[731e67e] | 1 | // $Id: |
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[373856] | 2 | //GMG, last modified 28.9.01 |
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| 3 | /////////////////////////////////////////////////////////////////////////////// |
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[823ff8] | 4 | version="$Id: teachstd.lib,v 1.10 2009-02-12 13:57:00 motsak 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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[823ff8] | 26 | prodcrit(f,g[,o]); test for product criterion |
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[373856] | 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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[836e389] | 193 | |
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[373856] | 194 | if( o == 0) //can only happen, if vectors f and g involve |
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| 195 | { //different module components |
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| 196 | vector sp; |
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| 197 | return(sp); |
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| 198 | } |
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[731e67e] | 199 | |
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[836e389] | 200 | poly m=(o/leadmonomial(f)); //compute the leading monomial as poly |
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| 201 | poly n=(o/leadmonomial(g)); |
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[7f3ad4] | 202 | |
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[836e389] | 203 | f = m * f; |
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[7f3ad4] | 204 | g = n * g; // now they have the same LM! |
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[373856] | 205 | |
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| 206 | if (#[1]==0) //the asymmetric s-poly |
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| 207 | { |
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[836e389] | 208 | def sp = f - (leadcoef(f)/leadcoef(g))*g; |
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[373856] | 209 | } |
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| 210 | else //the symmetric s-poly, avoiding division |
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| 211 | { |
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[836e389] | 212 | def sp = leadcoef(g)*f - leadcoef(f)*g; |
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[731e67e] | 213 | } |
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[373856] | 214 | return(sp); |
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[731e67e] | 215 | } |
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[373856] | 216 | example |
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| 217 | { "EXAMPLE:"; echo = 2; |
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| 218 | ring r=0,(x,y,z),ls; |
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| 219 | spoly(2x2+x2y,3y3+xyz); |
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| 220 | ring s=0,(x,y,z),(c,dp); |
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| 221 | spoly(2x2+x2y,3y3+xyz); |
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| 222 | spoly(2x2+x2y,3y3+xyz,1); //symmetric s-poly without division |
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| 223 | spoly([5x2+x2y,z5],[x2,y3,y4]); //s-poly for vectors |
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[731e67e] | 224 | } |
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[373856] | 225 | /////////////////////////////////////////////////////////////////////////////// |
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| 226 | proc minEcart(T,h) |
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| 227 | "USAGE: minEcart(T,h); T ideal or module, h poly or vector |
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| 228 | RETURN: element g from T such that leadmonom(g) divides leadmonom(h) |
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| 229 | ecart(g) is minimal with this property (if T != 0); |
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| 230 | return 0 if T is 0 or h = 0 |
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| 231 | EXAMPLE: example minEcart; shows an example |
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| 232 | " |
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| 233 | { |
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| 234 | int i,k,e; |
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| 235 | if (size(T)==0 or h==0 ) //trivial cases |
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[731e67e] | 236 | { |
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[373856] | 237 | h = 0; |
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| 238 | return(h); |
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| 239 | } |
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| 240 | //---- check whether some element in T has the same module component as h ---- |
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| 241 | int v; |
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| 242 | intvec w; |
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| 243 | T = simplify(T,2); |
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[731e67e] | 244 | |
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| 245 | if (typeof(h) == "vector") |
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[373856] | 246 | { |
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| 247 | e=1; |
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| 248 | v = leadexp(h)[nvars(basering)+1]; |
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| 249 | for( i=1; i<=size(T); i++) |
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| 250 | { |
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| 251 | w[i]=leadexp(T[i])[nvars(basering)+1]; |
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| 252 | if(v == w[i]) |
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| 253 | { |
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| 254 | e=0; //some element in T involves the same component as h |
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| 255 | } |
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| 256 | } |
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| 257 | } |
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| 258 | if ( e == 1 ) //no element in T involves the same component as h |
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[731e67e] | 259 | { |
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[373856] | 260 | h = 0; |
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| 261 | return(h); |
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| 262 | } |
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| 263 | |
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[731e67e] | 264 | if (typeof(h) == "poly") //for polys v=w[i] for all i |
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[373856] | 265 | { |
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| 266 | v = 1; |
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| 267 | w[size(T)]=0; |
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| 268 | w=w+1; |
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| 269 | } |
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[731e67e] | 270 | |
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[373856] | 271 | //------ check whether for some g in T leadmonom(g) divides leadmonom(h) ----- |
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| 272 | def g = T[1]; |
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| 273 | poly f = leadmonomial(h); |
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| 274 | |
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| 275 | for( i=1; i<=size(T); i++) |
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| 276 | { |
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| 277 | if( f/leadmonomial(T[i]) != 0 and v==w[i] ) |
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| 278 | { |
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| 279 | g=T[i]; |
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| 280 | k=i; |
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| 281 | break; |
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| 282 | } |
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| 283 | } |
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[731e67e] | 284 | if (k == 0) //no leadmonom(g) divides leadmonom(h) |
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| 285 | { |
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[373856] | 286 | g = 0; |
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| 287 | return(g); |
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| 288 | } |
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| 289 | //--look for T[i] with minimal ecart s.t.leadmonom(T[i]) divides leadmonom(h)-- |
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| 290 | |
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| 291 | for( i=k+1; i<=size(T); i++) |
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| 292 | { |
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| 293 | if( f/leadmonomial(T[i]) != 0 and v==w[i] ) |
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| 294 | { |
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| 295 | if (ecart(T[i]) < ecart(g)) |
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| 296 | { |
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| 297 | g=T[i]; |
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| 298 | } |
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| 299 | } |
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| 300 | } |
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| 301 | return(g); |
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[731e67e] | 302 | } |
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[373856] | 303 | example |
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| 304 | { "EXAMPLE:"; echo = 2; |
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| 305 | ring R=0,(x,y,z),dp; |
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| 306 | ideal T = x2y+x2,y3+xyz,xyz2+z4; |
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| 307 | poly h = x2y2z2+x5+yx3+z6; |
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| 308 | minEcart(T,h);""; |
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| 309 | ring S=0,(x,y,z),(c,ds); |
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| 310 | module T = [x2+x2y,y2],[y3+xyz,x3-z3],[x3y+z4,0,x2]; |
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| 311 | vector h = [x3y+x5+x2y2z2+z6,x3]; |
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| 312 | minEcart(T,h); |
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[731e67e] | 313 | } |
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[373856] | 314 | /////////////////////////////////////////////////////////////////////////////// |
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| 315 | proc NFMora(f,G,list #) |
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| 316 | "USAGE: NFMora(f,G[,s]); f poly or vector,G ideal or module, s int |
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| 317 | RETURN: the Mora normal form of f w.r.t. G, same type as f |
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| 318 | if s!=0 the symmetric s-polynomial (without division) is used |
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| 319 | NOTE: Show comments if printlevel > 0, pauses computation if printlevel > 1 |
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| 320 | EXAMPLE: example NFMora; shows an example |
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| 321 | " |
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| 322 | { |
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| 323 | if(size(#) == 0) |
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| 324 | { |
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| 325 | #[1]=0; |
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| 326 | } |
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| 327 | |
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| 328 | int y = printlevel - voice + 2; |
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| 329 | if( f==0 or size(G) ==0 ) |
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| 330 | { |
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[731e67e] | 331 | if (y>0) |
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[373856] | 332 | { |
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| 333 | "// 1st or 2nd argument 0"; |
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| 334 | } |
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| 335 | return(f); |
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| 336 | } |
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| 337 | |
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| 338 | int i,e; |
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| 339 | def h = f; |
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| 340 | def T = G; |
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| 341 | // -------------------- start with f to be reduced by G -------------------- |
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[731e67e] | 342 | if (y>0) |
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[373856] | 343 | {""; |
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| 344 | "// Input for NFMora is (f,T):"; |
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| 345 | "// f:"; f; |
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| 346 | "// T:"; T; |
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| 347 | "// Reduce f with T, eventually enlarging T for local ordering"; |
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| 348 | } |
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| 349 | |
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| 350 | // ----------------------- enter the reduction loop ------------------------ |
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| 351 | def g = minEcart(T,h); |
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| 352 | while (h!=0 and g!=0) |
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| 353 | { |
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[731e67e] | 354 | if (y>0) |
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[373856] | 355 | { ""; |
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| 356 | "// Reduction-step in NFMora:",i; |
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| 357 | "// h = (f after",i,"reductions) reduction with g from T:"; |
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[731e67e] | 358 | "// g = element of minimal ecart in T s.t. LM(g)|LM(h):"; |
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[373856] | 359 | "// h:";h; |
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[731e67e] | 360 | "// g:";g; |
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[373856] | 361 | } |
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[731e67e] | 362 | if (y>1) |
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| 363 | { |
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[373856] | 364 | pause("press <return> to continue"); |
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| 365 | "// T, set used for reduction:"; T; |
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| 366 | pause("press <return> to continue"); |
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| 367 | } |
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| 368 | e=0; |
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| 369 | if( ecart(g) > ecart(h) ) |
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| 370 | { |
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| 371 | T=T,h; e=1; |
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| 372 | } |
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[731e67e] | 373 | if (y>0 ) |
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| 374 | { |
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[373856] | 375 | "// T-set enlarged for next reduction? (yes/no = 1/0): ", e; |
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| 376 | if( e==1 ) |
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| 377 | { |
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| 378 | "// T-set for next reduction got enlarged by h:"; |
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| 379 | "// h:";h; |
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[731e67e] | 380 | if (y>1) |
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| 381 | { |
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[373856] | 382 | pause("press <return> to continue"); |
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| 383 | } |
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| 384 | } |
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| 385 | } |
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| 386 | h = spoly(h,g,#[1]); |
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| 387 | g = minEcart(T,h); |
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| 388 | i=i+1; |
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| 389 | } |
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| 390 | if(y>0) |
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| 391 | { ""; |
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| 392 | "// normal form is:"; |
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| 393 | } |
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| 394 | return(h); |
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[731e67e] | 395 | } |
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[373856] | 396 | example |
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| 397 | { "EXAMPLE:"; echo = 2; |
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| 398 | ring r=0,(x,y,z),dp; |
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| 399 | poly f = x2y2z2+x5+yx3+z6-3y3; |
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| 400 | ideal G = x2y+x2,y3+xyz,xyz2+z6; |
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| 401 | NFMora(f,G);""; |
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| 402 | ring s=0,(x,y,z),ds; |
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| 403 | poly f = x3y+x5+x2y2z2+z6; |
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| 404 | ideal G = x2+x2y,y3+xyz,x3y2+z4; |
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| 405 | NFMora(f,G);""; |
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| 406 | vector v = [f,x2+x2y]; |
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| 407 | module M = [x2+x2y,f],[y3+xyz,y3],[x3y2+z4,z2]; |
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| 408 | NFMora(v,M); |
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[731e67e] | 409 | } |
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[373856] | 410 | /////////////////////////////////////////////////////////////////////////////// |
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[823ff8] | 411 | proc prodcrit(f,g,list #) |
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| 412 | "USAGE: prodcrit(f,g[,o]); f,g poly or vector, and optional int argument o |
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[373856] | 413 | RETURN: 1 if product criterion applies in the same module component, |
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| 414 | 2 if lead(f) and lead(g) involve different components, 0 else |
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| 415 | NOTE: if product criterion applies we can delete (f,g) from pairset |
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[823ff8] | 416 | This procedure returns 0 if o is given and is a positive integer, or |
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| 417 | you may set the attribute \"default_arg\" for prodcrit to 1. |
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[373856] | 418 | EXAMPLE: example prodcrit; shows an example |
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| 419 | " |
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| 420 | { |
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[823ff8] | 421 | // ------------------ check for optional disabling argument ------------- |
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| 422 | if( size(#) > 0 ) |
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| 423 | {// "size(#): ", size(#); "typeof(#[1]): ", typeof(#[1]); |
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| 424 | |
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| 425 | if( typeof(#[1]) == "int" ) |
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| 426 | {// "#[1] = int ", #[1]; |
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| 427 | if( #[1] > 0 ) |
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| 428 | { |
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| 429 | return(0); |
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| 430 | } |
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| 431 | } |
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[25778b] | 432 | } |
---|
[7f3ad4] | 433 | |
---|
[823ff8] | 434 | // ------------------- product criterion for polynomials --------------------- |
---|
| 435 | if(typeof(f)=="poly") |
---|
[373856] | 436 | { |
---|
| 437 | if( monomialLcm(f,g)==leadmonom(f)*leadmonom(g)) |
---|
| 438 | { |
---|
| 439 | return(1); |
---|
| 440 | } |
---|
| 441 | return(0); |
---|
| 442 | } |
---|
[731e67e] | 443 | // ------------------- product criterion for modules --------------------- |
---|
[373856] | 444 | if(sameComponent(f,g)==1) |
---|
| 445 | { |
---|
| 446 | if( monomialLcm(f,g)==leadmonomial(f)*leadmonomial(g) ) |
---|
| 447 | { |
---|
| 448 | int c = leadexp(f)[nvars(basering)+1]; //component involving lead(f) |
---|
| 449 | if((f-f[c]*gen(c))-(g-g[c]*gen(c))==0) //other components are 0 |
---|
| 450 | { |
---|
| 451 | return(1); |
---|
| 452 | } |
---|
| 453 | } |
---|
| 454 | return(0); |
---|
| 455 | } |
---|
| 456 | return(2); |
---|
| 457 | } |
---|
| 458 | example |
---|
| 459 | { "EXAMPLE:"; echo = 2; |
---|
| 460 | ring r=0,(x,y,z),dp; |
---|
| 461 | poly f = y3z3+x5+yx3+z6; |
---|
| 462 | poly g = x5+yx3; |
---|
| 463 | prodcrit(f,g); |
---|
| 464 | vector v = x3z2*gen(1)+x3y*gen(1)+x2y*gen(2); |
---|
| 465 | vector w = y4*gen(1)+y3*gen(2)+xyz*gen(1); |
---|
| 466 | prodcrit(v,w); |
---|
| 467 | } |
---|
| 468 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 469 | proc chaincrit(f,g,h) |
---|
| 470 | "USAGE: chaincrit(f,g,h); f,g,h poly or module |
---|
| 471 | RETURN: 1 if chain criterion applies, 0 else |
---|
| 472 | NOTE: if chain criterion applies to f,g,h we can delete (g,h) from pairset |
---|
| 473 | EXAMPLE: example chaincrit; shows an example |
---|
| 474 | " |
---|
| 475 | { |
---|
| 476 | if(sameComponent(f,g) and sameComponent(f,h)) |
---|
| 477 | { |
---|
| 478 | if( monomialLcm(g,h)/leadmonomial(f) !=0 ) |
---|
| 479 | { |
---|
| 480 | return(1); |
---|
| 481 | } |
---|
| 482 | } |
---|
| 483 | return(0); |
---|
| 484 | } |
---|
| 485 | example |
---|
| 486 | { "EXAMPLE:"; echo = 2; |
---|
| 487 | ring r=0,(x,y,z),dp; |
---|
| 488 | poly f = x2y2z2+x5+yx3+z6; |
---|
| 489 | poly g = x5+yx3; |
---|
| 490 | poly h = y2z5+x5+yx3; |
---|
| 491 | chaincrit(f,g,h); |
---|
| 492 | vector u = [x2y3-z2,x2y]; |
---|
| 493 | vector v = [x2y2+z2,x2-y2]; |
---|
| 494 | vector w = [x2y4+z3,x2+y2]; |
---|
| 495 | chaincrit(u,v,w); |
---|
| 496 | } |
---|
| 497 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 498 | proc pairset(G) |
---|
| 499 | "USAGE: pairset(G); G ideal or module |
---|
[731e67e] | 500 | RETURN: list L, |
---|
| 501 | L[1] = the pairset of G as list (not containing pairs for |
---|
[373856] | 502 | which the product or the chain criterion applies) |
---|
| 503 | L[2] = intvec v, v[1]= # product criterion, v[2]= # chain criterion |
---|
| 504 | EXAMPLE: example pairset; shows an example |
---|
| 505 | " |
---|
| 506 | { |
---|
| 507 | int i,j,k,s,c,ccrit,pcrit,pr; |
---|
| 508 | int y = printlevel - voice + 2; |
---|
| 509 | G = simplify(G,10); |
---|
| 510 | def g = G; |
---|
| 511 | ideal pair; |
---|
| 512 | list P,I; //P=pairlist of G, I=list of corresponding indices of pairs |
---|
| 513 | for (i=1; i<=size(G); i++) |
---|
| 514 | { |
---|
| 515 | for(j = i+1; j<=size(G); j++) |
---|
| 516 | { |
---|
| 517 | pr = prodcrit(G[i],G[j]); //check first product criterion |
---|
| 518 | if( pr != 0 ) |
---|
[731e67e] | 519 | { |
---|
| 520 | pcrit=pcrit+(pr==1); |
---|
[373856] | 521 | } |
---|
| 522 | else |
---|
| 523 | { |
---|
| 524 | s = size(P); //now check chain criterion |
---|
| 525 | for(k=1; k<=s; k++) |
---|
| 526 | { |
---|
| 527 | if( i==I[k][2] ) |
---|
| 528 | { |
---|
| 529 | if ( chaincrit(P[k][1],P[k][2],G[j]) ) |
---|
| 530 | { //need not include (G[i],G[j]) in P |
---|
[731e67e] | 531 | c=1; ccrit=ccrit+1; |
---|
[373856] | 532 | break; |
---|
| 533 | } |
---|
| 534 | } |
---|
| 535 | if( j==I[k][1] and c==0 ) |
---|
| 536 | { |
---|
| 537 | "########### enter pairset2 #############"; |
---|
| 538 | if ( chaincrit(G[i],P[k][1],P[k][2]) ) |
---|
| 539 | { //can delete P[k]=(P[k][1],P[k][2]) |
---|
[731e67e] | 540 | ccrit=ccrit+1; |
---|
[373856] | 541 | P = delete(P,k); |
---|
| 542 | s = s-1; |
---|
| 543 | } |
---|
| 544 | } |
---|
| 545 | } |
---|
| 546 | if ( c==0 ) |
---|
| 547 | { |
---|
| 548 | g = G[i],G[j]; |
---|
| 549 | P[s+1]=g; |
---|
| 550 | I[s+1]=intvec(i,j); |
---|
| 551 | } |
---|
| 552 | c=0; |
---|
| 553 | } |
---|
| 554 | } |
---|
| 555 | } |
---|
[731e67e] | 556 | if (y>0) |
---|
[373856] | 557 | { ""; |
---|
| 558 | "// product criterion:",pcrit," chain criterion:",ccrit; |
---|
| 559 | } |
---|
| 560 | intvec v = pcrit,ccrit; |
---|
| 561 | P=P,v; |
---|
| 562 | return(P); |
---|
| 563 | } |
---|
| 564 | example |
---|
| 565 | { "EXAMPLE:"; echo = 2; |
---|
| 566 | ring r=0,(x,y,z),dp; |
---|
| 567 | ideal G = x2y+x2,y3+xyz,xyz2+z4; |
---|
| 568 | pairset(G);""; |
---|
| 569 | module T = [x2y3-z2,x2y],[x2y2+z2,x2-y2],[x2y4+z3,x2+y2]; |
---|
| 570 | pairset(T); |
---|
[731e67e] | 571 | } |
---|
[373856] | 572 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 573 | proc updatePairs(P,S,h) |
---|
| 574 | "USAGE: updatePairs(P,S,h); P list, S ideal or module, h poly or vector |
---|
| 575 | P a list of pairs of polys or vectors (obtained from pairset) |
---|
[731e67e] | 576 | RETURN: list Q, |
---|
[373856] | 577 | Q[1] = the pairset P enlarged by all pairs (f,h), f from S, |
---|
| 578 | without pairs for which the product or the chain criterion applies |
---|
| 579 | Q[2] = intvec v, v[1]= # product criterion, v[2]= # chain criterion |
---|
| 580 | EXAMPLE: example updatePairs; shows an example |
---|
| 581 | " |
---|
| 582 | { |
---|
| 583 | int i,j,k,s,r,c,ccrit,pcrit,pr; |
---|
| 584 | int y = printlevel - voice + 2; |
---|
| 585 | ideal pair; |
---|
| 586 | list Q = P; //Q will become enlarged pairset |
---|
| 587 | s = size(P); |
---|
| 588 | r = size(Q); //r will grow with Q |
---|
| 589 | list R; |
---|
| 590 | def g = S; //give g the correct type ideal/module |
---|
| 591 | for (i=1; i<=size(S); i++) |
---|
| 592 | { |
---|
| 593 | pr = prodcrit(h,S[i]); |
---|
| 594 | if( pr != 0 ) //check product criterion |
---|
[731e67e] | 595 | { |
---|
[373856] | 596 | pcrit=pcrit+(pr==1); //count product criterion in same component |
---|
| 597 | } |
---|
| 598 | else |
---|
| 599 | { //prodcrit did not apply, check for chaincrit |
---|
| 600 | r=size(Q); |
---|
| 601 | for(k=1; k<=r; k++) |
---|
| 602 | { |
---|
| 603 | if( Q[k][2]==S[i] ) //S[i]=Q[k][2] |
---|
| 604 | { |
---|
| 605 | if( chaincrit(Q[k][1],S[i],h) ) |
---|
| 606 | { //can forget (S[i],h) |
---|
| 607 | c=1; ccrit=ccrit+1; |
---|
| 608 | break; |
---|
| 609 | } |
---|
| 610 | } |
---|
| 611 | } |
---|
| 612 | if ( c==0 ) |
---|
| 613 | { |
---|
| 614 | g = S[i],h; //add pair (S[i],h) |
---|
| 615 | Q[r+1] = g; |
---|
| 616 | } |
---|
| 617 | c=0; |
---|
| 618 | } |
---|
| 619 | } |
---|
[731e67e] | 620 | if (y>0) |
---|
[373856] | 621 | { "";; |
---|
| 622 | "// product criterion:",pcrit," chain criterion:",ccrit; |
---|
| 623 | } |
---|
| 624 | intvec v = pcrit,ccrit; |
---|
| 625 | Q = Q,v; |
---|
| 626 | return(Q); |
---|
| 627 | } |
---|
| 628 | example |
---|
| 629 | { "EXAMPLE:"; echo = 2; |
---|
| 630 | ring R1=0,(x,y,z),(c,dp); |
---|
| 631 | ideal S = x2y+x2,y3+xyz; |
---|
| 632 | poly h = x2y+xyz; |
---|
| 633 | list P = pairset(S)[1]; |
---|
| 634 | P;""; |
---|
| 635 | updatePairs(P,S,h);""; |
---|
| 636 | module T = [x2y3-z2,x2y],[x2y4+z3,x2+y2]; |
---|
| 637 | P = pairset(T)[1]; |
---|
| 638 | P;""; |
---|
| 639 | updatePairs(P,T,[x2+x2y,y3+xyz]); |
---|
[731e67e] | 640 | } |
---|
[373856] | 641 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 642 | proc standard(id, list #) |
---|
| 643 | "USAGE: standard(i[,s]); id ideal or module, s int |
---|
| 644 | RETURN: a standard basis of id, using generalized Mora's algorithm |
---|
| 645 | which is Buchberger's algorithm for global monomial orderings. |
---|
| 646 | If s!=0 the symmetric s-polynomial (without division) is used |
---|
| 647 | NOTE: Show comments if printlevel > 0, pauses computation if printlevel > 1 |
---|
| 648 | EXAMPLE: example standard; shows an example |
---|
| 649 | " |
---|
| 650 | { |
---|
| 651 | if(size(#) == 0) |
---|
| 652 | { |
---|
| 653 | #[1]=0; |
---|
| 654 | } |
---|
| 655 | |
---|
| 656 | def S = id; //S will become the standard basis of id |
---|
| 657 | def h = S[1]; |
---|
| 658 | int i,z; |
---|
| 659 | int y = printlevel - voice + 2; |
---|
| 660 | if(y>0) |
---|
| 661 | { ""; |
---|
| 662 | "// the set S, to become a standard basis:"; S; |
---|
| 663 | if(y>1) |
---|
| 664 | { |
---|
| 665 | "// create pairset, i.e. pairs from S,"; |
---|
| 666 | "// after application of product and chain criterion"; |
---|
| 667 | } |
---|
[731e67e] | 668 | } |
---|
[373856] | 669 | list P = pairset(S); //create pairset of S=id |
---|
| 670 | intvec v = P[2]; |
---|
| 671 | P = P[1]; |
---|
| 672 | //-------------------------- Main loop in SB lgorithm ---------------------- |
---|
| 673 | while (size(P) !=0) |
---|
| 674 | { z=z+1; |
---|
| 675 | if(y>0) |
---|
| 676 | { ""; |
---|
| 677 | "// Enter NFMora for next pair, count",z; |
---|
| 678 | "// size of partial standard basis S: (",size(S),")"; |
---|
| 679 | "// number of pairs of S after updating: (",size(P),")"; |
---|
| 680 | if(y>1) |
---|
| 681 | { |
---|
| 682 | "// 1st pair of new pairset:"; P[1]; |
---|
| 683 | "// set T=S used for reduction:";S; |
---|
| 684 | "// apply NFMora to (spoly,S), spoly = spoly(1st pair)"; |
---|
| 685 | } |
---|
| 686 | } |
---|
| 687 | //-------------------- apply NFMora = Mora's normal form ------------- |
---|
| 688 | h = spoly(P[1][1],P[1][2],#[1]); |
---|
| 689 | if(y>1) |
---|
[731e67e] | 690 | { |
---|
[373856] | 691 | "// spoly:";h; |
---|
| 692 | } |
---|
[731e67e] | 693 | h = NFMora(h,S,#[1]); |
---|
[373856] | 694 | if(h==0) //normal form is 0 |
---|
| 695 | { |
---|
| 696 | if(y==1) |
---|
| 697 | { |
---|
| 698 | "// pair has reduced to 0"; |
---|
| 699 | } |
---|
[731e67e] | 700 | if(y>1) |
---|
[373856] | 701 | { h;""; |
---|
| 702 | pause("press <return> to continue"); |
---|
[731e67e] | 703 | } |
---|
[373856] | 704 | } |
---|
| 705 | P = delete(P,1); //spoly of pair reduced to 0, pair can be deleted |
---|
| 706 | //--- spoly of pair did not reduce to 0, update S and paiset of S ---- |
---|
[731e67e] | 707 | if( h != 0) |
---|
[373856] | 708 | { |
---|
| 709 | if(y==1) |
---|
| 710 | { |
---|
| 711 | "// ** new spoly in degree **:", deg(h); |
---|
| 712 | } |
---|
[731e67e] | 713 | if(y>1) |
---|
[373856] | 714 | { h;""; |
---|
| 715 | pause("press <return> to continue"); |
---|
| 716 | "// update pairset"; |
---|
[731e67e] | 717 | } |
---|
[373856] | 718 | P=updatePairs(P,S,h); //update P (=paisert of S) |
---|
| 719 | v=v+P[2]; //with useful pairs (g,h), g from S |
---|
| 720 | P=P[1]; |
---|
| 721 | S=S,h; //update S, will become the standard basis |
---|
| 722 | } |
---|
[731e67e] | 723 | } |
---|
[373856] | 724 | //------------------------------ finished --------------------------------- |
---|
| 725 | if( find(option(),"prot") or y>0 ) |
---|
| 726 | { ""; //show how often prodcrit and chaincrit applied |
---|
| 727 | "// product criterion:",v[1]," chain criterion:",v[2]; |
---|
| 728 | ""; |
---|
| 729 | "// Final standard basis:"; |
---|
[731e67e] | 730 | } |
---|
[373856] | 731 | return(S); |
---|
| 732 | } |
---|
| 733 | example |
---|
| 734 | { "EXAMPLE:"; echo = 2; |
---|
| 735 | ring r=0,(x,y,z),dp; |
---|
| 736 | ideal G = x2y+x2,y3+xyz,xyz2+z4; |
---|
| 737 | standard(G);""; |
---|
| 738 | ring s=0,(x,y,z),(c,ds); |
---|
| 739 | ideal G = 2x2+x2y,y3+xyz,3x3y+z4; |
---|
| 740 | standard(G);""; |
---|
| 741 | standard(G,1);""; //use symmetric s-poly without division |
---|
| 742 | module M = [2x2,x3y+z4],[3y3+xyz,y3],[5z4,z2]; |
---|
| 743 | standard(M); |
---|
[731e67e] | 744 | } |
---|
[373856] | 745 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 746 | proc localstd (id) |
---|
| 747 | "USAGE: localstd (id); id = ideal |
---|
| 748 | RETURN: A standard basis for a local degree ordering, using Lazard's method. |
---|
[731e67e] | 749 | NOTE: The procedure homogenizes id w.r.t. a new 1st variable local@t, |
---|
[2c957af] | 750 | computes a SB w.r.t. (dp(1),dp) and substitutes local@t by 1. |
---|
[373856] | 751 | Hence the result is a SB with respect to an ordering which sorts |
---|
| 752 | first w.r.t. the subdegree of the original variables and then refines |
---|
| 753 | it with dp. This is the local degree ordering ds. |
---|
| 754 | localstd may be used in order to avoid cancellation of units and thus |
---|
| 755 | to be able to use option(contentSB) also for local orderings. |
---|
| 756 | EXAMPLE: example localstd; shows an example |
---|
| 757 | " |
---|
| 758 | { |
---|
| 759 | int ii; |
---|
| 760 | def bas = basering; |
---|
[731e67e] | 761 | execute("ring @r_locstd |
---|
[373856] | 762 | =("+charstr(bas)+"),(local@t,"+varstr(bas)+"),(dp(1),dp);"); |
---|
| 763 | ideal id = imap(bas,id); |
---|
| 764 | ideal hid = homog(id,local@t); |
---|
| 765 | hid = std(hid); |
---|
| 766 | hid = subst(hid,local@t,1); |
---|
| 767 | setring bas; |
---|
| 768 | def hid = imap(@r_locstd,hid); |
---|
| 769 | attrib(hid,"isSB",1); |
---|
| 770 | kill @r_locstd; |
---|
[731e67e] | 771 | return(hid); |
---|
[373856] | 772 | } |
---|
| 773 | example |
---|
| 774 | { "EXAMPLE:"; echo = 2; |
---|
| 775 | ring R = 0,(x,y,z),ds; |
---|
| 776 | ideal i = xyz+z5,2x2+y3+z7,3z5+y5; |
---|
| 777 | localstd(i); |
---|
[731e67e] | 778 | } |
---|
[373856] | 779 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 780 | |
---|
| 781 | /* |
---|
| 782 | // some examples: |
---|
| 783 | LIB"teachstd.lib"; |
---|
| 784 | option(prot); printlevel=3; |
---|
| 785 | ring r0 = 0,(t,x,y),lp; |
---|
| 786 | ideal i = x-t2,y-t3; |
---|
| 787 | standard(i); |
---|
| 788 | |
---|
| 789 | printlevel=1; |
---|
| 790 | standard(i); |
---|
| 791 | |
---|
| 792 | option(prot); printlevel =1; |
---|
| 793 | ring r1 = (0,a,b),(x,y,z),(c,ds); |
---|
| 794 | module M = [ax2,bx3y+z4],[a3y3+xyz,by3],[5az4,(a+b)*z2]; |
---|
| 795 | module N1= std(M); |
---|
| 796 | module N2 = standard(M,1); |
---|
| 797 | NF(lead(N2),lead(N1)); |
---|
| 798 | NF(lead(N1),lead(N2));rom T |
---|
| 799 | ring r2 = 0,(x,y,z),dp; |
---|
| 800 | ideal I = x2y+x2,y3+xyz,xyz2+z4; |
---|
| 801 | option(prot); |
---|
| 802 | int tt = timer; |
---|
| 803 | ideal J = standard(I); |
---|
| 804 | timer -tt; //4sec, product criterion: 9 chain criterion: 6 |
---|
| 805 | ideal J1 = std(I); |
---|
| 806 | NF(lead(J),lead(J1)); |
---|
| 807 | NF(lead(J1),lead(J)); |
---|
| 808 | |
---|
| 809 | ring r3 = 0,(x,y,z),ds; |
---|
| 810 | poly f = x3*y4+z5+xyz; |
---|
| 811 | ideal I = f,jacob(f); |
---|
| 812 | option(prot); |
---|
| 813 | int tt = timer; |
---|
| 814 | ideal J = standard(I); |
---|
| 815 | timer -tt; //3sec, product criterion: 1 chain criterion: 3 |
---|
| 816 | ideal J1 = std(I); |
---|
| 817 | NF(lead(J),lead(J1)); |
---|
| 818 | NF(lead(J1),lead(J)); |
---|
[731e67e] | 819 | |
---|
[373856] | 820 | //Becker: |
---|
| 821 | ring r4 = 32003,(x,y,z),lp; |
---|
| 822 | ideal I = x3-1, y3-1, |
---|
[731e67e] | 823 | -27x3-243x2y+27x2z-729xy2+162xyz-9xz2-729y3+243y2z-27yz2+z3-27; |
---|
[373856] | 824 | option(prot); |
---|
| 825 | int tt = timer; |
---|
| 826 | ideal J = standard(I); |
---|
| 827 | timer -tt; //201sec, product criterion: 42 chain criterion: 33 |
---|
| 828 | ideal J1 = std(I); |
---|
| 829 | NF(lead(J),lead(J1)); |
---|
| 830 | NF(lead(J1),lead(J)); |
---|
[731e67e] | 831 | |
---|
[373856] | 832 | //Alex |
---|
| 833 | ring r5 = 32003,(x,y,z,t),dp; |
---|
[731e67e] | 834 | ideal I = |
---|
[373856] | 835 | 2t3xy2z+x2ty+2x2y, |
---|
| 836 | 2tz+y3x2t+z2t3y2x; |
---|
| 837 | option(prot); printlevel =1; |
---|
| 838 | ideal J1 = std(I); |
---|
| 839 | int tt = timer; |
---|
| 840 | ideal J = standard(I); |
---|
| 841 | timer -tt; //15sec product criterion: 0 chain criterion: 12 |
---|
| 842 | NF(lead(J),lead(J1)); |
---|
| 843 | NF(lead(J1),lead(J)); |
---|
| 844 | |
---|
| 845 | ring r6 = 32003,(x,y,z,t),dp; //is already SB for ds, for dp too long |
---|
| 846 | ideal I= |
---|
| 847 | 9x8+y7t3z4+5x4y2t2+2xy2z3t2, |
---|
| 848 | 9y8+7xy6t+2x5y4t2+2x2yz3t2, |
---|
| 849 | 9z8+3x2y3z2t4; |
---|
| 850 | option(prot); |
---|
| 851 | int tt = timer; |
---|
| 852 | ideal J = standard(I); |
---|
| 853 | timer -tt; //0sec, product criterion: 3 chain criterion: 0 |
---|
| 854 | ideal J1 = std(I); |
---|
| 855 | NF(lead(J),lead(J1)); |
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| 856 | NF(lead(J1),lead(J)); |
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| 857 | |
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| 858 | |
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| 859 | */ |
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| 860 | |
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