1 | // -*- Mode: C++; tab-width: 2; indent-tabs-mode: nil; c-basic-offset: 2 -*- |
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2 | // Vi-modeline: vim: filetype=c:syntax:shiftwidth=2:tabstop=8:textwidth=0:expandtab |
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3 | /////////////////////////////////////////////////////////////////// |
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4 | version="$Id$"; |
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5 | category="Combinatorial Commutative Algebra"; |
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6 | info=" |
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7 | LIBRARY: multigrading.lib Multigraded Rings |
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8 | |
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9 | AUTHORS: Benjamin Bechtold, benjamin.bechtold@googlemail.com |
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10 | @* Rene Birkner, rbirkner@math.fu-berlin.de |
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11 | @* Lars Kastner, lkastner@math.fu-berlin.de |
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12 | @* Simon Keicher, keicher@mail.mathematik.uni-tuebingen.de |
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13 | @* Oleksandr Motsak, U@D, where U={motsak}, D={mathematik.uni-kl.de} |
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14 | |
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15 | |
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16 | OVERVIEW: This library allows one to virtually add multigradings to Singular. |
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17 | For more see http://code.google.com/p/convex-singular/wiki/Multigrading |
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18 | For theoretical references see: |
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19 | E. Miller, B. Sturmfels: 'Combinatorial Commutative Algebra' and |
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20 | M. Kreuzer, L. Robbiano: 'Computational Commutative Algebra'. |
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21 | |
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22 | NOTE: 'mDegBasis' relies on 4ti2 for computing Hilbert Bases. |
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23 | All groups are finitely generated Abelian |
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24 | |
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25 | PROCEDURES: |
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26 | setBaseMultigrading(M,L); attach multiweights/grading group matrices to the basering |
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27 | getVariableWeights([R]); get matrix of multidegrees of vars attached to a ring |
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28 | |
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29 | getGradingGroup([R]); get grading group attached to a ring |
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30 | getLattice([R[,choice]]); get grading group' lattice attached to a ring (or its NF) |
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31 | |
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32 | createGroup(S,L); create a group generated by S, with relations L |
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33 | createQuotientGroup(L); create a group generated by the unit matrix whith relations L |
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34 | createTorsionFreeGroup(S); create a group generated by S which is torsionfree |
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35 | printGroup(G); print a group |
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36 | |
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37 | areIsomorphicGroups(G,H); test wheter G an H are isomorphic groups (TODO Tuebingen) |
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38 | isGroup(G); test whether G is a valid group |
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39 | isGroupHomomorphism(L1,L2,A); test wheter A defines a group homomrphism from L1 to L2 |
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40 | |
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41 | isGradedRingHomomorphism(R,f,A); test graded ring homomorph |
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42 | createGradedRingHomomorphism(R,f,A); create a graded ring homomorph |
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43 | |
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44 | setModuleGrading(M,v); attach multiweights of units to a module and return it |
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45 | getModuleGrading(M); get multiweights of module units (attached to M) |
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46 | |
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47 | isSublattice(A,B); test whether A is a sublattice of B |
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48 | imageLattice(P,L); computes an integral basis for the image of the lattice L under the linear map P. |
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49 | intRank(A); computes the rank of the intmat A |
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50 | kernelLattice(P); computes an integral basis for the kernel of the linear map P. |
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51 | latticeBasis(B); compute an integral basis of the lattice B |
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52 | preimageLattice(P,L); computes an integral basis for the preimage of the lattice L under the linear map P. |
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53 | projectLattices(B); compute a linear map of lattices having the primitive span of B as its kernel. |
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54 | intersectLattices(A,B); compute an integral basis for the intersection of the lattices A and B. |
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55 | isIntegralSurjective(P); test whether the map P of lattices is surjective. |
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56 | isPrimitiveSublattice(A); test whether A generates a primitive sublattice. |
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57 | intInverse(A); compute the integral inverse matrix of the intmat A |
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58 | intAdjoint(A,i,j); delete row i and column j of the intmat A. |
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59 | integralSection(P); for a given linear surjective map P of lattices this procedure returns an integral section of P. |
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60 | primitiveSpan(A); compute a basis for the minimal primitive sublattice that contains the given vectors (by A). |
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61 | |
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62 | factorgroup(G,H); create the group G mod H |
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63 | productgroup(G,H); create the group G x H |
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64 | |
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65 | mDeg(A); compute the multidegree of A |
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66 | mDegBasis(d); compute all monomials of multidegree d |
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67 | mDegPartition(p); compute the multigraded-homogeneous components of p |
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68 | |
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69 | isTorsionFree(); test whether the current multigrading is free |
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70 | isPositive(); test whether the current multigrading is positive |
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71 | isZeroElement(p); test whether p has zero multidegree |
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72 | areZeroElements(M); test whether an integer matrix M considered as a collection of columns has zero multidegree |
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73 | isHomogeneous(a); test whether 'a' is multigraded-homogeneous |
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74 | |
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75 | equalMDeg(e1,e2[,V]); test whether e1==e2 in the current multigrading |
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76 | |
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77 | mDegGroebner(M); compute the multigraded GB/SB of M |
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78 | mDegSyzygy(M); compute the multigraded syzygies of M |
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79 | mDegModulo(I,J); compute the multigraded 'modulo' module of I and J |
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80 | mDegResolution(M,l[,m]); compute the multigraded resolution of M |
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81 | mDegTensor(m,n); compute the tensor product of multigraded modules m,n |
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82 | mDegTor(i,m,n); compute the Tor_i(m,n) for multigraded modules m,n |
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83 | |
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84 | defineHomogeneous(p); get a grading group wrt which p becomes homogeneous |
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85 | pushForward(f); find the finest grading on the image ring, homogenizing f |
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86 | gradiator(h); coarsens grading of the ring until h becomes homogeneous |
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87 | |
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88 | hermiteNormalForm(A); compute the Hermite Normal Form of a matrix |
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89 | smithNormalForm(A,#); compute matrices D,P,Q with D=P*A*Q and D is the smith normal form of A |
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90 | |
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91 | hilbertSeries(M); compute the multigraded Hilbert Series of M |
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92 | evalHilbertSeries(h,v); evaluate hilberts series h by substituting v[i] for t_(i) |
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93 | |
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94 | lll(A); applies LLL(.) of lll.lib which only works for lists on a matrix A |
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95 | |
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96 | (parameters in square brackets [] are optional) |
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97 | |
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98 | KEYWORDS: multigrading, multidegree, multiweights, multigraded-homogeneous, integral linear algebra |
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99 | "; |
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100 | |
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101 | // finestMDeg(def r) |
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102 | // newMap(map F, intmat Q, list #) |
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103 | |
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104 | LIB "standard.lib"; // for groebner |
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105 | LIB "lll.lib"; // for lll_matrix |
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106 | LIB "matrix.lib"; // for mDegTor |
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107 | |
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108 | /******************************************************/ |
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109 | |
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110 | static proc concatintmat(intmat A, intmat B) |
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111 | { |
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112 | |
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113 | if ( nrows(A) != nrows(B) ) |
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114 | { |
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115 | ERROR("matrices A and B have different number of rows."); |
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116 | } |
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117 | |
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118 | intmat At = transpose(A); |
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119 | intmat Bt = transpose(B); |
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120 | |
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121 | intmat Ct[nrows(At) + nrows(Bt)][ncols(At)] = At, Bt; |
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122 | |
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123 | return(transpose(Ct)); |
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124 | } |
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125 | |
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126 | |
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127 | /******************************************************/ |
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128 | proc createGradedRingHomomorphism(def src, ideal Im, def A) |
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129 | "USAGE: createGradedRingHomomorphism(R, f, A); ring R, ideal f, group homomorphism A |
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130 | PURPOSE: create a multigraded group ring homomorphism defined by |
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131 | a ring map from R to the current ring, given by generators images f |
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132 | and a group homomorphism A between grading groups |
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133 | RETURN: graded ring homorphism |
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134 | EXAMPLE: example createGradedRingHomomorphism; shows an example |
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135 | " |
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136 | { |
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137 | string isGRH = "isGRH"; |
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138 | |
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139 | if( !isGradedRingHomomorphism(def src, ideal Im, def A) ) |
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140 | { |
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141 | ERROR("Input data is wrong"); |
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142 | } |
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143 | |
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144 | list h; |
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145 | h[3] = A; |
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146 | |
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147 | // map f = src, Im; |
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148 | h[2] = Im; // f? |
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149 | h[1] = src; |
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150 | |
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151 | attrib(h, isGRH, (1==1)); // mark it "a graded ring homomorphism" |
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152 | |
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153 | return(h); |
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154 | } |
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155 | example |
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156 | { |
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157 | "EXAMPLE:"; echo=2; |
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158 | |
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159 | // TODO! |
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160 | |
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161 | } |
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162 | |
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163 | |
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164 | /******************************************************/ |
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165 | proc isGradedRingHomomorphism(def src, ideal Im, def A) |
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166 | "USAGE: isGradedRingHomomorphism(R, f, A); ring R, ideal f, group homomorphism A |
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167 | PURPOSE: test a multigraded group ring homomorphism defined by |
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168 | a ring map from R to the current ring, given by generators images f |
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169 | and a group homomorphism A between grading groups |
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170 | RETURN: int, 1 for TRUE, 0 otherwise |
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171 | EXAMPLE: example isGradedRingHomomorphism; shows an example |
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172 | " |
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173 | { |
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174 | def dst = basering; |
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175 | |
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176 | intmat result_degs = mDeg(Im); |
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177 | print(result_degs); |
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178 | |
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179 | setring src; |
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180 | |
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181 | intmat input_degs = mDeg(maxideal(1)); |
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182 | print(input_degs); |
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183 | |
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184 | def image_degs = A * input_degs; |
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185 | print( image_degs ); |
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186 | |
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187 | def df = image_degs - result_degs; |
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188 | print(df); |
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189 | |
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190 | setring dst; |
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191 | |
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192 | return (areZeroElements( df )); |
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193 | } |
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194 | example |
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195 | { |
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196 | "EXAMPLE:"; echo=2; |
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197 | |
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198 | ring r = 0, (x, y, z), dp; |
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199 | intmat S1[3][3] = |
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200 | 1, 0, 0, |
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201 | 0, 1, 0, |
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202 | 0, 0, 1; |
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203 | intmat L1[3][1] = |
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204 | 0, |
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205 | 0, |
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206 | 0; |
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207 | |
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208 | def G1 = createGroup(S1, L1); // (S1 + L1)/L1 |
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209 | printGroup(G1); |
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210 | |
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211 | setBaseMultigrading(S1, L1); // to change... |
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212 | |
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213 | ring R = 0, (a, b, c), dp; |
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214 | intmat S2[2][3] = |
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215 | 1, 0, |
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216 | 0, 1; |
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217 | intmat L2[2][1] = |
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218 | 0, |
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219 | 2; |
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220 | |
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221 | def G2 = createGroup(S2, L2); |
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222 | printGroup(G2); |
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223 | |
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224 | setBaseMultigrading(S2, L2); // to change... |
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225 | |
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226 | |
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227 | map F = r, a, b, c; |
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228 | intmat A[nrows(L2)][nrows(L1)] = |
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229 | 1, 0, 0, |
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230 | 3, 2, -6; |
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231 | |
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232 | // graded ring homomorphism is given by (compatible): |
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233 | print(F); |
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234 | print(A); |
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235 | |
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236 | isGradedRingHomomorphism(r, ideal(F), A); |
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237 | def h = createGradedRingHomomorphism(r, ideal(F), A); |
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238 | |
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239 | print(h); |
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240 | |
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241 | // not a homo.. |
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242 | intmat B[nrows(L2)][nrows(L1)] = |
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243 | 1, 1, 1, |
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244 | 0, 0, 0; |
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245 | print(B); |
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246 | |
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247 | isGradedRingHomomorphism(r, ideal(F), B); |
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248 | def hh = createGradedRingHomomorphism(r, ideal(F), B); |
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249 | |
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250 | if( defined(hh) ) { ERROR("That input was not valid"); } |
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251 | } |
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252 | |
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253 | |
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254 | proc createQuotientGroup(intmat L) |
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255 | " |
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256 | L - relations |
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257 | TODO: bad name |
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258 | " |
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259 | { |
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260 | int r = nrows(L); int i; |
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261 | intmat S[r][r]; // SQUARE!!! |
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262 | for(i = r; i > 0; i--){ S[i, i] = 1; } |
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263 | return (createGroup(S,L)); |
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264 | } |
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265 | |
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266 | proc createTorsionFreeGroup(intmat S) |
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267 | " |
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268 | S - generators |
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269 | TODO: bad name |
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270 | " |
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271 | { |
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272 | int r = nrows(S); int i; |
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273 | intmat L[r][1] = 0; |
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274 | return (createGroup(S,L)); |
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275 | } |
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276 | |
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277 | |
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278 | /******************************************************/ |
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279 | proc createGroup(intmat S, intmat L) |
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280 | "USAGE: createGroup(S, L); S, L are integer matrices |
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281 | PURPOSE: create the group of the form (S+L)/L, i.e. |
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282 | S specifies generators, L specifies relations. |
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283 | RETURN: group |
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284 | EXAMPLE: example createGroup; shows an example |
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285 | " |
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286 | { |
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287 | string isGroup = "isGroup"; |
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288 | string attrGroupHNF = "hermite"; |
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289 | string attrGroupSNF = "smith"; |
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290 | |
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291 | |
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292 | /* |
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293 | if( size(#) > 0 ) |
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294 | { |
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295 | if( typeof(#[1]) == "intmat" ) |
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296 | { |
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297 | intmat S = #[1]; |
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298 | } else { ERROR("Wrong optional argument: 1"); } |
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299 | |
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300 | if( size(#) > 1 ) |
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301 | { |
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302 | if( typeof(#[2]) == "intmat" ) |
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303 | { |
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304 | intmat L = #[2]; |
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305 | } else { ERROR("Wrong optional argument: 2"); } |
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306 | } |
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307 | } |
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308 | if( !defined(S) ) |
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309 | {} |
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310 | */ |
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311 | |
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312 | if( nrows(L) != nrows(S) ) |
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313 | { |
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314 | ERROR("Incompatible matrices!"); |
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315 | } |
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316 | |
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317 | def H = attrib(L, attrGroupHNF); |
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318 | if( !defined(H) || typeof(H) != "intmat") |
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319 | { |
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320 | attrib(L, attrGroupHNF, hermiteNormalForm(L)); |
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321 | } else { kill H; } |
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322 | |
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323 | def HH = attrib(L, attrGroupSNF); |
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324 | if( !defined(HH) || typeof(HH) != "intmat") |
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325 | { |
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326 | attrib(L, attrGroupSNF, smithNormalForm(L)); |
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327 | } else { kill HH; } |
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328 | |
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329 | list G; // Please, note the order: Generators + Relations: |
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330 | G[1] = S; G[2] = L; |
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331 | |
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332 | attrib(G, isGroup, (1==1)); // mark it "a group" |
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333 | |
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334 | return (G); |
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335 | } |
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336 | example |
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337 | { |
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338 | "EXAMPLE:"; echo=2; |
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339 | |
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340 | intmat S[3][3] = |
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341 | 1, 0, 0, |
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342 | 0, 1, 0, |
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343 | 0, 0, 1; |
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344 | |
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345 | intmat L[3][2] = |
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346 | 1, 1, |
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347 | 1, 3, |
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348 | 1, 5; |
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349 | |
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350 | def G = createGroup(S, L); // (S+L)/L |
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351 | |
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352 | printGroup(G); |
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353 | |
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354 | kill S, L, G; |
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355 | |
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356 | ///////////////////////////////////////////////// |
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357 | intmat S[2][3] = |
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358 | 1, -2, 1, |
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359 | 1, 1, 0; |
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360 | |
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361 | intmat L[2][1] = |
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362 | 0, |
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363 | 2; |
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364 | |
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365 | def G = createGroup(S, L); // (S+L)/L |
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366 | |
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367 | printGroup(G); |
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368 | |
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369 | kill S, L, G; |
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370 | |
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371 | // ----------- extreme case ------------ // |
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372 | intmat S[1][3] = |
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373 | 1, -1, 10; |
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374 | |
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375 | // Torsion: |
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376 | intmat L[1][1] = |
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377 | 0; |
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378 | |
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379 | def G = createGroup(S, L); // (S+L)/L |
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380 | |
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381 | printGroup(G); |
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382 | } |
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383 | |
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384 | |
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385 | /******************************************************/ |
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386 | proc printGroup(def G) |
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387 | "USAGE: printGroup(G); G is a group |
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388 | PURPOSE: prints the group G |
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389 | RETURN: nothing |
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390 | EXAMPLE: example printGroup; shows an example |
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391 | " |
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392 | { |
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393 | "Generators: "; |
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394 | print(G[1]); |
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395 | |
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396 | "Relations: "; |
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397 | print(G[2]); |
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398 | |
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399 | // attrib(G[2]); |
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400 | } |
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401 | example |
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402 | { |
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403 | "EXAMPLE:"; echo=2; |
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404 | |
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405 | } |
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406 | |
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407 | /******************************************************/ |
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408 | proc areIsomorphicGroups(def G, def H) |
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409 | "USAGE: areIsomorphicGroups(G, H); G and H are groups |
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410 | PURPOSE: ? |
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411 | RETURN: int, 1 for TRUE, 0 otherwise |
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412 | EXAMPLE: example areIsomorphicGroups; shows an example |
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413 | " |
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414 | { |
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415 | return (1); // TRUE |
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416 | } |
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417 | example |
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418 | { |
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419 | "EXAMPLE:"; echo=2; |
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420 | |
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421 | } |
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422 | |
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423 | |
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424 | proc isGroup(def G) |
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425 | "test whether G is a valid group" |
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426 | { |
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427 | string isGroup = "isGroup"; |
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428 | |
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429 | // valid? |
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430 | if( typeof(G) != "list" ){ return(0); } |
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431 | |
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432 | def a = attrib(G, isGroup); |
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433 | |
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434 | ///// TODO for Hans: fix attr^2 bug in Singular! |
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435 | |
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436 | // if( !defined(a) ) { return(0); } |
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437 | // if( typeof(a) != "int" ) { return(0); } |
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438 | // if( !a ){ return(0); } |
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439 | |
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440 | |
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441 | if( size(G) != 2 ){ return(0); } |
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442 | if( typeof(G[1]) != "intmat" ){ return(0); } |
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443 | if( typeof(G[2]) != "intmat" ){ return(0); } |
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444 | if( nrows(G[1]) != nrows(G[2]) ){ return(0); } |
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445 | |
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446 | return(1==1); |
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447 | } |
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448 | |
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449 | |
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450 | |
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451 | /******************************************************/ |
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452 | proc setBaseMultigrading(intmat M, list #) |
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453 | "USAGE: setBaseMultigrading(M[, G]); M is an intege matrix, G is a group (or lattice) |
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454 | PURPOSE: attaches weights of variables and grading group to the basering. |
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455 | NOTE: M encodes the weights of variables column-wise. |
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456 | RETURN: nothing |
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457 | EXAMPLE: example setBaseMultigrading; shows an example |
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458 | " |
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459 | { |
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460 | string attrMgrad = "mgrad"; |
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461 | string attrGradingGroup = "gradingGroup"; |
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462 | |
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463 | if( size(#) > 0 ) |
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464 | { |
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465 | if( typeof(#[1]) == "intmat" ) |
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466 | { |
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467 | def L = createGroup(M, #[1]); |
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468 | } |
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469 | |
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470 | if( isGroup(#[1]) ) |
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471 | { |
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472 | def L = #[1]; |
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473 | |
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474 | if( !isSublattice(M, L[1]) ) |
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475 | { |
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476 | ERROR("Multigrading is not contained in the grading group!"); |
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477 | } |
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478 | } |
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479 | } |
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480 | else |
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481 | { |
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482 | def L = createTorsionFreeGroup(M); |
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483 | } |
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484 | |
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485 | if( !defined(L) ){ ERROR("Wrong arguments: no group given?"); } |
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486 | |
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487 | |
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488 | |
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489 | attrib(basering, attrMgrad, M); |
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490 | attrib(basering, attrGradingGroup, L); |
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491 | |
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492 | ideal Q = ideal(basering); |
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493 | if( !isHomogeneous(Q) ) // easy now, but would be hard before setting ring attributes! |
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494 | { |
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495 | "Warning: your quotient ideal is not homogenous (multigrading was set anyway)!"; |
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496 | } |
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497 | |
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498 | |
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499 | |
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500 | |
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501 | } |
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502 | example |
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503 | { |
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504 | "EXAMPLE:"; echo=2; |
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505 | |
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506 | ring R = 0, (x, y, z), dp; |
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507 | |
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508 | // Weights of variables |
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509 | intmat M[3][3] = |
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510 | 1, 0, 0, |
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511 | 0, 1, 0, |
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512 | 0, 0, 1; |
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513 | |
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514 | // GradingGroup: |
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515 | intmat L[3][2] = |
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516 | 1, 1, |
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517 | 1, 3, |
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518 | 1, 5; |
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519 | |
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520 | // attaches M & L to R (==basering): |
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521 | setBaseMultigrading(M, L); // Grading: Z^3/L |
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522 | |
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523 | // Weights are accessible via "getVariableWeights()": |
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524 | getVariableWeights(); |
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525 | |
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526 | // Test all possible usages: |
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527 | (getVariableWeights() == M) && (getVariableWeights(R) == M) && (getVariableWeights(basering) == M); |
---|
528 | |
---|
529 | // Grading group is accessible via "getLattice()": |
---|
530 | getLattice(); |
---|
531 | |
---|
532 | // Test all possible usages: |
---|
533 | (getLattice() == L) && (getLattice(R) == L) && (getLattice(basering) == L); |
---|
534 | |
---|
535 | // And its hermite NF via getLattice("hermite"): |
---|
536 | getLattice("hermite"); |
---|
537 | |
---|
538 | // Test all possible usages: |
---|
539 | intmat H = hermiteNormalForm(L); |
---|
540 | (getLattice("hermite") == H) && (getLattice(R, "hermite") == H) && (getLattice(basering, "hermite") == H); |
---|
541 | |
---|
542 | kill L, M; |
---|
543 | |
---|
544 | // ----------- isomorphic multigrading -------- // |
---|
545 | |
---|
546 | // Weights of variables |
---|
547 | intmat M[2][3] = |
---|
548 | 1, -2, 1, |
---|
549 | 1, 1, 0; |
---|
550 | |
---|
551 | // Torsion: |
---|
552 | intmat L[2][1] = |
---|
553 | 0, |
---|
554 | 2; |
---|
555 | |
---|
556 | // attaches M & L to R (==basering): |
---|
557 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
558 | |
---|
559 | // Weights are accessible via "getVariableWeights()": |
---|
560 | getVariableWeights() == M; |
---|
561 | |
---|
562 | // Torsion is accessible via "getLattice()": |
---|
563 | getLattice() == L; |
---|
564 | |
---|
565 | kill L, M; |
---|
566 | // ----------- extreme case ------------ // |
---|
567 | |
---|
568 | // Weights of variables |
---|
569 | intmat M[1][3] = |
---|
570 | 1, -1, 10; |
---|
571 | |
---|
572 | // Torsion: |
---|
573 | intmat L[1][1] = |
---|
574 | 0; |
---|
575 | |
---|
576 | // attaches M & L to R (==basering): |
---|
577 | setBaseMultigrading(M); // Grading: Z^3 |
---|
578 | |
---|
579 | // Weights are accessible via "getVariableWeights()": |
---|
580 | getVariableWeights() == M; |
---|
581 | |
---|
582 | // Torsion is accessible via "getLattice()": |
---|
583 | getLattice() == L; |
---|
584 | } |
---|
585 | |
---|
586 | |
---|
587 | /******************************************************/ |
---|
588 | proc getVariableWeights(list #) |
---|
589 | "USAGE: getVariableWeights([R]) |
---|
590 | PURPOSE: get associated multigrading matrix for the basering [or R] |
---|
591 | RETURN: intmat, matrix of multidegrees of variables |
---|
592 | EXAMPLE: example getVariableWeights; shows an example |
---|
593 | " |
---|
594 | { |
---|
595 | string attrMgrad = "mgrad"; |
---|
596 | |
---|
597 | |
---|
598 | if( size(#) > 0 ) |
---|
599 | { |
---|
600 | if(( typeof(#[1]) == "ring" ) || ( typeof(#[1]) == "qring" )) |
---|
601 | { |
---|
602 | def R = #[1]; |
---|
603 | } |
---|
604 | else |
---|
605 | { |
---|
606 | ERROR("Optional argument must be a ring!"); |
---|
607 | } |
---|
608 | } |
---|
609 | else |
---|
610 | { |
---|
611 | def R = basering; |
---|
612 | } |
---|
613 | |
---|
614 | def M = attrib(R, attrMgrad); |
---|
615 | if( typeof(M) == "intmat"){ return (M); } |
---|
616 | ERROR( "Sorry no multigrading matrix!" ); |
---|
617 | } |
---|
618 | example |
---|
619 | { |
---|
620 | "EXAMPLE:"; echo=2; |
---|
621 | |
---|
622 | ring R = 0, (x, y, z), dp; |
---|
623 | |
---|
624 | // Weights of variables |
---|
625 | intmat M[3][3] = |
---|
626 | 1, 0, 0, |
---|
627 | 0, 1, 0, |
---|
628 | 0, 0, 1; |
---|
629 | |
---|
630 | // Grading group: |
---|
631 | intmat L[3][2] = |
---|
632 | 1, 1, |
---|
633 | 1, 3, |
---|
634 | 1, 5; |
---|
635 | |
---|
636 | // attaches M & L to R (==basering): |
---|
637 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
638 | |
---|
639 | // Weights are accessible via "getVariableWeights()": |
---|
640 | getVariableWeights() == M; |
---|
641 | |
---|
642 | kill L, M; |
---|
643 | |
---|
644 | // ----------- isomorphic multigrading -------- // |
---|
645 | |
---|
646 | // Weights of variables |
---|
647 | intmat M[2][3] = |
---|
648 | 1, -2, 1, |
---|
649 | 1, 1, 0; |
---|
650 | |
---|
651 | // Grading group: |
---|
652 | intmat L[2][1] = |
---|
653 | 0, |
---|
654 | 2; |
---|
655 | |
---|
656 | // attaches M & L to R (==basering): |
---|
657 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
658 | |
---|
659 | // Weights are accessible via "getVariableWeights()": |
---|
660 | getVariableWeights() == M; |
---|
661 | |
---|
662 | kill L, M; |
---|
663 | |
---|
664 | // ----------- extreme case ------------ // |
---|
665 | |
---|
666 | // Weights of variables |
---|
667 | intmat M[1][3] = |
---|
668 | 1, -1, 10; |
---|
669 | |
---|
670 | // Grading group: |
---|
671 | intmat L[1][1] = |
---|
672 | 0; |
---|
673 | |
---|
674 | // attaches M & L to R (==basering): |
---|
675 | setBaseMultigrading(M); // Grading: Z^3 |
---|
676 | |
---|
677 | // Weights are accessible via "getVariableWeights()": |
---|
678 | getVariableWeights() == M; |
---|
679 | } |
---|
680 | |
---|
681 | |
---|
682 | proc getGradingGroup(list #) |
---|
683 | "USAGE: getGradingGroup([R]) |
---|
684 | PURPOSE: get associated grading group |
---|
685 | RETURN: group, the grading group |
---|
686 | EXAMPLE: example getGradingGroup; shows an example |
---|
687 | " |
---|
688 | { |
---|
689 | string attrGradingGroup = "gradingGroup"; |
---|
690 | |
---|
691 | int i = 1; |
---|
692 | |
---|
693 | if( size(#) >= i ) |
---|
694 | { |
---|
695 | if( ( typeof(#[i]) == "ring" ) or ( typeof(#[i]) == "qring" ) ) |
---|
696 | { |
---|
697 | def R = #[i]; |
---|
698 | i++; |
---|
699 | } |
---|
700 | } |
---|
701 | |
---|
702 | if( !defined(R) ) |
---|
703 | { |
---|
704 | def R = basering; |
---|
705 | } |
---|
706 | |
---|
707 | def G = attrib(R, attrGradingGroup); |
---|
708 | |
---|
709 | if( !isGroup(G) ) |
---|
710 | { |
---|
711 | ERROR("Sorry no grading group!"); |
---|
712 | } |
---|
713 | |
---|
714 | return(G); |
---|
715 | } |
---|
716 | example |
---|
717 | { |
---|
718 | "EXAMPLE:"; echo=2; |
---|
719 | |
---|
720 | ring R = 0, (x, y, z), dp; |
---|
721 | |
---|
722 | // Weights of variables |
---|
723 | intmat M[3][3] = |
---|
724 | 1, 0, 0, |
---|
725 | 0, 1, 0, |
---|
726 | 0, 0, 1; |
---|
727 | |
---|
728 | // Torsion: |
---|
729 | intmat L[3][2] = |
---|
730 | 1, 1, |
---|
731 | 1, 3, |
---|
732 | 1, 5; |
---|
733 | |
---|
734 | // attaches M & L to R (==basering): |
---|
735 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
736 | |
---|
737 | def G = getGradingGroup(); |
---|
738 | |
---|
739 | printGroup( G ); |
---|
740 | |
---|
741 | G[1] == M; G[2] == L; |
---|
742 | |
---|
743 | kill L, M, G; |
---|
744 | |
---|
745 | // ----------- isomorphic multigrading -------- // |
---|
746 | |
---|
747 | // Weights of variables |
---|
748 | intmat M[2][3] = |
---|
749 | 1, -2, 1, |
---|
750 | 1, 1, 0; |
---|
751 | |
---|
752 | // Torsion: |
---|
753 | intmat L[2][1] = |
---|
754 | 0, |
---|
755 | 2; |
---|
756 | |
---|
757 | // attaches M & L to R (==basering): |
---|
758 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
759 | |
---|
760 | def G = getGradingGroup(); |
---|
761 | |
---|
762 | printGroup( G ); |
---|
763 | |
---|
764 | G[1] == M; G[2] == L; |
---|
765 | |
---|
766 | kill L, M, G; |
---|
767 | // ----------- extreme case ------------ // |
---|
768 | |
---|
769 | // Weights of variables |
---|
770 | intmat M[1][3] = |
---|
771 | 1, -1, 10; |
---|
772 | |
---|
773 | // Torsion: |
---|
774 | intmat L[1][1] = |
---|
775 | 0; |
---|
776 | |
---|
777 | // attaches M & L to R (==basering): |
---|
778 | setBaseMultigrading(M); // Grading: Z^3 |
---|
779 | |
---|
780 | def G = getGradingGroup(); |
---|
781 | |
---|
782 | printGroup( G ); |
---|
783 | |
---|
784 | G[1] == M; G[2] == L; |
---|
785 | |
---|
786 | kill L, M, G; |
---|
787 | } |
---|
788 | |
---|
789 | |
---|
790 | /******************************************************/ |
---|
791 | proc getLattice(list #) |
---|
792 | "USAGE: getLattice([R[,opt]]) |
---|
793 | PURPOSE: get associated grading group matrix, i.e. generators (cols) of the grading group |
---|
794 | RETURN: intmat, the grading group matrix, or |
---|
795 | its hermite normal form if an optional argument (\"hermiteNormalForm\") is given or |
---|
796 | smith normal form if an optional argument (\"smith\") is given |
---|
797 | EXAMPLE: example getLattice; shows an example |
---|
798 | " |
---|
799 | { |
---|
800 | int i = 1; |
---|
801 | if( size(#) >= i ) |
---|
802 | { |
---|
803 | if( ( typeof(#[i]) == "ring" ) or ( typeof(#[i]) == "qring" ) ) |
---|
804 | { |
---|
805 | i++; |
---|
806 | } |
---|
807 | } |
---|
808 | |
---|
809 | string attrGradingGroupHNF = "hermite"; |
---|
810 | string attrGradingGroupSNF = "smith"; |
---|
811 | |
---|
812 | def G = getGradingGroup(#); |
---|
813 | |
---|
814 | // printGroup(G); |
---|
815 | |
---|
816 | |
---|
817 | |
---|
818 | def T = G[2]; |
---|
819 | |
---|
820 | if( size(#) >= i ) |
---|
821 | { |
---|
822 | if( #[i] == "hermite" ) |
---|
823 | { |
---|
824 | def M = attrib(T, attrGradingGroupHNF); |
---|
825 | if( (!defined(M)) or (typeof(M) != "intmat") ) |
---|
826 | { |
---|
827 | M = hermiteNormalForm(T); |
---|
828 | } |
---|
829 | return (M); |
---|
830 | } |
---|
831 | |
---|
832 | if( #[i] == "smith" ) |
---|
833 | { |
---|
834 | def M = attrib(T, attrGradingGroupSNF); |
---|
835 | if( (!defined(M)) or (typeof(M) != "intmat") ) |
---|
836 | { |
---|
837 | M = smithNormalForm(T); |
---|
838 | } |
---|
839 | return (M); |
---|
840 | } |
---|
841 | } |
---|
842 | |
---|
843 | return(T); |
---|
844 | } |
---|
845 | example |
---|
846 | { |
---|
847 | "EXAMPLE:"; echo=2; |
---|
848 | |
---|
849 | ring R = 0, (x, y, z), dp; |
---|
850 | |
---|
851 | // Weights of variables |
---|
852 | intmat M[3][3] = |
---|
853 | 1, 0, 0, |
---|
854 | 0, 1, 0, |
---|
855 | 0, 0, 1; |
---|
856 | |
---|
857 | // Torsion: |
---|
858 | intmat L[3][2] = |
---|
859 | 1, 1, |
---|
860 | 1, 3, |
---|
861 | 1, 5; |
---|
862 | |
---|
863 | // attaches M & L to R (==basering): |
---|
864 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
865 | |
---|
866 | // Torsion is accessible via "getLattice()": |
---|
867 | getLattice() == L; |
---|
868 | |
---|
869 | // its hermite NF: |
---|
870 | print(getLattice("hermite")); |
---|
871 | |
---|
872 | kill L, M; |
---|
873 | |
---|
874 | // ----------- isomorphic multigrading -------- // |
---|
875 | |
---|
876 | // Weights of variables |
---|
877 | intmat M[2][3] = |
---|
878 | 1, -2, 1, |
---|
879 | 1, 1, 0; |
---|
880 | |
---|
881 | // Torsion: |
---|
882 | intmat L[2][1] = |
---|
883 | 0, |
---|
884 | 2; |
---|
885 | |
---|
886 | // attaches M & L to R (==basering): |
---|
887 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
888 | |
---|
889 | // Torsion is accessible via "getLattice()": |
---|
890 | getLattice() == L; |
---|
891 | |
---|
892 | // its hermite NF: |
---|
893 | print(getLattice("hermite")); |
---|
894 | |
---|
895 | kill L, M; |
---|
896 | |
---|
897 | // ----------- extreme case ------------ // |
---|
898 | |
---|
899 | // Weights of variables |
---|
900 | intmat M[1][3] = |
---|
901 | 1, -1, 10; |
---|
902 | |
---|
903 | // Torsion: |
---|
904 | intmat L[1][1] = |
---|
905 | 0; |
---|
906 | |
---|
907 | // attaches M & L to R (==basering): |
---|
908 | setBaseMultigrading(M); // Grading: Z^3 |
---|
909 | |
---|
910 | // Torsion is accessible via "getLattice()": |
---|
911 | getLattice() == L; |
---|
912 | |
---|
913 | // its hermite NF: |
---|
914 | print(getLattice("hermite")); |
---|
915 | } |
---|
916 | |
---|
917 | proc getGradedGenerator(def m, int i) |
---|
918 | " |
---|
919 | returns m[i], but with grading |
---|
920 | " |
---|
921 | { |
---|
922 | if( typeof(m) == "ideal" ) |
---|
923 | { |
---|
924 | return (m[i]); |
---|
925 | } |
---|
926 | |
---|
927 | if( typeof(m) == "module" ) |
---|
928 | { |
---|
929 | def v = getModuleGrading(m); |
---|
930 | |
---|
931 | return ( setModuleGrading(m[i],v) ); |
---|
932 | } |
---|
933 | |
---|
934 | ERROR("m is expected to be an ideal or a module"); |
---|
935 | } |
---|
936 | |
---|
937 | |
---|
938 | /******************************************************/ |
---|
939 | proc getModuleGrading(def m) |
---|
940 | "USAGE: getModuleGrading(m), 'm' module/vector |
---|
941 | RETURN: integer matrix of the multiweights of free module generators attached to 'm' |
---|
942 | EXAMPLE: example getModuleGrading; shows an example |
---|
943 | " |
---|
944 | { |
---|
945 | string attrModuleGrading = "genWeights"; |
---|
946 | |
---|
947 | // print(m); typeof(m); attrib(m); |
---|
948 | |
---|
949 | def V = attrib(m, attrModuleGrading); |
---|
950 | |
---|
951 | if( typeof(V) != "intmat" ) |
---|
952 | { |
---|
953 | if( (typeof(m) == "ideal") or (typeof(m) == "poly") ) |
---|
954 | { |
---|
955 | intmat M = getVariableWeights(); |
---|
956 | intmat VV[nrows(M)][1]; |
---|
957 | return (VV); |
---|
958 | } |
---|
959 | |
---|
960 | ERROR("Sorry: vector or module need module-grading-matrix! See 'getModuleGrading'."); |
---|
961 | } |
---|
962 | |
---|
963 | if( nrows(V) != nrows(getVariableWeights()) ) |
---|
964 | { |
---|
965 | ERROR("Sorry wrong height of V: " + string(nrows(V))); |
---|
966 | } |
---|
967 | |
---|
968 | if( ncols(V) < nrows(m) ) |
---|
969 | { |
---|
970 | ERROR("Sorry wrong width of V: " + string(ncols(V))); |
---|
971 | } |
---|
972 | |
---|
973 | return (V); |
---|
974 | } |
---|
975 | example |
---|
976 | { |
---|
977 | "EXAMPLE:"; echo=2; |
---|
978 | |
---|
979 | ring R = 0, (x,y), dp; |
---|
980 | intmat M[2][2]= |
---|
981 | 1, 1, |
---|
982 | 0, 2; |
---|
983 | intmat T[2][5]= |
---|
984 | 1, 2, 3, 4, 0, |
---|
985 | 0, 10, 20, 30, 1; |
---|
986 | |
---|
987 | setBaseMultigrading(M, T); |
---|
988 | |
---|
989 | ideal I = x, y, xy^5; |
---|
990 | isHomogeneous(I); |
---|
991 | |
---|
992 | intmat V = mDeg(I); print(V); |
---|
993 | |
---|
994 | module S = syz(I); print(S); |
---|
995 | |
---|
996 | S = setModuleGrading(S, V); |
---|
997 | |
---|
998 | getModuleGrading(S) == V; |
---|
999 | |
---|
1000 | vector v = getGradedGenerator(S, 1); |
---|
1001 | getModuleGrading(v) == V; |
---|
1002 | isHomogeneous(v); |
---|
1003 | print( mDeg(v) ); |
---|
1004 | |
---|
1005 | isHomogeneous(S); |
---|
1006 | print( mDeg(S) ); |
---|
1007 | } |
---|
1008 | |
---|
1009 | /******************************************************/ |
---|
1010 | proc setModuleGrading(def m, intmat G) |
---|
1011 | "USAGE: setModuleGrading(m, G), m module/vector, G intmat |
---|
1012 | PURPOSE: attaches the multiweights of free module generators to 'm' |
---|
1013 | WARNING: The method does not verify whether the multigrading makes the |
---|
1014 | module/vector homogeneous. One can do that using isHomogeneous(m). |
---|
1015 | EXAMPLE: example setModuleGrading; shows an example |
---|
1016 | " |
---|
1017 | { |
---|
1018 | string attrModuleGrading = "genWeights"; |
---|
1019 | |
---|
1020 | intmat R = getVariableWeights(); |
---|
1021 | |
---|
1022 | if(nrows(G) != nrows(R)){ ERROR("Incompatible gradings.");} |
---|
1023 | if(ncols(G) < nrows(m)){ ERROR("Multigrading does not fit to module.");} |
---|
1024 | |
---|
1025 | attrib(m, attrModuleGrading, G); |
---|
1026 | return(m); |
---|
1027 | } |
---|
1028 | example |
---|
1029 | { |
---|
1030 | "EXAMPLE:"; echo=2; |
---|
1031 | |
---|
1032 | ring R = 0, (x,y), dp; |
---|
1033 | intmat M[2][2]= |
---|
1034 | 1, 1, |
---|
1035 | 0, 2; |
---|
1036 | intmat T[2][5]= |
---|
1037 | 1, 2, 3, 4, 0, |
---|
1038 | 0, 10, 20, 30, 1; |
---|
1039 | |
---|
1040 | setBaseMultigrading(M, T); |
---|
1041 | |
---|
1042 | ideal I = x, y, xy^5; |
---|
1043 | intmat V = mDeg(I); |
---|
1044 | |
---|
1045 | // V == M; modulo T |
---|
1046 | print(V); |
---|
1047 | |
---|
1048 | module S = syz(I); |
---|
1049 | |
---|
1050 | S = setModuleGrading(S, V); |
---|
1051 | getModuleGrading(S) == V; |
---|
1052 | |
---|
1053 | print(S); |
---|
1054 | |
---|
1055 | vector v = getGradedGenerator(S, 1); |
---|
1056 | getModuleGrading(v) == V; |
---|
1057 | |
---|
1058 | print( mDeg(v) ); |
---|
1059 | |
---|
1060 | isHomogeneous(S); |
---|
1061 | |
---|
1062 | print( mDeg(S) ); |
---|
1063 | } |
---|
1064 | |
---|
1065 | |
---|
1066 | proc mDegTensor(module m, module n){ |
---|
1067 | matrix M = m; |
---|
1068 | matrix N = n; |
---|
1069 | intmat gm = getModuleGrading(m); |
---|
1070 | intmat gn = getModuleGrading(n); |
---|
1071 | int grows = nrows(gm); |
---|
1072 | int mr = nrows(M); |
---|
1073 | int mc = ncols(M); |
---|
1074 | if(rank(M) == 0){ mc = 0;} |
---|
1075 | int nr = nrows(N); |
---|
1076 | int nc = ncols(N); |
---|
1077 | if(rank(N) == 0){ nc = 0;} |
---|
1078 | intmat gresult[nrows(gm)][mr*nr]; |
---|
1079 | matrix result[mr*nr][mr*nc+mc*nr]; |
---|
1080 | int i, j; |
---|
1081 | int column = 1; |
---|
1082 | for(i = 1; i<=mr; i++){ |
---|
1083 | for(j = 1; j<=nr; j++){ |
---|
1084 | gresult[1..grows,(i-1)*nr+j] = gm[1..grows,i]+gn[1..grows,j]; |
---|
1085 | } |
---|
1086 | } |
---|
1087 | //gresult; |
---|
1088 | if( nc!=0 ){ |
---|
1089 | for(i = 1; i<=mr; i++) |
---|
1090 | { |
---|
1091 | result[((i-1)*nr+1)..(i*nr),((i-1)*nc+1)..(i*nc)] = N[1..nr,1..nc]; |
---|
1092 | } |
---|
1093 | } |
---|
1094 | list rownumbers, colnumbers; |
---|
1095 | //print(result); |
---|
1096 | if( mc!=0 ){ |
---|
1097 | for(j = 1; j<=nr; j++) |
---|
1098 | { |
---|
1099 | rownumbers = nr*(0..(mr-1))+j*(1:mr); |
---|
1100 | colnumbers = ((mr*nc+j):mc)+nr*(0..(mc-1)); |
---|
1101 | result[rownumbers[1..mr],colnumbers[1..mc] ] = M[1..mr,1..mc]; |
---|
1102 | } |
---|
1103 | } |
---|
1104 | module res = result; |
---|
1105 | res = setModuleGrading(res, gresult); |
---|
1106 | //getModuleGrading(res); |
---|
1107 | return(res); |
---|
1108 | } |
---|
1109 | example |
---|
1110 | { |
---|
1111 | "EXAMPLE: ";echo=2; |
---|
1112 | ring r = 0,(x),dp; |
---|
1113 | intmat g[2][1]=1,1; |
---|
1114 | setBaseMultigrading(g); |
---|
1115 | matrix m[5][3]=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15; |
---|
1116 | matrix n[3][2]=x,x2,x3,x4,x5,x6; |
---|
1117 | module mm = m; |
---|
1118 | module nn = n; |
---|
1119 | intmat gm[2][5]=1,2,3,4,5,0,0,0,0,0; |
---|
1120 | intmat gn[2][3]=0,0,0,1,2,3; |
---|
1121 | mm = setModuleGrading(mm, gm); |
---|
1122 | nn = setModuleGrading(nn, gn); |
---|
1123 | module mmtnn = mDegTensor(mm, nn); |
---|
1124 | print(mmtnn); |
---|
1125 | getModuleGrading(mmtnn); |
---|
1126 | LIB "homolog.lib"; |
---|
1127 | module tt = tensorMod(mm,nn); |
---|
1128 | print(tt); |
---|
1129 | |
---|
1130 | kill m, mm, n, nn, gm, gn; |
---|
1131 | |
---|
1132 | matrix m[7][3] = x, x-1,x+2, 3x, 4x, x5, x6, x-7, x-8, 9, 10, 11x, 12 -x, 13x, 14x, x15, (x-4)^2, x17, 18x, 19x, 20x, 21x; |
---|
1133 | matrix n[2][4] = 1, 2, 3, 4, x, x2, x3, x4; |
---|
1134 | module mm = m; |
---|
1135 | module nn = n; |
---|
1136 | print(mm); |
---|
1137 | print(nn); |
---|
1138 | intmat gm[2][7] = 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0; |
---|
1139 | intmat gn[2][2] = 0, 0, 1, 2; |
---|
1140 | mm = setModuleGrading(mm, gm); |
---|
1141 | nn = setModuleGrading(nn, gn); |
---|
1142 | module mmtnn = mDegTensor(mm, nn); |
---|
1143 | print(mmtnn); |
---|
1144 | getModuleGrading(mmtnn); |
---|
1145 | matrix a = mmtnn; |
---|
1146 | matrix b = tensorMod(mm, nn); |
---|
1147 | print(a-b); |
---|
1148 | |
---|
1149 | } |
---|
1150 | |
---|
1151 | proc mDegTor(int i, module m, module n) |
---|
1152 | { |
---|
1153 | def res = mDegResolution(n, 0, 1); |
---|
1154 | //print(res); |
---|
1155 | list l = res; |
---|
1156 | if(size(l)<i){ return(0);} |
---|
1157 | else |
---|
1158 | { |
---|
1159 | |
---|
1160 | matrix fd[nrows(m)][0]; |
---|
1161 | matrix fd2[nrows(l[i+1])][0]; |
---|
1162 | matrix fd3[nrows(l[i])][0]; |
---|
1163 | |
---|
1164 | module freedim = fd; |
---|
1165 | module freedim2 = fd2; |
---|
1166 | module freedim3 = fd3; |
---|
1167 | |
---|
1168 | freedim = setModuleGrading(freedim,getModuleGrading(m)); |
---|
1169 | freedim2 = setModuleGrading(freedim2,getModuleGrading(l[i+1])); |
---|
1170 | freedim3 = setModuleGrading(freedim3, getModuleGrading(l[i])); |
---|
1171 | |
---|
1172 | module mimag = mDegTensor(freedim3, m); |
---|
1173 | //"mimag ok."; |
---|
1174 | module mf = mDegTensor(l[i], freedim); |
---|
1175 | //"mf ok."; |
---|
1176 | module mim1 = mDegTensor(freedim2 ,m); |
---|
1177 | module mim2 = mDegTensor(l[i+1],freedim); |
---|
1178 | //"mim1+2 ok."; |
---|
1179 | module mker = mDegModulo(mf,mimag); |
---|
1180 | //"mker ok."; |
---|
1181 | module mim = mim1,mim2; |
---|
1182 | mim = setModuleGrading(mim, getModuleGrading(mim1)); |
---|
1183 | //"mim: r: ",nrows(mim)," c: ",ncols(mim); |
---|
1184 | //"mim1: r: ",nrows(mim1)," c: ",ncols(mim1); |
---|
1185 | //"mim2: r: ",nrows(mim2)," c: ",ncols(mim2); |
---|
1186 | //matrix mimmat = mim; |
---|
1187 | //matrix mimmat1[16][4]=mimmat[1..16,25..28]; |
---|
1188 | //print(mimmat1-matrix(mim2)); |
---|
1189 | return(mDegModulo(mker,mim)); |
---|
1190 | //return(0); |
---|
1191 | } |
---|
1192 | return(0); |
---|
1193 | } |
---|
1194 | example |
---|
1195 | { |
---|
1196 | "EXAMPLE: ";echo=2; |
---|
1197 | LIB "homolog.lib"; |
---|
1198 | ring r = 0,(x_(1..4)),dp; |
---|
1199 | intmat g[2][4]=1,1,0,0,0,1,1,-1; |
---|
1200 | setBaseMultigrading(g); |
---|
1201 | ideal i = maxideal(1); |
---|
1202 | module m = mDegSyzygy(i); |
---|
1203 | module rt = Tor(2,m,m); |
---|
1204 | module mDegT = mDegTor(2,m,m); |
---|
1205 | print(matrix(rt)-matrix(mDegT)); |
---|
1206 | /* |
---|
1207 | ring r = 0,(x),dp; |
---|
1208 | intmat g[2][1]=1,1; |
---|
1209 | setBaseMultigrading(g); |
---|
1210 | matrix m[5][3]=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15; |
---|
1211 | matrix n[3][2]=x,x2,x3,x4,x5,x6; |
---|
1212 | module mm = m; |
---|
1213 | module nn = n; |
---|
1214 | intmat gm[2][5]=1,1,1,1,1,1,1,1,1,1,1; |
---|
1215 | intmat gn[2][3]=0,-2,-4,0,-2,-4; |
---|
1216 | mm = setModuleGrading(mm, gm); |
---|
1217 | nn = setModuleGrading(nn, gn); |
---|
1218 | isHomogeneous(mm,"checkGens"); |
---|
1219 | isHomogeneous(nn,"checkGens"); |
---|
1220 | mDegTor(1,mm, nn); |
---|
1221 | |
---|
1222 | kill m, mm, n, nn, gm, gn; |
---|
1223 | |
---|
1224 | matrix m[7][3] = x, x-1,x+2, 3x, 4x, x5, x6, x-7, x-8, 9, 10, 11x, 12 -x, 13x, 14x, x15, (x-4)^2, x17, 18x, 19x, 20x, 21x; |
---|
1225 | matrix n[2][4] = 1, 2, 3, 4, x, x2, x3, x4; |
---|
1226 | module mm = m; |
---|
1227 | module nn = n; |
---|
1228 | print(mm); |
---|
1229 | print(nn); |
---|
1230 | intmat gm[2][7] = 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0; |
---|
1231 | intmat gn[2][2] = 0, 0, 1, 2; |
---|
1232 | mm = setModuleGrading(mm, gm); |
---|
1233 | nn = setModuleGrading(nn, gn); |
---|
1234 | module mmtnn = mDegTensor(mm, nn); |
---|
1235 | */ |
---|
1236 | } |
---|
1237 | |
---|
1238 | |
---|
1239 | /******************************************************/ |
---|
1240 | proc isGroupHomomorphism(def L1, def L2, intmat A) |
---|
1241 | "USAGE: gisGoupHomomorphism(L1,L2,A); L1 and L2 are groups, A is an integer matrix |
---|
1242 | PURPOSE: checks whether A defines a group homomorphism phi: L1 --> L2 |
---|
1243 | RETURN: int, 1 if A defines the homomorphism and 0 otherwise |
---|
1244 | EXAMPLE: example isGroupHomomorphism; shows an example |
---|
1245 | " |
---|
1246 | { |
---|
1247 | // TODO: L1, L2 |
---|
1248 | if( (ncols(A) != nrows(L1)) or (nrows(A) != nrows(L2)) ) |
---|
1249 | { |
---|
1250 | ERROR("Incompatible sizes!"); |
---|
1251 | } |
---|
1252 | |
---|
1253 | intmat im = A * L1; |
---|
1254 | |
---|
1255 | return (areZeroElements(im, L2)); |
---|
1256 | } |
---|
1257 | example |
---|
1258 | { |
---|
1259 | "EXAMPLE:"; echo=2; |
---|
1260 | |
---|
1261 | intmat L1[4][1]= |
---|
1262 | 0, |
---|
1263 | 0, |
---|
1264 | 0, |
---|
1265 | 2; |
---|
1266 | |
---|
1267 | intmat L2[3][2]= |
---|
1268 | 0, 0, |
---|
1269 | 2, 0, |
---|
1270 | 0, 3; |
---|
1271 | |
---|
1272 | intmat A[3][4] = |
---|
1273 | 1, 2, 3, 0, |
---|
1274 | 7, 0, 0, 0, |
---|
1275 | 1, 2, 0, 3; |
---|
1276 | print( A ); |
---|
1277 | |
---|
1278 | isGroupHomomorphism(L1, L2, A); |
---|
1279 | |
---|
1280 | intmat B[3][4] = |
---|
1281 | 1, 2, 3, 0, |
---|
1282 | 7, 0, 0, 0, |
---|
1283 | 1, 2, 0, 2; |
---|
1284 | print( B ); |
---|
1285 | |
---|
1286 | isGroupHomomorphism(L1, L2, B); // Not a homomorphism! |
---|
1287 | } |
---|
1288 | |
---|
1289 | /******************************************************/ |
---|
1290 | proc isTorsionFree() |
---|
1291 | "USAGE: isTorsionFree() |
---|
1292 | PURPOSE: Determines whether the multigrading attached to the current ring is free. |
---|
1293 | RETURN: boolean, the result of the test |
---|
1294 | EXAMPLE: example isTorsionFree; shows an example |
---|
1295 | " |
---|
1296 | { |
---|
1297 | intmat H = smithNormalForm(getLattice()); // TODO: ?cache it? //****** |
---|
1298 | |
---|
1299 | int i, j; |
---|
1300 | int r = nrows(H); |
---|
1301 | int c = ncols(H); |
---|
1302 | int d = 1; |
---|
1303 | for( i = 1; (i <= c) && (i <= r); i++ ) |
---|
1304 | { |
---|
1305 | for( j = i; (H[j, i] == 0)&&(j < r); j++ ) |
---|
1306 | { |
---|
1307 | } |
---|
1308 | |
---|
1309 | if(H[j, i]!=0) |
---|
1310 | { |
---|
1311 | d=d*H[j, i]; |
---|
1312 | } |
---|
1313 | } |
---|
1314 | |
---|
1315 | if( (d*d)==1 ) |
---|
1316 | { |
---|
1317 | return(1==1); |
---|
1318 | } |
---|
1319 | return(0==1); |
---|
1320 | } |
---|
1321 | example |
---|
1322 | { |
---|
1323 | "EXAMPLE:"; echo=2; |
---|
1324 | |
---|
1325 | ring R = 0,(x,y),dp; |
---|
1326 | intmat M[2][2]= |
---|
1327 | 1,0, |
---|
1328 | 0,1; |
---|
1329 | intmat T[2][5]= |
---|
1330 | 1, 2, 3, 4, 0, |
---|
1331 | 0,10,20,30, 1; |
---|
1332 | |
---|
1333 | setBaseMultigrading(M,T); |
---|
1334 | |
---|
1335 | // Is the resulting group free? |
---|
1336 | isTorsionFree(); |
---|
1337 | |
---|
1338 | kill R, M, T; |
---|
1339 | /////////////////////////////////////////// |
---|
1340 | |
---|
1341 | ring R=0,(x,y,z),dp; |
---|
1342 | intmat A[3][3] = |
---|
1343 | 1,0,0, |
---|
1344 | 0,1,0, |
---|
1345 | 0,0,1; |
---|
1346 | intmat B[3][4]= |
---|
1347 | 3,3,3,3, |
---|
1348 | 2,1,3,0, |
---|
1349 | 1,2,0,3; |
---|
1350 | setBaseMultigrading(A,B); |
---|
1351 | // Is the resulting group free? |
---|
1352 | isTorsionFree(); |
---|
1353 | |
---|
1354 | kill R, A, B; |
---|
1355 | } |
---|
1356 | |
---|
1357 | |
---|
1358 | static proc gcdcomb(int a, int b) |
---|
1359 | { |
---|
1360 | // a; |
---|
1361 | // b; |
---|
1362 | intvec av = a,1,0; |
---|
1363 | intvec bv = b,0,1; |
---|
1364 | intvec save; |
---|
1365 | while(av[1]*bv[1] != 0) |
---|
1366 | { |
---|
1367 | bv = bv - (bv[1] - bv[1]%av[1])/av[1] * av; |
---|
1368 | save = bv; |
---|
1369 | bv = av; |
---|
1370 | av = save; |
---|
1371 | } |
---|
1372 | if(bv[1] < 0) |
---|
1373 | { |
---|
1374 | bv = -bv; |
---|
1375 | } |
---|
1376 | return(bv); |
---|
1377 | } |
---|
1378 | |
---|
1379 | |
---|
1380 | proc lll(def A) |
---|
1381 | " |
---|
1382 | The lll algorithm of lll.lib only works for lists of vectors. |
---|
1383 | Maybe one should rescript it for matrices. This method will |
---|
1384 | convert a matrix to a list, plug it into lll and make the result |
---|
1385 | a matrix and return it. |
---|
1386 | " |
---|
1387 | { |
---|
1388 | if(typeof(A) == "list") |
---|
1389 | { |
---|
1390 | int sizeA= size (A); |
---|
1391 | if (sizeA == 0) |
---|
1392 | { |
---|
1393 | return (A); |
---|
1394 | } |
---|
1395 | if (typeof (A [1]) != "intvec") |
---|
1396 | { |
---|
1397 | ERROR("Unrecognized type."); |
---|
1398 | } |
---|
1399 | int columns= size (A [1]); |
---|
1400 | int i; |
---|
1401 | for (i= 2; i <= sizeA; i++) |
---|
1402 | { |
---|
1403 | if (typeof (A[i]) != "intvec") |
---|
1404 | { |
---|
1405 | ERROR("Unrecognized type."); |
---|
1406 | } |
---|
1407 | if (size (A [i]) != columns) |
---|
1408 | { |
---|
1409 | ERROR ("expected equal dimension"); |
---|
1410 | } |
---|
1411 | } |
---|
1412 | int j; |
---|
1413 | intmat m [columns] [sizeA]; |
---|
1414 | for (i= 1; i <= sizeA; i++) |
---|
1415 | { |
---|
1416 | for (j= 1; j <= columns; j++) |
---|
1417 | { |
---|
1418 | m[i,j]= A[i] [j]; |
---|
1419 | } |
---|
1420 | } |
---|
1421 | m= system ("LLL", m); |
---|
1422 | list result= list(); |
---|
1423 | |
---|
1424 | for (i= 1; i <= sizeA; i++) |
---|
1425 | { |
---|
1426 | intvec buf= intvec (m[i , 1]); |
---|
1427 | for (j= 2; j <= columns; j++) |
---|
1428 | { |
---|
1429 | buf= buf,intvec (m [i, j]); |
---|
1430 | } |
---|
1431 | result= result+ list (buf); |
---|
1432 | |
---|
1433 | } |
---|
1434 | return(result); |
---|
1435 | } |
---|
1436 | else |
---|
1437 | { |
---|
1438 | if(typeof(A) == "intmat") |
---|
1439 | { |
---|
1440 | A= system ("LLL", A); |
---|
1441 | return(A); |
---|
1442 | } |
---|
1443 | else |
---|
1444 | { |
---|
1445 | ERROR("Unrecognized type."); |
---|
1446 | } |
---|
1447 | } |
---|
1448 | } |
---|
1449 | |
---|
1450 | example |
---|
1451 | { |
---|
1452 | |
---|
1453 | "EXAMPLE:"; |
---|
1454 | |
---|
1455 | ring R = 0,x,dp; |
---|
1456 | intmat m[5][5]=13,25,37,83,294,12,-33,9,0,64,77,12,34,6,1,43,2,88,91,100,-46,32,37,42,15; |
---|
1457 | lll(m); |
---|
1458 | list l=intvec(13,25,37, 83, 294),intvec(12, -33, 9,0,64), intvec (77,12,34,6,1), intvec (43,2,88,91,100), intvec (-46,32,37,42,15); |
---|
1459 | lll(l); |
---|
1460 | } |
---|
1461 | |
---|
1462 | |
---|
1463 | proc smithNormalForm(intmat A, list #) |
---|
1464 | " |
---|
1465 | This method returns 3 Matrices P, D and Q such that D = P*A*Q. |
---|
1466 | WARNING: This might not be what you expect. |
---|
1467 | " |
---|
1468 | { |
---|
1469 | list l1 = hermiteNormalForm(A, 5); |
---|
1470 | // l1; |
---|
1471 | intmat B = transpose(l1[1]); |
---|
1472 | list l2 = hermiteNormalForm(B, 5); |
---|
1473 | // l2; |
---|
1474 | intmat P = transpose(l2[2]); |
---|
1475 | intmat D = transpose(l2[1]); |
---|
1476 | intmat Q = l1[2]; |
---|
1477 | int cc = ncols(D); |
---|
1478 | int rr = nrows(D); |
---|
1479 | intmat transform; |
---|
1480 | int k = 1; |
---|
1481 | int a, b, c; |
---|
1482 | // D; |
---|
1483 | intvec v; |
---|
1484 | if((cc==1)||(rr==1)){ |
---|
1485 | if(size(#)==0) |
---|
1486 | { |
---|
1487 | return(D); |
---|
1488 | } else |
---|
1489 | { |
---|
1490 | return(list(P,D,Q)); |
---|
1491 | } |
---|
1492 | } |
---|
1493 | while(D[k+1,k+1] !=0){ |
---|
1494 | if(D[k+1,k+1]%D[k,k]!=0){ |
---|
1495 | b = D[k, k]; c = D[k+1, k+1]; |
---|
1496 | v = gcdcomb(D[k,k],D[k+1,k+1]); |
---|
1497 | transform = unitMatrix(cc); |
---|
1498 | transform[k+1,k] = 1; |
---|
1499 | a = -v[3]*D[k+1,k+1]/v[1]; |
---|
1500 | transform[k, k+1] = a; |
---|
1501 | transform[k+1, k+1] = a+1; |
---|
1502 | //det(transform); |
---|
1503 | D = D*transform; |
---|
1504 | Q = Q*transform; |
---|
1505 | //D; |
---|
1506 | transform = unitMatrix(rr); |
---|
1507 | transform[k,k] = v[2]; |
---|
1508 | transform[k,k+1] = v[3]; |
---|
1509 | transform[k+1,k] = -c/v[1]; |
---|
1510 | transform[k+1,k+1] = b/v[1]; |
---|
1511 | D = transform * D; |
---|
1512 | P = transform * P; |
---|
1513 | //" "; |
---|
1514 | //D; |
---|
1515 | //"small transform: ", det(transform); |
---|
1516 | //transform; |
---|
1517 | k=0; |
---|
1518 | } |
---|
1519 | k++; |
---|
1520 | if((k==rr) || (k==cc)){ |
---|
1521 | break; |
---|
1522 | } |
---|
1523 | } |
---|
1524 | //"here is the size ",size(#); |
---|
1525 | if(size(#) == 0){ |
---|
1526 | return(D); |
---|
1527 | } else { |
---|
1528 | return(list(P, D, Q)); |
---|
1529 | } |
---|
1530 | } |
---|
1531 | example |
---|
1532 | { |
---|
1533 | "EXAMPLE: "; echo=2; |
---|
1534 | |
---|
1535 | intmat A[5][7] = |
---|
1536 | 1,0,1,0,-2,9,-71, |
---|
1537 | 0,-24,248,-32,-96,448,-3496, |
---|
1538 | 0,4,-42,4,-8,30,-260, |
---|
1539 | 0,0,0,18,-90,408,-3168, |
---|
1540 | 0,0,0,-32,224,-1008,7872; |
---|
1541 | |
---|
1542 | list l = smithNormalForm(A, 5); |
---|
1543 | |
---|
1544 | l; |
---|
1545 | l[1]*A*l[3]; |
---|
1546 | det(l[1]); |
---|
1547 | det(l[3]); |
---|
1548 | } |
---|
1549 | |
---|
1550 | |
---|
1551 | /******************************************************/ |
---|
1552 | proc hermiteNormalForm(intmat A, list #) |
---|
1553 | "USAGE: hermiteNormalForm( A ); |
---|
1554 | PURPOSE: Computes the (lower triangular) Hermite Normal Form |
---|
1555 | of the matrix A by column operations. |
---|
1556 | RETURN: intmat, the Hermite Normal Form of A |
---|
1557 | EXAMPLE: example hermiteNormalForm; shows an example |
---|
1558 | " |
---|
1559 | { |
---|
1560 | |
---|
1561 | int row, column, i, j; |
---|
1562 | int rr = nrows(A); |
---|
1563 | int cc = ncols(A); |
---|
1564 | intvec savev, gcdvec, v1, v2; |
---|
1565 | intmat q = unitMatrix(cc); |
---|
1566 | intmat transform; |
---|
1567 | column = 1; |
---|
1568 | for(row = 1; (row<=rr)&&(column<=cc); row++) |
---|
1569 | { |
---|
1570 | if(A[row,column]==0) |
---|
1571 | { |
---|
1572 | for(j = column; j<=cc; j++) |
---|
1573 | { |
---|
1574 | if(A[row, j]!=0) |
---|
1575 | { |
---|
1576 | transform = unitMatrix(cc); |
---|
1577 | transform[j,j] = 0; |
---|
1578 | transform[column, column] = 0; |
---|
1579 | transform[column,j] = 1; |
---|
1580 | transform[j,column] = 1; |
---|
1581 | q = q*transform; |
---|
1582 | A = A*transform; |
---|
1583 | break; |
---|
1584 | } |
---|
1585 | } |
---|
1586 | } |
---|
1587 | if(A[row,column] == 0) |
---|
1588 | { |
---|
1589 | row++; |
---|
1590 | continue; |
---|
1591 | } |
---|
1592 | for(j = column+1; j<=cc; j++) |
---|
1593 | { |
---|
1594 | if(A[row, j]!=0) |
---|
1595 | { |
---|
1596 | gcdvec = gcdcomb(A[row,column],A[row,j]); |
---|
1597 | // gcdvec; |
---|
1598 | // typeof(A[1..rr,column]); |
---|
1599 | v1 = A[1..rr,column]; |
---|
1600 | v2 = A[1..rr,j]; |
---|
1601 | transform = unitMatrix(cc); |
---|
1602 | transform[j,j] = v1[row]/gcdvec[1]; |
---|
1603 | transform[column, column] = gcdvec[2]; |
---|
1604 | transform[column,j] = -v2[row]/gcdvec[1]; |
---|
1605 | transform[j,column] = gcdvec[3]; |
---|
1606 | q = q*transform; |
---|
1607 | A = A*transform; |
---|
1608 | // A; |
---|
1609 | } |
---|
1610 | } |
---|
1611 | if(A[row,column]<0) |
---|
1612 | { |
---|
1613 | transform = unitMatrix(cc); |
---|
1614 | transform[column,column] = -1; |
---|
1615 | q = q*transform; |
---|
1616 | A = A*transform; |
---|
1617 | } |
---|
1618 | for( j=1; j<column; j++){ |
---|
1619 | if(A[row, j]!=0){ |
---|
1620 | transform = unitMatrix(cc); |
---|
1621 | transform[column, j] = (-A[row,j]+A[row, j]%A[row, column])/A[row, column]; |
---|
1622 | if(A[row,j]<0){ |
---|
1623 | transform[column,j]=transform[column,j]+1;} |
---|
1624 | q = q*transform; |
---|
1625 | A = A*transform; |
---|
1626 | } |
---|
1627 | } |
---|
1628 | column++; |
---|
1629 | } |
---|
1630 | if(size(#) > 0){ |
---|
1631 | return(list(A, q)); |
---|
1632 | } |
---|
1633 | return(A); |
---|
1634 | } |
---|
1635 | example |
---|
1636 | { |
---|
1637 | "EXAMPLE:"; echo=2; |
---|
1638 | |
---|
1639 | intmat M[2][5] = |
---|
1640 | 1, 2, 3, 4, 0, |
---|
1641 | 0,10,20,30, 1; |
---|
1642 | |
---|
1643 | // Hermite Normal Form of M: |
---|
1644 | print(hermiteNormalForm(M)); |
---|
1645 | |
---|
1646 | intmat T[3][4] = |
---|
1647 | 3,3,3,3, |
---|
1648 | 2,1,3,0, |
---|
1649 | 1,2,0,3; |
---|
1650 | |
---|
1651 | // Hermite Normal Form of T: |
---|
1652 | print(hermiteNormalForm(T)); |
---|
1653 | |
---|
1654 | intmat A[4][5] = |
---|
1655 | 1,2,3,2,2, |
---|
1656 | 1,2,3,4,0, |
---|
1657 | 0,5,4,2,1, |
---|
1658 | 3,2,4,0,2; |
---|
1659 | |
---|
1660 | // Hermite Normal Form of A: |
---|
1661 | print(hermiteNormalForm(A)); |
---|
1662 | } |
---|
1663 | |
---|
1664 | proc areZeroElements(intmat m, list #) |
---|
1665 | "same as isZeroElement but for an integer matrix considered as a collection of columns" |
---|
1666 | { |
---|
1667 | int r = nrows(m); |
---|
1668 | int i = ncols(m); |
---|
1669 | |
---|
1670 | intvec v; |
---|
1671 | |
---|
1672 | for( ; i > 0; i-- ) |
---|
1673 | { |
---|
1674 | v = m[1..r, i]; |
---|
1675 | if( !isZeroElement(v, #) ) |
---|
1676 | { |
---|
1677 | return (0); |
---|
1678 | } |
---|
1679 | } |
---|
1680 | return(1); |
---|
1681 | } |
---|
1682 | |
---|
1683 | example |
---|
1684 | { |
---|
1685 | "EXAMPLE:"; echo=2; |
---|
1686 | |
---|
1687 | ring r = 0,(x,y,z),dp; |
---|
1688 | |
---|
1689 | intmat g[2][3]= |
---|
1690 | 1,0,1, |
---|
1691 | 0,1,1; |
---|
1692 | intmat t[2][1]= |
---|
1693 | -2, |
---|
1694 | 1; |
---|
1695 | |
---|
1696 | intmat tt[2][1]= |
---|
1697 | 1, |
---|
1698 | -1; |
---|
1699 | |
---|
1700 | setBaseMultigrading(g,t); |
---|
1701 | |
---|
1702 | poly a = x10yz; |
---|
1703 | poly b = x8y2z; |
---|
1704 | poly c = x4z2; |
---|
1705 | poly d = y5; |
---|
1706 | poly e = x2y2; |
---|
1707 | poly f = z2; |
---|
1708 | |
---|
1709 | intmat m[5][2]=mDeg(a)-mDeg(b),mDeg(b)-mDeg(c),mDeg(c)-mDeg(d),mDeg(d)-mDeg(e),mDeg(e)-mDeg(f); |
---|
1710 | m=transpose(m); |
---|
1711 | areZeroElementes(m); |
---|
1712 | areZeroElementes(m,tt); |
---|
1713 | } |
---|
1714 | |
---|
1715 | |
---|
1716 | /******************************************************/ |
---|
1717 | proc isZeroElement(intvec mdeg, list #) |
---|
1718 | "USAGE: isZeroElement(d, [T]); intvec d, group T |
---|
1719 | PURPOSE: For a integer vector mdeg representing the multidegree of some polynomial |
---|
1720 | or vector this method computes if the multidegree is contained in the grading group |
---|
1721 | group (either set globally or given as an optional argument), i.e. if it is zero in the multigrading. |
---|
1722 | EXAMPLE: example isZeroElement; shows an example |
---|
1723 | " |
---|
1724 | { |
---|
1725 | if( size(#) > 0 ) |
---|
1726 | { |
---|
1727 | if( typeof(#[1]) == "intmat" ) |
---|
1728 | { |
---|
1729 | intmat H = hermiteNormalForm(#[1]); |
---|
1730 | } else |
---|
1731 | { |
---|
1732 | if( typeof(#[1]) == "list" ) |
---|
1733 | { |
---|
1734 | list L = #[1]; |
---|
1735 | intmat H = attrib(L, "hermite"); // todo |
---|
1736 | } |
---|
1737 | } |
---|
1738 | |
---|
1739 | } |
---|
1740 | if( !defined(H) ) |
---|
1741 | { |
---|
1742 | intmat H = getLattice("hermite"); |
---|
1743 | } |
---|
1744 | |
---|
1745 | int x, k, i, row; |
---|
1746 | |
---|
1747 | int r = nrows(H); |
---|
1748 | int c = ncols(H); |
---|
1749 | |
---|
1750 | int rr = nrows(mdeg); |
---|
1751 | row = 1; |
---|
1752 | intvec v; |
---|
1753 | for(i=1; (i<=r)&&(row<=r)&&(i<=c); i++) |
---|
1754 | { |
---|
1755 | while((H[row,i]==0)&&(row<=r)) |
---|
1756 | { |
---|
1757 | row++; |
---|
1758 | if(row == (r+1)){ |
---|
1759 | break; |
---|
1760 | } |
---|
1761 | } |
---|
1762 | if(row<=r){ |
---|
1763 | if(H[row,i]!=0) |
---|
1764 | { |
---|
1765 | v = H[1..r,i]; |
---|
1766 | mdeg = mdeg-(mdeg[row]-mdeg[row]%v[row])/v[row]*v; |
---|
1767 | } |
---|
1768 | } |
---|
1769 | } |
---|
1770 | return( mdeg == 0 ); |
---|
1771 | |
---|
1772 | } |
---|
1773 | example |
---|
1774 | { |
---|
1775 | "EXAMPLE:"; echo=2; |
---|
1776 | |
---|
1777 | ring r = 0,(x,y,z),dp; |
---|
1778 | |
---|
1779 | intmat g[2][3]= |
---|
1780 | 1,0,1, |
---|
1781 | 0,1,1; |
---|
1782 | intmat t[2][1]= |
---|
1783 | -2, |
---|
1784 | 1; |
---|
1785 | |
---|
1786 | intmat tt[2][1]= |
---|
1787 | 1, |
---|
1788 | -1; |
---|
1789 | |
---|
1790 | setBaseMultigrading(g,t); |
---|
1791 | |
---|
1792 | poly a = x10yz; |
---|
1793 | poly b = x8y2z; |
---|
1794 | poly c = x4z2; |
---|
1795 | poly d = y5; |
---|
1796 | poly e = x2y2; |
---|
1797 | poly f = z2; |
---|
1798 | |
---|
1799 | intvec v1 = mDeg(a) - mDeg(b); |
---|
1800 | v1; |
---|
1801 | isZeroElement(v1); |
---|
1802 | isZeroElement(v1, tt); |
---|
1803 | |
---|
1804 | intvec v2 = mDeg(a) - mDeg(c); |
---|
1805 | v2; |
---|
1806 | isZeroElement(v2); |
---|
1807 | isZeroElement(v2, tt); |
---|
1808 | |
---|
1809 | intvec v3 = mDeg(e) - mDeg(f); |
---|
1810 | v3; |
---|
1811 | isZeroElement(v3); |
---|
1812 | isZeroElement(v3, tt); |
---|
1813 | |
---|
1814 | intvec v4 = mDeg(c) - mDeg(d); |
---|
1815 | v4; |
---|
1816 | isZeroElement(v4); |
---|
1817 | isZeroElement(v4, tt); |
---|
1818 | } |
---|
1819 | |
---|
1820 | |
---|
1821 | /******************************************************/ |
---|
1822 | proc defineHomogeneous(poly f, list #) |
---|
1823 | "USAGE: defineHomogeneous(f[, G]); polynomial f, integer matrix G |
---|
1824 | PURPOSE: Yields a matrix which has to be appended to the grading group matrix to make the |
---|
1825 | polynomial f homogeneous in the grading by grad. |
---|
1826 | EXAMPLE: example defineHomogeneous; shows an example |
---|
1827 | " |
---|
1828 | { |
---|
1829 | if( size(#) > 0 ) |
---|
1830 | { |
---|
1831 | if( typeof(#[1]) == "intmat" ) |
---|
1832 | { |
---|
1833 | intmat grad = #[1]; |
---|
1834 | } |
---|
1835 | } |
---|
1836 | |
---|
1837 | if( !defined(grad) ) |
---|
1838 | { |
---|
1839 | intmat grad = getVariableWeights(); |
---|
1840 | } |
---|
1841 | |
---|
1842 | intmat newgg[nrows(grad)][size(f)-1]; |
---|
1843 | int i,j; |
---|
1844 | intvec l = grad*leadexp(f); |
---|
1845 | intvec v; |
---|
1846 | for(i=2; i <= size(f); i++) |
---|
1847 | { |
---|
1848 | v = grad * leadexp(f[i]) - l; |
---|
1849 | for( j=1; j<=size(v); j++) |
---|
1850 | { |
---|
1851 | newgg[j,i-1] = v[j]; |
---|
1852 | } |
---|
1853 | } |
---|
1854 | return(newgg); |
---|
1855 | } |
---|
1856 | example |
---|
1857 | { |
---|
1858 | "EXAMPLE:"; echo=2; |
---|
1859 | |
---|
1860 | ring r =0,(x,y,z),dp; |
---|
1861 | intmat grad[2][3] = |
---|
1862 | 1,0,1, |
---|
1863 | 0,1,1; |
---|
1864 | |
---|
1865 | setBaseMultigrading(grad); |
---|
1866 | |
---|
1867 | poly f = x2y3-z5+x-3zx; |
---|
1868 | |
---|
1869 | intmat M = defineHomogeneous(f); |
---|
1870 | M; |
---|
1871 | defineHomogeneous(f, grad) == M; |
---|
1872 | |
---|
1873 | isHomogeneous(f); |
---|
1874 | setBaseMultigrading(grad, M); |
---|
1875 | isHomogeneous(f); |
---|
1876 | } |
---|
1877 | |
---|
1878 | |
---|
1879 | proc gradiator(def h) |
---|
1880 | PURPOSE: coarsens the grading of the basering until the polynom or ideal h becomes homogeneous. |
---|
1881 | |
---|
1882 | { |
---|
1883 | if(typeof(h)=="poly"){ |
---|
1884 | intmat W = getVariableWeights(); |
---|
1885 | intmat L = getLattice(); |
---|
1886 | intmat toadd = defineHomogeneous(h); |
---|
1887 | //h; |
---|
1888 | //toadd; |
---|
1889 | if(ncols(toadd) == 0) |
---|
1890 | { |
---|
1891 | return(1==1); |
---|
1892 | } |
---|
1893 | int rr = nrows(W); |
---|
1894 | intmat newL[rr][ncols(L)+ncols(toadd)]; |
---|
1895 | newL[1..rr,1..ncols(L)] = L[1..rr,1..ncols(L)]; |
---|
1896 | newL[1..rr,(ncols(L)+1)..(ncols(L)+ncols(toadd))] = toadd[1..rr,1..ncols(toadd)]; |
---|
1897 | setBaseMultigrading(W,newL); |
---|
1898 | return(1==1); |
---|
1899 | } |
---|
1900 | if(typeof(h)=="ideal"){ |
---|
1901 | int i; |
---|
1902 | def s = (1==1); |
---|
1903 | for(i=1;i<=size(h);i++){ |
---|
1904 | s = s && gradiator(h[i]); |
---|
1905 | } |
---|
1906 | return(s); |
---|
1907 | } |
---|
1908 | return(1==0); |
---|
1909 | } |
---|
1910 | example |
---|
1911 | { |
---|
1912 | "EXAMPLE:"; echo=2; |
---|
1913 | ring r = 0,(x,y,z),dp; |
---|
1914 | intmat g[2][3] = 1,0,1,0,1,1; |
---|
1915 | intmat l[2][1] = 3,0; |
---|
1916 | |
---|
1917 | setBaseMultigrading(g,l); |
---|
1918 | |
---|
1919 | getLattice(); |
---|
1920 | |
---|
1921 | ideal i = -y5+x4, |
---|
1922 | y6+xz, |
---|
1923 | x2y; |
---|
1924 | gradiator(i); |
---|
1925 | getLattice(); |
---|
1926 | isHomogeneous(i); |
---|
1927 | } |
---|
1928 | |
---|
1929 | |
---|
1930 | proc pushForward(map f) |
---|
1931 | "USAGE: pushForward(f); |
---|
1932 | PURPOSE: Computes the finest grading of the image ring which makes the map f |
---|
1933 | a map of graded rings. The group map between the two grading groups is given |
---|
1934 | by transpose( (Id, 0) ). Pay attention that the group spanned by the columns of |
---|
1935 | the grading group matrix may not be a subgroup of the grading group. Still all columns |
---|
1936 | are needed to find the correct image of the preimage gradings. |
---|
1937 | EXAMPLE: example pushForward; shows an example |
---|
1938 | " |
---|
1939 | { |
---|
1940 | |
---|
1941 | int k,i,j; |
---|
1942 | // f; |
---|
1943 | |
---|
1944 | // listvar(); |
---|
1945 | def pre = preimage(f); |
---|
1946 | |
---|
1947 | // "pre: "; pre; |
---|
1948 | |
---|
1949 | intmat oldgrad=getVariableWeights(pre); |
---|
1950 | intmat oldtor=getLattice(pre); |
---|
1951 | |
---|
1952 | int n=nvars(pre); |
---|
1953 | int np=nvars(basering); |
---|
1954 | int p=nrows(oldgrad); |
---|
1955 | int pp=p+np; |
---|
1956 | |
---|
1957 | intmat newgrad[pp][np]; |
---|
1958 | |
---|
1959 | for(i=1;i<=np;i++){ newgrad[p+i,i]=1;} |
---|
1960 | |
---|
1961 | //newgrad; |
---|
1962 | |
---|
1963 | |
---|
1964 | |
---|
1965 | list newtor; |
---|
1966 | intmat toadd; |
---|
1967 | int columns=0; |
---|
1968 | |
---|
1969 | intmat toadd1[pp][n]; |
---|
1970 | intvec v; |
---|
1971 | poly im; |
---|
1972 | |
---|
1973 | for(i=1;i<=p;i++){ |
---|
1974 | for(j=1;j<=n;j++){ toadd1[i,j]=oldgrad[i,j];} |
---|
1975 | } |
---|
1976 | |
---|
1977 | for(i=1;i<=n;i++){ |
---|
1978 | im=f[i]; |
---|
1979 | //im; |
---|
1980 | toadd = defineHomogeneous(im, newgrad); |
---|
1981 | newtor=insert(newtor,toadd); |
---|
1982 | columns=columns+ncols(toadd); |
---|
1983 | |
---|
1984 | v=leadexp(f[i]); |
---|
1985 | for(j=p+1;j<=p+np;j++){ toadd1[j,i]=-v[j-p];} |
---|
1986 | } |
---|
1987 | |
---|
1988 | newtor=insert(newtor,toadd1); |
---|
1989 | columns=columns+ncols(toadd1); |
---|
1990 | |
---|
1991 | |
---|
1992 | if(typeof(basering)=="qring"){ |
---|
1993 | //"Entering qring"; |
---|
1994 | ideal a=ideal(basering); |
---|
1995 | for(i=1;i<=size(a);i++){ |
---|
1996 | toadd = defineHomogeneous(a[i], newgrad); |
---|
1997 | //toadd; |
---|
1998 | columns=columns+ncols(toadd); |
---|
1999 | newtor=insert(newtor,toadd); |
---|
2000 | } |
---|
2001 | } |
---|
2002 | |
---|
2003 | //newtor; |
---|
2004 | intmat imofoldtor[pp][ncols(oldtor)]; |
---|
2005 | for(i=1; i<=nrows(oldtor);i++){ |
---|
2006 | for(j=1; j<=ncols(oldtor); j++){ |
---|
2007 | imofoldtor[i,j]=oldtor[i,j]; |
---|
2008 | } |
---|
2009 | } |
---|
2010 | |
---|
2011 | columns=columns+ncols(oldtor); |
---|
2012 | newtor=insert(newtor, imofoldtor); |
---|
2013 | |
---|
2014 | intmat gragr[pp][columns]; |
---|
2015 | columns=0; |
---|
2016 | for(k=1;k<=size(newtor);k++){ |
---|
2017 | for(i=1;i<=pp;i++){ |
---|
2018 | for(j=1;j<=ncols(newtor[k]);j++){gragr[i,j+columns]=newtor[k][i,j];} |
---|
2019 | } |
---|
2020 | columns=columns+ncols(newtor[k]); |
---|
2021 | } |
---|
2022 | |
---|
2023 | gragr=hermiteNormalForm(gragr); |
---|
2024 | intmat result[pp][pp]; |
---|
2025 | for(i=1;i<=pp;i++){ |
---|
2026 | for(j=1;j<=pp;j++){result[i,j]=gragr[i,j];} |
---|
2027 | } |
---|
2028 | |
---|
2029 | setBaseMultigrading(newgrad, result); |
---|
2030 | |
---|
2031 | } |
---|
2032 | example |
---|
2033 | { |
---|
2034 | "EXAMPLE:"; echo=2; |
---|
2035 | |
---|
2036 | ring r = 0,(x,y,z),dp; |
---|
2037 | |
---|
2038 | |
---|
2039 | |
---|
2040 | // Setting degrees for preimage ring.; |
---|
2041 | intmat grad[3][3] = |
---|
2042 | 1,0,0, |
---|
2043 | 0,1,0, |
---|
2044 | 0,0,1; |
---|
2045 | |
---|
2046 | setBaseMultigrading(grad); |
---|
2047 | |
---|
2048 | // grading on r: |
---|
2049 | getVariableWeights(); |
---|
2050 | getLattice(); |
---|
2051 | |
---|
2052 | // only for the purpose of this example |
---|
2053 | if( voice > 1 ){ /*keepring(r);*/ export(r); } |
---|
2054 | |
---|
2055 | ring R = 0,(a,b),dp; |
---|
2056 | ideal i = a2-b2+a6-b5+ab3,a7b+b15-ab6+a6b6; |
---|
2057 | |
---|
2058 | // The quotient ring by this ideal will become our image ring.; |
---|
2059 | qring Q = std(i); |
---|
2060 | |
---|
2061 | listvar(); |
---|
2062 | |
---|
2063 | map f = r,-a2b6+b5+a3b+a2+ab,-a2b7-3a2b5+b4+a,a6-b6-b3+a2; f; |
---|
2064 | |
---|
2065 | |
---|
2066 | // TODO: Unfortunately this is not a very spectacular example...: |
---|
2067 | // Pushing forward f: |
---|
2068 | pushForward(f); |
---|
2069 | |
---|
2070 | // due to pushForward we have got new grading on Q |
---|
2071 | getVariableWeights(); |
---|
2072 | getLattice(); |
---|
2073 | |
---|
2074 | |
---|
2075 | // only for the purpose of this example |
---|
2076 | if( voice > 1 ){ kill r; } |
---|
2077 | |
---|
2078 | } |
---|
2079 | |
---|
2080 | |
---|
2081 | /******************************************************/ |
---|
2082 | proc equalMDeg(intvec exp1, intvec exp2, list #) |
---|
2083 | "USAGE: equalMDeg(exp1, exp2[, V]); intvec exp1, exp2, intmat V |
---|
2084 | PURPOSE: Tests if the exponent vectors of two monomials (given by exp1 and exp2) |
---|
2085 | represent the same multidegree. |
---|
2086 | NOTE: the integer matrix V encodes multidegrees of module components, |
---|
2087 | if module component is present in exp1 and exp2 |
---|
2088 | EXAMPLE: example equalMDeg; shows an example |
---|
2089 | " |
---|
2090 | { |
---|
2091 | if( size(exp1) != size(exp2) ) |
---|
2092 | { |
---|
2093 | ERROR("Sorry: we cannot compare exponents comming from a polynomial and a vector yet!"); |
---|
2094 | } |
---|
2095 | |
---|
2096 | if( exp1 == exp2) |
---|
2097 | { |
---|
2098 | return (1==1); |
---|
2099 | } |
---|
2100 | |
---|
2101 | |
---|
2102 | |
---|
2103 | intmat M = getVariableWeights(); |
---|
2104 | |
---|
2105 | if( nrows(exp1) > ncols(M) ) // vectors => last exponent is the module component! |
---|
2106 | { |
---|
2107 | if( (size(#) == 0) or (typeof(#[1])!="intmat") ) |
---|
2108 | { |
---|
2109 | ERROR("Sorry: wrong or missing module-unit-weights-matrix V!"); |
---|
2110 | } |
---|
2111 | intmat V = #[1]; |
---|
2112 | |
---|
2113 | // typeof(V); print(V); |
---|
2114 | |
---|
2115 | int N = ncols(M); |
---|
2116 | int r = nrows(M); |
---|
2117 | |
---|
2118 | intvec d = intvec(exp1[1..N]) - intvec(exp2[1..N]); |
---|
2119 | intvec dm = intvec(V[1..r, exp1[N+1]]) - intvec(V[1..r, exp2[N+1]]); |
---|
2120 | |
---|
2121 | intvec difference = M * d + dm; |
---|
2122 | } |
---|
2123 | else |
---|
2124 | { |
---|
2125 | intvec d = (exp1 - exp2); |
---|
2126 | intvec difference = M * d; |
---|
2127 | } |
---|
2128 | |
---|
2129 | if (isFreeRepresented()) // no grading group!? |
---|
2130 | { |
---|
2131 | return ( difference == 0); |
---|
2132 | } |
---|
2133 | return ( isZeroElement( difference ) ); |
---|
2134 | } |
---|
2135 | example |
---|
2136 | { |
---|
2137 | "EXAMPLE:"; echo=2;printlevel=3; |
---|
2138 | |
---|
2139 | ring r = 0,(x,y,z),dp; |
---|
2140 | |
---|
2141 | intmat g[2][3]= |
---|
2142 | 1,0,1, |
---|
2143 | 0,1,1; |
---|
2144 | |
---|
2145 | intmat t[2][1]= |
---|
2146 | -2, |
---|
2147 | 1; |
---|
2148 | |
---|
2149 | setBaseMultigrading(g,t); |
---|
2150 | |
---|
2151 | poly a = x10yz; |
---|
2152 | poly b = x8y2z; |
---|
2153 | poly c = x4z2; |
---|
2154 | poly d = y5; |
---|
2155 | poly e = x2y2; |
---|
2156 | poly f = z2; |
---|
2157 | |
---|
2158 | |
---|
2159 | equalMDeg(leadexp(a), leadexp(b)); |
---|
2160 | equalMDeg(leadexp(a), leadexp(c)); |
---|
2161 | equalMDeg(leadexp(a), leadexp(d)); |
---|
2162 | equalMDeg(leadexp(a), leadexp(e)); |
---|
2163 | equalMDeg(leadexp(a), leadexp(f)); |
---|
2164 | |
---|
2165 | equalMDeg(leadexp(b), leadexp(c)); |
---|
2166 | equalMDeg(leadexp(b), leadexp(d)); |
---|
2167 | equalMDeg(leadexp(b), leadexp(e)); |
---|
2168 | equalMDeg(leadexp(b), leadexp(f)); |
---|
2169 | |
---|
2170 | equalMDeg(leadexp(c), leadexp(d)); |
---|
2171 | equalMDeg(leadexp(c), leadexp(e)); |
---|
2172 | equalMDeg(leadexp(c), leadexp(f)); |
---|
2173 | |
---|
2174 | equalMDeg(leadexp(d), leadexp(e)); |
---|
2175 | equalMDeg(leadexp(d), leadexp(f)); |
---|
2176 | |
---|
2177 | equalMDeg(leadexp(e), leadexp(f)); |
---|
2178 | |
---|
2179 | } |
---|
2180 | |
---|
2181 | |
---|
2182 | |
---|
2183 | /******************************************************/ |
---|
2184 | static proc isFreeRepresented() |
---|
2185 | "check whether the base muligrading is free (it is zero). |
---|
2186 | " |
---|
2187 | { |
---|
2188 | intmat T = getLattice(); |
---|
2189 | |
---|
2190 | intmat Z[nrows(T)][ncols(T)]; |
---|
2191 | |
---|
2192 | return (T == Z); // no grading group! |
---|
2193 | } |
---|
2194 | |
---|
2195 | |
---|
2196 | /******************************************************/ |
---|
2197 | proc isHomogeneous(def a, list #) |
---|
2198 | "USAGE: isHomogeneous(a[, f]); a polynomial/vector/ideal/module |
---|
2199 | RETURN: boolean, TRUE if a is (multi)homogeneous, and FALSE otherwise |
---|
2200 | EXAMPLE: example isHomogeneous; shows an example |
---|
2201 | " |
---|
2202 | { |
---|
2203 | if( (typeof(a) == "poly") or (typeof(a) == "vector") ) |
---|
2204 | { |
---|
2205 | return ( size(mDegPartition(a)) <= 1 ) |
---|
2206 | } |
---|
2207 | |
---|
2208 | if( (typeof(a) == "ideal") or (typeof(a) == "module") ) |
---|
2209 | { |
---|
2210 | if(size(#) > 0) |
---|
2211 | { |
---|
2212 | if (#[1] == "checkGens") |
---|
2213 | { |
---|
2214 | def aa; |
---|
2215 | for( int i = ncols(a); i > 0; i-- ) |
---|
2216 | { |
---|
2217 | aa = getGradedGenerator(a, i); |
---|
2218 | |
---|
2219 | if(!isHomogeneous(aa)) |
---|
2220 | { |
---|
2221 | return(0==1); |
---|
2222 | } |
---|
2223 | } |
---|
2224 | return(1==1); |
---|
2225 | } |
---|
2226 | } |
---|
2227 | |
---|
2228 | def g = groebner(a); // !!!! |
---|
2229 | |
---|
2230 | def b, aa; int j; |
---|
2231 | for( int i = ncols(a); i > 0; i-- ) |
---|
2232 | { |
---|
2233 | aa = getGradedGenerator(a, i); |
---|
2234 | |
---|
2235 | b = mDegPartition(aa); |
---|
2236 | for( j = ncols(b); j > 0; j-- ) |
---|
2237 | { |
---|
2238 | if(NF(b[j],g) != 0) |
---|
2239 | { |
---|
2240 | return(0==1); |
---|
2241 | } |
---|
2242 | } |
---|
2243 | } |
---|
2244 | return(1==1); |
---|
2245 | } |
---|
2246 | } |
---|
2247 | example |
---|
2248 | { |
---|
2249 | "EXAMPLE:"; echo=2; |
---|
2250 | |
---|
2251 | ring r = 0,(x,y,z),dp; |
---|
2252 | |
---|
2253 | //Grading and Torsion matrices: |
---|
2254 | intmat M[3][3] = |
---|
2255 | 1,0,0, |
---|
2256 | 0,1,0, |
---|
2257 | 0,0,1; |
---|
2258 | |
---|
2259 | intmat T[3][1] = |
---|
2260 | 1,2,3; |
---|
2261 | |
---|
2262 | setBaseMultigrading(M,T); |
---|
2263 | |
---|
2264 | attrib(r); |
---|
2265 | |
---|
2266 | poly f = x-yz; |
---|
2267 | |
---|
2268 | mDegPartition(f); |
---|
2269 | print(mDeg(_)); |
---|
2270 | |
---|
2271 | isHomogeneous(f); // f: is not homogeneous |
---|
2272 | |
---|
2273 | poly g = 1-xy2z3; |
---|
2274 | isHomogeneous(g); // g: is homogeneous |
---|
2275 | mDegPartition(g); |
---|
2276 | |
---|
2277 | kill T; |
---|
2278 | ///////////////////////////////////////////////////////// |
---|
2279 | // new Torsion matrix: |
---|
2280 | intmat T[3][4] = |
---|
2281 | 3,3,3,3, |
---|
2282 | 2,1,3,0, |
---|
2283 | 1,2,0,3; |
---|
2284 | |
---|
2285 | setBaseMultigrading(M,T); |
---|
2286 | |
---|
2287 | f; |
---|
2288 | isHomogeneous(f); |
---|
2289 | mDegPartition(f); |
---|
2290 | |
---|
2291 | // --------------------- |
---|
2292 | g; |
---|
2293 | isHomogeneous(g); |
---|
2294 | mDegPartition(g); |
---|
2295 | |
---|
2296 | kill r, T, M; |
---|
2297 | |
---|
2298 | ring R = 0, (x,y,z), dp; |
---|
2299 | |
---|
2300 | intmat A[2][3] = |
---|
2301 | 0,0,1, |
---|
2302 | 3,2,1; |
---|
2303 | intmat T[2][1] = |
---|
2304 | -1, |
---|
2305 | 4; |
---|
2306 | setBaseMultigrading(A, T); |
---|
2307 | |
---|
2308 | isHomogeneous(ideal(x2 - y3 -xy +z, x*y-z, x^3 - y^2*z + x^2 -y^3)); // 1 |
---|
2309 | isHomogeneous(ideal(x2 - y3 -xy +z, x*y-z, x^3 - y^2*z + x^2 -y^3), "checkGens"); |
---|
2310 | isHomogeneous(ideal(x+y, x2 - y2)); // 0 |
---|
2311 | |
---|
2312 | // Degree partition: |
---|
2313 | mDegPartition(x2 - y3 -xy +z); |
---|
2314 | mDegPartition(x3 -y2z + x2 -y3 + z + 1); |
---|
2315 | |
---|
2316 | |
---|
2317 | module N = gen(1) + (x+y) * gen(2), z*gen(3); |
---|
2318 | |
---|
2319 | intmat V[2][3] = 0; // 1, 2, 3, 4, 5, 6; // column-wise weights of components!!?? |
---|
2320 | |
---|
2321 | vector v1, v2; |
---|
2322 | |
---|
2323 | v1 = setModuleGrading(N[1], V); v1; |
---|
2324 | mDegPartition(v1); |
---|
2325 | print( mDeg(_) ); |
---|
2326 | |
---|
2327 | v2 = setModuleGrading(N[2], V); v2; |
---|
2328 | mDegPartition(v2); |
---|
2329 | print( mDeg(_) ); |
---|
2330 | |
---|
2331 | N = setModuleGrading(N, V); |
---|
2332 | isHomogeneous(N); |
---|
2333 | print( mDeg(N) ); |
---|
2334 | |
---|
2335 | /////////////////////////////////////// |
---|
2336 | |
---|
2337 | V = |
---|
2338 | 1, 2, 3, |
---|
2339 | 4, 5, 6; |
---|
2340 | |
---|
2341 | v1 = setModuleGrading(N[1], V); v1; |
---|
2342 | mDegPartition(v1); |
---|
2343 | print( mDeg(_) ); |
---|
2344 | |
---|
2345 | v2 = setModuleGrading(N[2], V); v2; |
---|
2346 | mDegPartition(v2); |
---|
2347 | print( mDeg(_) ); |
---|
2348 | |
---|
2349 | N = setModuleGrading(N, V); |
---|
2350 | isHomogeneous(N); |
---|
2351 | print( mDeg(N) ); |
---|
2352 | |
---|
2353 | /////////////////////////////////////// |
---|
2354 | |
---|
2355 | V = |
---|
2356 | 0, 0, 0, |
---|
2357 | 4, 1, 0; |
---|
2358 | |
---|
2359 | N = gen(1) + x * gen(2), z*gen(3); |
---|
2360 | N = setModuleGrading(N, V); print(N); |
---|
2361 | isHomogeneous(N); |
---|
2362 | print( mDeg(N) ); |
---|
2363 | v1 = getGradedGenerator(N,1); print(v1); |
---|
2364 | mDegPartition(v1); |
---|
2365 | print( mDeg(_) ); |
---|
2366 | N = setModuleGrading(N, V); print(N); |
---|
2367 | isHomogeneous(N); |
---|
2368 | print( mDeg(N) ); |
---|
2369 | } |
---|
2370 | |
---|
2371 | /******************************************************/ |
---|
2372 | proc mDeg(def A) |
---|
2373 | "USAGE: mDeg(A); polynomial/vector/ideal/module A |
---|
2374 | PURPOSE: compute multidegree |
---|
2375 | EXAMPLE: example mDeg; shows an example |
---|
2376 | " |
---|
2377 | { |
---|
2378 | if( defined(attrib(A, "grad")) > 0 ) |
---|
2379 | { |
---|
2380 | return (attrib(A, "grad")); |
---|
2381 | } |
---|
2382 | |
---|
2383 | intmat M = getVariableWeights(); |
---|
2384 | int N = nvars(basering); |
---|
2385 | |
---|
2386 | if( ncols(M) != N ) |
---|
2387 | { |
---|
2388 | ERROR("Sorry wrong mgrad-size of M: " + string(ncols(M))); |
---|
2389 | } |
---|
2390 | |
---|
2391 | int r = nrows(M); |
---|
2392 | |
---|
2393 | if( (typeof(A) == "vector") or (typeof(A) == "module") ) |
---|
2394 | { |
---|
2395 | intmat V = getModuleGrading(A); |
---|
2396 | |
---|
2397 | if( nrows(V) != r ) |
---|
2398 | { |
---|
2399 | ERROR("Sorry wrong mgrad-size of V: " + string(nrows(V))); |
---|
2400 | } |
---|
2401 | } |
---|
2402 | |
---|
2403 | if( A == 0 ) |
---|
2404 | { |
---|
2405 | intvec v; v[r] = 0; |
---|
2406 | return (v); |
---|
2407 | } |
---|
2408 | |
---|
2409 | intvec m; m[r] = 0; |
---|
2410 | |
---|
2411 | if( typeof(A) == "poly" ) |
---|
2412 | { |
---|
2413 | intvec v = leadexp(A); // v; |
---|
2414 | m = M * v; |
---|
2415 | |
---|
2416 | // We assume homogeneous input! |
---|
2417 | return(m); |
---|
2418 | |
---|
2419 | A = A - lead(A); |
---|
2420 | while( size(A) > 0 ) |
---|
2421 | { |
---|
2422 | v = leadexp(A); // v; |
---|
2423 | m = max( m, M * v, r ); // ???? |
---|
2424 | A = A - lead(A); |
---|
2425 | } |
---|
2426 | |
---|
2427 | return(m); |
---|
2428 | } |
---|
2429 | |
---|
2430 | |
---|
2431 | if( typeof(A) == "vector" ) |
---|
2432 | { |
---|
2433 | intvec v; |
---|
2434 | v = leadexp(A); // v; |
---|
2435 | m = intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]); |
---|
2436 | |
---|
2437 | // We assume homogeneous input! |
---|
2438 | return(m); |
---|
2439 | |
---|
2440 | A = A - lead(A); |
---|
2441 | while( size(A) > 0 ) |
---|
2442 | { |
---|
2443 | v = leadexp(A); // v; |
---|
2444 | |
---|
2445 | // intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]); |
---|
2446 | |
---|
2447 | m = max( m, intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]), r ); // ??? |
---|
2448 | |
---|
2449 | A = A - lead(A); |
---|
2450 | } |
---|
2451 | |
---|
2452 | return(m); |
---|
2453 | } |
---|
2454 | |
---|
2455 | int i, j; intvec d; |
---|
2456 | |
---|
2457 | if( typeof(A) == "ideal" ) |
---|
2458 | { |
---|
2459 | intmat G[ r ] [ ncols(A)]; |
---|
2460 | for( i = ncols(A); i > 0; i-- ) |
---|
2461 | { |
---|
2462 | d = mDeg( A[i] ); |
---|
2463 | |
---|
2464 | for( j = 1; j <= r; j++ ) // see ticket: 253 |
---|
2465 | { |
---|
2466 | G[j, i] = d[j]; |
---|
2467 | } |
---|
2468 | } |
---|
2469 | return(G); |
---|
2470 | } |
---|
2471 | |
---|
2472 | if( typeof(A) == "module" ) |
---|
2473 | { |
---|
2474 | intmat G[ r ] [ ncols(A)]; |
---|
2475 | vector v; |
---|
2476 | |
---|
2477 | for( i = ncols(A); i > 0; i-- ) |
---|
2478 | { |
---|
2479 | v = getGradedGenerator(A, i); |
---|
2480 | |
---|
2481 | // G[1..r, i] |
---|
2482 | d = mDeg(v); |
---|
2483 | |
---|
2484 | for( j = 1; j <= r; j++ ) // see ticket: 253 |
---|
2485 | { |
---|
2486 | G[j, i] = d[j]; |
---|
2487 | } |
---|
2488 | |
---|
2489 | } |
---|
2490 | |
---|
2491 | return(G); |
---|
2492 | } |
---|
2493 | |
---|
2494 | } |
---|
2495 | example |
---|
2496 | { |
---|
2497 | "EXAMPLE:"; echo=2; |
---|
2498 | |
---|
2499 | ring r = 0,(x, y), dp; |
---|
2500 | |
---|
2501 | intmat A[2][2] = 1, 0, 0, 1; |
---|
2502 | print(A); |
---|
2503 | |
---|
2504 | intmat Ta[2][1] = 0, 3; |
---|
2505 | print(Ta); |
---|
2506 | |
---|
2507 | // attrib(A, "gradingGroup", Ta); // to think about |
---|
2508 | |
---|
2509 | // "poly:"; |
---|
2510 | setBaseMultigrading(A); |
---|
2511 | |
---|
2512 | |
---|
2513 | mDeg( x*x, A ); |
---|
2514 | mDeg( y*y*y, A ); |
---|
2515 | |
---|
2516 | setBaseMultigrading(A, Ta); |
---|
2517 | |
---|
2518 | mDeg( x*x*y ); |
---|
2519 | |
---|
2520 | mDeg( y*y*y*x ); |
---|
2521 | |
---|
2522 | mDeg( x*y + x + 1 ); |
---|
2523 | |
---|
2524 | mDegPartition(x*y + x + 1); |
---|
2525 | |
---|
2526 | print ( mDeg(0) ); |
---|
2527 | poly zero = 0; |
---|
2528 | print ( mDeg(zero) ); |
---|
2529 | |
---|
2530 | // "ideal:"; |
---|
2531 | |
---|
2532 | ideal I = y*x*x, x*y*y*y; |
---|
2533 | print( mDeg(I) ); |
---|
2534 | |
---|
2535 | print ( mDeg(ideal(0)) ); |
---|
2536 | print ( mDeg(ideal(0,0,0)) ); |
---|
2537 | |
---|
2538 | // "vectors:"; |
---|
2539 | |
---|
2540 | intmat B[2][2] = 0, 1, 1, 0; |
---|
2541 | print(B); |
---|
2542 | |
---|
2543 | mDeg( setModuleGrading(y*y*y*gen(2), B )); |
---|
2544 | mDeg( setModuleGrading(x*x*gen(1), B )); |
---|
2545 | |
---|
2546 | |
---|
2547 | vector V = x*gen(1) + y*gen(2); |
---|
2548 | V = setModuleGrading(V, B); |
---|
2549 | mDeg( V ); |
---|
2550 | |
---|
2551 | vector v1 = setModuleGrading([0, 0, 0], B); |
---|
2552 | print( mDeg( v1 ) ); |
---|
2553 | |
---|
2554 | vector v2 = setModuleGrading([0], B); |
---|
2555 | print( mDeg( v2 ) ); |
---|
2556 | |
---|
2557 | // "module:"; |
---|
2558 | |
---|
2559 | module D = x*gen(1), y*gen(2); |
---|
2560 | D; |
---|
2561 | D = setModuleGrading(D, B); |
---|
2562 | print( mDeg( D ) ); |
---|
2563 | |
---|
2564 | |
---|
2565 | module DD = [0, 0],[0, 0, 0]; |
---|
2566 | DD = setModuleGrading(DD, B); |
---|
2567 | print( mDeg( DD ) ); |
---|
2568 | |
---|
2569 | module DDD = [0, 0]; |
---|
2570 | DDD = setModuleGrading(DDD, B); |
---|
2571 | print( mDeg( DDD ) ); |
---|
2572 | |
---|
2573 | }; |
---|
2574 | |
---|
2575 | |
---|
2576 | |
---|
2577 | |
---|
2578 | |
---|
2579 | /******************************************************/ |
---|
2580 | proc mDegPartition(def p) |
---|
2581 | "USAGE: mDegPartition(def p), p polynomial/vector |
---|
2582 | RETURNS: an ideal/module consisting of multigraded-homogeneous parts of p |
---|
2583 | EXAMPLE: example mDegPartition; shows an example |
---|
2584 | " |
---|
2585 | { // TODO: What about an ideal or module??? |
---|
2586 | |
---|
2587 | if( typeof(p) == "poly" ) |
---|
2588 | { |
---|
2589 | ideal I; |
---|
2590 | poly mp, t, tt; |
---|
2591 | } |
---|
2592 | else |
---|
2593 | { |
---|
2594 | if( typeof(p) == "vector" ) |
---|
2595 | { |
---|
2596 | module I; |
---|
2597 | vector mp, t, tt; |
---|
2598 | } |
---|
2599 | else |
---|
2600 | { |
---|
2601 | ERROR("Wrong ARGUMENT type!"); |
---|
2602 | } |
---|
2603 | } |
---|
2604 | |
---|
2605 | if( typeof(p) == "vector" ) |
---|
2606 | { |
---|
2607 | intmat V = getModuleGrading(p); |
---|
2608 | } |
---|
2609 | else |
---|
2610 | { |
---|
2611 | intmat V; |
---|
2612 | } |
---|
2613 | |
---|
2614 | if( size(p) > 1) |
---|
2615 | { |
---|
2616 | intvec m; |
---|
2617 | |
---|
2618 | while( p != 0 ) |
---|
2619 | { |
---|
2620 | m = leadexp(p); |
---|
2621 | mp = lead(p); |
---|
2622 | p = p - lead(p); |
---|
2623 | tt = p; t = 0; |
---|
2624 | |
---|
2625 | while( size(tt) > 0 ) |
---|
2626 | { |
---|
2627 | // TODO: we do not cache matrices (M,T,H,V), which remain the same :( |
---|
2628 | // TODO: we need some low-level procedure with all these arguments...! |
---|
2629 | if( equalMDeg( leadexp(tt), m, V ) ) |
---|
2630 | { |
---|
2631 | mp = mp + lead(tt); // "mp", mp; |
---|
2632 | } |
---|
2633 | else |
---|
2634 | { |
---|
2635 | t = t + lead(tt); // "t", t; |
---|
2636 | } |
---|
2637 | |
---|
2638 | tt = tt - lead(tt); |
---|
2639 | } |
---|
2640 | |
---|
2641 | I[size(I)+1] = mp; |
---|
2642 | |
---|
2643 | p = t; |
---|
2644 | } |
---|
2645 | } |
---|
2646 | else |
---|
2647 | { |
---|
2648 | I[1] = p; // single monom |
---|
2649 | } |
---|
2650 | |
---|
2651 | if( typeof(I) == "module" ) |
---|
2652 | { |
---|
2653 | I = setModuleGrading(I, V); |
---|
2654 | } |
---|
2655 | |
---|
2656 | return (I); |
---|
2657 | } |
---|
2658 | example |
---|
2659 | { |
---|
2660 | "EXAMPLE:"; echo=2; |
---|
2661 | |
---|
2662 | ring r = 0,(x,y,z),dp; |
---|
2663 | |
---|
2664 | intmat g[2][3]= |
---|
2665 | 1,0,1, |
---|
2666 | 0,1,1; |
---|
2667 | intmat t[2][1]= |
---|
2668 | -2, |
---|
2669 | 1; |
---|
2670 | |
---|
2671 | setBaseMultigrading(g,t); |
---|
2672 | |
---|
2673 | poly f = x10yz+x8y2z-x4z2+y5+x2y2-z2+x17z3-y6; |
---|
2674 | |
---|
2675 | mDegPartition(f); |
---|
2676 | |
---|
2677 | vector v = xy*gen(1)-x3y2*gen(2)+x4y*gen(3); |
---|
2678 | intmat B[2][3]=1,-1,-2,0,0,1; |
---|
2679 | v = setModuleGrading(v,B); |
---|
2680 | getModuleGrading(v); |
---|
2681 | |
---|
2682 | mDegPartition(v, B); |
---|
2683 | } |
---|
2684 | |
---|
2685 | |
---|
2686 | |
---|
2687 | /******************************************************/ |
---|
2688 | static proc unitMatrix(int n) |
---|
2689 | { |
---|
2690 | intmat A[n][n]; |
---|
2691 | |
---|
2692 | for( int i = n; i > 0; i-- ) |
---|
2693 | { |
---|
2694 | A[i,i] = 1; |
---|
2695 | } |
---|
2696 | |
---|
2697 | return (A); |
---|
2698 | } |
---|
2699 | |
---|
2700 | |
---|
2701 | |
---|
2702 | /******************************************************/ |
---|
2703 | static proc finestMDeg(def r) |
---|
2704 | " |
---|
2705 | USAGE: finestMDeg(r); ring r |
---|
2706 | RETURN: ring, r endowed with the finest multigrading |
---|
2707 | TODO: not yet... |
---|
2708 | " |
---|
2709 | { |
---|
2710 | def save = basering; |
---|
2711 | setring (r); |
---|
2712 | |
---|
2713 | // in basering |
---|
2714 | ideal I = ideal(basering); |
---|
2715 | |
---|
2716 | int n = 0; int i; poly p; |
---|
2717 | for( i = ncols(I); i > 0; i-- ) |
---|
2718 | { |
---|
2719 | p = I[i]; |
---|
2720 | if( size(p) > 1 ) |
---|
2721 | { |
---|
2722 | n = n + (size(p) - 1); |
---|
2723 | } |
---|
2724 | else |
---|
2725 | { |
---|
2726 | I[i] = 0; |
---|
2727 | } |
---|
2728 | } |
---|
2729 | |
---|
2730 | int N = nvars(basering); |
---|
2731 | intmat A = unitMatrix(N); |
---|
2732 | |
---|
2733 | |
---|
2734 | |
---|
2735 | if( n > 0) |
---|
2736 | { |
---|
2737 | |
---|
2738 | intmat L[N][n]; |
---|
2739 | // list L; |
---|
2740 | int j = n; |
---|
2741 | |
---|
2742 | for( i = ncols(I); i > 0; i-- ) |
---|
2743 | { |
---|
2744 | p = I[i]; |
---|
2745 | |
---|
2746 | if( size(p) > 1 ) |
---|
2747 | { |
---|
2748 | intvec m0 = leadexp(p); |
---|
2749 | p = p - lead(p); |
---|
2750 | |
---|
2751 | while( size(p) > 0 ) |
---|
2752 | { |
---|
2753 | L[ 1..N, j ] = leadexp(p) - m0; |
---|
2754 | p = p - lead(p); |
---|
2755 | j--; |
---|
2756 | } |
---|
2757 | } |
---|
2758 | } |
---|
2759 | |
---|
2760 | print(L); |
---|
2761 | setBaseMultigrading(A, L); |
---|
2762 | } |
---|
2763 | else |
---|
2764 | { |
---|
2765 | setBaseMultigrading(A); |
---|
2766 | } |
---|
2767 | |
---|
2768 | // ERROR("nope"); |
---|
2769 | |
---|
2770 | // ring T = integer, (x), (C, dp); |
---|
2771 | |
---|
2772 | setring(save); |
---|
2773 | return (r); |
---|
2774 | } |
---|
2775 | example |
---|
2776 | { |
---|
2777 | "EXAMPLE:"; echo=2; |
---|
2778 | |
---|
2779 | ring r = 0,(x, y), dp; |
---|
2780 | qring q = std(x^2 - y); |
---|
2781 | |
---|
2782 | finestMDeg(q); |
---|
2783 | |
---|
2784 | } |
---|
2785 | |
---|
2786 | |
---|
2787 | |
---|
2788 | |
---|
2789 | /******************************************************/ |
---|
2790 | static proc newMap(map F, intmat Q, list #) |
---|
2791 | " |
---|
2792 | USAGE: newMap(F, Q[, P]); map F, intmat Q[, intmat P] |
---|
2793 | PURPOSE: endowe the map F with the integer matrices P [and Q] |
---|
2794 | " |
---|
2795 | { |
---|
2796 | attrib(F, "Q", Q); |
---|
2797 | |
---|
2798 | if( size(#) > 0 and typeof(#[1]) == "intmat" ) |
---|
2799 | { |
---|
2800 | attrib(F, "P", #[1]); |
---|
2801 | } |
---|
2802 | return (F); |
---|
2803 | } |
---|
2804 | |
---|
2805 | /******************************************************/ |
---|
2806 | static proc matrix2intmat( matrix M ) |
---|
2807 | { |
---|
2808 | execute( "intmat A[ "+ string(nrows(M)) + "]["+ string(ncols(M)) + "] = " + string(M) + ";" ); |
---|
2809 | return (A); |
---|
2810 | } |
---|
2811 | |
---|
2812 | |
---|
2813 | /******************************************************/ |
---|
2814 | static proc leftKernelZ(intmat M) |
---|
2815 | "USAGE: leftKernel(M); M a matrix |
---|
2816 | RETURN: module |
---|
2817 | PURPOSE: computes left kernel of matrix M (a module of all elements v such that vM=0) |
---|
2818 | EXAMPLE: example leftKernel; shows an example |
---|
2819 | " |
---|
2820 | { |
---|
2821 | if( nameof(basering) != "basering" ) |
---|
2822 | { |
---|
2823 | def save = basering; |
---|
2824 | } |
---|
2825 | |
---|
2826 | ring r = integer, (x), dp; |
---|
2827 | |
---|
2828 | |
---|
2829 | // basering; |
---|
2830 | module N = matrix((M)); // transpose |
---|
2831 | // print(N); |
---|
2832 | |
---|
2833 | def MM = modulo( N, std(0) ) ; |
---|
2834 | // print(MM); |
---|
2835 | |
---|
2836 | intmat R = ( matrix2intmat( MM ) ); // transpose |
---|
2837 | |
---|
2838 | if( defined(save) > 0 ) |
---|
2839 | { |
---|
2840 | setring save; |
---|
2841 | } |
---|
2842 | |
---|
2843 | kill r; |
---|
2844 | return( R ); |
---|
2845 | } |
---|
2846 | example |
---|
2847 | { |
---|
2848 | "EXAMPLE:"; echo=2; |
---|
2849 | |
---|
2850 | ring r= 0,(x,y,z),dp; |
---|
2851 | matrix M[3][1] = x,y,z; |
---|
2852 | print(M); |
---|
2853 | matrix L = leftKernel(M); |
---|
2854 | print(L); |
---|
2855 | // check: |
---|
2856 | print(L*M); |
---|
2857 | }; |
---|
2858 | |
---|
2859 | |
---|
2860 | |
---|
2861 | /******************************************************/ |
---|
2862 | // the following is taken from "sing4ti2.lib" as we need 'hilbert' from 4ti2 |
---|
2863 | |
---|
2864 | static proc hilbert4ti2intmat(intmat A, list #) |
---|
2865 | "USAGE: hilbert4ti2(A[,i]); |
---|
2866 | @* A=intmat |
---|
2867 | @* i=int |
---|
2868 | ASSUME: - A is a matrix with integer entries which describes the lattice |
---|
2869 | @* as ker(A), if second argument is not present, and |
---|
2870 | @* as the left image Im(A) = {zA : z \in ZZ^k}, if second argument is a positive integer |
---|
2871 | @* - number of variables of basering equals number of columns of A |
---|
2872 | @* (for ker(A)) resp. of rows of A (for Im(A)) |
---|
2873 | CREATE: temporary files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar |
---|
2874 | @* in the current directory (I/O files for communication with 4ti2) |
---|
2875 | NOTE: input rules for 4ti2 also apply to input to this procedure |
---|
2876 | @* hence ker(A)={x|Ax=0} and Im(A)={xA} |
---|
2877 | RETURN: toric ideal specified by Hilbert basis thereof |
---|
2878 | EXAMPLE: example graver4ti2; shows an example |
---|
2879 | " |
---|
2880 | { |
---|
2881 | if( system("sh","which hilbert 2> /dev/null 1> /dev/null") != 0 ) |
---|
2882 | { |
---|
2883 | ERROR("Sorry: cannot find 'hilbert' command from 4ti2. Please install 4ti2!"); |
---|
2884 | } |
---|
2885 | |
---|
2886 | //-------------------------------------------------------------------------- |
---|
2887 | // Initialization and Sanity Checks |
---|
2888 | //-------------------------------------------------------------------------- |
---|
2889 | int i,j; |
---|
2890 | int nr=nrows(A); |
---|
2891 | int nc=ncols(A); |
---|
2892 | string fileending="mat"; |
---|
2893 | if (size(#)!=0) |
---|
2894 | { |
---|
2895 | //--- default behaviour: use ker(A) as lattice |
---|
2896 | //--- if #[1]!=0 use Im(A) as lattice |
---|
2897 | if(typeof(#[1])!="int") |
---|
2898 | { |
---|
2899 | ERROR("optional parameter needs to be integer value"); |
---|
2900 | } |
---|
2901 | if(#[1]!=0) |
---|
2902 | { |
---|
2903 | fileending="lat"; |
---|
2904 | } |
---|
2905 | } |
---|
2906 | //--- we should also be checking whether all entries are indeed integers |
---|
2907 | //--- or whether there are fractions, but in this case the error message |
---|
2908 | //--- of 4ti2 is printed directly |
---|
2909 | |
---|
2910 | //-------------------------------------------------------------------------- |
---|
2911 | // preparing input file for 4ti2 |
---|
2912 | //-------------------------------------------------------------------------- |
---|
2913 | link eing=":w sing4ti2."+fileending; |
---|
2914 | string eingstring=string(nr)+" "+string(nc); |
---|
2915 | write(eing,eingstring); |
---|
2916 | for(i=1;i<=nr;i++) |
---|
2917 | { |
---|
2918 | kill eingstring; |
---|
2919 | string eingstring; |
---|
2920 | for(j=1;j<=nc;j++) |
---|
2921 | { |
---|
2922 | // if(g(A[i,j])>0)||(char(basering)!=0)||(npars(basering)>0)) |
---|
2923 | // { |
---|
2924 | // ERROR("Input to hilbert4ti2 needs to be a matrix with integer entries"); |
---|
2925 | // } |
---|
2926 | eingstring=eingstring+string(A[i,j])+" "; |
---|
2927 | } |
---|
2928 | write(eing, eingstring); |
---|
2929 | } |
---|
2930 | close(eing); |
---|
2931 | |
---|
2932 | //---------------------------------------------------------------------- |
---|
2933 | // calling 4ti2 and converting output |
---|
2934 | // Singular's string is too clumsy for this, hence we first prepare |
---|
2935 | // using standard unix commands |
---|
2936 | //---------------------------------------------------------------------- |
---|
2937 | |
---|
2938 | |
---|
2939 | j=system("sh","hilbert -q -n sing4ti2"); ////////// be quiet + no loggin!!! |
---|
2940 | |
---|
2941 | j=system("sh", "awk \'BEGIN{ORS=\",\";}{print $0;}\' sing4ti2.hil " + |
---|
2942 | "| sed s/[\\\ \\\t\\\v\\\f]/,/g " + |
---|
2943 | "| sed s/,+/,/g|sed s/,,/,/g " + |
---|
2944 | "| sed s/,,/,/g " + |
---|
2945 | "> sing4ti2.converted" ); |
---|
2946 | |
---|
2947 | |
---|
2948 | //---------------------------------------------------------------------- |
---|
2949 | // reading output of 4ti2 |
---|
2950 | //---------------------------------------------------------------------- |
---|
2951 | link ausg=":r sing4ti2.converted"; |
---|
2952 | //--- last entry ideal(0) is used to tie the list to the basering |
---|
2953 | //--- it will not be processed any further |
---|
2954 | |
---|
2955 | string s = read(ausg); |
---|
2956 | |
---|
2957 | if( defined(keepfiles) <= 0) |
---|
2958 | { |
---|
2959 | j=system("sh",("rm -f sing4ti2.hil sing4ti2.converted sing4ti2."+fileending)); |
---|
2960 | } |
---|
2961 | |
---|
2962 | string ergstr = "intvec erglist = " + s + "0;"; |
---|
2963 | execute(ergstr); |
---|
2964 | |
---|
2965 | // print(erglist); |
---|
2966 | |
---|
2967 | int Rnc = erglist[1]; |
---|
2968 | int Rnr = erglist[2]; |
---|
2969 | |
---|
2970 | intmat R[Rnr][Rnc]; |
---|
2971 | |
---|
2972 | int k = 3; |
---|
2973 | |
---|
2974 | for(i=1;i<=Rnc;i++) |
---|
2975 | { |
---|
2976 | for(j=1;j<=Rnr;j++) |
---|
2977 | { |
---|
2978 | // "i: ", i, ", j: ", j, ", v: ", erglist[k]; |
---|
2979 | R[j, i] = erglist[k]; |
---|
2980 | k = k + 1; |
---|
2981 | } |
---|
2982 | } |
---|
2983 | |
---|
2984 | |
---|
2985 | |
---|
2986 | return (R); |
---|
2987 | //--- get rid of leading entry 0; |
---|
2988 | // toric=toric[2..ncols(toric)]; |
---|
2989 | // return(toric); |
---|
2990 | } |
---|
2991 | // A nice example here is the 3x3 Magic Squares |
---|
2992 | example |
---|
2993 | { |
---|
2994 | "EXAMPLE:"; echo=2; |
---|
2995 | |
---|
2996 | ring r=0,(x1,x2,x3,x4,x5,x6,x7,x8,x9),dp; |
---|
2997 | intmat M[7][9]= |
---|
2998 | 1, 1, 1, -1, -1, -1, 0, 0, 0, |
---|
2999 | 1, 1, 1, 0, 0, 0,-1,-1,-1, |
---|
3000 | 0, 1, 1, -1, 0, 0,-1, 0, 0, |
---|
3001 | 1, 0, 1, 0, -1, 0, 0,-1, 0, |
---|
3002 | 1, 1, 0, 0, 0, -1, 0, 0,-1, |
---|
3003 | 0, 1, 1, 0, -1, 0, 0, 0,-1, |
---|
3004 | 1, 1, 0, 0, -1, 0,-1, 0, 0; |
---|
3005 | hilbert4ti2intmat(M); |
---|
3006 | hermiteNormalForm(M); |
---|
3007 | } |
---|
3008 | |
---|
3009 | ///////////////////////////////////////////////////////////////////////////// |
---|
3010 | static proc getMonomByExponent(intvec exp) |
---|
3011 | { |
---|
3012 | int n = nvars(basering); |
---|
3013 | |
---|
3014 | if( nrows(exp) < n ) |
---|
3015 | { |
---|
3016 | n = nrows(exp); |
---|
3017 | } |
---|
3018 | |
---|
3019 | poly m = 1; int e; |
---|
3020 | |
---|
3021 | for( int i = 1; i <= n; i++ ) |
---|
3022 | { |
---|
3023 | e = exp[i]; |
---|
3024 | if( e < 0 ) |
---|
3025 | { |
---|
3026 | ERROR("Negative exponent!!!"); |
---|
3027 | } |
---|
3028 | |
---|
3029 | m = m * (var(i)^e); |
---|
3030 | } |
---|
3031 | |
---|
3032 | return (m); |
---|
3033 | |
---|
3034 | } |
---|
3035 | |
---|
3036 | /******************************************************/ |
---|
3037 | proc mDegBasis(intvec d) |
---|
3038 | " |
---|
3039 | USAGE: multidegree d |
---|
3040 | ASSUME: current ring is multigraded, monomial ordering is global |
---|
3041 | PURPOSE: compute all monomials of multidegree d |
---|
3042 | EXAMPLE: example mDegBasis; shows an example |
---|
3043 | " |
---|
3044 | { |
---|
3045 | def R = basering; // setring R; |
---|
3046 | |
---|
3047 | intmat M = getVariableWeights(R); |
---|
3048 | |
---|
3049 | // print(M); |
---|
3050 | |
---|
3051 | int nr = nrows(M); |
---|
3052 | int nc = ncols(M); |
---|
3053 | |
---|
3054 | intmat A[nr][nc+1]; |
---|
3055 | A[1..nr, 1..nc] = M[1..nr, 1..nc]; |
---|
3056 | //typeof(A[1..nr, nc+1]); |
---|
3057 | if( nr==1) |
---|
3058 | { |
---|
3059 | A[1..nr, nc+1]=-d[1]; |
---|
3060 | } |
---|
3061 | else |
---|
3062 | { |
---|
3063 | A[1..nr, nc+1] = -d; |
---|
3064 | } |
---|
3065 | |
---|
3066 | intmat T = getLattice(R); |
---|
3067 | |
---|
3068 | if( isFreeRepresented() ) |
---|
3069 | { |
---|
3070 | intmat B = hilbert4ti2intmat(A); |
---|
3071 | |
---|
3072 | // matrix B = unitMatrix(nrows(T)); |
---|
3073 | } |
---|
3074 | else |
---|
3075 | { |
---|
3076 | int n = ncols(T); |
---|
3077 | |
---|
3078 | nc = ncols(A); |
---|
3079 | |
---|
3080 | intmat AA[nr][nc + 2 * n]; |
---|
3081 | AA[1..nr, 1.. nc] = A[1..nr, 1.. nc]; |
---|
3082 | AA[1..nr, nc + (1.. n)] = T[1..nr, 1.. n]; |
---|
3083 | AA[1..nr, nc + n + (1.. n)] = -T[1..nr, 1.. n]; |
---|
3084 | |
---|
3085 | |
---|
3086 | // print ( AA ); |
---|
3087 | |
---|
3088 | intmat K = leftKernelZ(( AA ) ); // |
---|
3089 | |
---|
3090 | // print(K); |
---|
3091 | |
---|
3092 | intmat KK[nc][ncols(K)] = K[ 1.. nc, 1.. ncols(K) ]; |
---|
3093 | |
---|
3094 | // print(KK); |
---|
3095 | // "!"; |
---|
3096 | |
---|
3097 | intmat B = hilbert4ti2intmat(transpose(KK), 1); |
---|
3098 | |
---|
3099 | // "!"; print(B); |
---|
3100 | |
---|
3101 | } |
---|
3102 | |
---|
3103 | |
---|
3104 | // print(A); |
---|
3105 | |
---|
3106 | |
---|
3107 | |
---|
3108 | int i; |
---|
3109 | int nnr = nrows(B); |
---|
3110 | int nnc = ncols(B); |
---|
3111 | ideal I, J; |
---|
3112 | if(nnc==0){ |
---|
3113 | I=0; |
---|
3114 | return(I); |
---|
3115 | } |
---|
3116 | I[nnc] = 0; |
---|
3117 | J[nnc] = 0; |
---|
3118 | |
---|
3119 | for( i = 1; i <= nnc; i++ ) |
---|
3120 | { |
---|
3121 | // "i: ", i; B[nnr, i]; |
---|
3122 | |
---|
3123 | if( B[nnr, i] == 1) |
---|
3124 | { |
---|
3125 | // intvec(B[1..nnr-1, i]); |
---|
3126 | I[i] = getMonomByExponent(intvec(B[1..nnr-1, i])); |
---|
3127 | } |
---|
3128 | else |
---|
3129 | { |
---|
3130 | if( B[nnr, i] == 0) |
---|
3131 | { |
---|
3132 | // intvec(B[1..nnr-1, i]); |
---|
3133 | J[i] = getMonomByExponent(intvec(B[1..nnr-1, i])); |
---|
3134 | } |
---|
3135 | } |
---|
3136 | // I[i]; |
---|
3137 | } |
---|
3138 | |
---|
3139 | ideal Q = (ideal(basering)); |
---|
3140 | |
---|
3141 | if ( size(Q) > 0 ) |
---|
3142 | { |
---|
3143 | I = NF( I, lead(Q) ); |
---|
3144 | J = NF( J, lead(Q) ); // Global ordering!!! |
---|
3145 | } |
---|
3146 | |
---|
3147 | I = simplify(I, 2); // d |
---|
3148 | J = simplify(J, 2); // d |
---|
3149 | |
---|
3150 | attrib(I, "ZeroPart", J); |
---|
3151 | |
---|
3152 | return (I); |
---|
3153 | |
---|
3154 | // setring ; |
---|
3155 | } |
---|
3156 | example |
---|
3157 | { |
---|
3158 | "EXAMPLE:"; echo=2; |
---|
3159 | |
---|
3160 | ring R = 0, (x, y), dp; |
---|
3161 | |
---|
3162 | intmat g1[2][2]=1,0,0,1; |
---|
3163 | intmat l[2][1]=2,0; |
---|
3164 | intmat g2[2][2]=1,1,1,1; |
---|
3165 | intvec v1=4,0; |
---|
3166 | intvec v2=4,4; |
---|
3167 | |
---|
3168 | intmat g3[1][2]=1,1; |
---|
3169 | setBaseMultigrading(g3); |
---|
3170 | intvec v3=4:1; |
---|
3171 | v3; |
---|
3172 | mDegBasis(v3); |
---|
3173 | |
---|
3174 | setBaseMultigrading(g1,l); |
---|
3175 | mDegBasis(v1); |
---|
3176 | setBaseMultigrading(g2); |
---|
3177 | mDegBasis(v2); |
---|
3178 | |
---|
3179 | intmat M[2][2] = 1, -1, -1, 1; |
---|
3180 | intvec d = -2, 2; |
---|
3181 | |
---|
3182 | setBaseMultigrading(M); |
---|
3183 | |
---|
3184 | mDegBasis(d); |
---|
3185 | attrib(_, "ZeroPart"); |
---|
3186 | |
---|
3187 | kill R; |
---|
3188 | ring R = 0, (x, y, z), dp; |
---|
3189 | |
---|
3190 | intmat M[2][3] = 1, -2, 1, 1, 1, 0; |
---|
3191 | |
---|
3192 | intmat L[2][1] = 0, 2; |
---|
3193 | |
---|
3194 | intvec d = 4, 1; |
---|
3195 | |
---|
3196 | setBaseMultigrading(M, L); |
---|
3197 | |
---|
3198 | mDegBasis(d); |
---|
3199 | attrib(_, "ZeroPart"); |
---|
3200 | |
---|
3201 | |
---|
3202 | kill R; |
---|
3203 | |
---|
3204 | ring R = 0, (x, y, z), dp; |
---|
3205 | qring Q = std(ideal( y^6+ x*y^3*z-x^2*z^2 )); |
---|
3206 | |
---|
3207 | |
---|
3208 | intmat M[2][3] = 1, 1, 2, 2, 1, 1; |
---|
3209 | // intmat T[2][1] = 0, 2; |
---|
3210 | |
---|
3211 | setBaseMultigrading(M); |
---|
3212 | |
---|
3213 | intvec d = 6, 6; |
---|
3214 | mDegBasis(d); |
---|
3215 | attrib(_, "ZeroPart"); |
---|
3216 | |
---|
3217 | |
---|
3218 | |
---|
3219 | kill R; |
---|
3220 | ring R = 0, (x, y, z), dp; |
---|
3221 | qring Q = std(ideal( x*z^3 - y *z^6, x*y*z - x^4*y^2 )); |
---|
3222 | |
---|
3223 | |
---|
3224 | intmat M[2][3] = 1, -2, 1, 1, 1, 0; |
---|
3225 | intmat T[2][1] = 0, 2; |
---|
3226 | |
---|
3227 | intvec d = 4, 1; |
---|
3228 | |
---|
3229 | setBaseMultigrading(M, T); |
---|
3230 | |
---|
3231 | mDegBasis(d); |
---|
3232 | attrib(_, "ZeroPart"); |
---|
3233 | } |
---|
3234 | |
---|
3235 | |
---|
3236 | proc mDegSyzygy(def I) |
---|
3237 | "USAGE: mDegSyzygy(I); I is a ideal or a module |
---|
3238 | PURPOSE: computes the multigraded syzygy module of I |
---|
3239 | RETURNS: module, the syzygy module of I |
---|
3240 | NOTE: generators of I must be multigraded homogeneous |
---|
3241 | EXAMPLE: example mDegSyzygy; shows an example |
---|
3242 | " |
---|
3243 | { |
---|
3244 | if( isHomogeneous(I, "checkGens") == 0) |
---|
3245 | { |
---|
3246 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3247 | } |
---|
3248 | module S = syz(I); |
---|
3249 | S = setModuleGrading(S, mDeg(I)); |
---|
3250 | return (S); |
---|
3251 | } |
---|
3252 | example |
---|
3253 | { |
---|
3254 | "EXAMPLE:"; echo=2; |
---|
3255 | |
---|
3256 | |
---|
3257 | ring r = 0,(x,y,z,w),dp; |
---|
3258 | intmat MM[2][4]= |
---|
3259 | 1,1,1,1, |
---|
3260 | 0,1,3,4; |
---|
3261 | setBaseMultigrading(MM); |
---|
3262 | module M = ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3263 | |
---|
3264 | |
---|
3265 | intmat v[2][nrows(M)]= |
---|
3266 | 1, |
---|
3267 | 0; |
---|
3268 | |
---|
3269 | M = setModuleGrading(M, v); |
---|
3270 | |
---|
3271 | isHomogeneous(M); |
---|
3272 | "Multidegrees: "; print(mDeg(M)); |
---|
3273 | // Let's compute syzygies! |
---|
3274 | def S = mDegSyzygy(M); S; |
---|
3275 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3276 | "Multidegrees: "; print(mDeg(S)); |
---|
3277 | |
---|
3278 | isHomogeneous(S); |
---|
3279 | } |
---|
3280 | |
---|
3281 | |
---|
3282 | |
---|
3283 | proc mDegModulo(def I, def J) |
---|
3284 | "USAGE: mDegModulo(I); I, J are ideals or modules |
---|
3285 | PURPOSE: computes the multigraded 'modulo' module of I and J |
---|
3286 | RETURNS: module, see 'modulo' command |
---|
3287 | NOTE: I and J should have the same multigrading, and their |
---|
3288 | generators must be multigraded homogeneous |
---|
3289 | EXAMPLE: example mDegModulo; shows an example |
---|
3290 | " |
---|
3291 | { |
---|
3292 | if( (isHomogeneous(I, "checkGens") == 0) or (isHomogeneous(J, "checkGens") == 0) ) |
---|
3293 | { |
---|
3294 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3295 | } |
---|
3296 | module K = modulo(I, J); |
---|
3297 | K = setModuleGrading(K, mDeg(I)); |
---|
3298 | return (K); |
---|
3299 | } |
---|
3300 | example |
---|
3301 | { |
---|
3302 | "EXAMPLE:"; echo=2; |
---|
3303 | |
---|
3304 | ring r = 0,(x,y,z),dp; |
---|
3305 | intmat MM[2][3]= |
---|
3306 | -1,1,1, |
---|
3307 | 0,1,3; |
---|
3308 | setBaseMultigrading(MM); |
---|
3309 | |
---|
3310 | ideal h1 = x, y, z; |
---|
3311 | ideal h2 = x; |
---|
3312 | |
---|
3313 | "Multidegrees: "; print(mDeg(h1)); |
---|
3314 | |
---|
3315 | // Let's compute modulo(h1, h2): |
---|
3316 | def K = mDegModulo(h1, h2); K; |
---|
3317 | |
---|
3318 | "Module Units Multigrading: "; print( getModuleGrading(K) ); |
---|
3319 | "Multidegrees: "; print(mDeg(K)); |
---|
3320 | |
---|
3321 | isHomogeneous(K); |
---|
3322 | } |
---|
3323 | |
---|
3324 | |
---|
3325 | proc mDegGroebner(def I) |
---|
3326 | "USAGE: mDegGroebner(I); I is a poly/vector/ideal/module |
---|
3327 | PURPOSE: computes the multigraded standard/groebner basis of I |
---|
3328 | NOTE: I must be multigraded homogeneous |
---|
3329 | RETURNS: ideal/module, the computed basis |
---|
3330 | EXAMPLE: example mDegGroebner; shows an example |
---|
3331 | " |
---|
3332 | { |
---|
3333 | if( isHomogeneous(I) == 0) |
---|
3334 | { |
---|
3335 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3336 | } |
---|
3337 | |
---|
3338 | def S = groebner(I); |
---|
3339 | |
---|
3340 | if( typeof(I) == "module" or typeof(I) == "vector" ) |
---|
3341 | { |
---|
3342 | S = setModuleGrading(S, getModuleGrading(I)); |
---|
3343 | } |
---|
3344 | |
---|
3345 | return(S); |
---|
3346 | } |
---|
3347 | example |
---|
3348 | { |
---|
3349 | "EXAMPLE:"; echo=2; |
---|
3350 | |
---|
3351 | ring r = 0,(x,y,z,w),dp; |
---|
3352 | |
---|
3353 | intmat MM[2][4]= |
---|
3354 | 1,1,1,1, |
---|
3355 | 0,1,3,4; |
---|
3356 | |
---|
3357 | setBaseMultigrading(MM); |
---|
3358 | |
---|
3359 | |
---|
3360 | module M = ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3361 | |
---|
3362 | |
---|
3363 | intmat v[2][nrows(M)]= |
---|
3364 | 1, |
---|
3365 | 0; |
---|
3366 | |
---|
3367 | M = setModuleGrading(M, v); |
---|
3368 | |
---|
3369 | |
---|
3370 | ///////////////////////////////////////////////////////////////////////////// |
---|
3371 | // GB: |
---|
3372 | M = mDegGroebner(M); M; |
---|
3373 | "Module Units Multigrading: "; print( getModuleGrading(M) ); |
---|
3374 | "Multidegrees: "; print(mDeg(M)); |
---|
3375 | |
---|
3376 | isHomogeneous(M); |
---|
3377 | |
---|
3378 | ///////////////////////////////////////////////////////////////////////////// |
---|
3379 | // Let's compute Syzygy! |
---|
3380 | def S = mDegSyzygy(M); S; |
---|
3381 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3382 | "Multidegrees: "; print(mDeg(S)); |
---|
3383 | |
---|
3384 | isHomogeneous(S); |
---|
3385 | |
---|
3386 | ///////////////////////////////////////////////////////////////////////////// |
---|
3387 | // GB: |
---|
3388 | S = mDegGroebner(S); S; |
---|
3389 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3390 | "Multidegrees: "; print(mDeg(S)); |
---|
3391 | |
---|
3392 | isHomogeneous(S); |
---|
3393 | } |
---|
3394 | |
---|
3395 | |
---|
3396 | /******************************************************/ |
---|
3397 | proc mDegResolution(def I, int ll, list #) |
---|
3398 | "USAGE: mDegResolution(I,l,[f]); I is poly/vector/ideal/module; l,f are integers |
---|
3399 | PURPOSE: computes the multigraded resolution of I of the length l, |
---|
3400 | or the whole resolution if l is zero. Returns minimal resolution if an optional |
---|
3401 | argument 1 is supplied |
---|
3402 | NOTE: input must have multigraded-homogeneous generators. |
---|
3403 | The returned list is truncated beginning with the first zero differential. |
---|
3404 | RETURNS: list, the computed resolution |
---|
3405 | EXAMPLE: example mDegResolution; shows an example |
---|
3406 | " |
---|
3407 | { |
---|
3408 | if( isHomogeneous(I, "checkGens") == 0) |
---|
3409 | { |
---|
3410 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3411 | } |
---|
3412 | |
---|
3413 | def R = res(I, ll, #); list L = R; int l = size(L); |
---|
3414 | def V = getModuleGrading(I); |
---|
3415 | if( (typeof(I) == "module") or (typeof(I) == "vector") ) |
---|
3416 | { |
---|
3417 | L[1] = setModuleGrading(L[1], V); |
---|
3418 | } |
---|
3419 | |
---|
3420 | int i; |
---|
3421 | for( i = 2; i <= l; i++ ) |
---|
3422 | { |
---|
3423 | if( size(L[i]) > 0 ) |
---|
3424 | { |
---|
3425 | L[i] = setModuleGrading( L[i], mDeg(L[i-1]) ); |
---|
3426 | } else |
---|
3427 | { |
---|
3428 | return (L[1..(i-1)]); |
---|
3429 | } |
---|
3430 | } |
---|
3431 | |
---|
3432 | return (L); |
---|
3433 | |
---|
3434 | |
---|
3435 | } |
---|
3436 | example |
---|
3437 | { |
---|
3438 | "EXAMPLE:"; echo=2; |
---|
3439 | |
---|
3440 | ring r = 0,(x,y,z,w),dp; |
---|
3441 | |
---|
3442 | intmat M[2][4]= |
---|
3443 | 1,1,1,1, |
---|
3444 | 0,1,3,4; |
---|
3445 | |
---|
3446 | setBaseMultigrading(M); |
---|
3447 | |
---|
3448 | |
---|
3449 | module m= ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3450 | |
---|
3451 | isHomogeneous(ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3), "checkGens"); |
---|
3452 | |
---|
3453 | ideal A = xw-yz, x2z-y3, xz2-y2w, yw2-z3; |
---|
3454 | |
---|
3455 | int j; |
---|
3456 | |
---|
3457 | for(j=1; j<=ncols(A); j++) |
---|
3458 | { |
---|
3459 | mDegPartition(A[j]); |
---|
3460 | } |
---|
3461 | |
---|
3462 | intmat v[2][1]= |
---|
3463 | 1, |
---|
3464 | 0; |
---|
3465 | |
---|
3466 | m = setModuleGrading(m, v); |
---|
3467 | |
---|
3468 | // Let's compute Syzygy! |
---|
3469 | def S = mDegSyzygy(m); S; |
---|
3470 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3471 | "Multidegrees: "; print(mDeg(S)); |
---|
3472 | |
---|
3473 | ///////////////////////////////////////////////////////////////////////////// |
---|
3474 | |
---|
3475 | S = mDegGroebner(S); S; |
---|
3476 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3477 | "Multidegrees: "; print(mDeg(S)); |
---|
3478 | |
---|
3479 | ///////////////////////////////////////////////////////////////////////////// |
---|
3480 | |
---|
3481 | def L = mDegResolution(m, 0, 1); |
---|
3482 | |
---|
3483 | for( j =1; j<=size(L); j++) |
---|
3484 | { |
---|
3485 | "----------------------------------- ", j, " -----------------------------"; |
---|
3486 | L[j]; |
---|
3487 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
3488 | "Multigrading: "; print(mDeg(L[j])); |
---|
3489 | } |
---|
3490 | |
---|
3491 | ///////////////////////////////////////////////////////////////////////////// |
---|
3492 | |
---|
3493 | L = mDegResolution(maxideal(1), 0, 1); |
---|
3494 | |
---|
3495 | for( j =1; j<=size(L); j++) |
---|
3496 | { |
---|
3497 | "----------------------------------- ", j, " -----------------------------"; |
---|
3498 | L[j]; |
---|
3499 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
3500 | "Multigrading: "; print(mDeg(L[j])); |
---|
3501 | } |
---|
3502 | |
---|
3503 | kill v; |
---|
3504 | |
---|
3505 | |
---|
3506 | def h = hilbertSeries(m); |
---|
3507 | setring h; |
---|
3508 | |
---|
3509 | numerator1; |
---|
3510 | factorize(numerator1); |
---|
3511 | |
---|
3512 | denominator1; |
---|
3513 | factorize(denominator1); |
---|
3514 | |
---|
3515 | numerator2; |
---|
3516 | factorize(numerator2); |
---|
3517 | |
---|
3518 | denominator2; |
---|
3519 | factorize(denominator2); |
---|
3520 | } |
---|
3521 | |
---|
3522 | /******************************************************/ |
---|
3523 | proc hilbertSeries(def I) |
---|
3524 | "USAGE: hilbertSeries(I); I is poly/vector/ideal/module |
---|
3525 | PURPOSE: computes the multigraded Hilbert Series of M |
---|
3526 | NOTE: input must have multigraded-homogeneous generators. |
---|
3527 | Multigrading should be positive. |
---|
3528 | RETURNS: a ring in variables t_(i), s_(i), with polynomials |
---|
3529 | numerator1 and denominator1 and muturally prime numerator2 |
---|
3530 | and denominator2, quotients of which give the series. |
---|
3531 | EXAMPLE: example hilbertSeries; shows an example |
---|
3532 | " |
---|
3533 | { |
---|
3534 | |
---|
3535 | if( !isFreeRepresented() ) |
---|
3536 | { |
---|
3537 | "Things might happen, since we are not free."; |
---|
3538 | //ERROR("SORRY: ONLY TORSION-FREE CASE (POSITIVE GRADING)"); |
---|
3539 | } |
---|
3540 | |
---|
3541 | int i, j, k, v; |
---|
3542 | |
---|
3543 | intmat M = getVariableWeights(); |
---|
3544 | |
---|
3545 | int cc = ncols(M); |
---|
3546 | int n = nrows(M); |
---|
3547 | |
---|
3548 | if( n == 0 ) |
---|
3549 | { |
---|
3550 | ERROR("Error: wrong Variable Weights?"); |
---|
3551 | } |
---|
3552 | |
---|
3553 | list RES = mDegResolution(I,0,1); |
---|
3554 | |
---|
3555 | int l = size(RES); |
---|
3556 | |
---|
3557 | list L; L[l + 1] = 0; |
---|
3558 | |
---|
3559 | if(typeof(I) == "ideal") |
---|
3560 | { |
---|
3561 | intmat zeros[n][1]; |
---|
3562 | L[1] = zeros; |
---|
3563 | } |
---|
3564 | else |
---|
3565 | { |
---|
3566 | L[1] = getModuleGrading(RES[1]); |
---|
3567 | } |
---|
3568 | |
---|
3569 | for( j = 1; j <= l; j++) |
---|
3570 | { |
---|
3571 | L[j + 1] = mDeg(RES[j]); |
---|
3572 | } |
---|
3573 | |
---|
3574 | l++; |
---|
3575 | |
---|
3576 | ring R = 0,(t_(1..n),s_(1..n)),dp; |
---|
3577 | |
---|
3578 | ideal units; |
---|
3579 | for( i=n; i>=1; i--) |
---|
3580 | { |
---|
3581 | units[i] = (var(i) * var(n + i) - 1); |
---|
3582 | } |
---|
3583 | |
---|
3584 | qring Q = std(units); |
---|
3585 | |
---|
3586 | // TODO: should not it be a quotient ring depending on Torsion??? |
---|
3587 | // I am not sure about what to do in the torsion case, but since |
---|
3588 | // we want to evaluate the polynomial at certain points to get |
---|
3589 | // a dimension we need uniqueness for this. I think we would lose |
---|
3590 | // this uniqueness if switching to this torsion ring. |
---|
3591 | |
---|
3592 | poly monom, summand, numerator; |
---|
3593 | poly denominator = 1; |
---|
3594 | |
---|
3595 | for( i = 1; i <= cc; i++) |
---|
3596 | { |
---|
3597 | monom = 1; |
---|
3598 | for( k = 1; k <= n; k++) |
---|
3599 | { |
---|
3600 | v = M[k,i]; |
---|
3601 | |
---|
3602 | if(v >= 0) |
---|
3603 | { |
---|
3604 | monom = monom * (var(k)^(v)); |
---|
3605 | } |
---|
3606 | else |
---|
3607 | { |
---|
3608 | monom = monom * (var(n+k)^(-v)); |
---|
3609 | } |
---|
3610 | } |
---|
3611 | |
---|
3612 | if( monom == 1) |
---|
3613 | { |
---|
3614 | ERROR("Multigrading not positive."); |
---|
3615 | } |
---|
3616 | |
---|
3617 | denominator = denominator * (1 - monom); |
---|
3618 | } |
---|
3619 | |
---|
3620 | for( j = 1; j<= l; j++) |
---|
3621 | { |
---|
3622 | summand = 0; |
---|
3623 | M = L[j]; |
---|
3624 | |
---|
3625 | for( i = 1; i <= ncols(M); i++) |
---|
3626 | { |
---|
3627 | monom = 1; |
---|
3628 | for( k = 1; k <= n; k++) |
---|
3629 | { |
---|
3630 | v = M[k,i]; |
---|
3631 | if( v > 0 ) |
---|
3632 | { |
---|
3633 | monom = monom * (var(k)^v); |
---|
3634 | } |
---|
3635 | else |
---|
3636 | { |
---|
3637 | monom = monom * (var(n+k)^(-v)); |
---|
3638 | } |
---|
3639 | } |
---|
3640 | summand = summand + monom; |
---|
3641 | } |
---|
3642 | numerator = numerator - (-1)^j * summand; |
---|
3643 | } |
---|
3644 | |
---|
3645 | if( denominator == 0 ) |
---|
3646 | { |
---|
3647 | ERROR("Multigrading not positive."); |
---|
3648 | } |
---|
3649 | |
---|
3650 | poly denominator1 = denominator; |
---|
3651 | poly numerator1 = numerator; |
---|
3652 | |
---|
3653 | export denominator1; |
---|
3654 | export numerator1; |
---|
3655 | |
---|
3656 | if( numerator != 0 ) |
---|
3657 | { |
---|
3658 | poly d = gcd(denominator, numerator); |
---|
3659 | |
---|
3660 | poly denominator2 = denominator/d; |
---|
3661 | poly numerator2 = numerator/d; |
---|
3662 | |
---|
3663 | if( gcd(denominator2, numerator2) != 1 ) |
---|
3664 | { |
---|
3665 | ERROR("Sorry: gcd should be 1 (after dividing out gcd)! Something went wrong!"); |
---|
3666 | } |
---|
3667 | } |
---|
3668 | else |
---|
3669 | { |
---|
3670 | poly denominator2 = denominator; |
---|
3671 | poly numerator2 = numerator; |
---|
3672 | } |
---|
3673 | |
---|
3674 | |
---|
3675 | export denominator2; |
---|
3676 | export numerator2; |
---|
3677 | |
---|
3678 | " ------------ "; |
---|
3679 | "This proc returns a ring with polynomials called 'numerator1/2' and 'denominator1/2'!"; |
---|
3680 | "They represent the first and the second Hilbert Series."; |
---|
3681 | "The s_(i)-variables are defined to be the inverse of the t_(i)-variables."; |
---|
3682 | " ------------ "; |
---|
3683 | |
---|
3684 | return(Q); |
---|
3685 | } |
---|
3686 | example |
---|
3687 | { |
---|
3688 | "EXAMPLE:"; echo=2; |
---|
3689 | |
---|
3690 | ring r = 0,(x,y,z,w),dp; |
---|
3691 | intmat g[2][4]= |
---|
3692 | 1,1,1,1, |
---|
3693 | 0,1,3,4; |
---|
3694 | setBaseMultigrading(g); |
---|
3695 | |
---|
3696 | module M = ideal(xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3697 | intmat V[2][1]= |
---|
3698 | 1, |
---|
3699 | 0; |
---|
3700 | |
---|
3701 | M = setModuleGrading(M, V); |
---|
3702 | |
---|
3703 | def h = hilbertSeries(M); setring h; |
---|
3704 | |
---|
3705 | factorize(numerator2); |
---|
3706 | factorize(denominator2); |
---|
3707 | |
---|
3708 | kill g, h; setring r; |
---|
3709 | |
---|
3710 | intmat g[2][4]= |
---|
3711 | 1,2,3,4, |
---|
3712 | 0,0,5,8; |
---|
3713 | |
---|
3714 | setBaseMultigrading(g); |
---|
3715 | |
---|
3716 | ideal I = x^2, y, z^3; |
---|
3717 | I = std(I); |
---|
3718 | def L = mDegResolution(I, 0, 1); |
---|
3719 | |
---|
3720 | for( int j = 1; j<=size(L); j++) |
---|
3721 | { |
---|
3722 | "----------------------------------- ", j, " -----------------------------"; |
---|
3723 | L[j]; |
---|
3724 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
3725 | "Multigrading: "; print(mDeg(L[j])); |
---|
3726 | } |
---|
3727 | |
---|
3728 | mDeg(I); |
---|
3729 | def h = hilbertSeries(I); setring h; |
---|
3730 | |
---|
3731 | factorize(numerator2); |
---|
3732 | factorize(denominator2); |
---|
3733 | |
---|
3734 | kill r, h, g, V; |
---|
3735 | //////////////////////////////////////////////// |
---|
3736 | ring R = 0,(x,y,z),dp; |
---|
3737 | intmat W[2][3] = |
---|
3738 | 1,1, 1, |
---|
3739 | 0,0,-1; |
---|
3740 | setBaseMultigrading(W); |
---|
3741 | ideal I = x3y,yz2,y2z,z4; |
---|
3742 | |
---|
3743 | def h = hilbertSeries(I); setring h; |
---|
3744 | |
---|
3745 | factorize(numerator2); |
---|
3746 | factorize(denominator2); |
---|
3747 | |
---|
3748 | kill R, W, h; |
---|
3749 | //////////////////////////////////////////////// |
---|
3750 | ring R = 0,(x,y,z,a,b,c),dp; |
---|
3751 | intmat W[2][6] = |
---|
3752 | 1,1, 1,1,1,1, |
---|
3753 | 0,0,-1,0,0,0; |
---|
3754 | setBaseMultigrading(W); |
---|
3755 | ideal I = x3y,yz2,y2z,z4; |
---|
3756 | |
---|
3757 | def h = hilbertSeries(I); setring h; |
---|
3758 | |
---|
3759 | factorize(numerator2); |
---|
3760 | factorize(denominator2); |
---|
3761 | |
---|
3762 | kill R, W, h; |
---|
3763 | //////////////////////////////////////////////// |
---|
3764 | // This is example 5.3.9. from Robbianos book. |
---|
3765 | |
---|
3766 | ring R = 0,(x,y,z,w),dp; |
---|
3767 | intmat W[1][4] = |
---|
3768 | 1,1, 1,1; |
---|
3769 | setBaseMultigrading(W); |
---|
3770 | ideal I = z3,y3zw2,x2y4w2xyz2; |
---|
3771 | |
---|
3772 | hilb(std(I)); |
---|
3773 | |
---|
3774 | def h = hilbertSeries(I); setring h; |
---|
3775 | |
---|
3776 | numerator1; |
---|
3777 | denominator1; |
---|
3778 | |
---|
3779 | factorize(numerator2); |
---|
3780 | factorize(denominator2); |
---|
3781 | |
---|
3782 | |
---|
3783 | kill h; |
---|
3784 | //////////////////////////////////////////////// |
---|
3785 | setring R; |
---|
3786 | |
---|
3787 | ideal I2 = x2,y2,z2; I2; |
---|
3788 | |
---|
3789 | hilb(std(I2)); |
---|
3790 | |
---|
3791 | def h = hilbertSeries(I2); setring h; |
---|
3792 | |
---|
3793 | numerator1; |
---|
3794 | denominator1; |
---|
3795 | |
---|
3796 | |
---|
3797 | kill h; |
---|
3798 | //////////////////////////////////////////////// |
---|
3799 | setring R; |
---|
3800 | |
---|
3801 | W = 2,2,2,2; |
---|
3802 | |
---|
3803 | setBaseMultigrading(W); |
---|
3804 | |
---|
3805 | getVariableWeights(); |
---|
3806 | |
---|
3807 | intvec w = 2,2,2,2; |
---|
3808 | |
---|
3809 | hilb(std(I2), 1, w); |
---|
3810 | |
---|
3811 | kill w; |
---|
3812 | |
---|
3813 | |
---|
3814 | def h = hilbertSeries(I2); setring h; |
---|
3815 | |
---|
3816 | |
---|
3817 | numerator1; denominator1; |
---|
3818 | kill h; |
---|
3819 | |
---|
3820 | |
---|
3821 | kill R, W; |
---|
3822 | |
---|
3823 | //////////////////////////////////////////////// |
---|
3824 | ring R = 0,(x),dp; |
---|
3825 | intmat W[1][1] = |
---|
3826 | 1; |
---|
3827 | setBaseMultigrading(W); |
---|
3828 | |
---|
3829 | ideal I; |
---|
3830 | |
---|
3831 | I = 1; I; |
---|
3832 | |
---|
3833 | hilb(std(I)); |
---|
3834 | |
---|
3835 | def h = hilbertSeries(I); setring h; |
---|
3836 | |
---|
3837 | numerator1; denominator1; |
---|
3838 | |
---|
3839 | kill h; |
---|
3840 | //////////////////////////////////////////////// |
---|
3841 | setring R; |
---|
3842 | |
---|
3843 | I = x; I; |
---|
3844 | |
---|
3845 | hilb(std(I)); |
---|
3846 | |
---|
3847 | def h = hilbertSeries(I); setring h; |
---|
3848 | |
---|
3849 | numerator1; denominator1; |
---|
3850 | |
---|
3851 | kill h; |
---|
3852 | //////////////////////////////////////////////// |
---|
3853 | setring R; |
---|
3854 | |
---|
3855 | I = x^5; I; |
---|
3856 | |
---|
3857 | hilb(std(I)); |
---|
3858 | hilb(std(I), 1); |
---|
3859 | |
---|
3860 | def h = hilbertSeries(I); setring h; |
---|
3861 | |
---|
3862 | numerator1; denominator1; |
---|
3863 | |
---|
3864 | |
---|
3865 | kill h; |
---|
3866 | //////////////////////////////////////////////// |
---|
3867 | setring R; |
---|
3868 | |
---|
3869 | I = x^10; I; |
---|
3870 | |
---|
3871 | hilb(std(I)); |
---|
3872 | |
---|
3873 | def h = hilbertSeries(I); setring h; |
---|
3874 | |
---|
3875 | numerator1; denominator1; |
---|
3876 | |
---|
3877 | kill h; |
---|
3878 | //////////////////////////////////////////////// |
---|
3879 | setring R; |
---|
3880 | |
---|
3881 | module M = 1; |
---|
3882 | |
---|
3883 | M = setModuleGrading(M, W); |
---|
3884 | |
---|
3885 | |
---|
3886 | hilb(std(M)); |
---|
3887 | |
---|
3888 | def h = hilbertSeries(M); setring h; |
---|
3889 | |
---|
3890 | numerator1; denominator1; |
---|
3891 | |
---|
3892 | kill h; |
---|
3893 | //////////////////////////////////////////////// |
---|
3894 | setring R; |
---|
3895 | |
---|
3896 | kill M; module M = x^5*gen(1); |
---|
3897 | // intmat V[1][3] = 0; // TODO: this would lead to a wrong result!!!? |
---|
3898 | intmat V[1][1] = 0; // all gen(i) of degree 0! |
---|
3899 | |
---|
3900 | M = setModuleGrading(M, V); |
---|
3901 | |
---|
3902 | hilb(std(M)); |
---|
3903 | |
---|
3904 | def h = hilbertSeries(M); setring h; |
---|
3905 | |
---|
3906 | numerator1; denominator1; |
---|
3907 | |
---|
3908 | kill h; |
---|
3909 | //////////////////////////////////////////////// |
---|
3910 | setring R; |
---|
3911 | |
---|
3912 | module N = x^5*gen(3); |
---|
3913 | |
---|
3914 | kill V; |
---|
3915 | |
---|
3916 | intmat V[1][3] = 0; // all gen(i) of degree 0! |
---|
3917 | |
---|
3918 | N = setModuleGrading(N, V); |
---|
3919 | |
---|
3920 | hilb(std(N)); |
---|
3921 | |
---|
3922 | def h = hilbertSeries(N); setring h; |
---|
3923 | |
---|
3924 | numerator1; denominator1; |
---|
3925 | |
---|
3926 | kill h; |
---|
3927 | //////////////////////////////////////////////// |
---|
3928 | setring R; |
---|
3929 | |
---|
3930 | |
---|
3931 | module S = M + N; |
---|
3932 | |
---|
3933 | S = setModuleGrading(S, V); |
---|
3934 | |
---|
3935 | hilb(std(S)); |
---|
3936 | |
---|
3937 | def h = hilbertSeries(S); setring h; |
---|
3938 | |
---|
3939 | numerator1; denominator1; |
---|
3940 | |
---|
3941 | kill h; |
---|
3942 | |
---|
3943 | kill V; |
---|
3944 | kill R, W; |
---|
3945 | |
---|
3946 | } |
---|
3947 | |
---|
3948 | proc evalHilbertSeries(def h, intvec v) |
---|
3949 | " |
---|
3950 | evaluate hilbert series h by substibuting v[i] for t_(i) (1/v[i] for s_(i)) |
---|
3951 | return: int (h(v)) |
---|
3952 | " |
---|
3953 | { |
---|
3954 | if( 2*size(v) != nvars(h) ) |
---|
3955 | { |
---|
3956 | ERROR("Wrong input/size!"); |
---|
3957 | } |
---|
3958 | |
---|
3959 | setring h; |
---|
3960 | |
---|
3961 | if( defined(numerator2) and defined(denominator2) ) |
---|
3962 | { |
---|
3963 | poly n = numerator2; poly d = denominator2; |
---|
3964 | } else |
---|
3965 | { |
---|
3966 | poly n = numerator1; poly d = denominator1; |
---|
3967 | } |
---|
3968 | |
---|
3969 | int N = size(v); |
---|
3970 | int i; number k; |
---|
3971 | ideal V; |
---|
3972 | |
---|
3973 | for( i = N; i > 0; i -- ) |
---|
3974 | { |
---|
3975 | k = v[i]; |
---|
3976 | V[i] = var(i) - k; |
---|
3977 | } |
---|
3978 | |
---|
3979 | V = groebner(V); |
---|
3980 | |
---|
3981 | n = NF(n, V); |
---|
3982 | d = NF(d, V); |
---|
3983 | |
---|
3984 | n; |
---|
3985 | d; |
---|
3986 | |
---|
3987 | if( d == 0 ) |
---|
3988 | { |
---|
3989 | ERROR("Sorry: denominator is zero!"); |
---|
3990 | } |
---|
3991 | |
---|
3992 | if( n == 0 ) |
---|
3993 | { |
---|
3994 | return (0); |
---|
3995 | } |
---|
3996 | |
---|
3997 | poly g = gcd(n, d); |
---|
3998 | |
---|
3999 | if( g != leadcoef(g) ) |
---|
4000 | { |
---|
4001 | n = n / g; |
---|
4002 | d = d / g; |
---|
4003 | } |
---|
4004 | |
---|
4005 | n; |
---|
4006 | d; |
---|
4007 | |
---|
4008 | |
---|
4009 | for( i = N; i > 0; i -- ) |
---|
4010 | { |
---|
4011 | "i: ", i; |
---|
4012 | n; |
---|
4013 | d; |
---|
4014 | |
---|
4015 | k = v[i]; |
---|
4016 | k; |
---|
4017 | |
---|
4018 | n = subst(n, var(i), k); |
---|
4019 | d = subst(d, var(i), k); |
---|
4020 | |
---|
4021 | if( k != 0 ) |
---|
4022 | { |
---|
4023 | k = 1/k; |
---|
4024 | n = subst(n, var(N+i), k); |
---|
4025 | d = subst(d, var(N+i), k); |
---|
4026 | } |
---|
4027 | } |
---|
4028 | |
---|
4029 | n; |
---|
4030 | d; |
---|
4031 | |
---|
4032 | if( d == 0 ) |
---|
4033 | { |
---|
4034 | ERROR("Sorry: denominator is zero!"); |
---|
4035 | } |
---|
4036 | |
---|
4037 | if( n == 0 ) |
---|
4038 | { |
---|
4039 | return (0); |
---|
4040 | } |
---|
4041 | |
---|
4042 | poly g = gcd(n, d); |
---|
4043 | |
---|
4044 | if( g != leadcoef(g) ) |
---|
4045 | { |
---|
4046 | n = n / g; |
---|
4047 | d = d / g; |
---|
4048 | } |
---|
4049 | |
---|
4050 | n; |
---|
4051 | d; |
---|
4052 | |
---|
4053 | if( n != leadcoef(n) || d != leadcoef(d) ) |
---|
4054 | { |
---|
4055 | ERROR("Sorry cannot completely evaluate. Partial result: (" + string(n) + ")/(" + string(d) + ")"); |
---|
4056 | } |
---|
4057 | |
---|
4058 | n; |
---|
4059 | d; |
---|
4060 | |
---|
4061 | return (leadcoef(n)/leadcoef(d)); |
---|
4062 | } |
---|
4063 | example |
---|
4064 | { |
---|
4065 | "EXAMPLE:"; echo=2; |
---|
4066 | |
---|
4067 | // TODO! |
---|
4068 | |
---|
4069 | } |
---|
4070 | |
---|
4071 | |
---|
4072 | proc isPositive() |
---|
4073 | "USAGE: isPositive() |
---|
4074 | PURPOSE: Computes whether the multigrading of the ring is positive. |
---|
4075 | For computation theorem 8.6 of the Miller/Sturmfels book is used. |
---|
4076 | RETURNS: true if the multigrading is positive |
---|
4077 | EXAMPLE: example isPositive; shows an example |
---|
4078 | " |
---|
4079 | { |
---|
4080 | ideal I = mDegBasis(0); |
---|
4081 | ideal J = attrib(I,"ZeroPart"); |
---|
4082 | /* |
---|
4083 | I am not quite sure what this ZeroPart is anymore. I thought it |
---|
4084 | should contain all monomials of degree 0, but then apparently 1 should |
---|
4085 | be contained. It makes sense to exclude 1, but was this also the intention? |
---|
4086 | */ |
---|
4087 | return(J==0); |
---|
4088 | } |
---|
4089 | example |
---|
4090 | { |
---|
4091 | echo = 2; printlevel = 3; |
---|
4092 | ring r = 0,(x,y),dp; |
---|
4093 | intmat A[1][2]=-1,1; |
---|
4094 | setBaseMultigrading(A); |
---|
4095 | isPositive(); |
---|
4096 | |
---|
4097 | intmat A[1][2]=1,1; |
---|
4098 | setBaseMultigrading(A); |
---|
4099 | isPositive(A); |
---|
4100 | } |
---|
4101 | |
---|
4102 | /////////////////////////////////////////////////////////////////////////////// |
---|
4103 | // testing for consistency of the library: |
---|
4104 | proc testMultigradingLib () |
---|
4105 | { |
---|
4106 | example setBaseMultigrading; |
---|
4107 | example setModuleGrading; |
---|
4108 | |
---|
4109 | example getVariableWeights; |
---|
4110 | example getLattice; |
---|
4111 | example getGradingGroup; |
---|
4112 | example getModuleGrading; |
---|
4113 | |
---|
4114 | |
---|
4115 | example mDeg; |
---|
4116 | example mDegPartition; |
---|
4117 | |
---|
4118 | |
---|
4119 | example hermiteNormalForm; |
---|
4120 | example isHomogeneous; |
---|
4121 | example isTorsionFree; |
---|
4122 | example pushForward; |
---|
4123 | example defineHomogeneous; |
---|
4124 | |
---|
4125 | example equalMDeg; |
---|
4126 | example isZeroElement; |
---|
4127 | |
---|
4128 | example mDegResolution; |
---|
4129 | |
---|
4130 | "// ******************* example hilbertSeries ************************//"; |
---|
4131 | example hilbertSeries; |
---|
4132 | |
---|
4133 | |
---|
4134 | // example mDegBasis; // needs 4ti2! |
---|
4135 | |
---|
4136 | "The End!"; |
---|
4137 | } |
---|
4138 | |
---|
4139 | |
---|
4140 | proc mDegTruncate(def M, intvec md) |
---|
4141 | { |
---|
4142 | "d: "; |
---|
4143 | print(md); |
---|
4144 | |
---|
4145 | "M: "; |
---|
4146 | module LL = M; // + L for d+1 |
---|
4147 | LL; |
---|
4148 | print(mDeg(LL)); |
---|
4149 | |
---|
4150 | |
---|
4151 | intmat V = getModuleGrading(M); |
---|
4152 | intvec vi; |
---|
4153 | int s = nrows(M); |
---|
4154 | int r = nrows(V); |
---|
4155 | int i; |
---|
4156 | module L; def B; |
---|
4157 | for (i=s; i>0; i--) |
---|
4158 | { |
---|
4159 | "comp: ", i; |
---|
4160 | vi = V[1..r, i]; |
---|
4161 | "v[i]: "; vi; |
---|
4162 | |
---|
4163 | B = mDegBasis(md - vi); // ZeroPart is always the same... |
---|
4164 | "B: "; B; |
---|
4165 | |
---|
4166 | L = L, B*gen(i); |
---|
4167 | } |
---|
4168 | L = simplify(L, 2); |
---|
4169 | L = setModuleGrading(L,V); |
---|
4170 | |
---|
4171 | "L: "; L; |
---|
4172 | print(mDeg(L)); |
---|
4173 | |
---|
4174 | L = mDegModulo(L, LL); |
---|
4175 | L = mDegGroebner(L); |
---|
4176 | // L = minbase(prune(L)); |
---|
4177 | |
---|
4178 | "??????????"; |
---|
4179 | print(L); |
---|
4180 | print(mDeg(L)); |
---|
4181 | |
---|
4182 | V = getModuleGrading(L); |
---|
4183 | |
---|
4184 | // take out other degrees |
---|
4185 | for(i = ncols(L); i > 0; i-- ) |
---|
4186 | { |
---|
4187 | if( !equalMDeg( mDeg(getGradedGenerator(L, i)), md ) ) |
---|
4188 | { |
---|
4189 | L[i] = 0; |
---|
4190 | } |
---|
4191 | } |
---|
4192 | |
---|
4193 | L = simplify(L, 2); |
---|
4194 | L = setModuleGrading(L, V); |
---|
4195 | print(L); |
---|
4196 | print(mDeg(L)); |
---|
4197 | |
---|
4198 | return(L); |
---|
4199 | } |
---|
4200 | example |
---|
4201 | { |
---|
4202 | "EXAMPLE:"; echo=2; |
---|
4203 | |
---|
4204 | // TODO! |
---|
4205 | ring r = 32003, (x,y), dp; |
---|
4206 | |
---|
4207 | intmat M[2][2] = |
---|
4208 | 1, 0, |
---|
4209 | 0, 1; |
---|
4210 | |
---|
4211 | setBaseMultigrading(M); |
---|
4212 | |
---|
4213 | intmat V[2][1] = |
---|
4214 | 0, |
---|
4215 | 0; |
---|
4216 | |
---|
4217 | "X:"; |
---|
4218 | module h1 = x; |
---|
4219 | h1 = setModuleGrading(h1, V); |
---|
4220 | mDegTruncate(h1, mDeg(x)); |
---|
4221 | mDegTruncate(h1, mDeg(y)); |
---|
4222 | |
---|
4223 | "XY:"; |
---|
4224 | module h2 = ideal(x, y); |
---|
4225 | h2 = setModuleGrading(h2, V); |
---|
4226 | mDegTruncate(h2, mDeg(x)); |
---|
4227 | mDegTruncate(h2, mDeg(y)); |
---|
4228 | mDegTruncate(h2, mDeg(xy)); |
---|
4229 | } |
---|
4230 | |
---|
4231 | |
---|
4232 | /******************************************************/ |
---|
4233 | /* Some functions on lattices. |
---|
4234 | TODO Tuebingen: - add functionality (see wiki) and |
---|
4235 | - adjust them to work for groups as well.*/ |
---|
4236 | /******************************************************/ |
---|
4237 | |
---|
4238 | |
---|
4239 | |
---|
4240 | /******************************************************/ |
---|
4241 | proc imageLattice(intmat Q, intmat L) |
---|
4242 | "USAGE: imageLattice(Q,L); Q and L are of type intmat |
---|
4243 | PURPOSE: compute an integral basis for the image of the |
---|
4244 | lattice L under the homomorphism of lattices Q. |
---|
4245 | RETURN: intmat |
---|
4246 | EXAMPLE: example imageLattice; shows an example |
---|
4247 | " |
---|
4248 | { |
---|
4249 | intmat Mul = Q*L; |
---|
4250 | intmat L = latticeBasis(Mul); |
---|
4251 | |
---|
4252 | return(L); |
---|
4253 | } |
---|
4254 | example |
---|
4255 | { |
---|
4256 | "EXAMPLE:"; echo=2; |
---|
4257 | |
---|
4258 | intmat Q[2][3] = |
---|
4259 | 1,2,3, |
---|
4260 | 3,2,1; |
---|
4261 | |
---|
4262 | intmat L[3][2] = |
---|
4263 | 1,4, |
---|
4264 | 2,5, |
---|
4265 | 3,6; |
---|
4266 | |
---|
4267 | // should be a 2x2 matrix with columns |
---|
4268 | // [2,-14], [0,36] |
---|
4269 | imageLattice(Q,L); |
---|
4270 | |
---|
4271 | } |
---|
4272 | |
---|
4273 | /******************************************************/ |
---|
4274 | proc intRank(intmat A) |
---|
4275 | " |
---|
4276 | USAGE: intRank(A); intmat A |
---|
4277 | PURPOSE: compute the rank of the integral matrix A |
---|
4278 | by computing a hermite normalform. |
---|
4279 | RETURNS: int |
---|
4280 | EXAMPLE: example intRank; shows an example |
---|
4281 | " |
---|
4282 | { |
---|
4283 | intmat B = hermiteNormalForm(A); |
---|
4284 | |
---|
4285 | // get number of zero columns |
---|
4286 | int nzerocols = 0; |
---|
4287 | int j; |
---|
4288 | int i; |
---|
4289 | int iszero; |
---|
4290 | for ( j = 1; j <= ncols(B); j++ ) |
---|
4291 | { |
---|
4292 | iszero = 1; |
---|
4293 | |
---|
4294 | for ( i = 1; i <= nrows(B); i++ ) |
---|
4295 | { |
---|
4296 | if ( B[i,j] != 0 ) |
---|
4297 | { |
---|
4298 | iszero = 0; |
---|
4299 | break; |
---|
4300 | } |
---|
4301 | } |
---|
4302 | |
---|
4303 | if ( iszero == 1 ) |
---|
4304 | { |
---|
4305 | nzerocols++; |
---|
4306 | } |
---|
4307 | } |
---|
4308 | |
---|
4309 | // get number of zero rows |
---|
4310 | int nzerorows = 0; |
---|
4311 | |
---|
4312 | for ( i = 1; i <= nrows(B); i++ ) |
---|
4313 | { |
---|
4314 | iszero = 1; |
---|
4315 | |
---|
4316 | for ( j = 1; j <= ncols(B); j++ ) |
---|
4317 | { |
---|
4318 | if ( B[i,j] != 0 ) |
---|
4319 | { |
---|
4320 | iszero = 0; |
---|
4321 | break; |
---|
4322 | } |
---|
4323 | } |
---|
4324 | |
---|
4325 | if ( iszero == 1 ) |
---|
4326 | { |
---|
4327 | nzerorows++; |
---|
4328 | } |
---|
4329 | } |
---|
4330 | |
---|
4331 | int r = nrows(B) - nzerorows; |
---|
4332 | |
---|
4333 | if ( ncols(B) - nzerocols < r ) |
---|
4334 | { |
---|
4335 | r = ncols(B) - nzerocols; |
---|
4336 | } |
---|
4337 | |
---|
4338 | return(r); |
---|
4339 | } |
---|
4340 | example |
---|
4341 | { |
---|
4342 | |
---|
4343 | intmat A[3][4] = |
---|
4344 | 1,0,1,0, |
---|
4345 | 1,2,0,0, |
---|
4346 | 0,0,0,0; |
---|
4347 | |
---|
4348 | int r = intRank(A); |
---|
4349 | |
---|
4350 | print(A); |
---|
4351 | print(r); // Should be 2 |
---|
4352 | |
---|
4353 | kill A; |
---|
4354 | |
---|
4355 | } |
---|
4356 | |
---|
4357 | /*****************************************************/ |
---|
4358 | |
---|
4359 | proc isSublattice(intmat L, intmat S) |
---|
4360 | "USAGE: isSublattice(L, S); L, S are of tpye intmat |
---|
4361 | PURPOSE: checks whether the lattice created by L is a |
---|
4362 | sublattice of the lattice created by S. |
---|
4363 | The procedure checks whether each generator of L is |
---|
4364 | contained in S. |
---|
4365 | RETURN: 0 if false, 1 if true |
---|
4366 | EXAMPLE: example isSublattice; shows an example |
---|
4367 | " |
---|
4368 | { |
---|
4369 | int a,b,g,i,j,k; |
---|
4370 | intmat Ker; |
---|
4371 | |
---|
4372 | // check whether each column v of L is contained in |
---|
4373 | // the lattice generated by S |
---|
4374 | for ( i = 1; i <= ncols(L); i++ ) |
---|
4375 | { |
---|
4376 | |
---|
4377 | // v is the i-th column of L |
---|
4378 | intvec v; |
---|
4379 | for ( j = 1; j <= nrows(L); j++ ) |
---|
4380 | { |
---|
4381 | v[j] = L[j,i]; |
---|
4382 | } |
---|
4383 | |
---|
4384 | // concatenate B = [S,v] |
---|
4385 | intmat B[nrows(L)][ncols(S) + 1]; |
---|
4386 | |
---|
4387 | for ( a = 1; a <= nrows(S); a++ ) |
---|
4388 | { |
---|
4389 | for ( b = 1; b <= ncols(S); b++ ) |
---|
4390 | { |
---|
4391 | B[a,b] = S[a,b]; |
---|
4392 | } |
---|
4393 | } |
---|
4394 | |
---|
4395 | for ( a = 1; a <= size(v); a++ ) |
---|
4396 | { |
---|
4397 | B[a,ncols(B)] = v[a]; |
---|
4398 | } |
---|
4399 | |
---|
4400 | |
---|
4401 | // check gcd |
---|
4402 | Ker = kernelLattice(B); |
---|
4403 | k = nrows(Ker); |
---|
4404 | list R; // R is the last row |
---|
4405 | |
---|
4406 | for ( j = 1; j <= ncols(Ker); j++ ) |
---|
4407 | { |
---|
4408 | R[j] = Ker[k,j]; |
---|
4409 | } |
---|
4410 | |
---|
4411 | g = R[1]; |
---|
4412 | |
---|
4413 | for ( j = 2; j <= size(R); j++ ) |
---|
4414 | { |
---|
4415 | g = gcd(g,R[j]); |
---|
4416 | } |
---|
4417 | |
---|
4418 | if ( g != 1 ) |
---|
4419 | { |
---|
4420 | return(0); |
---|
4421 | } |
---|
4422 | |
---|
4423 | kill B, v, R; |
---|
4424 | |
---|
4425 | } |
---|
4426 | |
---|
4427 | return(1); |
---|
4428 | } |
---|
4429 | example |
---|
4430 | { |
---|
4431 | "EXAMPLE:"; echo=2; |
---|
4432 | |
---|
4433 | //ring R = 0,(x,y),dp; |
---|
4434 | intmat S2[3][2]= |
---|
4435 | 0, 2, 3, |
---|
4436 | 0, 1, 1, |
---|
4437 | 3, 0, 2; |
---|
4438 | |
---|
4439 | intmat S1[3][3]= |
---|
4440 | 0, 6, |
---|
4441 | 0, 2, |
---|
4442 | 3, 4; |
---|
4443 | |
---|
4444 | isSublattice(S1,S2); // Yes! |
---|
4445 | |
---|
4446 | intmat S3[3][1] = |
---|
4447 | 0, |
---|
4448 | 0, |
---|
4449 | 1; |
---|
4450 | |
---|
4451 | not(isSublattice(S3,S2)); // Yes! |
---|
4452 | |
---|
4453 | } |
---|
4454 | |
---|
4455 | /******************************************************/ |
---|
4456 | |
---|
4457 | proc latticeBasis(intmat B) |
---|
4458 | "USAGE: latticeBasis(B); intmat B |
---|
4459 | PURPOSE: compute an integral basis for the lattice defined by |
---|
4460 | the columns of B. |
---|
4461 | RETURNS: intmat |
---|
4462 | EXAMPLE: example latticeBasis; shows an example |
---|
4463 | " |
---|
4464 | { |
---|
4465 | int n = ncols(B); |
---|
4466 | int r = intRank(B); |
---|
4467 | |
---|
4468 | if ( r == 0 ) |
---|
4469 | { |
---|
4470 | intmat H[nrows(B)][1]; |
---|
4471 | int j; |
---|
4472 | |
---|
4473 | for ( j = 1; j <= nrows(B); j++ ) |
---|
4474 | { |
---|
4475 | H[j,1] = 0; |
---|
4476 | } |
---|
4477 | } |
---|
4478 | else |
---|
4479 | { |
---|
4480 | intmat H = hermiteNormalForm(B);; |
---|
4481 | |
---|
4482 | if (r < n) |
---|
4483 | { |
---|
4484 | // delete columns r+1 to n |
---|
4485 | // should be identical with the function |
---|
4486 | // H = submat(H,1..nrows(H),1..r); |
---|
4487 | // for matrices |
---|
4488 | intmat Hdel[nrows(H)][r]; |
---|
4489 | int k; |
---|
4490 | int m; |
---|
4491 | |
---|
4492 | for ( k = 1; k <= nrows(H); k++ ) |
---|
4493 | { |
---|
4494 | for ( m = 1; m <= r; m++ ) |
---|
4495 | { |
---|
4496 | Hdel[k,m] = H[k,m]; |
---|
4497 | } |
---|
4498 | } |
---|
4499 | |
---|
4500 | H = Hdel; |
---|
4501 | } |
---|
4502 | } |
---|
4503 | |
---|
4504 | return(H); |
---|
4505 | } |
---|
4506 | example |
---|
4507 | { |
---|
4508 | "EXAMPLE:"; echo=2; |
---|
4509 | |
---|
4510 | intmat L[3][3] = |
---|
4511 | 1,4,8, |
---|
4512 | 2,5,10, |
---|
4513 | 3,6,12; |
---|
4514 | |
---|
4515 | intmat B = latticeBasis(B); // should be a matrix with columns [1,2,3] and [0,3,6] |
---|
4516 | |
---|
4517 | kill B,L; |
---|
4518 | } |
---|
4519 | |
---|
4520 | /******************************************************/ |
---|
4521 | |
---|
4522 | proc kernelLattice(def P) |
---|
4523 | " |
---|
4524 | USAGE: kernelLattice(P); intmat P |
---|
4525 | PURPOSE: compute a integral basis for the kernel of the |
---|
4526 | homomorphism of lattices defined by the intmat P. |
---|
4527 | RETURNS: intmat |
---|
4528 | EXAMPLE: example kernelLattice; shows an example |
---|
4529 | " |
---|
4530 | { |
---|
4531 | int n = ncols(P); |
---|
4532 | int r = intRank(P); |
---|
4533 | |
---|
4534 | if ( r == 0 ) |
---|
4535 | { |
---|
4536 | intmat U = unitMatrix(n); |
---|
4537 | } |
---|
4538 | else |
---|
4539 | { |
---|
4540 | if ( r == n ) |
---|
4541 | { |
---|
4542 | intmat U[n][1]; // now all entries are zero. |
---|
4543 | } |
---|
4544 | else |
---|
4545 | { |
---|
4546 | list L = hermiteNormalForm(P, "transform"); //hermite(P, "transform"); // now, Hermite = L[1] = A*L[2] |
---|
4547 | intmat U = L[2]; |
---|
4548 | |
---|
4549 | // delete columns 1 to r |
---|
4550 | // should be identical with the function |
---|
4551 | // U = submat(U,1..nrows(U),r+1..); |
---|
4552 | // for matrices |
---|
4553 | intmat Udel[nrows(U)][ncols(U) - r]; |
---|
4554 | int k; |
---|
4555 | int m; |
---|
4556 | |
---|
4557 | for ( k = 1; k <= nrows(U); k++ ) |
---|
4558 | { |
---|
4559 | for ( m = r + 1; m <= ncols(U); m++ ) |
---|
4560 | { |
---|
4561 | Udel[k,m - r] = U[k,m]; |
---|
4562 | } |
---|
4563 | } |
---|
4564 | |
---|
4565 | U = Udel; |
---|
4566 | |
---|
4567 | } |
---|
4568 | } |
---|
4569 | |
---|
4570 | return(U); |
---|
4571 | } |
---|
4572 | example |
---|
4573 | { |
---|
4574 | "EXAMPLE"; echo = 2; |
---|
4575 | |
---|
4576 | intmat LL[3][4] = |
---|
4577 | 1,4,7,10, |
---|
4578 | 2,5,8,11, |
---|
4579 | 3,6,9,12; |
---|
4580 | |
---|
4581 | // should be a 4x2 matrix with colums |
---|
4582 | // [-1,2,-1,0],[2,-3,0,1] |
---|
4583 | intmat B = kernelLattice(LL); |
---|
4584 | |
---|
4585 | print(B); |
---|
4586 | |
---|
4587 | kill B; |
---|
4588 | |
---|
4589 | } |
---|
4590 | |
---|
4591 | /*****************************************************/ |
---|
4592 | |
---|
4593 | proc preimageLattice(def P, def B) |
---|
4594 | " |
---|
4595 | USAGE: preimageLattice(P, B); intmat P, intmat B |
---|
4596 | PURPOSE: compute an integral basis for the preimage of B under |
---|
4597 | the homomorphism of lattices defined by the intmat P. |
---|
4598 | RETURNS: intmat |
---|
4599 | EXAMPLE: example preimageLattice; shows an example |
---|
4600 | " |
---|
4601 | { |
---|
4602 | // concatenate matrices: Con = [P,-B] |
---|
4603 | intmat Con[nrows(P)][ncols(P) + ncols(B)]; |
---|
4604 | int i; |
---|
4605 | int j; |
---|
4606 | |
---|
4607 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4608 | { |
---|
4609 | for ( j = 1; j <= ncols(P); j++ ) // P first |
---|
4610 | { |
---|
4611 | Con[i,j] = P[i,j]; |
---|
4612 | } |
---|
4613 | } |
---|
4614 | |
---|
4615 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4616 | { |
---|
4617 | for ( j = 1; j <= ncols(B); j++ ) // now -B |
---|
4618 | { |
---|
4619 | Con[i,ncols(P) + j] = - B[i,j]; |
---|
4620 | } |
---|
4621 | } |
---|
4622 | |
---|
4623 | |
---|
4624 | print(Con); |
---|
4625 | |
---|
4626 | intmat L = kernelLattice(Con); |
---|
4627 | |
---|
4628 | print(L); |
---|
4629 | print(ncols(P)); |
---|
4630 | print(ncols(L)); |
---|
4631 | |
---|
4632 | // delete rows ncols(P)+1 to nrows(L) out of L |
---|
4633 | intmat Del[ncols(P)][ncols(L)]; |
---|
4634 | int k; |
---|
4635 | int m; |
---|
4636 | |
---|
4637 | for ( k = 1; k <= nrows(Del); k++ ) |
---|
4638 | { |
---|
4639 | for ( m = 1; m <= ncols(Del); m++ ) |
---|
4640 | { |
---|
4641 | Del[k,m] = L[k,m]; |
---|
4642 | } |
---|
4643 | } |
---|
4644 | |
---|
4645 | L = latticeBasis(Del); |
---|
4646 | |
---|
4647 | return(L); |
---|
4648 | |
---|
4649 | } |
---|
4650 | example |
---|
4651 | { |
---|
4652 | "EXAMPLE"; echo = 2; |
---|
4653 | |
---|
4654 | intmat P[2][3] = |
---|
4655 | 2,6,10, |
---|
4656 | 4,8,12; |
---|
4657 | |
---|
4658 | intmat B[2][1] = |
---|
4659 | 1, |
---|
4660 | 0; |
---|
4661 | |
---|
4662 | // should be a 2x2 matrix with columns |
---|
4663 | // [1,1,-1] and [-2,1,0] |
---|
4664 | intmat L = preimageLattice(P,B); |
---|
4665 | |
---|
4666 | kill B, P, L; |
---|
4667 | |
---|
4668 | } |
---|
4669 | |
---|
4670 | /******************************************************/ |
---|
4671 | proc isPrimitiveSublattice(intmat A); |
---|
4672 | "USAGE: isPrimitiveSublattice(A); intmat A |
---|
4673 | PURPOSE: check whether the given set of integral vectors in ZZ^m, |
---|
4674 | i.e. the columns of A, generate a primitive sublattice in ZZ^m |
---|
4675 | (a direct summand of ZZ^m). |
---|
4676 | RETURNS: int, where 0 is false and 1 is true. |
---|
4677 | EXAMPLE: example isPrimitiveSublattice; shows an example |
---|
4678 | " |
---|
4679 | { |
---|
4680 | intmat B = smithNormalForm(A); |
---|
4681 | int r = intRank(B); |
---|
4682 | |
---|
4683 | if ( r == 0 ) |
---|
4684 | { |
---|
4685 | return(1); |
---|
4686 | } |
---|
4687 | |
---|
4688 | if ( 1 < B[r,r] ) |
---|
4689 | { |
---|
4690 | return(0); |
---|
4691 | } |
---|
4692 | |
---|
4693 | return(1); |
---|
4694 | } |
---|
4695 | example |
---|
4696 | { |
---|
4697 | "EXAMPLE"; echo = 2; |
---|
4698 | |
---|
4699 | intmat A[3][2] = |
---|
4700 | 1,4, |
---|
4701 | 2,5, |
---|
4702 | 3,6; |
---|
4703 | |
---|
4704 | // should be 0 |
---|
4705 | int b = isPrimitiveSublattice(A); |
---|
4706 | |
---|
4707 | kill A,b; |
---|
4708 | } |
---|
4709 | |
---|
4710 | /******************************************************/ |
---|
4711 | proc isIntegralSurjective(intmat P); |
---|
4712 | "USAGE: isIntegralSurjective(P); intmat P |
---|
4713 | PURPOSE: test whether the given linear map P of lattices is |
---|
4714 | surjective. |
---|
4715 | RETURNS: int, where 0 is false and 1 is true. |
---|
4716 | EXAMPLE: example isIntegralSurjective; shows an example |
---|
4717 | " |
---|
4718 | { |
---|
4719 | int r = intRank(P); |
---|
4720 | |
---|
4721 | if ( r < nrows(P) ) |
---|
4722 | { |
---|
4723 | return(0); |
---|
4724 | } |
---|
4725 | |
---|
4726 | if ( isPrimitiveSublattice(A) == 1 ) |
---|
4727 | { |
---|
4728 | return(1); |
---|
4729 | } |
---|
4730 | |
---|
4731 | return(0); |
---|
4732 | } |
---|
4733 | example |
---|
4734 | { |
---|
4735 | "EXAMPLE"; echo = 2; |
---|
4736 | |
---|
4737 | intmat A[3][2] = |
---|
4738 | 1,3,5, |
---|
4739 | 2,4,6; |
---|
4740 | |
---|
4741 | // should be 0 |
---|
4742 | int b = isIntegralSurjective(A); |
---|
4743 | print(b); |
---|
4744 | |
---|
4745 | kill A,b; |
---|
4746 | } |
---|
4747 | |
---|
4748 | /******************************************************/ |
---|
4749 | proc projectLattice(intmat B) |
---|
4750 | "USAGE: projectLattice(B); intmat B |
---|
4751 | PURPOSE: A set of vectors in ZZ^m is given as the columns of B. |
---|
4752 | Then this function provides a linear map ZZ^m --> ZZ^n |
---|
4753 | having the primitive span of B its kernel. |
---|
4754 | RETURNS: intmat |
---|
4755 | EXAMPLE: example projectLattice; shows an example |
---|
4756 | " |
---|
4757 | { |
---|
4758 | int n = nrows(B); |
---|
4759 | int r = intRank(B); |
---|
4760 | |
---|
4761 | if ( r == 0 ) |
---|
4762 | { |
---|
4763 | intmat U = unitMatrix(n); |
---|
4764 | } |
---|
4765 | else |
---|
4766 | { |
---|
4767 | if ( r == n ) |
---|
4768 | { |
---|
4769 | intmat U[n][1]; // U now is the n-dim zero-vector |
---|
4770 | } |
---|
4771 | else |
---|
4772 | { |
---|
4773 | // we want a matrix with column operations so we transpose |
---|
4774 | list L = hermiteNormalForm(B, "transform"); //hermite(transpose(B), "transform"); |
---|
4775 | intmat U = transpose(L[2]); |
---|
4776 | |
---|
4777 | // delete rows 1 to r |
---|
4778 | intmat Udel[nrows(U) - r][ncols(U)]; |
---|
4779 | int k; |
---|
4780 | int m; |
---|
4781 | |
---|
4782 | for ( k = 1; k <= nrows(U) - r ; k++ ) |
---|
4783 | { |
---|
4784 | for ( m = 1; m <= ncols(U); m++ ) |
---|
4785 | { |
---|
4786 | Udel[k,m] = U[k + r,m]; |
---|
4787 | } |
---|
4788 | } |
---|
4789 | |
---|
4790 | U = Udel; |
---|
4791 | |
---|
4792 | } |
---|
4793 | } |
---|
4794 | |
---|
4795 | return(U); |
---|
4796 | } |
---|
4797 | example |
---|
4798 | { |
---|
4799 | "EXAMPLE"; echo = 2; |
---|
4800 | |
---|
4801 | intmat B[4][2] = |
---|
4802 | 1,5, |
---|
4803 | 2,6, |
---|
4804 | 3,7, |
---|
4805 | 4,8; |
---|
4806 | |
---|
4807 | // should result in a (2x4)-matrix with columns |
---|
4808 | // [-1, 2], [2, â3], [-1, 0] and [0, 1]. |
---|
4809 | intmat U = projectLattice(B); |
---|
4810 | |
---|
4811 | } |
---|
4812 | |
---|
4813 | /******************************************************/ |
---|
4814 | proc intersectLattices(intmat A, intmat B) |
---|
4815 | "USAGE: intersectLattices(A, B); intmat A, intmat B |
---|
4816 | PURPOSE: compute an integral basis for the intersection of the |
---|
4817 | lattices A and B. |
---|
4818 | RETURNS: intmat |
---|
4819 | EXAMPLE: example intersectLattices; shows an example |
---|
4820 | " |
---|
4821 | { |
---|
4822 | // concatenate matrices: Con = [A,-B] |
---|
4823 | intmat Con[nrows(A)][ncols(A) + ncols(B)]; |
---|
4824 | int i; |
---|
4825 | int j; |
---|
4826 | |
---|
4827 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4828 | { |
---|
4829 | for ( j = 1; j <= ncols(A); j++ ) // A first |
---|
4830 | { |
---|
4831 | Con[i,j] = A[i,j]; |
---|
4832 | } |
---|
4833 | } |
---|
4834 | |
---|
4835 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4836 | { |
---|
4837 | for ( j = 1; j <= ncols(B); j++ ) // now -B |
---|
4838 | { |
---|
4839 | Con[i,ncols(A) + j] = - B[i,j]; |
---|
4840 | } |
---|
4841 | } |
---|
4842 | |
---|
4843 | intmat K = kernelLattice(Con); |
---|
4844 | |
---|
4845 | // delete all rows in K from ncols(A)+1 onwards |
---|
4846 | intmat Bas[ncols(A)][ncols(K)]; |
---|
4847 | |
---|
4848 | for ( i = 1; i <= nrows(Bas); i++ ) |
---|
4849 | { |
---|
4850 | for ( j = 1; j <= ncols(Bas); j++ ) |
---|
4851 | { |
---|
4852 | Bas[i,j] = K[i,j]; |
---|
4853 | } |
---|
4854 | } |
---|
4855 | |
---|
4856 | // take product in order to obtain the intersection |
---|
4857 | intmat S = A * Bas; |
---|
4858 | intmat Cut = hermiteNormalForm(S); //hermite(S); |
---|
4859 | int r = intRank(Cut); |
---|
4860 | |
---|
4861 | if ( r == 0 ) |
---|
4862 | { |
---|
4863 | intmat Cutdel[nrows(Cut)][1]; // is now the zero-vector |
---|
4864 | |
---|
4865 | Cut = Cutdel; |
---|
4866 | } |
---|
4867 | else |
---|
4868 | { |
---|
4869 | // delte columns from r+1 onwards |
---|
4870 | intmat Cutdel[nrows(Cut)][r]; |
---|
4871 | |
---|
4872 | for ( i = 1; i <= nrows(Cutdel); i++ ) |
---|
4873 | { |
---|
4874 | for ( j = 1; j <= r; j++ ) |
---|
4875 | { |
---|
4876 | Cutdel[i,j] = Cut[i,j]; |
---|
4877 | } |
---|
4878 | } |
---|
4879 | |
---|
4880 | Cut = Cutdel; |
---|
4881 | } |
---|
4882 | |
---|
4883 | return(Cut); |
---|
4884 | } |
---|
4885 | example |
---|
4886 | { |
---|
4887 | "EXAMPLE"; echo = 2; |
---|
4888 | |
---|
4889 | intmat A[3][2] = |
---|
4890 | 1,4, |
---|
4891 | 2,5, |
---|
4892 | 3,6; |
---|
4893 | |
---|
4894 | intmat B[3][2] = |
---|
4895 | 6,9, |
---|
4896 | 7,10, |
---|
4897 | 8,11; |
---|
4898 | |
---|
4899 | // should result in a (2x4)-matrix with columns |
---|
4900 | // [3, 3, 3], [0, 3, 6] |
---|
4901 | intmat U = intersectLattices(A,B); |
---|
4902 | |
---|
4903 | } |
---|
4904 | |
---|
4905 | proc intInverse(intmat A); |
---|
4906 | "USAGE: intInverse(A); intmat A |
---|
4907 | PURPOSE: compute the integral inverse of the intmat A. |
---|
4908 | If det(A) is neither 1 nor -1 an error is returned. |
---|
4909 | RETURNS: intmat |
---|
4910 | EXAMPLE: example intInverse; shows an example |
---|
4911 | " |
---|
4912 | { |
---|
4913 | int d = det(A); |
---|
4914 | |
---|
4915 | if ( d * d != 1 ) // is d = 1 or -1? Else: error |
---|
4916 | { |
---|
4917 | ERROR("determinant of the given intmat has to be 1 or -1."); |
---|
4918 | } |
---|
4919 | |
---|
4920 | int c; |
---|
4921 | int i,j; |
---|
4922 | intmat C[nrows(A)][ncols(A)]; |
---|
4923 | intmat Ad; |
---|
4924 | int s; |
---|
4925 | |
---|
4926 | for ( i = 1; i <= nrows(C); i++ ) |
---|
4927 | { |
---|
4928 | for ( j = 1; j <= ncols(C); j++ ) |
---|
4929 | { |
---|
4930 | Ad = intAdjoint(A,i,j); |
---|
4931 | s = 1; |
---|
4932 | |
---|
4933 | if ( ((i + j) % 2) > 0 ) |
---|
4934 | { |
---|
4935 | s = -1; |
---|
4936 | } |
---|
4937 | |
---|
4938 | C[i,j] = d * s * intDet(Ad); // mult by d is equal to div by det |
---|
4939 | } |
---|
4940 | } |
---|
4941 | |
---|
4942 | C = transpose(C); |
---|
4943 | |
---|
4944 | return(C); |
---|
4945 | } |
---|
4946 | example |
---|
4947 | { |
---|
4948 | "EXAMPLE"; echo = 2; |
---|
4949 | |
---|
4950 | intmat A[3][3] = |
---|
4951 | 1,1,3, |
---|
4952 | 3,2,0, |
---|
4953 | 0,0,1; |
---|
4954 | |
---|
4955 | intmat B = intInverse(A); |
---|
4956 | |
---|
4957 | // should be the unit matrix |
---|
4958 | print(A * B); |
---|
4959 | |
---|
4960 | kill A,B; |
---|
4961 | } |
---|
4962 | |
---|
4963 | |
---|
4964 | /******************************************************/ |
---|
4965 | proc intAdjoint(intmat A, int indrow, int indcol) |
---|
4966 | "USAGE: intAdjoint(A); intmat A |
---|
4967 | PURPOSE: return the matrix where the given row and column are deleted. |
---|
4968 | RETURNS: intmat |
---|
4969 | EXAMPLE: example intAdjoint; shows an example |
---|
4970 | " |
---|
4971 | { |
---|
4972 | int n = nrows(A); |
---|
4973 | int m = ncols(A); |
---|
4974 | int i, j; |
---|
4975 | intmat B[n - 1][m - 1]; |
---|
4976 | int a, b; |
---|
4977 | |
---|
4978 | for ( i = 1; i < indrow; i++ ) |
---|
4979 | { |
---|
4980 | for ( j = 1; j < indcol; j++ ) |
---|
4981 | { |
---|
4982 | B[i,j] = A[i,j]; |
---|
4983 | } |
---|
4984 | for ( j = indcol + 1; j <= ncols(A); j++ ) |
---|
4985 | { |
---|
4986 | B[i,j - 1] = A[i,j]; |
---|
4987 | } |
---|
4988 | } |
---|
4989 | |
---|
4990 | for ( i = indrow + 1; i <= nrows(A); i++ ) |
---|
4991 | { |
---|
4992 | for ( j = 1; j < indcol; j++ ) |
---|
4993 | { |
---|
4994 | B[i - 1,j] = A[i,j]; |
---|
4995 | } |
---|
4996 | for ( j = indcol+1; j <= ncols(A); j++ ) |
---|
4997 | { |
---|
4998 | B[i - 1,j - 1] = A[i,j]; |
---|
4999 | } |
---|
5000 | } |
---|
5001 | |
---|
5002 | return(B); |
---|
5003 | } |
---|
5004 | example |
---|
5005 | { |
---|
5006 | "EXAMPLE"; echo = 2; |
---|
5007 | |
---|
5008 | intmat A[2][3] = |
---|
5009 | 1,3,5, |
---|
5010 | 2,4,6; |
---|
5011 | |
---|
5012 | intmat B = intAdjoint(A,2,2); |
---|
5013 | print(B); |
---|
5014 | |
---|
5015 | kill A,B; |
---|
5016 | } |
---|
5017 | |
---|
5018 | /******************************************************/ |
---|
5019 | proc integralSection(intmat P); |
---|
5020 | "USAGE: integralSection(P); intmat P |
---|
5021 | PURPOSE: for a given linear surjective map P of lattices |
---|
5022 | this procedure returns an integral section of P. |
---|
5023 | RETURNS: intmat |
---|
5024 | EXAMPLE: example integralSection; shows an example |
---|
5025 | " |
---|
5026 | { |
---|
5027 | int m = nrows(P); |
---|
5028 | int n = ncols(P); |
---|
5029 | |
---|
5030 | if ( m == n ) |
---|
5031 | { |
---|
5032 | intmat U = intInverse(P); |
---|
5033 | } |
---|
5034 | else |
---|
5035 | { |
---|
5036 | intmat U = (hermiteNormalForm(P, "transform"))[2]; |
---|
5037 | |
---|
5038 | // delete columns m+1 to n |
---|
5039 | intmat Udel[nrows(U)][ncols(U) - (n - m)]; |
---|
5040 | int k; |
---|
5041 | int z; |
---|
5042 | |
---|
5043 | for ( k = 1; k <= nrows(U); k++ ) |
---|
5044 | { |
---|
5045 | for ( z = 1; z <= m; z++ ) |
---|
5046 | { |
---|
5047 | Udel[k,z] = U[k,z]; |
---|
5048 | } |
---|
5049 | } |
---|
5050 | |
---|
5051 | U = Udel; |
---|
5052 | } |
---|
5053 | |
---|
5054 | return(U); |
---|
5055 | } |
---|
5056 | example |
---|
5057 | { |
---|
5058 | "EXAMPLE"; echo = 2; |
---|
5059 | |
---|
5060 | intmat P[2][4] = |
---|
5061 | 1,3,4,6, |
---|
5062 | 2,4,5,7; |
---|
5063 | |
---|
5064 | // should be a matrix with two columns |
---|
5065 | // for example: [â2, 1, 0, 0], [3, â3, 0, 1] |
---|
5066 | intmat U = integralSection(P); |
---|
5067 | |
---|
5068 | print(U); |
---|
5069 | print(P * U); |
---|
5070 | |
---|
5071 | kill U; |
---|
5072 | } |
---|
5073 | |
---|
5074 | |
---|
5075 | |
---|
5076 | /******************************************************/ |
---|
5077 | proc factorgroup(G,H) |
---|
5078 | "USAGE: factorgroup(G,H); list G, list H |
---|
5079 | PURPOSE: returns a representation of the factor group G mod H using the first isomorphism thm |
---|
5080 | RETURNS: list |
---|
5081 | EXAMPLE: example factorgroup(G,H); shows an example |
---|
5082 | " |
---|
5083 | { |
---|
5084 | intmat S1 = G[1]; |
---|
5085 | intmat L1 = G[2]; |
---|
5086 | intmat S2 = H[1]; |
---|
5087 | intmat L2 = H[2]; |
---|
5088 | |
---|
5089 | // check whether G,H are subgroups of a common group, i.e. whether L1 and L2 span the same lattice |
---|
5090 | if ( !isSublattice(L1,L2) || !isSublattice(L2,L1)) |
---|
5091 | { |
---|
5092 | ERROR("G and H are not subgroups of a common group."); |
---|
5093 | } |
---|
5094 | |
---|
5095 | // check whether H is a subgroup of G, i.e. whether S2 is a sublattice of S1+L1 |
---|
5096 | intmat B = concatintmat(S1,L1); // check whether this gives the concatinated matrix |
---|
5097 | if ( !isSublattice(S2,B) ) |
---|
5098 | { |
---|
5099 | ERROR("H is not a subgroup of G"); |
---|
5100 | } |
---|
5101 | // use first isomorphism thm to get the factor group |
---|
5102 | intmat L = concatintmat(L1,S2); // check whether this gives the concatinated matrix |
---|
5103 | list GmodH; |
---|
5104 | GmodH[1]=S1; |
---|
5105 | GmodH[2]=L; |
---|
5106 | return(GmodH); |
---|
5107 | } |
---|
5108 | example |
---|
5109 | { |
---|
5110 | "EXAMPLE"; echo = 2; |
---|
5111 | |
---|
5112 | intmat S1[2][2] = |
---|
5113 | 1,0, |
---|
5114 | 0,1; |
---|
5115 | intmat L1[2][1] = |
---|
5116 | 2, |
---|
5117 | 0; |
---|
5118 | |
---|
5119 | intmat S2[2][1] = |
---|
5120 | 1, |
---|
5121 | 0; |
---|
5122 | intmat L2[2][1] = |
---|
5123 | 2, |
---|
5124 | 0; |
---|
5125 | |
---|
5126 | list G = createGroup(S1,L1); |
---|
5127 | list H = createGroup(S2,L2); |
---|
5128 | |
---|
5129 | list N = factorgroup(G,H); |
---|
5130 | print(N); |
---|
5131 | |
---|
5132 | kill G,H,N,S1,L1,S2,L2; |
---|
5133 | |
---|
5134 | } |
---|
5135 | |
---|
5136 | /******************************************************/ |
---|
5137 | proc productgroup(G,H) |
---|
5138 | "USAGE: productgroup(G,H); list G, list H |
---|
5139 | PURPOSE: returns a representation of the G x H |
---|
5140 | RETURNS: list |
---|
5141 | EXAMPLE: example productgroup(G,H); shows an example |
---|
5142 | " |
---|
5143 | { |
---|
5144 | intmat S1 = G[1]; |
---|
5145 | intmat L1 = G[2]; |
---|
5146 | intmat S2 = H[1]; |
---|
5147 | intmat L2 = H[2]; |
---|
5148 | intmat OS1[nrows(S1)][ncols(S2)]; |
---|
5149 | intmat OS2[nrows(S2)][ncols(S1)]; |
---|
5150 | intmat OL1[nrows(L1)][ncols(L2)]; |
---|
5151 | intmat OL2[nrows(L2)][ncols(L1)]; |
---|
5152 | |
---|
5153 | // concatinate matrices to get S |
---|
5154 | intmat A = concatintmat(S1,OS1); |
---|
5155 | intmat B = concatintmat(OS2,S2); |
---|
5156 | intmat At = transpose(A); |
---|
5157 | intmat Bt = transpose(B); |
---|
5158 | intmat St = concatintmat(At,Bt); |
---|
5159 | intmat S = transpose(St); |
---|
5160 | |
---|
5161 | // concatinate matrices to get L |
---|
5162 | intmat C = concatintmat(L1,OL1); |
---|
5163 | intmat D = concatintmat(OL2,L2); |
---|
5164 | intmat Ct = transpose(C); |
---|
5165 | intmat Dt = transpose(D); |
---|
5166 | intmat Lt = concatintmat(Ct,Dt); |
---|
5167 | intmat L = transpose(Lt); |
---|
5168 | |
---|
5169 | list GxH; |
---|
5170 | GxH[1]=S; |
---|
5171 | GxH[2]=L; |
---|
5172 | return(GxH); |
---|
5173 | } |
---|
5174 | example |
---|
5175 | { |
---|
5176 | "EXAMPLE"; echo = 2; |
---|
5177 | |
---|
5178 | intmat S1[2][2] = |
---|
5179 | 1,0, |
---|
5180 | 0,1; |
---|
5181 | intmat L1[2][1] = |
---|
5182 | 2, |
---|
5183 | 0; |
---|
5184 | |
---|
5185 | intmat S2[2][2] = |
---|
5186 | 1,0, |
---|
5187 | 0,2; |
---|
5188 | intmat L2[2][1] = |
---|
5189 | 0, |
---|
5190 | 3; |
---|
5191 | |
---|
5192 | list G = createGroup(S1,L1); |
---|
5193 | list H = createGroup(S2,L2); |
---|
5194 | |
---|
5195 | list N = productgroup(G,H); |
---|
5196 | print(N); |
---|
5197 | |
---|
5198 | kill G,H,N,S1,L1,S2,L2; |
---|
5199 | |
---|
5200 | } |
---|
5201 | |
---|
5202 | /******************************************************/ |
---|
5203 | proc primitiveSpan(intmat V); |
---|
5204 | "USAGE: isIntegralSurjective(V); intmat V |
---|
5205 | PURPOSE: compute an integral basis for the minimal primitive |
---|
5206 | sublattice that contains the given vectors, i.e. the columns of V. |
---|
5207 | RETURNS: int, where 0 is false and 1 is true. |
---|
5208 | EXAMPLE: example isIntegralSurjective; shows an example |
---|
5209 | " |
---|
5210 | { |
---|
5211 | int n = ncols(V); |
---|
5212 | int m = nrows(V); |
---|
5213 | int r = intRank(V); |
---|
5214 | |
---|
5215 | |
---|
5216 | if ( r == 0 ) |
---|
5217 | { |
---|
5218 | intmat P[m][1]; // this is the m-zero-vector now |
---|
5219 | } |
---|
5220 | else |
---|
5221 | { |
---|
5222 | list L = smithNormalForm(V, "transform"); // L = [A,S,B] where S is the smith-NF and S = A*S*B |
---|
5223 | intmat P = intInverse(L[1]); |
---|
5224 | |
---|
5225 | print(L); |
---|
5226 | |
---|
5227 | if ( r < m ) |
---|
5228 | { |
---|
5229 | // delete columns r+1 to m in P: |
---|
5230 | intmat Pdel[nrows(P)][r]; |
---|
5231 | int i,j; |
---|
5232 | |
---|
5233 | for ( i = 1; i <= nrows(Pdel); i++ ) |
---|
5234 | { |
---|
5235 | for ( j = 1; j <= ncols(Pdel); j++ ) |
---|
5236 | { |
---|
5237 | Pdel[i,j] = P[i,j]; |
---|
5238 | } |
---|
5239 | } |
---|
5240 | |
---|
5241 | P = Pdel; |
---|
5242 | } |
---|
5243 | } |
---|
5244 | |
---|
5245 | return(P); |
---|
5246 | } |
---|
5247 | example |
---|
5248 | { |
---|
5249 | "EXAMPLE"; echo = 2; |
---|
5250 | |
---|
5251 | intmat V[2][4] = |
---|
5252 | 1,4, |
---|
5253 | 2,5, |
---|
5254 | 3,6; |
---|
5255 | |
---|
5256 | // should return a (4x2)-matrix with columns |
---|
5257 | // [1, 2, 3] and [1, 1, 1] (or similar) |
---|
5258 | intmat R = primitiveSpan(V); |
---|
5259 | print(R); |
---|
5260 | |
---|
5261 | kill V,R; |
---|
5262 | } |
---|
5263 | |
---|
5264 | /***********************************************************/ |
---|