[858cae] | 1 | ////////////////////////////////////////////////////////////////////////////// |
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| 2 | version="$Id$"; |
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| 3 | category="Commutative Algebra"; |
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| 4 | info=" |
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| 5 | LIBRARY: sagbi.lib Compute SAGBI basis (subalgebra bases analogous to Groebner bases for ideals) of a subalgebra |
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| 6 | AUTHORS: Jan Hackfeld, Jan.Hackfeld@rwth-aachen.de |
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| 7 | Gerhard Pfister, pfister@mathematik.uni-kl.de |
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| 8 | Viktor Levandovskyy, levandov@math.rwth-aachen.de |
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| 9 | |
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| 10 | OVERVIEW: |
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| 11 | SAGBI stands for 'subalgebra bases analogous to Groebner bases for ideals'. |
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| 12 | SAGBI bases provide important tools for working with finitely presented |
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| 13 | subalgebras of a polynomial ring. Note, that in contrast to Groebner |
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| 14 | bases, SAGBI bases may be infinite. |
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| 15 | |
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| 16 | REFERENCES: |
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| 17 | Ana Bravo: Some Facts About Canonical Subalgebra Bases, |
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| 18 | MSRI Publications 51, p. 247-254 |
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| 19 | |
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| 20 | PROCEDURES: |
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| 21 | sagbiSPoly(A [,r,m]); computes SAGBI S-polynomials of A |
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| 22 | sagbiReduce(I,A [,t,mt]); performs subalgebra reduction of I by A |
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| 23 | sagbi(A [,m,t]); computes SAGBI basis for A |
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| 24 | sagbiPart(A,k[,m]); computes partial SAGBI basis for A |
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| 25 | algebraicDependence(I,it); performs iterations of SAGBI for algebraic dependencies of I |
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| 26 | |
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| 27 | SEE ALSO: algebra_lib |
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| 28 | "; |
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| 29 | |
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| 30 | LIB "elim.lib"; |
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| 31 | LIB "toric.lib"; |
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| 32 | LIB "algebra.lib"; |
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| 33 | ////////////////////////////////////////////////////////////////////////////// |
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| 34 | |
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| 35 | static proc assumeQring() |
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| 36 | { |
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| 37 | if (ideal(basering) != 0) |
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| 38 | { |
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| 39 | ERROR("This function has not yet been implemented over qrings."); |
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| 40 | } |
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| 41 | } |
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| 42 | |
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| 43 | |
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| 44 | static proc uniqueVariableName (string variableName) |
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| 45 | { |
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| 46 | //Adds character "@" at the beginning of variableName until this name ist unique |
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| 47 | //(not contained in the names of the ring variables or description of the coefficient field) |
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| 48 | string ringVars = charstr(basering) + "," + varstr(basering); |
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| 49 | while (find(ringVars,variableName) <> 0) |
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| 50 | { |
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| 51 | variableName="@"+variableName; |
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| 52 | } |
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| 53 | return(variableName); |
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| 54 | } |
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| 55 | |
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| 56 | static proc extendRing(r, ideal leadTermsAlgebra, int method) { |
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| 57 | /* Extends ring r with additional variables. If k=ncols(leadTermsAlgebra) and |
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| 58 | * r contains already m additional variables @y, the procedure adds k-m variables |
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| 59 | * @y(m+1)...@y(k) to the ring. |
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| 60 | * The monomial ordering of the extended ring depends on method. |
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| 61 | * Important: When calling this function, the basering (where algebra is defined) has to be active |
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| 62 | */ |
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| 63 | def br=basering; |
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| 64 | int i; |
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| 65 | ideal varsBasering=maxideal(1); |
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| 66 | int numTotalAdditionalVars=ncols(leadTermsAlgebra); |
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| 67 | string variableName=uniqueVariableName("@y"); |
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| 68 | //get a variable name different from existing variables |
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| 69 | |
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| 70 | //-------- extend current baserring r with new variables @y, |
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| 71 | // one for each new element in ideal algebra ------------- |
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[cc50e4] | 72 | setring r; |
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[858cae] | 73 | list l = ringlist(r); |
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| 74 | for (i=nvars(r)-nvars(br)+1; i<=numTotalAdditionalVars;i++) |
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| 75 | { |
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| 76 | l[2][i+nvars(br)]=string(variableName,"(",i,")"); |
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| 77 | } |
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| 78 | if (method>=0 && method<=1) |
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| 79 | { |
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| 80 | if (nvars(r)==nvars(br)) |
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| 81 | { //first run of spolynomialGB in sagbi construction algorithms |
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| 82 | l[3][size(l[3])+1]=l[3][size(l[3])]; //save module ordering |
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| 83 | l[3][size(l[3])-1]=list("dp",intvec(1:numTotalAdditionalVars)); |
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| 84 | } |
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| 85 | else |
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| 86 | { //overwrite existing order for @y(i) to only get one block for the @y |
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| 87 | l[3][size(l[3])-1]=list("dp",intvec(1:numTotalAdditionalVars)); |
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| 88 | } |
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| 89 | } |
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| 90 | // VL : todo noncomm case: correctly use l[5] and l[6] |
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| 91 | // that is update matrices |
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| 92 | // at the moment this is troublesome, so use nc_algebra call |
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| 93 | // see how it done in algebraicDependence proc // VL |
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| 94 | def rNew=ring(l); |
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| 95 | setring br; |
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| 96 | return(rNew); |
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| 97 | } |
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| 98 | |
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| 99 | |
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| 100 | static proc stdKernPhi(ideal kernNew, ideal kernOld, ideal leadTermsAlgebra,int method) |
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| 101 | { |
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| 102 | /* Computes Groebner basis of kernNew+kernOld, where kernOld already is a GB |
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| 103 | * and kernNew contains elements of the form @y(i)-leadTermsAlgebra[i] added to it. |
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| 104 | * The techniques chosen is specified by the integer method |
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| 105 | */ |
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| 106 | ideal kern; |
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| 107 | attrib(kernOld,"isSB",1); |
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| 108 | if (method==0) |
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| 109 | { |
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| 110 | kernNew=reduce(kernNew,kernOld); |
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| 111 | kern=kernOld+kernNew; |
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| 112 | kern=std(kern); |
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| 113 | //kern=std(kernOld,kernNew); //Found bug using this method. |
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| 114 | // TODO Change if bug is removed |
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| 115 | //this call of std return Groebner Basis of ideal kernNew+kernOld |
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| 116 | // given that kernOld is a Groebner basis |
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| 117 | } |
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| 118 | if (method==1) |
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| 119 | { |
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| 120 | kernNew=reduce(kernNew,kernOld); |
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| 121 | kern=slimgb(kernNew+kernOld); |
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| 122 | } |
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| 123 | return(kern); |
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| 124 | } |
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| 125 | |
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| 126 | |
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| 127 | static proc spolynomialsGB(ideal algebra,r,int method) |
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| 128 | { |
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| 129 | /* This procedure does the actual S-polynomial calculation using Groebner basis methods and is |
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| 130 | * called by the procedures sagbiSPoly,sagbi and sagbiPart. As this procedure is called |
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| 131 | * at each step of the SAGBI construction algorithm, we can reuse the information already calculated |
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| 132 | * which is contained in the ring r. This is done in the following order |
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| 133 | * 1. If r already contain m additional variables and m'=number of elements in algebra, extend r with variables @y(m+1),...,@y(m') |
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| 134 | * 2. Transfer all objects to this ring, kernOld=kern is the Groebnerbasis already computed |
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| 135 | * 3. Define ideal kernNew containing elements of the form leadTermsAlgebra(m+1)-@y(m+1),...,leadTermsAlgebra(m')-@y(m') |
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| 136 | * 4. Compute Groebnerbasis of kernOld+kernNew |
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| 137 | * 5. Compute the new algebraic relations |
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| 138 | */ |
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| 139 | int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information |
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| 140 | dbprint(ppl,"//Spoly-1- initialisation and precomputation"); |
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| 141 | def br=basering; |
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| 142 | ideal varsBasering=maxideal(1); |
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| 143 | ideal leadTermsAlgebra=lead(algebra); |
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| 144 | //save leading terms as ordering in ring extension |
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| 145 | //may not be compatible with ordering in basering |
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| 146 | int numGenerators=ncols(algebra); |
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| 147 | |
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| 148 | def rNew=extendRing(r,leadTermsAlgebra,method); |
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| 149 | // important: br has to be active here |
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| 150 | setring r; |
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| 151 | if (!defined(kern)) |
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| 152 | //only true for first run of spolynomialGB in sagbi construction algorithms |
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| 153 | { |
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| 154 | ideal kern=0; |
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| 155 | ideal algebraicRelations=0; |
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| 156 | } |
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| 157 | setring rNew; |
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| 158 | //-------------------------- transfer object to new ring rNew ---------------------- |
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| 159 | ideal varsBasering=fetch(br,varsBasering); |
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| 160 | ideal kernOld,algebraicRelationsOld; |
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| 161 | kernOld=fetch(r,kern); //kern is Groebner basis of the kernel of the map Phi:r->K[x_1,...,x_n], x(i)->x(i), @y(i)->leadTermsAlgebra(i) |
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| 162 | algebraicRelationsOld=fetch(r,algebraicRelations); |
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| 163 | ideal leadTermsAlgebra=fetch(br,leadTermsAlgebra); |
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| 164 | ideal listOfVariables=maxideal(1); |
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| 165 | //---------define kernNew containing elements to be added to the ideal kern -------- |
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| 166 | ideal kernNew; |
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| 167 | for (int i=nvars(r)-nvars(br)+1; i<=numGenerators; i++) |
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| 168 | { |
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| 169 | kernNew[i-nvars(r)+nvars(br)]=leadTermsAlgebra[i]-listOfVariables[i+nvars(br)]; |
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| 170 | } |
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[1e1ec4] | 171 | //--------------- calculate kernel of Phi depending on method chosen --------------- |
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[858cae] | 172 | dbprint(ppl,"//Spoly-2- Groebner basis computation"); |
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| 173 | attrib(kernOld,"isSB",1); |
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| 174 | ideal kern=stdKernPhi(kernNew,kernOld,leadTermsAlgebra,method); |
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| 175 | dbprint(ppl-2,"//Spoly-2-1- ideal kern",kern); |
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| 176 | //-------------------------- calulate algebraic relations ----------------------- |
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| 177 | dbprint(ppl,"//Spoly-3- computing new algebraic relations"); |
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| 178 | ideal algebraicRelations=nselect(kern,1..nvars(br)); |
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| 179 | attrib(algebraicRelationsOld,"isSB",1); |
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| 180 | ideal algebraicRelationsNew=reduce(algebraicRelations,algebraicRelationsOld); |
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| 181 | /* canonicalizing: */ |
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| 182 | algebraicRelationsNew=canonicalform(algebraicRelationsNew); |
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| 183 | dbprint(ppl-2,"//Spoly-3-1- ideal of new algebraic relations",algebraicRelationsNew); |
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| 184 | /* algebraicRelationsOld is a groebner basis by construction (as variable |
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| 185 | * ordering is |
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| 186 | * block ordering we have an elemination ordering for the varsBasering) |
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| 187 | * Therefore, to only get the new algebraic relations, calculate |
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| 188 | * <algebraicRelations>\<algebraicRelationsOld> using groebner reduction |
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| 189 | */ |
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| 190 | kill kernOld,kernNew,algebraicRelationsOld,listOfVariables; |
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| 191 | export algebraicRelationsNew,algebraicRelations,kern; |
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| 192 | setring br; |
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| 193 | return(rNew); |
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| 194 | } |
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| 195 | |
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| 196 | static proc spolynomialsToric(ideal algebra) { |
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| 197 | /* This procedure does the actual S-polynomial calculation using toric.lib for |
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| 198 | * computation of a Groebner basis for the toric ideal kern(phi), where |
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| 199 | * phi:K[y_1,...,y_m]->K[x_1,...,x_n], y_i->leadmonom(algebra[i]) |
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| 200 | * By suitable substitutions we obtain the kernel of the map |
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| 201 | * K[y_1,...,y_m]->K[x_1,...,x_n], x(i)->x(i), @y(i)->leadterm(algebra[i]) |
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| 202 | */ |
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| 203 | int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information |
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| 204 | dbprint(ppl,"//Spoly-1- initialisation and precomputation"); |
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| 205 | def br=basering; |
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| 206 | int m=ncols(algebra); |
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| 207 | int n=nvars(basering); |
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| 208 | intvec tempVec; |
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| 209 | int i,j; |
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| 210 | ideal leadCoefficients; |
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| 211 | for (i=1;i<=m; i++) |
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| 212 | { |
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| 213 | leadCoefficients[i]=leadcoef(algebra[i]); |
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| 214 | } |
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| 215 | dbprint(ppl-2,"//Spoly-1-1- Vector of leading coefficients",leadCoefficients); |
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| 216 | int k=1; |
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| 217 | for (i=1;i<=n;i++) |
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| 218 | { |
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| 219 | for (j=1; j<=m; j++) |
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| 220 | { |
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| 221 | tempVec[k]=leadexp(algebra[j])[i]; |
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| 222 | k++; |
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| 223 | } |
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| 224 | } |
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| 225 | //The columns of the matrix A are now the exponent vectors |
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| 226 | //of the leadings monomials in algebra. |
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| 227 | intmat A[n][m]=intmat(tempVec,n,m); |
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| 228 | dbprint(ppl-2,"//Spoly-1-2- Matrix A",A); |
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| 229 | //Create the preimage ring K[@y(1),...,@y(m)], where m=ncols(algebra). |
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| 230 | string variableName=uniqueVariableName("@y"); |
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| 231 | list l = ringlist(basering); |
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| 232 | for (i=1; i<=m;i++) |
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| 233 | { |
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| 234 | l[2][i]=string(variableName,"(",i,")"); |
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| 235 | } |
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| 236 | l[3][2]=l[3][size(l[3])]; |
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| 237 | l[3][1]=list("dp",intvec(1:m)); |
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| 238 | def rNew=ring(l); |
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| 239 | setring rNew; |
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| 240 | //Use toric_ideal to compute the kernel |
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| 241 | dbprint(ppl,"//Spoly-2- call of toric_ideal"); |
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| 242 | ideal algebraicRelations=toric_ideal(A,"ect"); |
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| 243 | //Suitable substitution |
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| 244 | dbprint(ppl,"//Spoly-3- substitutions"); |
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| 245 | ideal leadCoefficients=fetch(br,leadCoefficients); |
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| 246 | for (i=1; i<=m; i++) |
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| 247 | { |
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| 248 | if (leadCoefficients[i]!=0) |
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| 249 | { |
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| 250 | algebraicRelations=subst(algebraicRelations,var(i),1/leadCoefficients[i]*var(i)); |
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| 251 | } |
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| 252 | } |
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| 253 | dbprint(ppl-2,"//Spoly-3-1- algebraic relations",algebraicRelations); |
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| 254 | export algebraicRelations; |
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| 255 | return(rNew); |
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| 256 | } |
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| 257 | |
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| 258 | |
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| 259 | static proc reductionGB(ideal F, ideal algebra,r, int tailreduction,int method,int parRed) |
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| 260 | { |
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| 261 | /* This procedure does the actual SAGBI/subalgebra reduction using GB methods and is |
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| 262 | * called by the procedures sagbiReduce,sagbi and sagbiPart |
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| 263 | * If r already is an extension of the basering |
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| 264 | * and contains the ideal kern needed for the subalgebra reduction, |
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| 265 | * the reduction can be started directly, at each reduction step using the fact that |
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| 266 | * p=reduce(leadF,kern) in K[@y(1),...,@y(m)] <=> leadF in K[lead(algebra)] |
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| 267 | * Otherwise some precomputation has to be done, outlined below. |
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| 268 | * When using sagbiReduce,sagbi and sagbiPart the integer parRed will always be zero. Only the procedure |
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| 269 | * algebraicDependence causes this procedure to be called with parRed<>0. The only difference when parRed<>0 |
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| 270 | * is that the reduction algorithms returns the non-zero constants it attains (instead of just returning zero as the |
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| 271 | * correct remainder), as they will be expressions in parameters for an algebraic dependence. |
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| 272 | */ |
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| 273 | int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information |
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| 274 | dbprint(ppl,"//Red-1- initialisation and precomputation"); |
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| 275 | def br=basering; |
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| 276 | int numVarsBasering=nvars(br); |
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| 277 | ideal varsBasering=maxideal(1); |
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| 278 | int i; |
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| 279 | |
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| 280 | if (numVarsBasering==nvars(r)) |
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| 281 | { |
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| 282 | dbprint(ppl-1,"//Red-1-1- Groebner basis computation"); |
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| 283 | /* Case that ring r is the same ring as the basering. Using proc extendRing, |
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| 284 | * stdKernPhi |
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| 285 | * one construct the extension of the current baserring with new variables @y, one for each element |
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| 286 | * in ideal algebra and calculates the kernel of Phi, where |
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| 287 | * Phi: r---->br, x_i-->x_i, y_i-->f_i, |
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| 288 | * algebra={f_1,...f_m}, br=K[x1,...,x_n] und r=K[x1,...x_n,@y1,...@y_m] |
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| 289 | * This is similarly dones |
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| 290 | * (however step by step for each run of the SAGBI construction algorithm) |
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| 291 | * in the procedure spolynomialsGB |
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| 292 | */ |
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| 293 | ideal leadTermsAlgebra=lead(algebra); |
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| 294 | kill r; |
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| 295 | def r=extendRing(br,leadTermsAlgebra,method); |
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| 296 | setring r; |
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| 297 | ideal listOfVariables=maxideal(1); |
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| 298 | ideal leadTermsAlgebra=fetch(br,leadTermsAlgebra); |
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| 299 | ideal kern; |
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| 300 | for (i=1; i<=ncols(leadTermsAlgebra); i++) |
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| 301 | { |
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| 302 | kern[i]=leadTermsAlgebra[i]-listOfVariables[numVarsBasering+i]; |
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| 303 | } |
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| 304 | kern=stdKernPhi(kern,0,leadTermsAlgebra,method); |
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| 305 | dbprint(ppl-2,"//Red-1-1-1- Ideal kern",kern); |
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| 306 | } |
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| 307 | setring r; |
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| 308 | poly p,leadF; |
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| 309 | ideal varsBasering=fetch(br,varsBasering); |
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| 310 | setring br; |
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| 311 | map phi=r,varsBasering,algebra; |
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| 312 | poly p,normalform,leadF; |
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| 313 | intvec tempExp; |
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| 314 | //-------------algebraic reduction for each polynomial F[i] ------------------------ |
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| 315 | dbprint(ppl,"//Red-2- reduction, polynomial by polynomial"); |
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| 316 | for (i=1; i<=ncols(F);i++) |
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| 317 | { |
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| 318 | dbprint(ppl-1,"//Red-2-"+string(i)+"- starting with new polynomial"); |
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| 319 | dbprint(ppl-2,"//Red-2-"+string(i)+"-1- Polynomial before reduction",F[i]); |
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| 320 | normalform=0; |
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| 321 | while (F[i]!=0) |
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| 322 | { |
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| 323 | leadF=lead(F[i]); |
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| 324 | if(leadmonom(leadF)==1) |
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| 325 | { |
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| 326 | //K is always contained in the subalgebra, |
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| 327 | //thus the remainder is zero in this case |
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| 328 | if (parRed) |
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| 329 | { |
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| 330 | //If parRed<>0 save non-zero constants the reduction algorithms attains. |
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| 331 | break; |
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| 332 | } |
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| 333 | else |
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| 334 | { |
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| 335 | F[i]=0; |
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| 336 | break; |
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| 337 | } |
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| 338 | } |
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| 339 | //note: as the ordering in br and r might not be compatible |
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| 340 | //it can be that lead(F[i]) in r is |
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| 341 | //different from lead(F[i]) in br. |
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| 342 | //To take the "correct" leading term therefore take lead(F[i]) |
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| 343 | //in br and transfer it to the extension r |
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| 344 | setring r; |
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| 345 | leadF=fetch(br,leadF); |
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| 346 | p=reduce(leadF,kern); |
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| 347 | if (leadmonom(p)<varsBasering[numVarsBasering]) |
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| 348 | { |
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[1e1ec4] | 349 | //as chosen ordering is a block ordering, |
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[858cae] | 350 | //lm(p) in K[y_1...y_m] is equivalent to lm(p)<x_n |
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| 351 | //Needs to be changed, if no block ordering is used! |
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| 352 | setring br; |
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| 353 | F[i]=F[i]-phi(p); |
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| 354 | } |
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| 355 | else |
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| 356 | { |
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| 357 | if (tailreduction) |
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| 358 | { |
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| 359 | setring br; |
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| 360 | normalform=normalform+lead(F[i]); |
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| 361 | F[i]=F[i]-lead(F[i]); |
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| 362 | } |
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| 363 | else |
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| 364 | { |
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| 365 | setring br; |
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| 366 | break; |
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| 367 | } |
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| 368 | } |
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| 369 | } |
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| 370 | if (tailreduction) |
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| 371 | { |
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| 372 | F[i] = normalform; |
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| 373 | } |
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| 374 | dbprint(ppl-2,"//Red-2-"+string(i)+"-2- Polynomial after reduction",F[i]); |
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| 375 | } |
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| 376 | return(F); |
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| 377 | } |
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| 378 | |
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| 379 | static proc reduceByMonomials(ideal algebra) |
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| 380 | /*This procedures uses the sagbiReduce procedure to reduce all polynomials in algebra, |
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| 381 | * which are not monomials, by the subset of all monomials. |
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| 382 | */ |
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| 383 | { |
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| 384 | ideal monomials; |
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| 385 | int i; |
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| 386 | for (i=1; i<=ncols(algebra);i++) |
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| 387 | { |
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| 388 | if(size(algebra[i])==1) |
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| 389 | { |
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| 390 | monomials[i]=algebra[i]; |
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| 391 | algebra[i]=0; |
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| 392 | } |
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| 393 | else |
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| 394 | { |
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| 395 | monomials[i]=0; |
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| 396 | } |
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| 397 | } |
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| 398 | //Monomials now contains the subset of all monomials in algebra, |
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| 399 | //algebra contains the non-monomials. |
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| 400 | if(size(monomials)>0) |
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| 401 | { |
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| 402 | algebra=sagbiReduce(algebra,monomials,1); |
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| 403 | for (i=1; i<=ncols(algebra);i++) |
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| 404 | { |
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| 405 | if(size(monomials[i])==1) |
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| 406 | { |
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| 407 | //Put back monomials into algebra. |
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| 408 | algebra[i]=monomials[i]; |
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| 409 | } |
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| 410 | } |
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| 411 | } |
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| 412 | return(algebra); |
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| 413 | } |
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| 414 | |
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| 415 | |
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| 416 | static proc sagbiConstruction(ideal algebra, int iterations, int tailreduction, int method,int parRed) |
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| 417 | /* This procedure is the SAGBI construction algorithm and does the actual computation |
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| 418 | * both for the procedure sagbi and sagbiPart. |
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| 419 | * - If the sagbi procedure calls this procedure, iterations==-1 |
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| 420 | * and this procedure only stops |
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| 421 | * if all S-Polynomials reduce to zero |
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| 422 | * (criterion for termination of SAGBI construction algorithm). |
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| 423 | * - If the sagbiPart procedure calls this procedure, iterations>=0 |
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| 424 | * and iterations specifies the |
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| 425 | * number of iterations. A degree boundary is not used here. |
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| 426 | * When this method is called via the procedures sagbi and sagbiPart the integer parRed |
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| 427 | * will always be zero. Only the procedure algebraicDependence calls this procedure with |
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| 428 | * parRed<>0. The only difference when parRed<>0 is that the reduction algorithms returns |
---|
| 429 | * the non-zero constants it attains (instead of just returning zero as the correct |
---|
| 430 | * remainder), as they will be expressions in parameters for an algebraic dependence. |
---|
| 431 | * These constants are saved in the ideal reducedParameters. |
---|
| 432 | */ |
---|
| 433 | { |
---|
| 434 | int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information |
---|
| 435 | dbprint(ppl,"// -0- initialisation and precomputation"); |
---|
| 436 | def br=basering; |
---|
| 437 | int i=1; |
---|
| 438 | |
---|
| 439 | ideal reducedParameters; |
---|
| 440 | int numReducedParameters=1; //number of elements plus one in reducedParameters |
---|
| 441 | int j; |
---|
| 442 | if (parRed==0) //if parRed<>0 the algebra does not contain monomials and normalisation should be avoided |
---|
| 443 | { |
---|
| 444 | algebra=reduceByMonomials(algebra); |
---|
| 445 | algebra=simplify(simplify(algebra,3),4); |
---|
| 446 | } |
---|
| 447 | // canonicalizing the gen's: |
---|
| 448 | algebra = canonicalform(algebra); |
---|
| 449 | ideal P=1; |
---|
| 450 | //note: P is initialized this way, so that the while loop is entered. |
---|
| 451 | //P gets overriden there, anyhow. |
---|
| 452 | ideal varsBasering=maxideal(1); |
---|
| 453 | map phi; |
---|
| 454 | ideal spolynomialsNew; |
---|
| 455 | def r=br; |
---|
| 456 | while (size(P)>0) |
---|
| 457 | { |
---|
| 458 | dbprint(ppl,"// -"+string(i)+"- interation of SAGBI construction algorithm"); |
---|
| 459 | dbprint(ppl-1,"// -"+string(i)+"-1- Computing algebraic relations"); |
---|
| 460 | def rNew=spolynomialsGB(algebra,r,method); /* canonicalizing inside! */ |
---|
| 461 | kill r; |
---|
| 462 | def r=rNew; |
---|
| 463 | kill rNew; |
---|
| 464 | phi=r,varsBasering,algebra; |
---|
| 465 | dbprint(ppl-1,"// -"+string(i)+"-2- Substituting into algebraic relations"); |
---|
| 466 | spolynomialsNew=simplify(phi(algebraicRelationsNew),6); |
---|
| 467 | //By construction spolynomialsNew only contains the spolynomials, |
---|
| 468 | //that have not already |
---|
| 469 | //been calculated in the steps before. |
---|
| 470 | dbprint(ppl-1,"// -"+string(i)+"-3- SAGBI reduction"); |
---|
| 471 | dbprint(ppl-2,"// -"+string(i)+"-3-1- new S-polynomials before reduction",spolynomialsNew); |
---|
| 472 | P=reductionGB(spolynomialsNew,algebra,r,tailreduction,method,parRed); |
---|
| 473 | if (parRed) |
---|
| 474 | { |
---|
| 475 | for(j=1; j<=ncols(P); j++) |
---|
| 476 | { |
---|
| 477 | if (leadmonom(P[j])==1) |
---|
| 478 | { |
---|
| 479 | reducedParameters[numReducedParameters]=P[j]; |
---|
| 480 | P[j]=0; |
---|
| 481 | numReducedParameters++; |
---|
| 482 | } |
---|
| 483 | } |
---|
| 484 | } |
---|
| 485 | if (parRed==0) |
---|
| 486 | { |
---|
| 487 | P=reduceByMonomials(P); |
---|
| 488 | //Reducing with monomials is cheap and can only result in less terms |
---|
| 489 | P=simplify(simplify(P,3),4); |
---|
| 490 | //Avoid that zeros are added to the bases or one element in P more than once |
---|
| 491 | } |
---|
| 492 | else |
---|
| 493 | { |
---|
| 494 | P=simplify(P,6); |
---|
| 495 | } |
---|
| 496 | /* canonicalize ! */ |
---|
| 497 | P = canonicalform(P); |
---|
| 498 | dbprint(ppl-2,"// -"+string(i)+"-3-1- new S-polynomials after reduction",P); |
---|
| 499 | algebra=algebra,P; |
---|
| 500 | //Note that elements and order of elements must in algebra must not be changed, |
---|
| 501 | //otherwise the already calculated |
---|
| 502 | //ideal in r will give wrong results. Thus it is important to use a komma here. |
---|
| 503 | i=i+1; |
---|
| 504 | if (iterations!=-1 && i>iterations) //When iterations==-1 the number of iterations is unlimited |
---|
| 505 | { |
---|
| 506 | break; |
---|
| 507 | } |
---|
| 508 | } |
---|
| 509 | if (iterations!=-1) |
---|
| 510 | { //case that sagbiPart called this procedure |
---|
| 511 | if (size(P)==0) |
---|
| 512 | { |
---|
| 513 | dbprint(4-voice, |
---|
| 514 | "//SAGBI construction algorithm terminated after "+string(i-1) |
---|
| 515 | +" iterations, as all SAGBI S-polynomials reduced to 0. |
---|
| 516 | //Returned generators therefore are a SAGBI basis."); |
---|
| 517 | } |
---|
| 518 | else |
---|
| 519 | { |
---|
| 520 | dbprint(4-voice, |
---|
| 521 | "//SAGBI construction algorithm stopped as it reached the limit of " |
---|
| 522 | +string(iterations)+" iterations. |
---|
| 523 | //In general the returned generators are no SAGBI basis for the given algebra."); |
---|
| 524 | } |
---|
| 525 | } |
---|
| 526 | kill r; |
---|
| 527 | if (parRed) |
---|
| 528 | { |
---|
| 529 | algebra=algebra,reducedParameters; |
---|
| 530 | } |
---|
| 531 | algebra = simplify(algebra,6); |
---|
| 532 | algebra = canonicalform(algebra); |
---|
| 533 | return(algebra); |
---|
| 534 | } |
---|
| 535 | |
---|
| 536 | |
---|
| 537 | proc sagbiSPoly(ideal algebra,list #) |
---|
| 538 | "USAGE: sagbiSPoly(A[, returnRing, meth]); A is an ideal, returnRing and meth are integers. |
---|
| 539 | RETURN: ideal or ring |
---|
| 540 | ASSUME: basering is not a qring |
---|
| 541 | PURPOSE: Returns SAGBI S-polynomials of the leading terms of a given ideal A if returnRing=0. |
---|
| 542 | @* Otherwise returns a new ring containing the ideals algebraicRelations |
---|
| 543 | @* and spolynomials, where these objects are explained by their name. |
---|
| 544 | @* See the example on how to access these objects. |
---|
| 545 | @format The other optional argument meth determines which method is |
---|
| 546 | used for computing the algebraic relations. |
---|
| 547 | - If meth=0 (default), the procedure std is used. |
---|
| 548 | - If meth=1, the procedure slimgb is used. |
---|
| 549 | - If meth=2, the prodecure uses toric_ideal. |
---|
| 550 | @end format |
---|
| 551 | EXAMPLE: example sagbiSPoly; shows an example" |
---|
| 552 | { |
---|
| 553 | assumeQring(); |
---|
| 554 | int returnRing; |
---|
| 555 | int method=0; |
---|
| 556 | def br=basering; |
---|
| 557 | ideal spolynomials; |
---|
| 558 | if (size(#)>=1) |
---|
| 559 | { |
---|
| 560 | if (typeof(#[1])=="int") |
---|
| 561 | { |
---|
| 562 | returnRing=#[1]; |
---|
| 563 | } |
---|
| 564 | else |
---|
| 565 | { |
---|
| 566 | ERROR("Type of first optional argument needs to be int."); |
---|
| 567 | } |
---|
| 568 | } |
---|
| 569 | if (size(#)==2) |
---|
| 570 | { |
---|
| 571 | if (typeof(#[2])=="int") |
---|
| 572 | { |
---|
| 573 | if (#[2]<0 || #[2]>2) |
---|
| 574 | { |
---|
| 575 | ERROR("Type of second optional argument needs to be 0,1 or 2."); |
---|
| 576 | } |
---|
| 577 | else |
---|
| 578 | { |
---|
| 579 | method=#[2]; |
---|
| 580 | } |
---|
| 581 | } |
---|
| 582 | else |
---|
| 583 | { |
---|
| 584 | ERROR("Type of second optional argument needs to be int."); |
---|
| 585 | } |
---|
| 586 | } |
---|
| 587 | if (method>=0 and method<=1) |
---|
| 588 | { |
---|
| 589 | ideal varsBasering=maxideal(1); |
---|
| 590 | def rNew=spolynomialsGB(algebra,br,method); |
---|
| 591 | map phi=rNew,varsBasering,algebra; |
---|
| 592 | spolynomials=simplify(phi(algebraicRelationsNew),7); |
---|
| 593 | } |
---|
| 594 | if(method==2) |
---|
| 595 | { |
---|
| 596 | def r2=spolynomialsToric(algebra); |
---|
| 597 | map phi=r2,algebra; |
---|
| 598 | spolynomials=simplify(phi(algebraicRelations),7); |
---|
| 599 | def rNew=extendRing(br,lead(algebra),0); |
---|
| 600 | setring rNew; |
---|
| 601 | ideal algebraicRelations=imap(r2,algebraicRelations); |
---|
| 602 | export algebraicRelations; |
---|
| 603 | setring br; |
---|
| 604 | } |
---|
| 605 | |
---|
| 606 | if (returnRing==0) |
---|
| 607 | { |
---|
| 608 | return(spolynomials); |
---|
| 609 | } |
---|
| 610 | else |
---|
| 611 | { |
---|
| 612 | setring rNew; |
---|
| 613 | ideal spolynomials=fetch(br,spolynomials); |
---|
| 614 | export spolynomials; |
---|
| 615 | setring br; |
---|
| 616 | return(rNew); |
---|
| 617 | } |
---|
| 618 | } |
---|
| 619 | example |
---|
| 620 | { "EXAMPLE:"; echo = 2; |
---|
| 621 | ring r= 0,(x,y),dp; |
---|
| 622 | ideal A=x*y+x,x*y^2,y^2+y,x^2+x; |
---|
| 623 | //------------------ Compute the SAGBI S-polynomials only |
---|
| 624 | sagbiSPoly(A); |
---|
| 625 | //------------------ Extended ring is to be returned, which contains |
---|
| 626 | // the ideal of algebraic relations and the ideal of the S-polynomials |
---|
| 627 | def rNew=sagbiSPoly(A,1); setring rNew; |
---|
| 628 | spolynomials; |
---|
| 629 | algebraicRelations; |
---|
| 630 | //----------------- Now we verify that the substitution of A[i] into @y(i) |
---|
| 631 | // results in the spolynomials listed above |
---|
| 632 | ideal A=fetch(r,A); |
---|
| 633 | map phi=rNew,x,y,A; |
---|
| 634 | ideal spolynomials2=simplify(phi(algebraicRelations),1); |
---|
| 635 | spolynomials2; |
---|
| 636 | } |
---|
| 637 | |
---|
| 638 | |
---|
| 639 | proc sagbiReduce(idealORpoly, ideal algebra, list #) |
---|
| 640 | "USAGE: sagbiReduce(I, A[, tr, mt]); I, A ideals, tr, mt optional integers |
---|
| 641 | RETURN: ideal of remainders of I after SAGBI reduction by A |
---|
| 642 | ASSUME: basering is not a qring |
---|
| 643 | PURPOSE: |
---|
| 644 | @format |
---|
| 645 | The optional argument tr=tailred determines whether tail reduction will be performed. |
---|
| 646 | - If (tailred=0), no tail reduction is done. |
---|
| 647 | - If (tailred<>0), tail reduction is done. |
---|
| 648 | The other optional argument meth determines which method is |
---|
| 649 | used for Groebner basis computations. |
---|
| 650 | - If mt=0 (default), the procedure std is used. |
---|
| 651 | - If mt=1, the procedure slimgb is used. |
---|
| 652 | @end format |
---|
| 653 | EXAMPLE: example sagbiReduce; shows an example" |
---|
| 654 | { |
---|
| 655 | assumeQring(); |
---|
| 656 | int tailreduction=0; //Default |
---|
| 657 | int method=0; //Default |
---|
| 658 | ideal I; |
---|
| 659 | if(typeof(idealORpoly)=="ideal") |
---|
| 660 | { |
---|
| 661 | I=idealORpoly; |
---|
| 662 | } |
---|
| 663 | else |
---|
| 664 | { |
---|
| 665 | if(typeof(idealORpoly)=="poly") |
---|
| 666 | { |
---|
| 667 | I[1]=idealORpoly; |
---|
| 668 | } |
---|
| 669 | else |
---|
| 670 | { |
---|
| 671 | ERROR("Type of first argument needs to be an ideal or polynomial."); |
---|
| 672 | } |
---|
| 673 | } |
---|
| 674 | if (size(#)>=1) |
---|
| 675 | { |
---|
| 676 | if (typeof(#[1])=="int") |
---|
| 677 | { |
---|
| 678 | tailreduction=#[1]; |
---|
| 679 | } |
---|
| 680 | else |
---|
| 681 | { |
---|
| 682 | ERROR("Type of optional argument needs to be int."); |
---|
| 683 | } |
---|
| 684 | } |
---|
| 685 | if (size(#)>=2 ) |
---|
| 686 | { |
---|
| 687 | if (typeof(#[2])=="int") |
---|
| 688 | { |
---|
| 689 | if (#[2]<0 || #[2]>1) |
---|
| 690 | { |
---|
| 691 | ERROR("Type of second optional argument needs to be 0 or 1."); |
---|
| 692 | } |
---|
| 693 | else |
---|
| 694 | { |
---|
| 695 | method=#[2]; |
---|
| 696 | } |
---|
| 697 | } |
---|
| 698 | else |
---|
| 699 | { |
---|
| 700 | ERROR("Type of optional arguments needs to be int."); |
---|
| 701 | } |
---|
| 702 | } |
---|
| 703 | |
---|
| 704 | def r=basering; |
---|
| 705 | I=simplify(reductionGB(I,algebra,r,tailreduction,method,0),1); |
---|
| 706 | |
---|
| 707 | if(typeof(idealORpoly)=="ideal") |
---|
| 708 | { |
---|
| 709 | return(I); |
---|
| 710 | } |
---|
| 711 | else |
---|
| 712 | { |
---|
| 713 | if(typeof(idealORpoly)=="poly") |
---|
| 714 | { |
---|
| 715 | return(I[1]); |
---|
| 716 | } |
---|
| 717 | } |
---|
| 718 | } |
---|
| 719 | example |
---|
| 720 | { "EXAMPLE:"; echo = 2; |
---|
| 721 | ring r=0,(x,y,z),dp; |
---|
| 722 | ideal A=x2,2*x2y+y,x3y2; |
---|
| 723 | poly p1=x^5+x2y+y; |
---|
| 724 | poly p2=x^16+x^12*y^5+6*x^8*y^4+x^6+y^4+3; |
---|
| 725 | ideal P=p1,p2; |
---|
| 726 | //--------------------------------------------- |
---|
| 727 | //SAGBI reduction of polynomial p1 by algebra A. |
---|
| 728 | //Default call, that is, no tail-reduction is done. |
---|
| 729 | sagbiReduce(p1,A); |
---|
| 730 | //--------------------------------------------- |
---|
| 731 | //SAGBI reduction of set of polynomials P by algebra A, |
---|
| 732 | //now tail-reduction is done. |
---|
| 733 | sagbiReduce(P,A,1); |
---|
| 734 | } |
---|
| 735 | |
---|
| 736 | proc sagbi(ideal algebra, list #) |
---|
| 737 | "USAGE: sagbi(A[, tr, mt]); A ideal, tr, mt optional integers |
---|
| 738 | RETURN: ideal, a SAGBI basis for A |
---|
| 739 | ASSUME: basering is not a qring |
---|
| 740 | PURPOSE: Computes a SAGBI basis for the subalgebra given by the generators in A. |
---|
| 741 | @format |
---|
| 742 | The optional argument tr=tailred determines whether tail reduction will be performed. |
---|
| 743 | - If (tailred=0), no tail reduction is performed, |
---|
| 744 | - If (tailred<>0), tail reduction is performed. |
---|
| 745 | The other optional argument meth determines which method is |
---|
| 746 | used for Groebner basis computations. |
---|
| 747 | - If mt=0 (default), the procedure std is used. |
---|
| 748 | - If mt=1, the procedure slimgb is used. |
---|
| 749 | @end format |
---|
| 750 | EXAMPLE: example sagbi; shows an example" |
---|
| 751 | { |
---|
| 752 | assumeQring(); |
---|
| 753 | int tailreduction=0; //default value |
---|
| 754 | int method=0; //default value |
---|
| 755 | if (size(#)>=1) |
---|
| 756 | { |
---|
| 757 | if (typeof(#[1])=="int") |
---|
| 758 | { |
---|
| 759 | tailreduction=#[1]; |
---|
| 760 | } |
---|
| 761 | else |
---|
| 762 | { |
---|
| 763 | ERROR("Type of optional argument needs to be int."); |
---|
| 764 | } |
---|
| 765 | } |
---|
| 766 | if (size(#)>=2 ) |
---|
| 767 | { |
---|
| 768 | if (typeof(#[2])=="int") |
---|
| 769 | { |
---|
| 770 | if (#[2]<0 || #[2]>1) |
---|
| 771 | { |
---|
| 772 | ERROR("Type of second optional argument needs to be 0 or 1."); |
---|
| 773 | } |
---|
| 774 | else |
---|
| 775 | { |
---|
| 776 | method=#[2]; |
---|
| 777 | } |
---|
| 778 | } |
---|
| 779 | else |
---|
| 780 | { |
---|
| 781 | ERROR("Type of optional arguments needs to be int."); |
---|
| 782 | } |
---|
| 783 | } |
---|
| 784 | ideal a; |
---|
| 785 | a=sagbiConstruction(algebra,-1,tailreduction,method,0); |
---|
| 786 | a=simplify(a,7); |
---|
| 787 | // a=interreduced(a); |
---|
| 788 | return(a); |
---|
| 789 | } |
---|
| 790 | example |
---|
| 791 | { "EXAMPLE:"; echo = 2; |
---|
| 792 | ring r= 0,(x,y,z),dp; |
---|
| 793 | ideal A=x2,y2,xy+y; |
---|
| 794 | //Default call, no tail-reduction is done. |
---|
| 795 | sagbi(A); |
---|
| 796 | //--------------------------------------------- |
---|
| 797 | //Call with tail-reduction and method specified. |
---|
| 798 | sagbi(A,1,0); |
---|
| 799 | } |
---|
| 800 | |
---|
| 801 | proc sagbiPart(ideal algebra, int iterations, list #) |
---|
| 802 | "USAGE: sagbiPart(A, k,[tr, mt]); A is an ideal, k, tr and mt are integers |
---|
| 803 | RETURN: ideal |
---|
| 804 | ASSUME: basering is not a qring |
---|
| 805 | PURPOSE: Performs k iterations of the SAGBI construction algorithm for the subalgebra given by the generators given by A. |
---|
| 806 | @format |
---|
| 807 | The optional argument tr=tailred determines if tail reduction will be performed. |
---|
| 808 | - If (tailred=0), no tail reduction is performed, |
---|
| 809 | - If (tailred<>0), tail reduction is performed. |
---|
| 810 | The other optional argument meth determines which method is |
---|
| 811 | used for Groebner basis computations. |
---|
| 812 | - If mt=0 (default), the procedure std is used. |
---|
| 813 | - If mt=1, the procedure slimgb is used. |
---|
| 814 | @end format |
---|
| 815 | EXAMPLE: example sagbiPart; shows an example" |
---|
| 816 | { |
---|
| 817 | assumeQring(); |
---|
| 818 | int tailreduction=0; //default value |
---|
| 819 | int method=0; //default value |
---|
| 820 | if (size(#)>=1) |
---|
| 821 | { |
---|
| 822 | if (typeof(#[1])=="int") |
---|
| 823 | { |
---|
| 824 | tailreduction=#[1]; |
---|
| 825 | } |
---|
| 826 | else |
---|
| 827 | { |
---|
| 828 | ERROR("Type of optional argument needs to be int."); |
---|
| 829 | } |
---|
| 830 | } |
---|
| 831 | if (size(#)>=2 ) |
---|
| 832 | { |
---|
| 833 | if (typeof(#[2])=="int") |
---|
| 834 | { |
---|
| 835 | if (#[2]<0 || #[2]>3) |
---|
| 836 | { |
---|
| 837 | ERROR("Type of second optional argument needs to be 0 or 1."); |
---|
| 838 | } |
---|
| 839 | else |
---|
| 840 | { |
---|
| 841 | method=#[2]; |
---|
| 842 | } |
---|
| 843 | } |
---|
| 844 | else |
---|
| 845 | { |
---|
| 846 | ERROR("Type of optional arguments needs to be int."); |
---|
| 847 | } |
---|
| 848 | } |
---|
| 849 | if (iterations<0) |
---|
| 850 | { |
---|
| 851 | ERROR("Number of iterations needs to be non-negative."); |
---|
| 852 | } |
---|
| 853 | ideal a; |
---|
| 854 | a=sagbiConstruction(algebra,iterations,tailreduction,method,0); |
---|
| 855 | a=simplify(a,6); |
---|
| 856 | // a=interreduced(a); |
---|
| 857 | return(a); |
---|
| 858 | } |
---|
| 859 | example |
---|
| 860 | { "EXAMPLE:"; echo = 2; |
---|
| 861 | ring r= 0,(x,y,z),dp; |
---|
| 862 | //The following algebra does not have a finite SAGBI basis. |
---|
| 863 | ideal A=x,xy-y2,xy2; |
---|
| 864 | //--------------------------------------------------- |
---|
| 865 | //Call with two iterations, no tail-reduction is done. |
---|
| 866 | sagbiPart(A,2); |
---|
| 867 | //--------------------------------------------------- |
---|
| 868 | //Call with three iterations, tail-reduction and method 0. |
---|
| 869 | sagbiPart(A,3,1,0); |
---|
| 870 | } |
---|
| 871 | |
---|
| 872 | |
---|
| 873 | proc algebraicDependence(ideal I,int iterations) |
---|
| 874 | "USAGE: algebraicDependence(I,it); I an an ideal, it is an integer |
---|
| 875 | RETURN: ring |
---|
| 876 | ASSUME: basering is not a qring |
---|
| 877 | PURPOSE: Returns a ring containing the ideal @code{algDep}, which contains possibly |
---|
| 878 | @* some algebraic dependencies of the elements of I obtained through @code{it} |
---|
| 879 | @* iterations of the SAGBI construction algorithms. See the example on how |
---|
| 880 | @* to access these objects. |
---|
| 881 | EXAMPLE: example algebraicDependence; shows an example" |
---|
| 882 | { |
---|
| 883 | assumeQring(); |
---|
| 884 | int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information |
---|
| 885 | dbprint(ppl,"//AlgDep-1- initialisation and precomputation"); |
---|
| 886 | def br=basering; |
---|
| 887 | int i; |
---|
| 888 | I=simplify(I,2); //avoid that I contains zeros |
---|
| 889 | |
---|
| 890 | //Create two polynomial rings, which both are extensions of the current basering. |
---|
| 891 | //The first ring will contain the additional paramteres @c(1),...,@c(m), the second one |
---|
| 892 | //will contain the additional variables @c(1),...,@c(m), where m=ncols(I). |
---|
| 893 | string parameterName=uniqueVariableName("@c"); |
---|
| 894 | list l = ringlist(basering); |
---|
| 895 | list parList; |
---|
| 896 | for (i=1; i<=ncols(I);i++) |
---|
| 897 | { |
---|
| 898 | parList[i]=string(parameterName,"(",i,")"); |
---|
| 899 | } |
---|
| 900 | l[1]=list(l[1],parList,list(list("dp",1:ncols(I)))); //add @c(i) to the ring as paramteres |
---|
| 901 | ideal temp=0; |
---|
| 902 | l[1][4]=temp; |
---|
| 903 | // addition VL: noncomm case |
---|
| 904 | int isNCcase = 0; // default for comm |
---|
| 905 | // if (size(l)>4) |
---|
| 906 | // { |
---|
| 907 | // // that is we're in the noncomm algebra |
---|
| 908 | // isNCcase = 1; // noncomm |
---|
| 909 | // matrix @C@ = l[5]; |
---|
| 910 | // matrix @D@ = l[6]; |
---|
| 911 | // l = l[1],l[2],l[3],l[4]; |
---|
| 912 | // } |
---|
| 913 | def parameterRing=ring(l); |
---|
| 914 | |
---|
| 915 | string extendVarName=uniqueVariableName("@c"); |
---|
| 916 | list l2 = ringlist(basering); |
---|
| 917 | for (i=1; i<=ncols(I);i++) |
---|
| 918 | { |
---|
| 919 | l2[2][i+nvars(br)]=string(extendVarName,"(",i,")"); //add @c(i) to the rings as variables |
---|
| 920 | } |
---|
| 921 | l2[3][size(l2[3])+1]=l2[3][size(l2[3])]; |
---|
| 922 | l2[3][size(l2[3])-1]=list("dp",intvec(1:ncols(I))); |
---|
| 923 | // if (isNCcase) |
---|
| 924 | // { |
---|
| 925 | // // that is we're in the noncomm algebra |
---|
| 926 | // matrix @C@2 = l2[5]; |
---|
| 927 | // matrix @D@2 = l2[6]; |
---|
| 928 | // l2 = l2[1],l2[2],l2[3],l2[4]; |
---|
| 929 | // } |
---|
| 930 | |
---|
| 931 | def extendVarRing=ring(l2); |
---|
| 932 | setring extendVarRing; |
---|
| 933 | // VL : this requires extended matrices |
---|
| 934 | // let's forget it for the moment |
---|
| 935 | // since this holds only for showing the answer |
---|
| 936 | // if (isNCcase) |
---|
| 937 | // { |
---|
| 938 | // matrix C2=imap(br,@C@2); |
---|
| 939 | // matrix D2=imap(br,@D@2); |
---|
| 940 | // def er2 = nc_algebra(C2,D2); |
---|
| 941 | // setring er2; |
---|
| 942 | // def extendVarRing=er2; |
---|
| 943 | // } |
---|
| 944 | |
---|
| 945 | setring parameterRing; |
---|
| 946 | |
---|
| 947 | // if (isNCcase) |
---|
| 948 | // { |
---|
| 949 | // matrix C=imap(br,@C@); |
---|
| 950 | // matrix D=imap(br,@D@); |
---|
| 951 | // def pr = nc_algebra(C,D); |
---|
| 952 | // setring pr; |
---|
| 953 | // def parameterRing=pr; |
---|
| 954 | // } |
---|
| 955 | |
---|
| 956 | //Compute a partial SAGBI basis of the algebra generated by I[1]-@c(1),...,I[m]-@c(m), |
---|
| 957 | //where the @c(n) are parameters |
---|
| 958 | ideal I=fetch(br,I); |
---|
| 959 | ideal algebra; |
---|
| 960 | for (i=1; i<=ncols(I);i++) |
---|
| 961 | { |
---|
| 962 | algebra[i]=I[i]-par(i); |
---|
| 963 | } |
---|
| 964 | dbprint(ppl,"//AlgDep-2- call of SAGBI construction algorithm"); |
---|
| 965 | algebra=sagbiConstruction(algebra, iterations,0,0,1); |
---|
| 966 | dbprint(ppl,"//AlgDep-3- postprocessing of results"); |
---|
| 967 | int j=1; |
---|
| 968 | //If K[x_1,...,x_n] was the basering, then algebra is in K(@c(1),...,@c(m))[x_1,...x_n]. We intersect |
---|
| 969 | //elements in algebra with K(@c(1),..,@c(n)) to get algDep. Note that @c(i) can only appear in the numerator, |
---|
| 970 | //as the SAGBI construction algorithms just multiplies and substracts polynomials. So actually we have |
---|
| 971 | //algDep=algebra intersect K[@c(1),...,@c(m)] |
---|
| 972 | ideal algDep; |
---|
| 973 | for (i=1; i<= ncols(algebra); i++) |
---|
| 974 | { |
---|
| 975 | if (leadmonom(algebra[i])==1) //leadmonom(algebra[i])==1 iff algebra[i] in K[@c(1),...,@c(m)] |
---|
| 976 | { |
---|
| 977 | algDep[j]=algebra[i]; |
---|
| 978 | j++; |
---|
| 979 | } |
---|
| 980 | } |
---|
| 981 | //Transfer algebraic dependencies to ring where @c(i) are not parameters, but now variables. |
---|
| 982 | setring extendVarRing; |
---|
| 983 | ideal algDep=imap(parameterRing,algDep); |
---|
| 984 | ideal algebra=imap(parameterRing,algebra); |
---|
| 985 | //Now get rid of constants in K that may have been added to algDep. |
---|
| 986 | for (i=1; i<=ncols(algDep); i++) |
---|
| 987 | { |
---|
| 988 | if(leadmonom(algDep[i])==1) |
---|
| 989 | { |
---|
| 990 | algDep[i]=0; |
---|
| 991 | } |
---|
| 992 | } |
---|
| 993 | algDep=simplify(algDep,2); |
---|
| 994 | export algDep,algebra; |
---|
| 995 | setring br; |
---|
| 996 | return(extendVarRing); |
---|
| 997 | } |
---|
| 998 | example |
---|
| 999 | { "EXAMPLE:"; echo = 2; |
---|
| 1000 | ring r= 0,(x,y),dp; |
---|
| 1001 | //The following algebra does not have a finite SAGBI basis. |
---|
| 1002 | ideal I=x^2, xy-y2, xy2; |
---|
| 1003 | //--------------------------------------------------- |
---|
| 1004 | //Call with two iterations |
---|
| 1005 | def DI = algebraicDependence(I,2); |
---|
| 1006 | setring DI; algDep; |
---|
| 1007 | // we see that no dependency has been seen so far |
---|
| 1008 | //--------------------------------------------------- |
---|
| 1009 | //Call with two iterations |
---|
| 1010 | setring r; kill DI; |
---|
| 1011 | def DI = algebraicDependence(I,3); |
---|
| 1012 | setring DI; algDep; |
---|
| 1013 | map F = DI,x,y,x^2, xy-y2, xy2; |
---|
| 1014 | F(algDep); // we see that it is a dependence indeed |
---|
| 1015 | } |
---|
| 1016 | |
---|
| 1017 | static proc interreduced(ideal I) |
---|
| 1018 | { |
---|
| 1019 | /* performs subalgebra interreduction of a set of subalgebra generators */ |
---|
| 1020 | int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information |
---|
| 1021 | dbprint(ppl,"//Interred-1- starting interreduction"); |
---|
| 1022 | ideal J,B; |
---|
| 1023 | int i,j,k; |
---|
| 1024 | poly f; |
---|
| 1025 | for(k=1;k<=ncols(I);k++) |
---|
| 1026 | { |
---|
| 1027 | dbprint(ppl-1,"//Interred-1-"+string(k)+"- reducing next poly"); |
---|
| 1028 | f=I[k]; |
---|
| 1029 | I[k]=0; |
---|
| 1030 | f=sagbiReduce(f,I,1); |
---|
| 1031 | I[k]=f; |
---|
| 1032 | } |
---|
| 1033 | I=simplify(I,2); |
---|
| 1034 | dbprint(ppl,"//Interred-2- interreduction completed"); |
---|
| 1035 | return(I); |
---|
| 1036 | } |
---|
| 1037 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1038 | |
---|
| 1039 | proc sagbiReduction(poly p,ideal dom,list #) |
---|
| 1040 | "USAGE: sagbiReduction(p,dom[,n]); p poly , dom ideal |
---|
| 1041 | RETURN: polynomial, after one step of subalgebra reduction |
---|
| 1042 | PURPOSE: |
---|
| 1043 | @format |
---|
| 1044 | Three algorithm variants are used to perform subalgebra reduction. |
---|
| 1045 | The positive interger n determines which variant should be used. |
---|
| 1046 | n may take the values 0 (default), 1 or 2. |
---|
| 1047 | @end format |
---|
| 1048 | NOTE: works over both polynomial rings and their quotients |
---|
| 1049 | EXAMPLE: example sagbiReduction; shows an example" |
---|
| 1050 | { |
---|
| 1051 | def bsr=basering; |
---|
| 1052 | ideal B=ideal(bsr);//When the basering is quotient ring this type casting |
---|
| 1053 | // gives the quotient ideal. |
---|
| 1054 | int b=size(B); |
---|
| 1055 | int n=nvars(bsr); |
---|
| 1056 | |
---|
| 1057 | //In quotient rings, SINGULAR, usually does not reduce polynomials w.r.t the |
---|
| 1058 | //quotient ideal,therefore we should first reduce, |
---|
| 1059 | //when it is necessary for computations, |
---|
| 1060 | // to have a uniquely determined representant for each equivalent |
---|
| 1061 | //class,which is the case of this algorithm. |
---|
| 1062 | |
---|
| 1063 | if(b !=0) //means that the basering is a quotient ring |
---|
| 1064 | { |
---|
| 1065 | p=reduce(p,std(0)); |
---|
| 1066 | dom=reduce(dom,std(0)); |
---|
| 1067 | } |
---|
| 1068 | |
---|
| 1069 | int i,choose; |
---|
| 1070 | int z=ncols(dom); |
---|
| 1071 | |
---|
| 1072 | if((size(#)>0) && (typeof(#[1])=="int")) |
---|
| 1073 | { |
---|
| 1074 | choose = #[1]; |
---|
| 1075 | } |
---|
| 1076 | if (size(#)>1) |
---|
| 1077 | { |
---|
| 1078 | choose =#[2]; |
---|
| 1079 | } |
---|
| 1080 | |
---|
| 1081 | //=======================first algorithm(default)========================= |
---|
| 1082 | if ( choose == 0 ) |
---|
| 1083 | { |
---|
| 1084 | list L = algebra_containment(lead(p),lead(dom),1); |
---|
| 1085 | if( L[1]==1 ) |
---|
| 1086 | { |
---|
| 1087 | // the ring L[2] = char(bsr),(x(1..nvars(bsr)),y(1..z)),(dp(n),dp(m)), |
---|
| 1088 | // contains poly check s.t. LT(p) is of the form check(LT(f1),...,LT(fr)) |
---|
| 1089 | def s1 = L[2]; |
---|
| 1090 | map psi = s1,maxideal(1),dom; |
---|
| 1091 | poly re = p - psi(check); |
---|
| 1092 | // divide by the maximal power of #[1] |
---|
| 1093 | if ( (size(#)>0) && (typeof(#[1])=="poly") ) |
---|
| 1094 | { |
---|
| 1095 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
---|
| 1096 | { |
---|
| 1097 | re=re/#[1]; |
---|
| 1098 | } |
---|
| 1099 | } |
---|
| 1100 | return(re); |
---|
| 1101 | } |
---|
| 1102 | return(p); |
---|
| 1103 | } |
---|
| 1104 | //======================2end variant of algorithm========================= |
---|
| 1105 | //It uses two different commands for elimaination. |
---|
| 1106 | //if(choose==1):"elimainate"command. |
---|
| 1107 | //if (choose==2):"nselect" command. |
---|
| 1108 | else |
---|
| 1109 | { |
---|
| 1110 | poly v=product(maxideal(1)); |
---|
| 1111 | |
---|
| 1112 | //------------- change the basering bsr to bsr[@(0),...,@(z)] ---------- |
---|
| 1113 | execute("ring s=("+charstr(basering)+"),("+varstr(basering)+",@(0..z)),dp;"); |
---|
| 1114 | // Ev hier die Reihenfolge der Vars aendern. Dazu muss unten aber entsprechend |
---|
| 1115 | // geaendert werden: |
---|
| 1116 | // execute("ring s="+charstr(basering)+",(@(0..z),"+varstr(basering)+"),dp;"); |
---|
| 1117 | |
---|
| 1118 | //constructs the leading ideal of dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
---|
| 1119 | ideal dom=imap(bsr,dom); |
---|
| 1120 | for (i=1;i<=z;i++) |
---|
| 1121 | { |
---|
| 1122 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
---|
| 1123 | } |
---|
| 1124 | dom=lead(imap(bsr,p))-@(0),dom; |
---|
| 1125 | |
---|
| 1126 | //---------- eliminate the variables of the basering bsr -------------- |
---|
| 1127 | //i.e. computes dom intersected with K[@(0),...,@(z)]. |
---|
| 1128 | |
---|
| 1129 | if(choose==1) |
---|
| 1130 | { |
---|
| 1131 | ideal kern=eliminate(dom,imap(bsr,v));//eliminate does not need a |
---|
| 1132 | //standard basis as input. |
---|
| 1133 | } |
---|
| 1134 | if(choose==2) |
---|
| 1135 | { |
---|
| 1136 | ideal kern= nselect(groebner(dom),1..n);//"nselect" is combinatorial command |
---|
| 1137 | //which uses the internal command |
---|
| 1138 | // "simplify" |
---|
| 1139 | } |
---|
| 1140 | |
---|
| 1141 | //--------- test wether @(0)-h(@(1),...,@(z)) is in ker --------------- |
---|
| 1142 | // for some poly h and divide by maximal power of q=#[1] |
---|
| 1143 | poly h; |
---|
| 1144 | z=size(kern); |
---|
| 1145 | for (i=1;i<=z;i++) |
---|
| 1146 | { |
---|
| 1147 | h=kern[i]/@(0); |
---|
| 1148 | if (deg(h)==0) |
---|
| 1149 | { |
---|
| 1150 | h=(1/h)*kern[i]; |
---|
| 1151 | // define the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
---|
| 1152 | setring bsr; |
---|
| 1153 | map psi=s,maxideal(1),p,dom; |
---|
| 1154 | poly re=psi(h); |
---|
| 1155 | // divide by the maximal power of #[1] |
---|
| 1156 | if ((size(#)>0) && (typeof(#[1])== "poly") ) |
---|
| 1157 | { |
---|
| 1158 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
---|
| 1159 | { |
---|
| 1160 | re=re/#[1]; |
---|
| 1161 | } |
---|
| 1162 | } |
---|
| 1163 | return(re); |
---|
| 1164 | } |
---|
| 1165 | } |
---|
| 1166 | setring bsr; |
---|
| 1167 | return(p); |
---|
| 1168 | } |
---|
| 1169 | } |
---|
| 1170 | example |
---|
| 1171 | {"EXAMPLE:"; echo = 2; |
---|
| 1172 | ring r= 0,(x,y),dp; |
---|
| 1173 | ideal dom =x2,y2,xy-y; |
---|
| 1174 | poly p=x4+x3y+xy2-y2; |
---|
| 1175 | sagbiReduction(p,dom); |
---|
| 1176 | sagbiReduction(p,dom,2); |
---|
| 1177 | // now let us see the action over quotient ring |
---|
| 1178 | ideal I = xy; |
---|
| 1179 | qring Q = std(I); |
---|
| 1180 | ideal dom = imap(r,dom); poly p = imap(r,p); |
---|
| 1181 | sagbiReduction(p,dom,1); |
---|
| 1182 | } |
---|
| 1183 | |
---|
| 1184 | proc sagbiNF(id,ideal dom,int k,list#) |
---|
| 1185 | "USAGE: sagbiNF(id,dom,k[,n]); id either poly or ideal,dom ideal, k and n positive intergers. |
---|
| 1186 | RETURN: same as type of id; ideal or polynomial. |
---|
| 1187 | PURPOSE: |
---|
| 1188 | @format |
---|
| 1189 | The integer k determines what kind of s-reduction is performed: |
---|
| 1190 | - if (k=0) no tail s-reduction is performed. |
---|
| 1191 | - if (k=1) tail s-reduction is performed. |
---|
| 1192 | Three Algorithm variants are used to perform subalgebra reduction. |
---|
| 1193 | The positive integer n determines which variant should be used. |
---|
| 1194 | n may take the values (0 or default),1 or 2. |
---|
| 1195 | @end format |
---|
| 1196 | NOTE: sagbiNF works over both rings and quotient rings |
---|
| 1197 | EXAMPLE: example sagbiNF; show example " |
---|
| 1198 | { |
---|
| 1199 | ideal rs; |
---|
| 1200 | if (ideal(basering) == 0) |
---|
| 1201 | { |
---|
| 1202 | rs = sagbiReduce(id,dom,k) ; |
---|
| 1203 | } |
---|
| 1204 | else |
---|
| 1205 | { |
---|
| 1206 | rs = sagbiReduction(id,dom,k) ; |
---|
| 1207 | } |
---|
| 1208 | return(rs); |
---|
| 1209 | } |
---|
| 1210 | example |
---|
| 1211 | {"EXAMPLE:"; echo = 2; |
---|
| 1212 | ring r=0,(x,y),dp; |
---|
| 1213 | poly p=x4+x2y+y; |
---|
| 1214 | ideal dom =x2,x2y+y,x3y2; |
---|
| 1215 | sagbiNF(p,dom,1); |
---|
| 1216 | ideal I= x2-xy; |
---|
| 1217 | qring Q=std(I); // we go to the quotient ring |
---|
| 1218 | poly p=imap(r,p); |
---|
| 1219 | NF(p,std(0)); // the representative of p has changed |
---|
| 1220 | ideal dom = imap(r,dom); |
---|
| 1221 | print(matrix(NF(dom,std(0)))); // dom has changed as well |
---|
| 1222 | sagbiNF(p,dom,0); // no tail reduction |
---|
| 1223 | sagbiNF(p,dom,1);// tail subalgebra reduction is performed |
---|
| 1224 | } |
---|
| 1225 | |
---|
| 1226 | static proc canonicalform(ideal I) |
---|
| 1227 | { |
---|
| 1228 | /* placeholder for the canonical form of a set of gen's */ |
---|
| 1229 | /* for the time being we agree on content(p)=1; that is coeffs with no fractions */ |
---|
| 1230 | int i; ideal J=I; |
---|
| 1231 | for(i=ncols(I); i>=1; i--) |
---|
| 1232 | { |
---|
| 1233 | J[i] = canonicalform_poly(I[i]); |
---|
| 1234 | } |
---|
| 1235 | return(J); |
---|
| 1236 | } |
---|
| 1237 | |
---|
| 1238 | static proc canonicalform_poly(poly p) |
---|
| 1239 | { |
---|
| 1240 | /* placeholder for the canonical form of a poly */ |
---|
| 1241 | /* for the time being we agree on content(p)=1; that is coeffs with no fractions */ |
---|
| 1242 | number n = content(p); |
---|
| 1243 | return( p/content(p) ); |
---|
| 1244 | } |
---|
| 1245 | |
---|
| 1246 | /* |
---|
| 1247 | ring r= 0,(x,y),dp; |
---|
| 1248 | //The following algebra does not have a finite SAGBI basis. |
---|
| 1249 | ideal J=x^2, xy-y2, xy2, x^2*(x*y-y^2)^2 - (x*y^2)^2*x^4 + 11; |
---|
| 1250 | //--------------------------------------------------- |
---|
| 1251 | //Call with two iterations |
---|
| 1252 | def DI = algebraicDependence(J,2); |
---|
| 1253 | setring DI; algDep; |
---|
| 1254 | */ |
---|