////////////////////////////////////////////////////////////////////////////// |
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| 101 | proc sameComponent(f,g) |
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| 102 | "USAGE: sameComponent(f,g); f,g poly or vector |
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| 103 | RETURN: 1 if f and g are of type poly or if f and g are of type vector and |
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[731e67e] | 104 | their leading monomials involve the same module component, |
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[373856] | 105 | 0 if not |
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| 106 | EXAMPLE: example sameComponent; shows an example |
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| 107 | " |
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| 108 | { |
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| 109 | if(typeof(f) != typeof(g)) |
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| 110 | { |
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| 111 | ERROR("** arguments must be of same type"); |
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| 112 | } |
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| 113 | if(typeof(f) == "vector") |
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[731e67e] | 114 | { |
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[373856] | 115 | if( leadexp(f)[nvars(basering)+1] != leadexp(g)[nvars(basering)+1] ) |
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| 116 | { |
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| 117 | return(0); |
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| 118 | } |
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| 119 | } |
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| 120 | return(1); |
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[731e67e] | 121 | } |
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[373856] | 122 | example |
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| 123 | { "EXAMPLE:"; echo = 2; |
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| 124 | ring r=0,(x,y,z),dp; |
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| 125 | sameComponent([y+z+x,xyz],[z2,xyz]); |
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| 126 | sameComponent([y+z+x,xyz],[z4,xyz]); |
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[731e67e] | 127 | sameComponent(y+z+x+xyz, xy+z5); |
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| 128 | } |
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[373856] | 129 | /////////////////////////////////////////////////////////////////////////////// |
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| 130 | proc monomialLcm(m,n) |
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| 131 | "USAGE: monomialLcm(m,n); m,n of type poly or vector |
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| 132 | RETURN: least common multiple of leading monomials of m and n, of type poly |
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| 133 | NOTE: if m = (x1...xr)^(a1,...,ar)*gen(i) (gen(i)=1 if m is of type poly) |
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[731e67e] | 134 | and n = (x1...xr)^(b1,...,br)*gen(j), then the proc returns |
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[373856] | 135 | (x1,...,xr)^(max(a1,b1),...,max(ar,br)) if i=j and 0 if i!=j. |
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| 136 | EXAMPLE: example monomialLcm; shows an example |
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| 137 | " |
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| 138 | { |
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| 139 | if(typeof(n) != typeof(m)) |
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| 140 | { |
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| 141 | ERROR("** arguments must be of same type"); |
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| 142 | } |
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| 143 | |
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| 144 | poly u ; |
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[731e67e] | 145 | if(sameComponent(m,n) == 0) //leading term of vectors involve |
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[373856] | 146 | { //different module components |
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| 147 | return(u); |
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| 148 | } |
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[731e67e] | 149 | |
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[373856] | 150 | intvec v = leadexp(m); //now start to compute lcm |
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[731e67e] | 151 | intvec w = leadexp(n); |
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[373856] | 152 | u=1; |
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| 153 | int i; |
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| 154 | for (i=1; i<=nvars(basering); i++) |
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| 155 | { |
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[731e67e] | 156 | if(v[i]>=w[i]) |
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[373856] | 157 | { |
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| 158 | u = u*var(i)**v[i]; |
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| 159 | } |
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| 160 | else |
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| 161 | { |
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| 162 | u = u*var(i)**w[i]; |
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| 163 | } |
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| 164 | } |
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[731e67e] | 165 | return(u); |
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| 166 | } |
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[373856] | 167 | example |
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| 168 | { "EXAMPLE:"; echo = 2; |
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| 169 | ring r=0,(x,y,z),ds; |
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| 170 | monomialLcm(xy2,yz3); |
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| 171 | monomialLcm([xy2,xz],[yz3]); |
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| 172 | monomialLcm([xy2,xz3],[yz3]); |
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[731e67e] | 173 | } |
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[373856] | 174 | /////////////////////////////////////////////////////////////////////////////// |
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| 175 | proc spoly(f,g,list #) |
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| 176 | "USAGE: spoly(f,g[,s]); f,g poly or vector, s int |
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| 177 | RETURN: the s-polynomial of f and g, of type poly or vector |
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| 178 | if s!=0 the symmetric s-polynomial (without division) is returned |
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| 179 | EXAMPLE: example spoly; shows an example |
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| 180 | " |
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| 181 | { |
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| 182 | if(typeof(f) != typeof(g)) |
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| 183 | { |
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| 184 | ERROR("** arguments must be of same type"); |
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[731e67e] | 185 | } |
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[373856] | 186 | if(size(#) == 0) |
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| 187 | { |
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| 188 | #[1]=0; |
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| 189 | } |
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| 190 | |
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| 191 | int e; |
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[731e67e] | 192 | poly o = monomialLcm(f,g); |
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[373856] | 193 | if( o == 0) //can only happen, if vectors f and g involve |
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| 194 | { //different module components |
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| 195 | vector sp; |
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| 196 | return(sp); |
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| 197 | } |
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[731e67e] | 198 | |
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[373856] | 199 | poly m=leadmonomial(f); //compute the leading monomial as poly |
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| 200 | poly n=leadmonomial(g); |
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| 201 | |
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| 202 | if (#[1]==0) //the asymmetric s-poly |
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| 203 | { |
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| 204 | def sp = (o/m)*f - (leadcoef(f)/leadcoef(g))*(o/n)*g; |
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| 205 | } |
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| 206 | else //the symmetric s-poly, avoiding division |
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| 207 | { |
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| 208 | def sp = leadcoef(g)*(o/m)*f - leadcoef(f)*(o/n)*g; |
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[731e67e] | 209 | } |
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[373856] | 210 | return(sp); |
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[731e67e] | 211 | } |
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[373856] | 212 | example |
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| 213 | { "EXAMPLE:"; echo = 2; |
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| 214 | ring r=0,(x,y,z),ls; |
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| 215 | spoly(2x2+x2y,3y3+xyz); |
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| 216 | ring s=0,(x,y,z),(c,dp); |
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| 217 | spoly(2x2+x2y,3y3+xyz); |
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| 218 | spoly(2x2+x2y,3y3+xyz,1); //symmetric s-poly without division |
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| 219 | spoly([5x2+x2y,z5],[x2,y3,y4]); //s-poly for vectors |
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[731e67e] | 220 | } |
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[373856] | 221 | /////////////////////////////////////////////////////////////////////////////// |
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| 222 | proc minEcart(T,h) |
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| 223 | "USAGE: minEcart(T,h); T ideal or module, h poly or vector |
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| 224 | RETURN: element g from T such that leadmonom(g) divides leadmonom(h) |
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| 225 | ecart(g) is minimal with this property (if T != 0); |
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| 226 | return 0 if T is 0 or h = 0 |
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| 227 | EXAMPLE: example minEcart; shows an example |
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| 228 | " |
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| 229 | { |
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| 230 | int i,k,e; |
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| 231 | if (size(T)==0 or h==0 ) //trivial cases |
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[731e67e] | 232 | { |
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[373856] | 233 | h = 0; |
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| 234 | return(h); |
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| 235 | } |
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| 236 | //---- check whether some element in T has the same module component as h ---- |
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| 237 | int v; |
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| 238 | intvec w; |
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| 239 | T = simplify(T,2); |
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[731e67e] | 240 | |
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| 241 | if (typeof(h) == "vector") |
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[373856] | 242 | { |
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| 243 | e=1; |
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| 244 | v = leadexp(h)[nvars(basering)+1]; |
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| 245 | for( i=1; i<=size(T); i++) |
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| 246 | { |
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| 247 | w[i]=leadexp(T[i])[nvars(basering)+1]; |
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| 248 | if(v == w[i]) |
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| 249 | { |
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| 250 | e=0; //some element in T involves the same component as h |
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| 251 | } |
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| 252 | } |
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| 253 | } |
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| 254 | if ( e == 1 ) //no element in T involves the same component as h |
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[731e67e] | 255 | { |
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[373856] | 256 | h = 0; |
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| 257 | return(h); |
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| 258 | } |
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| 259 | |
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[731e67e] | 260 | if (typeof(h) == "poly") //for polys v=w[i] for all i |
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[373856] | 261 | { |
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| 262 | v = 1; |
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| 263 | w[size(T)]=0; |
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| 264 | w=w+1; |
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| 265 | } |
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[731e67e] | 266 | |
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[373856] | 267 | //------ check whether for some g in T leadmonom(g) divides leadmonom(h) ----- |
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| 268 | def g = T[1]; |
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| 269 | poly f = leadmonomial(h); |
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| 270 | |
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| 271 | for( i=1; i<=size(T); i++) |
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| 272 | { |
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| 273 | if( f/leadmonomial(T[i]) != 0 and v==w[i] ) |
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| 274 | { |
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| 275 | g=T[i]; |
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| 276 | k=i; |
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| 277 | break; |
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| 278 | } |
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| 279 | } |
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[731e67e] | 280 | if (k == 0) //no leadmonom(g) divides leadmonom(h) |
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| 281 | { |
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[373856] | 282 | g = 0; |
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| 283 | return(g); |
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| 284 | } |
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| 285 | //--look for T[i] with minimal ecart s.t.leadmonom(T[i]) divides leadmonom(h)-- |
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| 286 | |
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| 287 | for( i=k+1; i<=size(T); i++) |
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| 288 | { |
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| 289 | if( f/leadmonomial(T[i]) != 0 and v==w[i] ) |
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| 290 | { |
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| 291 | if (ecart(T[i]) < ecart(g)) |
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| 292 | { |
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| 293 | g=T[i]; |
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| 294 | } |
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| 295 | } |
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| 296 | } |
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| 297 | return(g); |
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[731e67e] | 298 | } |
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[373856] | 299 | example |
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| 300 | { "EXAMPLE:"; echo = 2; |
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| 301 | ring R=0,(x,y,z),dp; |
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| 302 | ideal T = x2y+x2,y3+xyz,xyz2+z4; |
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| 303 | poly h = x2y2z2+x5+yx3+z6; |
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| 304 | minEcart(T,h);""; |
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| 305 | ring S=0,(x,y,z),(c,ds); |
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| 306 | module T = [x2+x2y,y2],[y3+xyz,x3-z3],[x3y+z4,0,x2]; |
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| 307 | vector h = [x3y+x5+x2y2z2+z6,x3]; |
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| 308 | minEcart(T,h); |
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[731e67e] | 309 | } |
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[373856] | 310 | /////////////////////////////////////////////////////////////////////////////// |
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| 311 | proc NFMora(f,G,list #) |
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| 312 | "USAGE: NFMora(f,G[,s]); f poly or vector,G ideal or module, s int |
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| 313 | RETURN: the Mora normal form of f w.r.t. G, same type as f |
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| 314 | if s!=0 the symmetric s-polynomial (without division) is used |
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| 315 | NOTE: Show comments if printlevel > 0, pauses computation if printlevel > 1 |
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| 316 | EXAMPLE: example NFMora; shows an example |
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| 317 | " |
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| 318 | { |
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| 319 | if(size(#) == 0) |
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| 320 | { |
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| 321 | #[1]=0; |
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| 322 | } |
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| 323 | |
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| 324 | int y = printlevel - voice + 2; |
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| 325 | if( f==0 or size(G) ==0 ) |
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| 326 | { |
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[731e67e] | 327 | if (y>0) |
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[373856] | 328 | { |
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| 329 | "// 1st or 2nd argument 0"; |
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| 330 | } |
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| 331 | return(f); |
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| 332 | } |
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| 333 | |
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| 334 | int i,e; |
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| 335 | def h = f; |
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| 336 | def T = G; |
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| 337 | // -------------------- start with f to be reduced by G -------------------- |
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[731e67e] | 338 | if (y>0) |
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[373856] | 339 | {""; |
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| 340 | "// Input for NFMora is (f,T):"; |
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| 341 | "// f:"; f; |
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| 342 | "// T:"; T; |
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| 343 | "// Reduce f with T, eventually enlarging T for local ordering"; |
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| 344 | } |
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| 345 | |
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| 346 | // ----------------------- enter the reduction loop ------------------------ |
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| 347 | def g = minEcart(T,h); |
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| 348 | while (h!=0 and g!=0) |
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| 349 | { |
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[731e67e] | 350 | if (y>0) |
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[373856] | 351 | { ""; |
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| 352 | "// Reduction-step in NFMora:",i; |
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| 353 | "// h = (f after",i,"reductions) reduction with g from T:"; |
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[731e67e] | 354 | "// g = element of minimal ecart in T s.t. LM(g)|LM(h):"; |
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[373856] | 355 | "// h:";h; |
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[731e67e] | 356 | "// g:";g; |
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[373856] | 357 | } |
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[731e67e] | 358 | if (y>1) |
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| 359 | { |
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[373856] | 360 | pause("press <return> to continue"); |
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| 361 | "// T, set used for reduction:"; T; |
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| 362 | pause("press <return> to continue"); |
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| 363 | } |
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| 364 | e=0; |
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| 365 | if( ecart(g) > ecart(h) ) |
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| 366 | { |
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| 367 | T=T,h; e=1; |
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| 368 | } |
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[731e67e] | 369 | if (y>0 ) |
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| 370 | { |
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[373856] | 371 | "// T-set enlarged for next reduction? (yes/no = 1/0): ", e; |
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| 372 | if( e==1 ) |
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| 373 | { |
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| 374 | "// T-set for next reduction got enlarged by h:"; |
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| 375 | "// h:";h; |
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[731e67e] | 376 | if (y>1) |
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| 377 | { |
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[373856] | 378 | pause("press <return> to continue"); |
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| 379 | } |
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| 380 | } |
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| 381 | } |
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| 382 | h = spoly(h,g,#[1]); |
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| 383 | g = minEcart(T,h); |
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| 384 | i=i+1; |
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| 385 | } |
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| 386 | if(y>0) |
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| 387 | { ""; |
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| 388 | "// normal form is:"; |
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| 389 | } |
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| 390 | return(h); |
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[731e67e] | 391 | } |
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[373856] | 392 | example |
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| 393 | { "EXAMPLE:"; echo = 2; |
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| 394 | ring r=0,(x,y,z),dp; |
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| 395 | poly f = x2y2z2+x5+yx3+z6-3y3; |
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| 396 | ideal G = x2y+x2,y3+xyz,xyz2+z6; |
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| 397 | NFMora(f,G);""; |
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| 398 | ring s=0,(x,y,z),ds; |
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| 399 | poly f = x3y+x5+x2y2z2+z6; |
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| 400 | ideal G = x2+x2y,y3+xyz,x3y2+z4; |
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| 401 | NFMora(f,G);""; |
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| 402 | vector v = [f,x2+x2y]; |
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| 403 | module M = [x2+x2y,f],[y3+xyz,y3],[x3y2+z4,z2]; |
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| 404 | NFMora(v,M); |
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[731e67e] | 405 | } |
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[373856] | 406 | /////////////////////////////////////////////////////////////////////////////// |
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| 407 | proc prodcrit(f,g) |
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| 408 | "USAGE: prodcrit(f,g); f,g poly or vector |
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| 409 | RETURN: 1 if product criterion applies in the same module component, |
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| 410 | 2 if lead(f) and lead(g) involve different components, 0 else |
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| 411 | NOTE: if product criterion applies we can delete (f,g) from pairset |
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| 412 | EXAMPLE: example prodcrit; shows an example |
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| 413 | " |
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| 414 | { |
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| 415 | if(typeof(f)=="poly") //product criterion for polynomials |
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| 416 | { |
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| 417 | if( monomialLcm(f,g)==leadmonom(f)*leadmonom(g)) |
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| 418 | { |
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| 419 | return(1); |
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| 420 | } |
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| 421 | return(0); |
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| 422 | } |
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[731e67e] | 423 | // ------------------- product criterion for modules --------------------- |
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[373856] | 424 | if(sameComponent(f,g)==1) |
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| 425 | { |
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| 426 | if( monomialLcm(f,g)==leadmonomial(f)*leadmonomial(g) ) |
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| 427 | { |
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| 428 | int c = leadexp(f)[nvars(basering)+1]; //component involving lead(f) |
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| 429 | if((f-f[c]*gen(c))-(g-g[c]*gen(c))==0) //other components are 0 |
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| 430 | { |
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| 431 | return(1); |
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| 432 | } |
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| 433 | } |
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| 434 | return(0); |
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| 435 | } |
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| 436 | return(2); |
---|
| 437 | } |
---|
| 438 | example |
---|
| 439 | { "EXAMPLE:"; echo = 2; |
---|
| 440 | ring r=0,(x,y,z),dp; |
---|
| 441 | poly f = y3z3+x5+yx3+z6; |
---|
| 442 | poly g = x5+yx3; |
---|
| 443 | prodcrit(f,g); |
---|
| 444 | vector v = x3z2*gen(1)+x3y*gen(1)+x2y*gen(2); |
---|
| 445 | vector w = y4*gen(1)+y3*gen(2)+xyz*gen(1); |
---|
| 446 | prodcrit(v,w); |
---|
| 447 | } |
---|
| 448 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 449 | proc chaincrit(f,g,h) |
---|
| 450 | "USAGE: chaincrit(f,g,h); f,g,h poly or module |
---|
| 451 | RETURN: 1 if chain criterion applies, 0 else |
---|
| 452 | NOTE: if chain criterion applies to f,g,h we can delete (g,h) from pairset |
---|
| 453 | EXAMPLE: example chaincrit; shows an example |
---|
| 454 | " |
---|
| 455 | { |
---|
| 456 | if(sameComponent(f,g) and sameComponent(f,h)) |
---|
| 457 | { |
---|
| 458 | if( monomialLcm(g,h)/leadmonomial(f) !=0 ) |
---|
| 459 | { |
---|
| 460 | return(1); |
---|
| 461 | } |
---|
| 462 | } |
---|
| 463 | return(0); |
---|
| 464 | } |
---|
| 465 | example |
---|
| 466 | { "EXAMPLE:"; echo = 2; |
---|
| 467 | ring r=0,(x,y,z),dp; |
---|
| 468 | poly f = x2y2z2+x5+yx3+z6; |
---|
| 469 | poly g = x5+yx3; |
---|
| 470 | poly h = y2z5+x5+yx3; |
---|
| 471 | chaincrit(f,g,h); |
---|
| 472 | vector u = [x2y3-z2,x2y]; |
---|
| 473 | vector v = [x2y2+z2,x2-y2]; |
---|
| 474 | vector w = [x2y4+z3,x2+y2]; |
---|
| 475 | chaincrit(u,v,w); |
---|
| 476 | } |
---|
| 477 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 478 | proc pairset(G) |
---|
| 479 | "USAGE: pairset(G); G ideal or module |
---|
[731e67e] | 480 | RETURN: list L, |
---|
| 481 | L[1] = the pairset of G as list (not containing pairs for |
---|
[373856] | 482 | which the product or the chain criterion applies) |
---|
| 483 | L[2] = intvec v, v[1]= # product criterion, v[2]= # chain criterion |
---|
| 484 | EXAMPLE: example pairset; shows an example |
---|
| 485 | " |
---|
| 486 | { |
---|
| 487 | int i,j,k,s,c,ccrit,pcrit,pr; |
---|
| 488 | int y = printlevel - voice + 2; |
---|
| 489 | G = simplify(G,10); |
---|
| 490 | def g = G; |
---|
| 491 | ideal pair; |
---|
| 492 | list P,I; //P=pairlist of G, I=list of corresponding indices of pairs |
---|
| 493 | for (i=1; i<=size(G); i++) |
---|
| 494 | { |
---|
| 495 | for(j = i+1; j<=size(G); j++) |
---|
| 496 | { |
---|
| 497 | pr = prodcrit(G[i],G[j]); //check first product criterion |
---|
| 498 | if( pr != 0 ) |
---|
[731e67e] | 499 | { |
---|
| 500 | pcrit=pcrit+(pr==1); |
---|
[373856] | 501 | } |
---|
| 502 | else |
---|
| 503 | { |
---|
| 504 | s = size(P); //now check chain criterion |
---|
| 505 | for(k=1; k<=s; k++) |
---|
| 506 | { |
---|
| 507 | if( i==I[k][2] ) |
---|
| 508 | { |
---|
| 509 | if ( chaincrit(P[k][1],P[k][2],G[j]) ) |
---|
| 510 | { //need not include (G[i],G[j]) in P |
---|
[731e67e] | 511 | c=1; ccrit=ccrit+1; |
---|
[373856] | 512 | break; |
---|
| 513 | } |
---|
| 514 | } |
---|
| 515 | if( j==I[k][1] and c==0 ) |
---|
| 516 | { |
---|
| 517 | "########### enter pairset2 #############"; |
---|
| 518 | if ( chaincrit(G[i],P[k][1],P[k][2]) ) |
---|
| 519 | { //can delete P[k]=(P[k][1],P[k][2]) |
---|
[731e67e] | 520 | ccrit=ccrit+1; |
---|
[373856] | 521 | P = delete(P,k); |
---|
| 522 | s = s-1; |
---|
| 523 | } |
---|
| 524 | } |
---|
| 525 | } |
---|
| 526 | if ( c==0 ) |
---|
| 527 | { |
---|
| 528 | g = G[i],G[j]; |
---|
| 529 | P[s+1]=g; |
---|
| 530 | I[s+1]=intvec(i,j); |
---|
| 531 | } |
---|
| 532 | c=0; |
---|
| 533 | } |
---|
| 534 | } |
---|
| 535 | } |
---|
[731e67e] | 536 | if (y>0) |
---|
[373856] | 537 | { ""; |
---|
| 538 | "// product criterion:",pcrit," chain criterion:",ccrit; |
---|
| 539 | } |
---|
| 540 | intvec v = pcrit,ccrit; |
---|
| 541 | P=P,v; |
---|
| 542 | return(P); |
---|
| 543 | } |
---|
| 544 | example |
---|
| 545 | { "EXAMPLE:"; echo = 2; |
---|
| 546 | ring r=0,(x,y,z),dp; |
---|
| 547 | ideal G = x2y+x2,y3+xyz,xyz2+z4; |
---|
| 548 | pairset(G);""; |
---|
| 549 | module T = [x2y3-z2,x2y],[x2y2+z2,x2-y2],[x2y4+z3,x2+y2]; |
---|
| 550 | pairset(T); |
---|
[731e67e] | 551 | } |
---|
[373856] | 552 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 553 | proc updatePairs(P,S,h) |
---|
| 554 | "USAGE: updatePairs(P,S,h); P list, S ideal or module, h poly or vector |
---|
| 555 | P a list of pairs of polys or vectors (obtained from pairset) |
---|
[731e67e] | 556 | RETURN: list Q, |
---|
[373856] | 557 | Q[1] = the pairset P enlarged by all pairs (f,h), f from S, |
---|
| 558 | without pairs for which the product or the chain criterion applies |
---|
| 559 | Q[2] = intvec v, v[1]= # product criterion, v[2]= # chain criterion |
---|
| 560 | EXAMPLE: example updatePairs; shows an example |
---|
| 561 | " |
---|
| 562 | { |
---|
| 563 | int i,j,k,s,r,c,ccrit,pcrit,pr; |
---|
| 564 | int y = printlevel - voice + 2; |
---|
| 565 | ideal pair; |
---|
| 566 | list Q = P; //Q will become enlarged pairset |
---|
| 567 | s = size(P); |
---|
| 568 | r = size(Q); //r will grow with Q |
---|
| 569 | list R; |
---|
| 570 | def g = S; //give g the correct type ideal/module |
---|
| 571 | for (i=1; i<=size(S); i++) |
---|
| 572 | { |
---|
| 573 | pr = prodcrit(h,S[i]); |
---|
| 574 | if( pr != 0 ) //check product criterion |
---|
[731e67e] | 575 | { |
---|
[373856] | 576 | pcrit=pcrit+(pr==1); //count product criterion in same component |
---|
| 577 | } |
---|
| 578 | else |
---|
| 579 | { //prodcrit did not apply, check for chaincrit |
---|
| 580 | r=size(Q); |
---|
| 581 | for(k=1; k<=r; k++) |
---|
| 582 | { |
---|
| 583 | if( Q[k][2]==S[i] ) //S[i]=Q[k][2] |
---|
| 584 | { |
---|
| 585 | if( chaincrit(Q[k][1],S[i],h) ) |
---|
| 586 | { //can forget (S[i],h) |
---|
| 587 | c=1; ccrit=ccrit+1; |
---|
| 588 | break; |
---|
| 589 | } |
---|
| 590 | } |
---|
| 591 | } |
---|
| 592 | if ( c==0 ) |
---|
| 593 | { |
---|
| 594 | g = S[i],h; //add pair (S[i],h) |
---|
| 595 | Q[r+1] = g; |
---|
| 596 | } |
---|
| 597 | c=0; |
---|
| 598 | } |
---|
| 599 | } |
---|
[731e67e] | 600 | if (y>0) |
---|
[373856] | 601 | { "";; |
---|
| 602 | "// product criterion:",pcrit," chain criterion:",ccrit; |
---|
| 603 | } |
---|
| 604 | intvec v = pcrit,ccrit; |
---|
| 605 | Q = Q,v; |
---|
| 606 | return(Q); |
---|
| 607 | } |
---|
| 608 | example |
---|
| 609 | { "EXAMPLE:"; echo = 2; |
---|
| 610 | ring R1=0,(x,y,z),(c,dp); |
---|
| 611 | ideal S = x2y+x2,y3+xyz; |
---|
| 612 | poly h = x2y+xyz; |
---|
| 613 | list P = pairset(S)[1]; |
---|
| 614 | P;""; |
---|
| 615 | updatePairs(P,S,h);""; |
---|
| 616 | module T = [x2y3-z2,x2y],[x2y4+z3,x2+y2]; |
---|
| 617 | P = pairset(T)[1]; |
---|
| 618 | P;""; |
---|
| 619 | updatePairs(P,T,[x2+x2y,y3+xyz]); |
---|
[731e67e] | 620 | } |
---|
[373856] | 621 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 622 | proc standard(id, list #) |
---|
| 623 | "USAGE: standard(i[,s]); id ideal or module, s int |
---|
| 624 | RETURN: a standard basis of id, using generalized Mora's algorithm |
---|
| 625 | which is Buchberger's algorithm for global monomial orderings. |
---|
| 626 | If s!=0 the symmetric s-polynomial (without division) is used |
---|
| 627 | NOTE: Show comments if printlevel > 0, pauses computation if printlevel > 1 |
---|
| 628 | EXAMPLE: example standard; shows an example |
---|
| 629 | " |
---|
| 630 | { |
---|
| 631 | if(size(#) == 0) |
---|
| 632 | { |
---|
| 633 | #[1]=0; |
---|
| 634 | } |
---|
| 635 | |
---|
| 636 | def S = id; //S will become the standard basis of id |
---|
| 637 | def h = S[1]; |
---|
| 638 | int i,z; |
---|
| 639 | int y = printlevel - voice + 2; |
---|
| 640 | if(y>0) |
---|
| 641 | { ""; |
---|
| 642 | "// the set S, to become a standard basis:"; S; |
---|
| 643 | if(y>1) |
---|
| 644 | { |
---|
| 645 | "// create pairset, i.e. pairs from S,"; |
---|
| 646 | "// after application of product and chain criterion"; |
---|
| 647 | } |
---|
[731e67e] | 648 | } |
---|
[373856] | 649 | list P = pairset(S); //create pairset of S=id |
---|
| 650 | intvec v = P[2]; |
---|
| 651 | P = P[1]; |
---|
| 652 | //-------------------------- Main loop in SB lgorithm ---------------------- |
---|
| 653 | while (size(P) !=0) |
---|
| 654 | { z=z+1; |
---|
| 655 | if(y>0) |
---|
| 656 | { ""; |
---|
| 657 | "// Enter NFMora for next pair, count",z; |
---|
| 658 | "// size of partial standard basis S: (",size(S),")"; |
---|
| 659 | "// number of pairs of S after updating: (",size(P),")"; |
---|
| 660 | if(y>1) |
---|
| 661 | { |
---|
| 662 | "// 1st pair of new pairset:"; P[1]; |
---|
| 663 | "// set T=S used for reduction:";S; |
---|
| 664 | "// apply NFMora to (spoly,S), spoly = spoly(1st pair)"; |
---|
| 665 | } |
---|
| 666 | } |
---|
| 667 | //-------------------- apply NFMora = Mora's normal form ------------- |
---|
| 668 | h = spoly(P[1][1],P[1][2],#[1]); |
---|
| 669 | if(y>1) |
---|
[731e67e] | 670 | { |
---|
[373856] | 671 | "// spoly:";h; |
---|
| 672 | } |
---|
[731e67e] | 673 | h = NFMora(h,S,#[1]); |
---|
[373856] | 674 | if(h==0) //normal form is 0 |
---|
| 675 | { |
---|
| 676 | if(y==1) |
---|
| 677 | { |
---|
| 678 | "// pair has reduced to 0"; |
---|
| 679 | } |
---|
[731e67e] | 680 | if(y>1) |
---|
[373856] | 681 | { h;""; |
---|
| 682 | pause("press <return> to continue"); |
---|
[731e67e] | 683 | } |
---|
[373856] | 684 | } |
---|
| 685 | P = delete(P,1); //spoly of pair reduced to 0, pair can be deleted |
---|
| 686 | //--- spoly of pair did not reduce to 0, update S and paiset of S ---- |
---|
[731e67e] | 687 | if( h != 0) |
---|
[373856] | 688 | { |
---|
| 689 | if(y==1) |
---|
| 690 | { |
---|
| 691 | "// ** new spoly in degree **:", deg(h); |
---|
| 692 | } |
---|
[731e67e] | 693 | if(y>1) |
---|
[373856] | 694 | { h;""; |
---|
| 695 | pause("press <return> to continue"); |
---|
| 696 | "// update pairset"; |
---|
[731e67e] | 697 | } |
---|
[373856] | 698 | P=updatePairs(P,S,h); //update P (=paisert of S) |
---|
| 699 | v=v+P[2]; //with useful pairs (g,h), g from S |
---|
| 700 | P=P[1]; |
---|
| 701 | S=S,h; //update S, will become the standard basis |
---|
| 702 | } |
---|
[731e67e] | 703 | } |
---|
[373856] | 704 | //------------------------------ finished --------------------------------- |
---|
| 705 | if( find(option(),"prot") or y>0 ) |
---|
| 706 | { ""; //show how often prodcrit and chaincrit applied |
---|
| 707 | "// product criterion:",v[1]," chain criterion:",v[2]; |
---|
| 708 | ""; |
---|
| 709 | "// Final standard basis:"; |
---|
[731e67e] | 710 | } |
---|
[373856] | 711 | return(S); |
---|
| 712 | } |
---|
| 713 | example |
---|
| 714 | { "EXAMPLE:"; echo = 2; |
---|
| 715 | ring r=0,(x,y,z),dp; |
---|
| 716 | ideal G = x2y+x2,y3+xyz,xyz2+z4; |
---|
| 717 | standard(G);""; |
---|
| 718 | ring s=0,(x,y,z),(c,ds); |
---|
| 719 | ideal G = 2x2+x2y,y3+xyz,3x3y+z4; |
---|
| 720 | standard(G);""; |
---|
| 721 | standard(G,1);""; //use symmetric s-poly without division |
---|
| 722 | module M = [2x2,x3y+z4],[3y3+xyz,y3],[5z4,z2]; |
---|
| 723 | standard(M); |
---|
[731e67e] | 724 | } |
---|
[373856] | 725 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 726 | proc localstd (id) |
---|
| 727 | "USAGE: localstd (id); id = ideal |
---|
| 728 | RETURN: A standard basis for a local degree ordering, using Lazard's method. |
---|
[731e67e] | 729 | NOTE: The procedure homogenizes id w.r.t. a new 1st variable local@t, |
---|
[2c957af] | 730 | computes a SB w.r.t. (dp(1),dp) and substitutes local@t by 1. |
---|
[373856] | 731 | Hence the result is a SB with respect to an ordering which sorts |
---|
| 732 | first w.r.t. the subdegree of the original variables and then refines |
---|
| 733 | it with dp. This is the local degree ordering ds. |
---|
| 734 | localstd may be used in order to avoid cancellation of units and thus |
---|
| 735 | to be able to use option(contentSB) also for local orderings. |
---|
| 736 | EXAMPLE: example localstd; shows an example |
---|
| 737 | " |
---|
| 738 | { |
---|
| 739 | int ii; |
---|
| 740 | def bas = basering; |
---|
[731e67e] | 741 | execute("ring @r_locstd |
---|
[373856] | 742 | =("+charstr(bas)+"),(local@t,"+varstr(bas)+"),(dp(1),dp);"); |
---|
| 743 | ideal id = imap(bas,id); |
---|
| 744 | ideal hid = homog(id,local@t); |
---|
| 745 | hid = std(hid); |
---|
| 746 | hid = subst(hid,local@t,1); |
---|
| 747 | setring bas; |
---|
| 748 | def hid = imap(@r_locstd,hid); |
---|
| 749 | attrib(hid,"isSB",1); |
---|
| 750 | kill @r_locstd; |
---|
[731e67e] | 751 | return(hid); |
---|
[373856] | 752 | } |
---|
| 753 | example |
---|
| 754 | { "EXAMPLE:"; echo = 2; |
---|
| 755 | ring R = 0,(x,y,z),ds; |
---|
| 756 | ideal i = xyz+z5,2x2+y3+z7,3z5+y5; |
---|
| 757 | localstd(i); |
---|
[731e67e] | 758 | } |
---|
[373856] | 759 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 760 | |
---|
| 761 | /* |
---|
| 762 | // some examples: |
---|
| 763 | LIB"teachstd.lib"; |
---|
| 764 | option(prot); printlevel=3; |
---|
| 765 | ring r0 = 0,(t,x,y),lp; |
---|
| 766 | ideal i = x-t2,y-t3; |
---|
| 767 | standard(i); |
---|
| 768 | |
---|
| 769 | printlevel=1; |
---|
| 770 | standard(i); |
---|
| 771 | |
---|
| 772 | option(prot); printlevel =1; |
---|
| 773 | ring r1 = (0,a,b),(x,y,z),(c,ds); |
---|
| 774 | module M = [ax2,bx3y+z4],[a3y3+xyz,by3],[5az4,(a+b)*z2]; |
---|
| 775 | module N1= std(M); |
---|
| 776 | module N2 = standard(M,1); |
---|
| 777 | NF(lead(N2),lead(N1)); |
---|
| 778 | NF(lead(N1),lead(N2));rom T |
---|
| 779 | ring r2 = 0,(x,y,z),dp; |
---|
| 780 | ideal I = x2y+x2,y3+xyz,xyz2+z4; |
---|
| 781 | option(prot); |
---|
| 782 | int tt = timer; |
---|
| 783 | ideal J = standard(I); |
---|
| 784 | timer -tt; //4sec, product criterion: 9 chain criterion: 6 |
---|
| 785 | ideal J1 = std(I); |
---|
| 786 | NF(lead(J),lead(J1)); |
---|
| 787 | NF(lead(J1),lead(J)); |
---|
| 788 | |
---|
| 789 | ring r3 = 0,(x,y,z),ds; |
---|
| 790 | poly f = x3*y4+z5+xyz; |
---|
| 791 | ideal I = f,jacob(f); |
---|
| 792 | option(prot); |
---|
| 793 | int tt = timer; |
---|
| 794 | ideal J = standard(I); |
---|
| 795 | timer -tt; //3sec, product criterion: 1 chain criterion: 3 |
---|
| 796 | ideal J1 = std(I); |
---|
| 797 | NF(lead(J),lead(J1)); |
---|
| 798 | NF(lead(J1),lead(J)); |
---|
[731e67e] | 799 | |
---|
[373856] | 800 | //Becker: |
---|
| 801 | ring r4 = 32003,(x,y,z),lp; |
---|
| 802 | ideal I = x3-1, y3-1, |
---|
[731e67e] | 803 | -27x3-243x2y+27x2z-729xy2+162xyz-9xz2-729y3+243y2z-27yz2+z3-27; |
---|
[373856] | 804 | option(prot); |
---|
| 805 | int tt = timer; |
---|
| 806 | ideal J = standard(I); |
---|
| 807 | timer -tt; //201sec, product criterion: 42 chain criterion: 33 |
---|
| 808 | ideal J1 = std(I); |
---|
| 809 | NF(lead(J),lead(J1)); |
---|
| 810 | NF(lead(J1),lead(J)); |
---|
[731e67e] | 811 | |
---|
[373856] | 812 | //Alex |
---|
| 813 | ring r5 = 32003,(x,y,z,t),dp; |
---|
[731e67e] | 814 | ideal I = |
---|
[373856] | 815 | 2t3xy2z+x2ty+2x2y, |
---|
| 816 | 2tz+y3x2t+z2t3y2x; |
---|
| 817 | option(prot); printlevel =1; |
---|
| 818 | ideal J1 = std(I); |
---|
| 819 | int tt = timer; |
---|
| 820 | ideal J = standard(I); |
---|
| 821 | timer -tt; //15sec product criterion: 0 chain criterion: 12 |
---|
| 822 | NF(lead(J),lead(J1)); |
---|
| 823 | NF(lead(J1),lead(J)); |
---|
| 824 | |
---|
| 825 | ring r6 = 32003,(x,y,z,t),dp; //is already SB for ds, for dp too long |
---|
| 826 | ideal I= |
---|
| 827 | 9x8+y7t3z4+5x4y2t2+2xy2z3t2, |
---|
| 828 | 9y8+7xy6t+2x5y4t2+2x2yz3t2, |
---|
| 829 | 9z8+3x2y3z2t4; |
---|
| 830 | option(prot); |
---|
| 831 | int tt = timer; |
---|
| 832 | ideal J = standard(I); |
---|
| 833 | timer -tt; //0sec, product criterion: 3 chain criterion: 0 |
---|
| 834 | ideal J1 = std(I); |
---|
| 835 | NF(lead(J),lead(J1)); |
---|
| 836 | NF(lead(J1),lead(J)); |
---|
| 837 | |
---|
| 838 | |
---|
| 839 | */ |
---|
| 840 | |
---|