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 | @* Anna-Lena Winz, anna-lena.winz@math.fu-berlin.de |
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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 | grade multivariate polynomial rings with arbitrary (fin. gen. Abelian) groups. |
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18 | For more see http://code.google.com/p/convex-singular/wiki/Multigrading |
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19 | For theoretical references see: |
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20 | @* E. Miller, B. Sturmfels: 'Combinatorial Commutative Algebra' |
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21 | and |
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22 | @* M. Kreuzer, L. Robbiano: 'Computational Commutative Algebra'. |
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23 | |
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24 | NOTE: 'multiDegBasis' relies on 4ti2 for computing Hilbert Bases. |
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25 | All groups are finitely generated Abelian |
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26 | |
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27 | PROCEDURES: |
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28 | setBaseMultigrading(M,L); attach multiweights/grading group matrices to the basering |
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29 | getVariableWeights([R]); get matrix of multidegrees of vars attached to a ring |
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30 | |
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31 | getGradingGroup([R]); get grading group attached to a ring |
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32 | getLattice([R[,choice]]); get grading group' lattice attached to a ring (or its NF) |
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33 | |
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34 | createGroup(S,L); create a group generated by S, with relations L |
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35 | createQuotientGroup(L); create a group generated by the unit matrix whith relations L |
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36 | createTorsionFreeGroup(S); create a group generated by S which is torsionfree |
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37 | printGroup(G); print a group |
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38 | |
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39 | areIsomorphicGroups(G,H); test wheter G an H are isomorphic groups (TODO Tuebingen) |
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40 | isGroup(G); test whether G is a valid group |
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41 | isGroupHomomorphism(L1,L2,A); test wheter A defines a group homomrphism from L1 to L2 |
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42 | |
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43 | isGradedRingHomomorphism(R,f,A); test graded ring homomorph |
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44 | createGradedRingHomomorphism(R,f,A); create a graded ring homomorph |
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45 | |
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46 | setModuleGrading(M,v); attach multiweights of units to a module and return it |
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47 | getModuleGrading(M); get multiweights of module units (attached to M) |
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48 | |
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49 | isSublattice(A,B); test whether A is a sublattice of B |
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50 | imageLattice(P,L); computes an integral basis for P(L) |
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51 | intRank(A); computes the rank of the intmat A |
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52 | kernelLattice(P); computes an integral basis for the kernel of the linear map P. |
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53 | latticeBasis(B); computes an integral basis of the lattice B |
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54 | preimageLattice(P,L); computes an integral basis for the preimage of the lattice L under the linear map P. |
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55 | projectLattice(B); computes a linear map of lattices having the primitive span of B as its kernel. |
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56 | intersectLattices(A,B); computes an integral basis for the intersection of the lattices A and B. |
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57 | isIntegralSurjective(P); test whether the map P of lattices is surjective. |
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58 | isPrimitiveSublattice(A); test whether A generates a primitive sublattice. |
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59 | intInverse(A); computes the integral inverse matrix of the intmat A |
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60 | intAdjoint(A,i,j); delete row i and column j of the intmat A. |
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61 | integralSection(P); for a given linear surjective map P of lattices this procedure returns an integral section of P. |
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62 | primitiveSpan(A); computes a basis for the minimal primitive sublattice that contains the given vectors (by A). |
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63 | |
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64 | factorgroup(G,H); create the group G mod H |
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65 | productgroup(G,H); create the group G x H |
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66 | |
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67 | multiDeg(A); compute the multidegree of A |
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68 | multiDegBasis(d); compute all monomials of multidegree d |
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69 | multiDegPartition(p); compute the multigraded-homogeneous components of p |
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70 | |
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71 | isTorsionFree(); test whether the current multigrading is free |
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72 | isPositive(); test whether the current multigrading is positive |
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73 | isZeroElement(p); test whether p has zero multidegree |
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74 | areZeroElements(M); test whether an integer matrix M considered as a collection of columns has zero multidegree |
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75 | isHomogeneous(a); test whether 'a' is multigraded-homogeneous |
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76 | |
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77 | equalMultiDeg(e1,e2[,V]); test whether e1==e2 in the current multigrading |
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78 | |
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79 | multiDegGroebner(M); compute the multigraded GB/SB of M |
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80 | multiDegSyzygy(M); compute the multigraded syzygies of M |
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81 | multiDegModulo(I,J); compute the multigraded 'modulo' module of I and J |
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82 | multiDegResolution(M,l[,m]); compute the multigraded resolution of M |
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83 | multiDegTensor(m,n); compute the tensor product of multigraded modules m,n |
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84 | multiDegTor(i,m,n); compute the Tor_i(m,n) for multigraded modules m,n |
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85 | |
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86 | defineHomogeneous(p); get a grading group wrt which p becomes homogeneous |
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87 | pushForward(f); find the finest grading on the image ring, homogenizing f |
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88 | gradiator(h); coarsens grading of the ring until h becomes homogeneous |
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89 | |
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90 | hermiteNormalForm(A); compute the Hermite Normal Form of a matrix |
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91 | 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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92 | |
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93 | hilbertSeries(M); compute the multigraded Hilbert Series of M |
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94 | evalHilbertSeries(h,v); evaluate hilberts series h by substituting v[i] for t_(i) |
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95 | |
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96 | lll(A); applies LLL(.) of lll.lib which only works for lists on a matrix A |
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97 | |
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98 | (parameters in square brackets [] are optional) |
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99 | |
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100 | KEYWORDS: multigrading, multidegree, multiweights, multigraded-homogeneous, integral linear algebra |
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101 | "; |
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102 | |
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103 | // finestMDeg(def r) |
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104 | // newMap(map F, intmat Q, list #) |
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105 | |
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106 | LIB "standard.lib"; // for groebner |
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107 | LIB "lll.lib"; // for lll_matrix |
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108 | LIB "matrix.lib"; // for multiDegTor |
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109 | |
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110 | /******************************************************/ |
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111 | |
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112 | static proc concatintmat(intmat A, intmat B) |
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113 | { |
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114 | |
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115 | if ( nrows(A) != nrows(B) ) |
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116 | { |
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117 | ERROR("matrices A and B have different number of rows."); |
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118 | } |
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119 | |
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120 | intmat At = transpose(A); |
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121 | intmat Bt = transpose(B); |
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122 | |
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123 | intmat Ct[nrows(At) + nrows(Bt)][ncols(At)] = At, Bt; |
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124 | |
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125 | return(transpose(Ct)); |
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126 | } |
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127 | |
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128 | |
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129 | /******************************************************/ |
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130 | proc createGradedRingHomomorphism(def src, ideal Im, def A) |
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131 | "USAGE: createGradedRingHomomorphism(R, f, A); ring R, ideal f, group homomorphism A |
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132 | PURPOSE: create a multigraded group ring homomorphism defined by |
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133 | a ring map from R to the current ring, given by generators images f |
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134 | and a group homomorphism A between grading groups |
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135 | RETURN: graded ring homorphism |
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136 | EXAMPLE: example createGradedRingHomomorphism; shows an example |
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137 | " |
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138 | { |
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139 | string isGRH = "isGRH"; |
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140 | |
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141 | if( !isGradedRingHomomorphism(src, Im, A) ) |
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142 | { |
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143 | ERROR("Input data is wrong"); |
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144 | } |
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145 | |
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146 | list h; |
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147 | h[3] = A; |
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148 | |
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149 | // map f = src, Im; |
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150 | h[2] = Im; // f? |
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151 | h[1] = src; |
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152 | |
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153 | attrib(h, isGRH, (1==1)); // mark it "a graded ring homomorphism" |
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154 | |
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155 | return(h); |
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156 | } |
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157 | example |
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158 | { |
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159 | "EXAMPLE:"; echo=2; |
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160 | |
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161 | // TODO! |
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162 | |
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163 | } |
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164 | |
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165 | |
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166 | /******************************************************/ |
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167 | proc isGradedRingHomomorphism(def src, ideal Im, def A) |
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168 | "USAGE: isGradedRingHomomorphism(R, f, A); ring R, ideal f, group homomorphism A |
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169 | PURPOSE: test a multigraded group ring homomorphism defined by |
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170 | a ring map from R to the current ring, given by generators images f |
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171 | and a group homomorphism A between grading groups |
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172 | RETURN: int, 1 for TRUE, 0 otherwise |
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173 | EXAMPLE: example isGradedRingHomomorphism; shows an example |
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174 | " |
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175 | { |
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176 | def dst = basering; |
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177 | |
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178 | intmat result_degs = multiDeg(Im); |
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179 | // print(result_degs); |
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180 | |
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181 | setring src; |
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182 | |
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183 | intmat input_degs = multiDeg(maxideal(1)); |
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184 | // print(input_degs); |
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185 | |
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186 | def image_degs = A * input_degs; |
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187 | // print( image_degs ); |
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188 | |
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189 | def df = image_degs - result_degs; |
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190 | // print(df); |
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191 | |
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192 | setring dst; |
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193 | |
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194 | return (areZeroElements( df )); |
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195 | } |
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196 | example |
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197 | { |
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198 | "EXAMPLE:"; echo=2; |
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199 | |
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200 | ring r = 0, (x, y, z), dp; |
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201 | intmat S1[3][3] = |
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202 | 1, 0, 0, |
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203 | 0, 1, 0, |
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204 | 0, 0, 1; |
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205 | intmat L1[3][1] = |
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206 | 0, |
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207 | 0, |
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208 | 0; |
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209 | |
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210 | def G1 = createGroup(S1, L1); // (S1 + L1)/L1 |
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211 | printGroup(G1); |
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212 | |
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213 | setBaseMultigrading(S1, L1); // to change... |
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214 | |
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215 | ring R = 0, (a, b, c), dp; |
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216 | intmat S2[2][3] = |
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217 | 1, 0, |
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218 | 0, 1; |
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219 | intmat L2[2][1] = |
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220 | 0, |
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221 | 2; |
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222 | |
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223 | def G2 = createGroup(S2, L2); |
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224 | printGroup(G2); |
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225 | |
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226 | setBaseMultigrading(S2, L2); // to change... |
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227 | |
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228 | |
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229 | map F = r, a, b, c; |
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230 | intmat A[nrows(L2)][nrows(L1)] = |
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231 | 1, 0, 0, |
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232 | 3, 2, -6; |
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233 | |
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234 | // graded ring homomorphism is given by (compatible): |
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235 | print(F); |
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236 | print(A); |
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237 | |
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238 | isGradedRingHomomorphism(r, ideal(F), A); |
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239 | def h = createGradedRingHomomorphism(r, ideal(F), A); |
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240 | |
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241 | print(h); |
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242 | |
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243 | // not a homo.. |
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244 | intmat B[nrows(L2)][nrows(L1)] = |
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245 | 1, 1, 1, |
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246 | 0, 0, 0; |
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247 | print(B); |
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248 | |
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249 | isGradedRingHomomorphism(r, ideal(F), B); // FALSE: there is no such homomorphism! |
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250 | // Therefore: the following command should return an error |
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251 | // createGradedRingHomomorphism(r, ideal(F), B); |
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252 | |
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253 | } |
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254 | |
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255 | |
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256 | proc createQuotientGroup(intmat L) |
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257 | " |
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258 | L - relations |
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259 | TODO: bad name |
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260 | " |
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261 | { |
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262 | int r = nrows(L); int i; |
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263 | intmat S[r][r]; // SQUARE!!! |
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264 | for(i = r; i > 0; i--){ S[i, i] = 1; } |
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265 | return (createGroup(S,L)); |
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266 | } |
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267 | |
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268 | proc createTorsionFreeGroup(intmat S) |
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269 | " |
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270 | S - generators |
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271 | TODO: bad name |
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272 | " |
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273 | { |
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274 | int r = nrows(S); int i; |
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275 | intmat L[r][1] = 0; |
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276 | return (createGroup(S,L)); |
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277 | } |
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278 | |
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279 | |
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280 | /******************************************************/ |
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281 | proc createGroup(intmat S, intmat L) |
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282 | "USAGE: createGroup(S, L); S, L are integer matrices |
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283 | PURPOSE: create the group of the form (S+L)/L, i.e. |
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284 | S specifies generators, L specifies relations. |
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285 | RETURN: group |
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286 | EXAMPLE: example createGroup; shows an example |
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287 | " |
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288 | { |
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289 | string isGroup = "isGroup"; |
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290 | string attrGroupHNF = "hermite"; |
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291 | string attrGroupSNF = "smith"; |
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292 | |
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293 | |
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294 | /* |
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295 | if( size(#) > 0 ) |
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296 | { |
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297 | if( typeof(#[1]) == "intmat" ) |
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298 | { |
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299 | intmat S = #[1]; |
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300 | } else { ERROR("Wrong optional argument: 1"); } |
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301 | |
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302 | if( size(#) > 1 ) |
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303 | { |
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304 | if( typeof(#[2]) == "intmat" ) |
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305 | { |
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306 | intmat L = #[2]; |
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307 | } else { ERROR("Wrong optional argument: 2"); } |
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308 | } |
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309 | } |
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310 | if( !defined(S) ) |
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311 | {} |
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312 | */ |
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313 | |
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314 | if( nrows(L) != nrows(S) ) |
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315 | { |
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316 | ERROR("Incompatible matrices!"); |
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317 | } |
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318 | |
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319 | def H = attrib(L, attrGroupHNF); |
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320 | if( !defined(H) || typeof(H) != "intmat") |
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321 | { |
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322 | attrib(L, attrGroupHNF, hermiteNormalForm(L)); |
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323 | } else { kill H; } |
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324 | |
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325 | def HH = attrib(L, attrGroupSNF); |
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326 | if( !defined(HH) || typeof(HH) != "intmat") |
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327 | { |
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328 | attrib(L, attrGroupSNF, smithNormalForm(L)); |
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329 | } else { kill HH; } |
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330 | |
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331 | list G; // Please, note the order: Generators + Relations: |
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332 | G[1] = S; |
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333 | G[2] = L; |
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334 | // And now a quick-and-dirty fix of Singular inability to handle attribs of attribs: |
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335 | // For the use of a group as an attribute for multigraded rings |
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336 | G[3] = attrib(L, attrGroupHNF); |
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337 | G[4] = attrib(L, attrGroupSNF); |
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338 | |
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339 | |
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340 | attrib(G, isGroup, (1==1)); // mark it "a group" |
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341 | |
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342 | return (G); |
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343 | } |
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344 | example |
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345 | { |
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346 | "EXAMPLE:"; echo=2; |
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347 | |
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348 | intmat S[3][3] = |
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349 | 1, 0, 0, |
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350 | 0, 1, 0, |
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351 | 0, 0, 1; |
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352 | |
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353 | intmat L[3][2] = |
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354 | 1, 1, |
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355 | 1, 3, |
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356 | 1, 5; |
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357 | |
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358 | def G = createGroup(S, L); // (S+L)/L |
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359 | |
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360 | printGroup(G); |
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361 | |
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362 | kill S, L, G; |
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363 | |
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364 | ///////////////////////////////////////////////// |
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365 | intmat S[2][3] = |
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366 | 1, -2, 1, |
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367 | 1, 1, 0; |
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368 | |
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369 | intmat L[2][1] = |
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370 | 0, |
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371 | 2; |
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372 | |
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373 | def G = createGroup(S, L); // (S+L)/L |
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374 | |
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375 | printGroup(G); |
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376 | |
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377 | kill S, L, G; |
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378 | |
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379 | // ----------- extreme case ------------ // |
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380 | intmat S[1][3] = |
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381 | 1, -1, 10; |
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382 | |
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383 | // Torsion: |
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384 | intmat L[1][1] = |
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385 | 0; |
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386 | |
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387 | def G = createGroup(S, L); // (S+L)/L |
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388 | |
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389 | printGroup(G); |
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390 | } |
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391 | |
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392 | |
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393 | /******************************************************/ |
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394 | proc printGroup(def G) |
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395 | "USAGE: printGroup(G); G is a group |
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396 | PURPOSE: prints the group G |
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397 | RETURN: nothing |
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398 | EXAMPLE: example printGroup; shows an example |
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399 | " |
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400 | { |
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401 | "Generators: "; |
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402 | print(G[1]); |
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403 | |
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404 | "Relations: "; |
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405 | print(G[2]); |
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406 | |
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407 | // attrib(G[2]); |
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408 | } |
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409 | example |
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410 | { |
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411 | "EXAMPLE:"; echo=2; |
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412 | |
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413 | } |
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414 | |
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415 | /******************************************************/ |
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416 | proc areIsomorphicGroups(def G, def H) |
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417 | "USAGE: areIsomorphicGroups(G, H); G and H are groups |
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418 | PURPOSE: ? |
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419 | RETURN: int, 1 for TRUE, 0 otherwise |
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420 | EXAMPLE: example areIsomorphicGroups; shows an example |
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421 | " |
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422 | { |
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423 | ERROR("areIsomorphicGroups: Not yet implemented!"); |
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424 | return (1); // TRUE |
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425 | } |
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426 | example |
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427 | { |
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428 | "EXAMPLE:"; echo=2; |
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429 | |
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430 | } |
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431 | |
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432 | |
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433 | proc isGroup(def G) |
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434 | "test whether G is a valid group" |
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435 | { |
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436 | string isGroup = "isGroup"; |
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437 | |
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438 | // valid? |
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439 | if( typeof(G) != "list" ){ return(0); } |
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440 | |
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441 | def a = attrib(G, isGroup); |
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442 | |
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443 | ///// TODO for Hans: fix attr^2 bug in Singular! |
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444 | |
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445 | // if( !defined(a) ) { return(0); } |
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446 | // if( typeof(a) != "int" ) { return(0); } |
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447 | if( defined(a) ){ if(typeof(a) == "int") { return(a); } } |
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448 | |
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449 | |
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450 | if( (size(G) != 2) && (size(G) != 4) ){ return(0); } |
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451 | if( typeof(G[1]) != "intmat" ){ return(0); } |
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452 | if( typeof(G[2]) != "intmat" ){ return(0); } |
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453 | if( nrows(G[1]) != nrows(G[2]) ){ return(0); } |
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454 | |
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455 | return(1); |
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456 | } |
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457 | |
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458 | |
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459 | |
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460 | /******************************************************/ |
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461 | proc setBaseMultigrading(intmat M, list #) |
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462 | "USAGE: setBaseMultigrading(M[, G]); M is an integer matrix, G is a group (or lattice) |
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463 | PURPOSE: attaches weights of variables and grading group to the basering. |
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464 | NOTE: M encodes the weights of variables column-wise. |
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465 | RETURN: nothing |
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466 | EXAMPLE: example setBaseMultigrading; shows an example |
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467 | " |
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468 | { |
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469 | string attrMgrad = "mgrad"; |
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470 | string attrGradingGroup = "gradingGroup"; |
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471 | |
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472 | if( size(#) > 0 ) |
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473 | { |
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474 | if( typeof(#[1]) == "intmat" ) |
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475 | { |
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476 | def L = createGroup(M, #[1]); |
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477 | } |
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478 | |
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479 | if( isGroup(#[1]) ) |
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480 | { |
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481 | def L = #[1]; |
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482 | |
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483 | if( !isSublattice(M, L[1]) ) |
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484 | { |
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485 | ERROR("Multigrading is not contained in the grading group!"); |
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486 | } |
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487 | } |
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488 | } |
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489 | else |
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490 | { |
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491 | def L = createTorsionFreeGroup(M); |
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492 | } |
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493 | |
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494 | if( !defined(L) ){ ERROR("Wrong arguments: no group given?"); } |
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495 | |
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496 | attrib(basering, attrMgrad, M); |
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497 | attrib(basering, attrGradingGroup, L); |
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498 | |
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499 | ideal Q = ideal(basering); |
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500 | if( !isHomogeneous(Q) ) // easy now, but would be hard before setting ring attributes! |
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501 | { |
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502 | "Warning: your quotient ideal is not homogenous (multigrading was set anyway)!"; |
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503 | } |
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504 | |
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505 | } |
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506 | example |
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507 | { |
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508 | "EXAMPLE:"; echo=2; |
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509 | |
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510 | ring R = 0, (x, y, z), dp; |
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511 | |
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512 | // Weights of variables |
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513 | intmat M[3][3] = |
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514 | 1, 0, 0, |
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515 | 0, 1, 0, |
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516 | 0, 0, 1; |
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517 | |
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518 | // GradingGroup: |
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519 | intmat L[3][2] = |
---|
520 | 1, 1, |
---|
521 | 1, 3, |
---|
522 | 1, 5; |
---|
523 | |
---|
524 | // attaches M & L to R (==basering): |
---|
525 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
526 | |
---|
527 | // Weights are accessible via "getVariableWeights()": |
---|
528 | getVariableWeights(); |
---|
529 | |
---|
530 | // Test all possible usages: |
---|
531 | (getVariableWeights() == M) && (getVariableWeights(R) == M) && (getVariableWeights(basering) == M); |
---|
532 | |
---|
533 | // Grading group is accessible via "getLattice()": |
---|
534 | getLattice(); |
---|
535 | |
---|
536 | // Test all possible usages: |
---|
537 | (getLattice() == L) && (getLattice(R) == L) && (getLattice(basering) == L); |
---|
538 | |
---|
539 | // And its hermite NF via getLattice("hermite"): |
---|
540 | getLattice("hermite"); |
---|
541 | |
---|
542 | // Test all possible usages: |
---|
543 | intmat H = hermiteNormalForm(L); |
---|
544 | (getLattice("hermite") == H) && (getLattice(R, "hermite") == H) && (getLattice(basering, "hermite") == H); |
---|
545 | |
---|
546 | kill L, M; |
---|
547 | |
---|
548 | // ----------- isomorphic multigrading -------- // |
---|
549 | |
---|
550 | // Weights of variables |
---|
551 | intmat M[2][3] = |
---|
552 | 1, -2, 1, |
---|
553 | 1, 1, 0; |
---|
554 | |
---|
555 | // Torsion: |
---|
556 | intmat L[2][1] = |
---|
557 | 0, |
---|
558 | 2; |
---|
559 | |
---|
560 | // attaches M & L to R (==basering): |
---|
561 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
562 | |
---|
563 | // Weights are accessible via "getVariableWeights()": |
---|
564 | getVariableWeights() == M; |
---|
565 | |
---|
566 | // Torsion is accessible via "getLattice()": |
---|
567 | getLattice() == L; |
---|
568 | |
---|
569 | kill L, M; |
---|
570 | // ----------- extreme case ------------ // |
---|
571 | |
---|
572 | // Weights of variables |
---|
573 | intmat M[1][3] = |
---|
574 | 1, -1, 10; |
---|
575 | |
---|
576 | // Torsion: |
---|
577 | intmat L[1][1] = |
---|
578 | 0; |
---|
579 | |
---|
580 | // attaches M & L to R (==basering): |
---|
581 | setBaseMultigrading(M); // Grading: Z^3 |
---|
582 | |
---|
583 | // Weights are accessible via "getVariableWeights()": |
---|
584 | getVariableWeights() == M; |
---|
585 | |
---|
586 | // Torsion is accessible via "getLattice()": |
---|
587 | getLattice() == L; |
---|
588 | } |
---|
589 | |
---|
590 | |
---|
591 | /******************************************************/ |
---|
592 | proc getVariableWeights(list #) |
---|
593 | "USAGE: getVariableWeights([R]) |
---|
594 | PURPOSE: get associated multigrading matrix for the basering [or R] |
---|
595 | RETURN: intmat, matrix of multidegrees of variables |
---|
596 | EXAMPLE: example getVariableWeights; shows an example |
---|
597 | " |
---|
598 | { |
---|
599 | string attrMgrad = "mgrad"; |
---|
600 | |
---|
601 | |
---|
602 | if( size(#) > 0 ) |
---|
603 | { |
---|
604 | if(( typeof(#[1]) == "ring" ) || ( typeof(#[1]) == "qring" )) |
---|
605 | { |
---|
606 | def R = #[1]; |
---|
607 | } |
---|
608 | else |
---|
609 | { |
---|
610 | ERROR("Optional argument must be a ring!"); |
---|
611 | } |
---|
612 | } |
---|
613 | else |
---|
614 | { |
---|
615 | def R = basering; |
---|
616 | } |
---|
617 | |
---|
618 | def M = attrib(R, attrMgrad); |
---|
619 | if( typeof(M) == "intmat"){ return (M); } |
---|
620 | ERROR( "Sorry no multigrading matrix!" ); |
---|
621 | } |
---|
622 | example |
---|
623 | { |
---|
624 | "EXAMPLE:"; echo=2; |
---|
625 | |
---|
626 | ring R = 0, (x, y, z), dp; |
---|
627 | |
---|
628 | // Weights of variables |
---|
629 | intmat M[3][3] = |
---|
630 | 1, 0, 0, |
---|
631 | 0, 1, 0, |
---|
632 | 0, 0, 1; |
---|
633 | |
---|
634 | // Grading group: |
---|
635 | intmat L[3][2] = |
---|
636 | 1, 1, |
---|
637 | 1, 3, |
---|
638 | 1, 5; |
---|
639 | |
---|
640 | // attaches M & L to R (==basering): |
---|
641 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
642 | |
---|
643 | // Weights are accessible via "getVariableWeights()": |
---|
644 | getVariableWeights() == M; |
---|
645 | |
---|
646 | kill L, M; |
---|
647 | |
---|
648 | // ----------- isomorphic multigrading -------- // |
---|
649 | |
---|
650 | // Weights of variables |
---|
651 | intmat M[2][3] = |
---|
652 | 1, -2, 1, |
---|
653 | 1, 1, 0; |
---|
654 | |
---|
655 | // Grading group: |
---|
656 | intmat L[2][1] = |
---|
657 | 0, |
---|
658 | 2; |
---|
659 | |
---|
660 | // attaches M & L to R (==basering): |
---|
661 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
662 | |
---|
663 | // Weights are accessible via "getVariableWeights()": |
---|
664 | getVariableWeights() == M; |
---|
665 | |
---|
666 | kill L, M; |
---|
667 | |
---|
668 | // ----------- extreme case ------------ // |
---|
669 | |
---|
670 | // Weights of variables |
---|
671 | intmat M[1][3] = |
---|
672 | 1, -1, 10; |
---|
673 | |
---|
674 | // Grading group: |
---|
675 | intmat L[1][1] = |
---|
676 | 0; |
---|
677 | |
---|
678 | // attaches M & L to R (==basering): |
---|
679 | setBaseMultigrading(M); // Grading: Z^3 |
---|
680 | |
---|
681 | // Weights are accessible via "getVariableWeights()": |
---|
682 | getVariableWeights() == M; |
---|
683 | } |
---|
684 | |
---|
685 | |
---|
686 | proc getGradingGroup(list #) |
---|
687 | "USAGE: getGradingGroup([R]) |
---|
688 | PURPOSE: get associated grading group |
---|
689 | RETURN: group, the grading group |
---|
690 | EXAMPLE: example getGradingGroup; shows an example |
---|
691 | " |
---|
692 | { |
---|
693 | string attrGradingGroup = "gradingGroup"; |
---|
694 | |
---|
695 | int i = 1; |
---|
696 | |
---|
697 | if( size(#) >= i ) |
---|
698 | { |
---|
699 | if( ( typeof(#[i]) == "ring" ) or ( typeof(#[i]) == "qring" ) ) |
---|
700 | { |
---|
701 | def R = #[i]; |
---|
702 | i++; |
---|
703 | } |
---|
704 | } |
---|
705 | |
---|
706 | if( !defined(R) ) |
---|
707 | { |
---|
708 | def R = basering; |
---|
709 | } |
---|
710 | |
---|
711 | def G = attrib(R, attrGradingGroup); |
---|
712 | |
---|
713 | if( !isGroup(G) ) |
---|
714 | { |
---|
715 | ERROR("Sorry no grading group!"); |
---|
716 | } |
---|
717 | |
---|
718 | return(G); |
---|
719 | } |
---|
720 | example |
---|
721 | { |
---|
722 | "EXAMPLE:"; echo=2; |
---|
723 | |
---|
724 | ring R = 0, (x, y, z), dp; |
---|
725 | |
---|
726 | // Weights of variables |
---|
727 | intmat M[3][3] = |
---|
728 | 1, 0, 0, |
---|
729 | 0, 1, 0, |
---|
730 | 0, 0, 1; |
---|
731 | |
---|
732 | // Torsion: |
---|
733 | intmat L[3][2] = |
---|
734 | 1, 1, |
---|
735 | 1, 3, |
---|
736 | 1, 5; |
---|
737 | |
---|
738 | // attaches M & L to R (==basering): |
---|
739 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
740 | |
---|
741 | def G = getGradingGroup(); |
---|
742 | |
---|
743 | printGroup( G ); |
---|
744 | |
---|
745 | G[1] == M; G[2] == L; |
---|
746 | |
---|
747 | kill L, M, G; |
---|
748 | |
---|
749 | // ----------- isomorphic multigrading -------- // |
---|
750 | |
---|
751 | // Weights of variables |
---|
752 | intmat M[2][3] = |
---|
753 | 1, -2, 1, |
---|
754 | 1, 1, 0; |
---|
755 | |
---|
756 | // Torsion: |
---|
757 | intmat L[2][1] = |
---|
758 | 0, |
---|
759 | 2; |
---|
760 | |
---|
761 | // attaches M & L to R (==basering): |
---|
762 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
763 | |
---|
764 | def G = getGradingGroup(); |
---|
765 | |
---|
766 | printGroup( G ); |
---|
767 | |
---|
768 | G[1] == M; G[2] == L; |
---|
769 | |
---|
770 | kill L, M, G; |
---|
771 | // ----------- extreme case ------------ // |
---|
772 | |
---|
773 | // Weights of variables |
---|
774 | intmat M[1][3] = |
---|
775 | 1, -1, 10; |
---|
776 | |
---|
777 | // Torsion: |
---|
778 | intmat L[1][1] = |
---|
779 | 0; |
---|
780 | |
---|
781 | // attaches M & L to R (==basering): |
---|
782 | setBaseMultigrading(M); // Grading: Z^3 |
---|
783 | |
---|
784 | def G = getGradingGroup(); |
---|
785 | |
---|
786 | printGroup( G ); |
---|
787 | |
---|
788 | G[1] == M; G[2] == L; |
---|
789 | |
---|
790 | kill L, M, G; |
---|
791 | } |
---|
792 | |
---|
793 | |
---|
794 | /******************************************************/ |
---|
795 | proc getLattice(list #) |
---|
796 | "USAGE: getLattice([R[,opt]]) |
---|
797 | PURPOSE: get associated grading group matrix, i.e. generators (cols) of the grading group |
---|
798 | RETURN: intmat, the grading group matrix, or |
---|
799 | its hermite normal form if an optional argument (\"hermiteNormalForm\") is given or |
---|
800 | smith normal form if an optional argument (\"smith\") is given |
---|
801 | EXAMPLE: example getLattice; shows an example |
---|
802 | " |
---|
803 | { |
---|
804 | int i = 1; |
---|
805 | if( size(#) >= i ) |
---|
806 | { |
---|
807 | if( ( typeof(#[i]) == "ring" ) or ( typeof(#[i]) == "qring" ) ) |
---|
808 | { |
---|
809 | i++; |
---|
810 | } |
---|
811 | } |
---|
812 | |
---|
813 | string attrGradingGroupHNF = "hermite"; |
---|
814 | string attrGradingGroupSNF = "smith"; |
---|
815 | |
---|
816 | def G = getGradingGroup(#); |
---|
817 | |
---|
818 | // printGroup(G); |
---|
819 | |
---|
820 | |
---|
821 | |
---|
822 | def T = G[2]; |
---|
823 | |
---|
824 | if( size(#) >= i ) |
---|
825 | { |
---|
826 | if( #[i] == "hermite" ) |
---|
827 | { |
---|
828 | def M = attrib(T, attrGradingGroupHNF); |
---|
829 | if( (!defined(M)) or (typeof(M) != "intmat") ) |
---|
830 | { |
---|
831 | if( size(G) > 2 ) |
---|
832 | { |
---|
833 | M = G[3]; |
---|
834 | } else |
---|
835 | { |
---|
836 | M = hermiteNormalForm(T); |
---|
837 | } |
---|
838 | } |
---|
839 | return (M); |
---|
840 | } |
---|
841 | |
---|
842 | if( #[i] == "smith" ) |
---|
843 | { |
---|
844 | def M = attrib(T, attrGradingGroupSNF); |
---|
845 | if( (!defined(M)) or (typeof(M) != "intmat") ) |
---|
846 | { |
---|
847 | if( size(G) > 2 ) |
---|
848 | { |
---|
849 | M = G[4]; |
---|
850 | } else |
---|
851 | { |
---|
852 | M = smithNormalForm(T); |
---|
853 | } |
---|
854 | } |
---|
855 | return (M); |
---|
856 | } |
---|
857 | } |
---|
858 | |
---|
859 | return(T); |
---|
860 | } |
---|
861 | example |
---|
862 | { |
---|
863 | "EXAMPLE:"; echo=2; |
---|
864 | |
---|
865 | ring R = 0, (x, y, z), dp; |
---|
866 | |
---|
867 | // Weights of variables |
---|
868 | intmat M[3][3] = |
---|
869 | 1, 0, 0, |
---|
870 | 0, 1, 0, |
---|
871 | 0, 0, 1; |
---|
872 | |
---|
873 | // Torsion: |
---|
874 | intmat L[3][2] = |
---|
875 | 1, 1, |
---|
876 | 1, 3, |
---|
877 | 1, 5; |
---|
878 | |
---|
879 | // attaches M & L to R (==basering): |
---|
880 | setBaseMultigrading(M, L); // Grading: Z^3/L |
---|
881 | |
---|
882 | // Torsion is accessible via "getLattice()": |
---|
883 | getLattice() == L; |
---|
884 | |
---|
885 | // its hermite NF: |
---|
886 | print(getLattice("hermite")); |
---|
887 | |
---|
888 | kill L, M; |
---|
889 | |
---|
890 | // ----------- isomorphic multigrading -------- // |
---|
891 | |
---|
892 | // Weights of variables |
---|
893 | intmat M[2][3] = |
---|
894 | 1, -2, 1, |
---|
895 | 1, 1, 0; |
---|
896 | |
---|
897 | // Torsion: |
---|
898 | intmat L[2][1] = |
---|
899 | 0, |
---|
900 | 2; |
---|
901 | |
---|
902 | // attaches M & L to R (==basering): |
---|
903 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
---|
904 | |
---|
905 | // Torsion is accessible via "getLattice()": |
---|
906 | getLattice() == L; |
---|
907 | |
---|
908 | // its hermite NF: |
---|
909 | print(getLattice("hermite")); |
---|
910 | |
---|
911 | kill L, M; |
---|
912 | |
---|
913 | // ----------- extreme case ------------ // |
---|
914 | |
---|
915 | // Weights of variables |
---|
916 | intmat M[1][3] = |
---|
917 | 1, -1, 10; |
---|
918 | |
---|
919 | // Torsion: |
---|
920 | intmat L[1][1] = |
---|
921 | 0; |
---|
922 | |
---|
923 | // attaches M & L to R (==basering): |
---|
924 | setBaseMultigrading(M); // Grading: Z^3 |
---|
925 | |
---|
926 | // Torsion is accessible via "getLattice()": |
---|
927 | getLattice() == L; |
---|
928 | |
---|
929 | // its hermite NF: |
---|
930 | print(getLattice("hermite")); |
---|
931 | } |
---|
932 | |
---|
933 | proc getGradedGenerator(def m, int i) |
---|
934 | " |
---|
935 | returns m[i], but with grading |
---|
936 | " |
---|
937 | { |
---|
938 | if( typeof(m) == "ideal" ) |
---|
939 | { |
---|
940 | return (m[i]); |
---|
941 | } |
---|
942 | |
---|
943 | if( typeof(m) == "module" ) |
---|
944 | { |
---|
945 | def v = getModuleGrading(m); |
---|
946 | |
---|
947 | return ( setModuleGrading(m[i],v) ); |
---|
948 | } |
---|
949 | |
---|
950 | ERROR("m is expected to be an ideal or a module"); |
---|
951 | } |
---|
952 | |
---|
953 | |
---|
954 | /******************************************************/ |
---|
955 | proc getModuleGrading(def m) |
---|
956 | "USAGE: getModuleGrading(m), 'm' module/vector |
---|
957 | RETURN: integer matrix of the multiweights of free module generators attached to 'm' |
---|
958 | EXAMPLE: example getModuleGrading; shows an example |
---|
959 | " |
---|
960 | { |
---|
961 | string attrModuleGrading = "genWeights"; |
---|
962 | |
---|
963 | // print(m); typeof(m); attrib(m); |
---|
964 | |
---|
965 | def V = attrib(m, attrModuleGrading); |
---|
966 | |
---|
967 | if( typeof(V) != "intmat" ) |
---|
968 | { |
---|
969 | if( (typeof(m) == "ideal") or (typeof(m) == "poly") ) |
---|
970 | { |
---|
971 | intmat M = getVariableWeights(); |
---|
972 | intmat VV[nrows(M)][1]; |
---|
973 | return (VV); |
---|
974 | } |
---|
975 | |
---|
976 | ERROR("Sorry: vector or module need module-grading-matrix! See 'getModuleGrading'."); |
---|
977 | } |
---|
978 | |
---|
979 | if( nrows(V) != nrows(getVariableWeights()) ) |
---|
980 | { |
---|
981 | ERROR("Sorry wrong height of V: " + string(nrows(V))); |
---|
982 | } |
---|
983 | |
---|
984 | if( ncols(V) < nrows(m) ) |
---|
985 | { |
---|
986 | ERROR("Sorry wrong width of V: " + string(ncols(V))); |
---|
987 | } |
---|
988 | |
---|
989 | return (V); |
---|
990 | } |
---|
991 | example |
---|
992 | { |
---|
993 | "EXAMPLE:"; echo=2; |
---|
994 | |
---|
995 | ring R = 0, (x,y), dp; |
---|
996 | intmat M[2][2]= |
---|
997 | 1, 1, |
---|
998 | 0, 2; |
---|
999 | intmat T[2][5]= |
---|
1000 | 1, 2, 3, 4, 0, |
---|
1001 | 0, 10, 20, 30, 1; |
---|
1002 | |
---|
1003 | setBaseMultigrading(M, T); |
---|
1004 | |
---|
1005 | ideal I = x, y, xy^5; |
---|
1006 | isHomogeneous(I); |
---|
1007 | |
---|
1008 | intmat V = multiDeg(I); print(V); |
---|
1009 | |
---|
1010 | module S = syz(I); print(S); |
---|
1011 | |
---|
1012 | S = setModuleGrading(S, V); |
---|
1013 | |
---|
1014 | getModuleGrading(S) == V; |
---|
1015 | |
---|
1016 | vector v = getGradedGenerator(S, 1); |
---|
1017 | getModuleGrading(v) == V; |
---|
1018 | isHomogeneous(v); |
---|
1019 | print( multiDeg(v) ); |
---|
1020 | |
---|
1021 | isHomogeneous(S); |
---|
1022 | print( multiDeg(S) ); |
---|
1023 | } |
---|
1024 | |
---|
1025 | /******************************************************/ |
---|
1026 | proc setModuleGrading(def m, intmat G) |
---|
1027 | "USAGE: setModuleGrading(m, G), m module/vector, G intmat |
---|
1028 | PURPOSE: attaches the multiweights of free module generators to 'm' |
---|
1029 | WARNING: The method does not verify whether the multigrading makes the |
---|
1030 | module/vector homogeneous. One can do that using isHomogeneous(m). |
---|
1031 | EXAMPLE: example setModuleGrading; shows an example |
---|
1032 | " |
---|
1033 | { |
---|
1034 | string attrModuleGrading = "genWeights"; |
---|
1035 | |
---|
1036 | intmat R = getVariableWeights(); |
---|
1037 | |
---|
1038 | if(nrows(G) != nrows(R)){ ERROR("Incompatible gradings.");} |
---|
1039 | if(ncols(G) < nrows(m)){ ERROR("Multigrading does not fit to module.");} |
---|
1040 | |
---|
1041 | attrib(m, attrModuleGrading, G); |
---|
1042 | return(m); |
---|
1043 | } |
---|
1044 | example |
---|
1045 | { |
---|
1046 | "EXAMPLE:"; echo=2; |
---|
1047 | |
---|
1048 | ring R = 0, (x,y), dp; |
---|
1049 | intmat M[2][2]= |
---|
1050 | 1, 1, |
---|
1051 | 0, 2; |
---|
1052 | intmat T[2][5]= |
---|
1053 | 1, 2, 3, 4, 0, |
---|
1054 | 0, 10, 20, 30, 1; |
---|
1055 | |
---|
1056 | setBaseMultigrading(M, T); |
---|
1057 | |
---|
1058 | ideal I = x, y, xy^5; |
---|
1059 | intmat V = multiDeg(I); |
---|
1060 | |
---|
1061 | // V == M; modulo T |
---|
1062 | print(V); |
---|
1063 | |
---|
1064 | module S = syz(I); |
---|
1065 | |
---|
1066 | S = setModuleGrading(S, V); |
---|
1067 | getModuleGrading(S) == V; |
---|
1068 | |
---|
1069 | print(S); |
---|
1070 | |
---|
1071 | vector v = getGradedGenerator(S, 1); |
---|
1072 | getModuleGrading(v) == V; |
---|
1073 | |
---|
1074 | print( multiDeg(v) ); |
---|
1075 | |
---|
1076 | isHomogeneous(S); |
---|
1077 | |
---|
1078 | print( multiDeg(S) ); |
---|
1079 | } |
---|
1080 | |
---|
1081 | |
---|
1082 | proc multiDegTensor(module m, module n){ |
---|
1083 | matrix M = m; |
---|
1084 | matrix N = n; |
---|
1085 | intmat gm = getModuleGrading(m); |
---|
1086 | intmat gn = getModuleGrading(n); |
---|
1087 | int grows = nrows(gm); |
---|
1088 | int mr = nrows(M); |
---|
1089 | int mc = ncols(M); |
---|
1090 | if(rank(M) == 0){ mc = 0;} |
---|
1091 | int nr = nrows(N); |
---|
1092 | int nc = ncols(N); |
---|
1093 | if(rank(N) == 0){ nc = 0;} |
---|
1094 | intmat gresult[nrows(gm)][mr*nr]; |
---|
1095 | matrix result[mr*nr][mr*nc+mc*nr]; |
---|
1096 | int i, j; |
---|
1097 | int column = 1; |
---|
1098 | for(i = 1; i<=mr; i++){ |
---|
1099 | for(j = 1; j<=nr; j++){ |
---|
1100 | gresult[1..grows,(i-1)*nr+j] = gm[1..grows,i]+gn[1..grows,j]; |
---|
1101 | } |
---|
1102 | } |
---|
1103 | //gresult; |
---|
1104 | if( nc!=0 ){ |
---|
1105 | for(i = 1; i<=mr; i++) |
---|
1106 | { |
---|
1107 | result[((i-1)*nr+1)..(i*nr),((i-1)*nc+1)..(i*nc)] = N[1..nr,1..nc]; |
---|
1108 | } |
---|
1109 | } |
---|
1110 | list rownumbers, colnumbers; |
---|
1111 | //print(result); |
---|
1112 | if( mc!=0 ){ |
---|
1113 | for(j = 1; j<=nr; j++) |
---|
1114 | { |
---|
1115 | rownumbers = nr*(0..(mr-1))+j*(1:mr); |
---|
1116 | colnumbers = ((mr*nc+j):mc)+nr*(0..(mc-1)); |
---|
1117 | result[rownumbers[1..mr],colnumbers[1..mc] ] = M[1..mr,1..mc]; |
---|
1118 | } |
---|
1119 | } |
---|
1120 | module res = result; |
---|
1121 | res = setModuleGrading(res, gresult); |
---|
1122 | //getModuleGrading(res); |
---|
1123 | return(res); |
---|
1124 | } |
---|
1125 | example |
---|
1126 | { |
---|
1127 | "EXAMPLE: ";echo=2; |
---|
1128 | ring r = 0,(x),dp; |
---|
1129 | intmat g[2][1]=1,1; |
---|
1130 | setBaseMultigrading(g); |
---|
1131 | matrix m[5][3]=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15; |
---|
1132 | matrix n[3][2]=x,x2,x3,x4,x5,x6; |
---|
1133 | module mm = m; |
---|
1134 | module nn = n; |
---|
1135 | intmat gm[2][5]=1,2,3,4,5,0,0,0,0,0; |
---|
1136 | intmat gn[2][3]=0,0,0,1,2,3; |
---|
1137 | mm = setModuleGrading(mm, gm); |
---|
1138 | nn = setModuleGrading(nn, gn); |
---|
1139 | module mmtnn = multiDegTensor(mm, nn); |
---|
1140 | print(mmtnn); |
---|
1141 | getModuleGrading(mmtnn); |
---|
1142 | LIB "homolog.lib"; |
---|
1143 | module tt = tensorMod(mm,nn); |
---|
1144 | print(tt); |
---|
1145 | |
---|
1146 | kill m, mm, n, nn, gm, gn; |
---|
1147 | |
---|
1148 | 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; |
---|
1149 | matrix n[2][4] = 1, 2, 3, 4, x, x2, x3, x4; |
---|
1150 | module mm = m; |
---|
1151 | module nn = n; |
---|
1152 | print(mm); |
---|
1153 | print(nn); |
---|
1154 | intmat gm[2][7] = 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0; |
---|
1155 | intmat gn[2][2] = 0, 0, 1, 2; |
---|
1156 | mm = setModuleGrading(mm, gm); |
---|
1157 | nn = setModuleGrading(nn, gn); |
---|
1158 | module mmtnn = multiDegTensor(mm, nn); |
---|
1159 | print(mmtnn); |
---|
1160 | getModuleGrading(mmtnn); |
---|
1161 | matrix a = mmtnn; |
---|
1162 | matrix b = tensorMod(mm, nn); |
---|
1163 | print(a-b); |
---|
1164 | |
---|
1165 | } |
---|
1166 | |
---|
1167 | proc multiDegTor(int i, module m, module n) |
---|
1168 | { |
---|
1169 | def res = multiDegResolution(n, 0, 1); |
---|
1170 | //print(res); |
---|
1171 | list l = res; |
---|
1172 | if(size(l)<i){ return(0);} |
---|
1173 | else |
---|
1174 | { |
---|
1175 | |
---|
1176 | matrix fd[nrows(m)][0]; |
---|
1177 | matrix fd2[nrows(l[i+1])][0]; |
---|
1178 | matrix fd3[nrows(l[i])][0]; |
---|
1179 | |
---|
1180 | module freedim = fd; |
---|
1181 | module freedim2 = fd2; |
---|
1182 | module freedim3 = fd3; |
---|
1183 | |
---|
1184 | freedim = setModuleGrading(freedim,getModuleGrading(m)); |
---|
1185 | freedim2 = setModuleGrading(freedim2,getModuleGrading(l[i+1])); |
---|
1186 | freedim3 = setModuleGrading(freedim3, getModuleGrading(l[i])); |
---|
1187 | |
---|
1188 | module mimag = multiDegTensor(freedim3, m); |
---|
1189 | //"mimag ok."; |
---|
1190 | module mf = multiDegTensor(l[i], freedim); |
---|
1191 | //"mf ok."; |
---|
1192 | module mim1 = multiDegTensor(freedim2 ,m); |
---|
1193 | module mim2 = multiDegTensor(l[i+1],freedim); |
---|
1194 | //"mim1+2 ok."; |
---|
1195 | module mker = multiDegModulo(mf,mimag); |
---|
1196 | //"mker ok."; |
---|
1197 | module mim = mim1,mim2; |
---|
1198 | mim = setModuleGrading(mim, getModuleGrading(mim1)); |
---|
1199 | //"mim: r: ",nrows(mim)," c: ",ncols(mim); |
---|
1200 | //"mim1: r: ",nrows(mim1)," c: ",ncols(mim1); |
---|
1201 | //"mim2: r: ",nrows(mim2)," c: ",ncols(mim2); |
---|
1202 | //matrix mimmat = mim; |
---|
1203 | //matrix mimmat1[16][4]=mimmat[1..16,25..28]; |
---|
1204 | //print(mimmat1-matrix(mim2)); |
---|
1205 | return(multiDegModulo(mker,mim)); |
---|
1206 | //return(0); |
---|
1207 | } |
---|
1208 | return(0); |
---|
1209 | } |
---|
1210 | example |
---|
1211 | { |
---|
1212 | "EXAMPLE: ";echo=2; |
---|
1213 | LIB "homolog.lib"; |
---|
1214 | ring r = 0,(x_(1..4)),dp; |
---|
1215 | intmat g[2][4]=1,1,0,0,0,1,1,-1; |
---|
1216 | setBaseMultigrading(g); |
---|
1217 | ideal i = maxideal(1); |
---|
1218 | module m = multiDegSyzygy(i); |
---|
1219 | module rt = Tor(2,m,m); |
---|
1220 | module multiDegT = multiDegTor(2,m,m); |
---|
1221 | print(matrix(rt)-matrix(multiDegT)); |
---|
1222 | /* |
---|
1223 | ring r = 0,(x),dp; |
---|
1224 | intmat g[2][1]=1,1; |
---|
1225 | setBaseMultigrading(g); |
---|
1226 | matrix m[5][3]=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15; |
---|
1227 | matrix n[3][2]=x,x2,x3,x4,x5,x6; |
---|
1228 | module mm = m; |
---|
1229 | module nn = n; |
---|
1230 | intmat gm[2][5]=1,1,1,1,1,1,1,1,1,1,1; |
---|
1231 | intmat gn[2][3]=0,-2,-4,0,-2,-4; |
---|
1232 | mm = setModuleGrading(mm, gm); |
---|
1233 | nn = setModuleGrading(nn, gn); |
---|
1234 | isHomogeneous(mm,"checkGens"); |
---|
1235 | isHomogeneous(nn,"checkGens"); |
---|
1236 | multiDegTor(1,mm, nn); |
---|
1237 | |
---|
1238 | kill m, mm, n, nn, gm, gn; |
---|
1239 | |
---|
1240 | 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; |
---|
1241 | matrix n[2][4] = 1, 2, 3, 4, x, x2, x3, x4; |
---|
1242 | module mm = m; |
---|
1243 | module nn = n; |
---|
1244 | print(mm); |
---|
1245 | print(nn); |
---|
1246 | intmat gm[2][7] = 1, 2, 3, 4, 5, 6, 7, 0, 0, 0, 0, 0, 0, 0; |
---|
1247 | intmat gn[2][2] = 0, 0, 1, 2; |
---|
1248 | mm = setModuleGrading(mm, gm); |
---|
1249 | nn = setModuleGrading(nn, gn); |
---|
1250 | module mmtnn = multiDegTensor(mm, nn); |
---|
1251 | */ |
---|
1252 | } |
---|
1253 | |
---|
1254 | |
---|
1255 | /******************************************************/ |
---|
1256 | proc isGroupHomomorphism(def L1, def L2, intmat A) |
---|
1257 | "USAGE: gisGoupHomomorphism(L1,L2,A); L1 and L2 are groups, A is an integer matrix |
---|
1258 | PURPOSE: checks whether A defines a group homomorphism phi: L1 --> L2 |
---|
1259 | RETURN: int, 1 if A defines the homomorphism and 0 otherwise |
---|
1260 | EXAMPLE: example isGroupHomomorphism; shows an example |
---|
1261 | " |
---|
1262 | { |
---|
1263 | // TODO: L1, L2 |
---|
1264 | if( (ncols(A) != nrows(L1)) or (nrows(A) != nrows(L2)) ) |
---|
1265 | { |
---|
1266 | ERROR("Incompatible sizes!"); |
---|
1267 | } |
---|
1268 | |
---|
1269 | intmat im = A * L1; |
---|
1270 | |
---|
1271 | return (areZeroElements(im, L2)); |
---|
1272 | } |
---|
1273 | example |
---|
1274 | { |
---|
1275 | "EXAMPLE:"; echo=2; |
---|
1276 | |
---|
1277 | intmat L1[4][1]= |
---|
1278 | 0, |
---|
1279 | 0, |
---|
1280 | 0, |
---|
1281 | 2; |
---|
1282 | |
---|
1283 | intmat L2[3][2]= |
---|
1284 | 0, 0, |
---|
1285 | 2, 0, |
---|
1286 | 0, 3; |
---|
1287 | |
---|
1288 | intmat A[3][4] = |
---|
1289 | 1, 2, 3, 0, |
---|
1290 | 7, 0, 0, 0, |
---|
1291 | 1, 2, 0, 3; |
---|
1292 | print( A ); |
---|
1293 | |
---|
1294 | isGroupHomomorphism(L1, L2, A); |
---|
1295 | |
---|
1296 | intmat B[3][4] = |
---|
1297 | 1, 2, 3, 0, |
---|
1298 | 7, 0, 0, 0, |
---|
1299 | 1, 2, 0, 2; |
---|
1300 | print( B ); |
---|
1301 | |
---|
1302 | isGroupHomomorphism(L1, L2, B); // Not a homomorphism! |
---|
1303 | } |
---|
1304 | |
---|
1305 | /******************************************************/ |
---|
1306 | proc isTorsionFree() |
---|
1307 | "USAGE: isTorsionFree() |
---|
1308 | PURPOSE: Determines whether the multigrading attached to the current ring is free. |
---|
1309 | RETURN: boolean, the result of the test |
---|
1310 | EXAMPLE: example isTorsionFree; shows an example |
---|
1311 | " |
---|
1312 | { |
---|
1313 | intmat H = smithNormalForm(getLattice()); // TODO: ?cache it? //****** |
---|
1314 | |
---|
1315 | int i, j; |
---|
1316 | int r = nrows(H); |
---|
1317 | int c = ncols(H); |
---|
1318 | int d = 1; |
---|
1319 | for( i = 1; (i <= c) && (i <= r); i++ ) |
---|
1320 | { |
---|
1321 | for( j = i; (H[j, i] == 0)&&(j < r); j++ ) |
---|
1322 | { |
---|
1323 | } |
---|
1324 | |
---|
1325 | if(H[j, i]!=0) |
---|
1326 | { |
---|
1327 | d=d*H[j, i]; |
---|
1328 | } |
---|
1329 | } |
---|
1330 | |
---|
1331 | if( (d*d)==1 ) |
---|
1332 | { |
---|
1333 | return(1==1); |
---|
1334 | } |
---|
1335 | return(0==1); |
---|
1336 | } |
---|
1337 | example |
---|
1338 | { |
---|
1339 | "EXAMPLE:"; echo=2; |
---|
1340 | |
---|
1341 | ring R = 0,(x,y),dp; |
---|
1342 | intmat M[2][2]= |
---|
1343 | 1,0, |
---|
1344 | 0,1; |
---|
1345 | intmat T[2][5]= |
---|
1346 | 1, 2, 3, 4, 0, |
---|
1347 | 0,10,20,30, 1; |
---|
1348 | |
---|
1349 | setBaseMultigrading(M,T); |
---|
1350 | |
---|
1351 | // Is the resulting group free? |
---|
1352 | isTorsionFree(); |
---|
1353 | |
---|
1354 | kill R, M, T; |
---|
1355 | /////////////////////////////////////////// |
---|
1356 | |
---|
1357 | ring R=0,(x,y,z),dp; |
---|
1358 | intmat A[3][3] = |
---|
1359 | 1,0,0, |
---|
1360 | 0,1,0, |
---|
1361 | 0,0,1; |
---|
1362 | intmat B[3][4]= |
---|
1363 | 3,3,3,3, |
---|
1364 | 2,1,3,0, |
---|
1365 | 1,2,0,3; |
---|
1366 | setBaseMultigrading(A,B); |
---|
1367 | // Is the resulting group free? |
---|
1368 | isTorsionFree(); |
---|
1369 | |
---|
1370 | kill R, A, B; |
---|
1371 | } |
---|
1372 | |
---|
1373 | |
---|
1374 | static proc gcdcomb(int a, int b) |
---|
1375 | { |
---|
1376 | // a; |
---|
1377 | // b; |
---|
1378 | intvec av = a,1,0; |
---|
1379 | intvec bv = b,0,1; |
---|
1380 | intvec save; |
---|
1381 | while(av[1]*bv[1] != 0) |
---|
1382 | { |
---|
1383 | bv = bv - (bv[1] - bv[1]%av[1])/av[1] * av; |
---|
1384 | save = bv; |
---|
1385 | bv = av; |
---|
1386 | av = save; |
---|
1387 | } |
---|
1388 | if(bv[1] < 0) |
---|
1389 | { |
---|
1390 | bv = -bv; |
---|
1391 | } |
---|
1392 | return(bv); |
---|
1393 | } |
---|
1394 | |
---|
1395 | |
---|
1396 | proc lll(def A) |
---|
1397 | " |
---|
1398 | The lll algorithm of lll.lib only works for lists of vectors. |
---|
1399 | Maybe one should rescript it for matrices. This method will |
---|
1400 | convert a matrix to a list, plug it into lll and make the result |
---|
1401 | a matrix and return it. |
---|
1402 | " |
---|
1403 | { |
---|
1404 | if(typeof(A) == "list") |
---|
1405 | { |
---|
1406 | int sizeA= size (A); |
---|
1407 | if (sizeA == 0) |
---|
1408 | { |
---|
1409 | return (A); |
---|
1410 | } |
---|
1411 | if (typeof (A [1]) != "intvec") |
---|
1412 | { |
---|
1413 | ERROR("Unrecognized type."); |
---|
1414 | } |
---|
1415 | int columns= size (A [1]); |
---|
1416 | int i; |
---|
1417 | for (i= 2; i <= sizeA; i++) |
---|
1418 | { |
---|
1419 | if (typeof (A[i]) != "intvec") |
---|
1420 | { |
---|
1421 | ERROR("Unrecognized type."); |
---|
1422 | } |
---|
1423 | if (size (A [i]) != columns) |
---|
1424 | { |
---|
1425 | ERROR ("expected equal dimension"); |
---|
1426 | } |
---|
1427 | } |
---|
1428 | int j; |
---|
1429 | intmat m [columns] [sizeA]; |
---|
1430 | for (i= 1; i <= sizeA; i++) |
---|
1431 | { |
---|
1432 | for (j= 1; j <= columns; j++) |
---|
1433 | { |
---|
1434 | m[i,j]= A[i] [j]; |
---|
1435 | } |
---|
1436 | } |
---|
1437 | m= system ("LLL", m); |
---|
1438 | list result= list(); |
---|
1439 | intvec buf; |
---|
1440 | |
---|
1441 | for (i= 1; i <= sizeA; i++) |
---|
1442 | { |
---|
1443 | buf = intvec (m[i , 1..columns]); |
---|
1444 | result= result+ list (buf); |
---|
1445 | |
---|
1446 | } |
---|
1447 | return(result); |
---|
1448 | } |
---|
1449 | else |
---|
1450 | { |
---|
1451 | if(typeof(A) == "intmat") |
---|
1452 | { |
---|
1453 | A= system ("LLL", A); |
---|
1454 | return(A); |
---|
1455 | } |
---|
1456 | else |
---|
1457 | { |
---|
1458 | ERROR("Unrecognized type."); |
---|
1459 | } |
---|
1460 | } |
---|
1461 | } |
---|
1462 | |
---|
1463 | example |
---|
1464 | { |
---|
1465 | |
---|
1466 | "EXAMPLE:"; |
---|
1467 | |
---|
1468 | ring R = 0,x,dp; |
---|
1469 | 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; |
---|
1470 | lll(m); |
---|
1471 | 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); |
---|
1472 | lll(l); |
---|
1473 | } |
---|
1474 | |
---|
1475 | |
---|
1476 | proc smithNormalForm(intmat A, list #) |
---|
1477 | " |
---|
1478 | This method returns 3 Matrices P, D and Q such that D = P*A*Q. |
---|
1479 | WARNING: This might not be what you expect. |
---|
1480 | " |
---|
1481 | { |
---|
1482 | list l1 = hermiteNormalForm(A, 5); |
---|
1483 | // l1; |
---|
1484 | intmat B = transpose(l1[1]); |
---|
1485 | list l2 = hermiteNormalForm(B, 5); |
---|
1486 | // l2; |
---|
1487 | intmat P = transpose(l2[2]); |
---|
1488 | intmat D = transpose(l2[1]); |
---|
1489 | intmat Q = l1[2]; |
---|
1490 | int cc = ncols(D); |
---|
1491 | int rr = nrows(D); |
---|
1492 | intmat transform; |
---|
1493 | int k = 1; |
---|
1494 | int a, b, c; |
---|
1495 | // D; |
---|
1496 | intvec v; |
---|
1497 | if((cc==1)||(rr==1)){ |
---|
1498 | if(size(#)==0) |
---|
1499 | { |
---|
1500 | return(D); |
---|
1501 | } else |
---|
1502 | { |
---|
1503 | return(list(P,D,Q)); |
---|
1504 | } |
---|
1505 | } |
---|
1506 | while(D[k+1,k+1] !=0){ |
---|
1507 | if(D[k+1,k+1]%D[k,k]!=0){ |
---|
1508 | b = D[k, k]; c = D[k+1, k+1]; |
---|
1509 | v = gcdcomb(D[k,k],D[k+1,k+1]); |
---|
1510 | transform = unitMatrix(cc); |
---|
1511 | transform[k+1,k] = 1; |
---|
1512 | a = -v[3]*D[k+1,k+1]/v[1]; |
---|
1513 | transform[k, k+1] = a; |
---|
1514 | transform[k+1, k+1] = a+1; |
---|
1515 | //det(transform); |
---|
1516 | D = D*transform; |
---|
1517 | Q = Q*transform; |
---|
1518 | //D; |
---|
1519 | transform = unitMatrix(rr); |
---|
1520 | transform[k,k] = v[2]; |
---|
1521 | transform[k,k+1] = v[3]; |
---|
1522 | transform[k+1,k] = -c/v[1]; |
---|
1523 | transform[k+1,k+1] = b/v[1]; |
---|
1524 | D = transform * D; |
---|
1525 | P = transform * P; |
---|
1526 | //" "; |
---|
1527 | //D; |
---|
1528 | //"small transform: ", det(transform); |
---|
1529 | //transform; |
---|
1530 | k=0; |
---|
1531 | } |
---|
1532 | k++; |
---|
1533 | if((k==rr) || (k==cc)){ |
---|
1534 | break; |
---|
1535 | } |
---|
1536 | } |
---|
1537 | //"here is the size ",size(#); |
---|
1538 | if(size(#) == 0){ |
---|
1539 | return(D); |
---|
1540 | } else { |
---|
1541 | return(list(P, D, Q)); |
---|
1542 | } |
---|
1543 | } |
---|
1544 | example |
---|
1545 | { |
---|
1546 | "EXAMPLE: "; echo=2; |
---|
1547 | |
---|
1548 | intmat A[5][7] = |
---|
1549 | 1,0,1,0,-2,9,-71, |
---|
1550 | 0,-24,248,-32,-96,448,-3496, |
---|
1551 | 0,4,-42,4,-8,30,-260, |
---|
1552 | 0,0,0,18,-90,408,-3168, |
---|
1553 | 0,0,0,-32,224,-1008,7872; |
---|
1554 | |
---|
1555 | list l = smithNormalForm(A, 5); |
---|
1556 | |
---|
1557 | l; |
---|
1558 | l[1]*A*l[3]; |
---|
1559 | det(l[1]); |
---|
1560 | det(l[3]); |
---|
1561 | } |
---|
1562 | |
---|
1563 | |
---|
1564 | /******************************************************/ |
---|
1565 | proc hermiteNormalForm(intmat A, list #) |
---|
1566 | "USAGE: hermiteNormalForm( A ); |
---|
1567 | PURPOSE: Computes the (lower triangular) Hermite Normal Form |
---|
1568 | of the matrix A by column operations. |
---|
1569 | RETURN: intmat, the Hermite Normal Form of A |
---|
1570 | EXAMPLE: example hermiteNormalForm; shows an example |
---|
1571 | " |
---|
1572 | { |
---|
1573 | |
---|
1574 | int row, column, i, j; |
---|
1575 | int rr = nrows(A); |
---|
1576 | int cc = ncols(A); |
---|
1577 | intvec savev, gcdvec, v1, v2; |
---|
1578 | intmat q = unitMatrix(cc); |
---|
1579 | intmat transform; |
---|
1580 | column = 1; |
---|
1581 | for(row = 1; (row<=rr)&&(column<=cc); row++) |
---|
1582 | { |
---|
1583 | if(A[row,column]==0) |
---|
1584 | { |
---|
1585 | for(j = column; j<=cc; j++) |
---|
1586 | { |
---|
1587 | if(A[row, j]!=0) |
---|
1588 | { |
---|
1589 | transform = unitMatrix(cc); |
---|
1590 | transform[j,j] = 0; |
---|
1591 | transform[column, column] = 0; |
---|
1592 | transform[column,j] = 1; |
---|
1593 | transform[j,column] = 1; |
---|
1594 | q = q*transform; |
---|
1595 | A = A*transform; |
---|
1596 | break; |
---|
1597 | } |
---|
1598 | } |
---|
1599 | } |
---|
1600 | if(A[row,column] == 0) |
---|
1601 | { |
---|
1602 | row++; |
---|
1603 | continue; |
---|
1604 | } |
---|
1605 | for(j = column+1; j<=cc; j++) |
---|
1606 | { |
---|
1607 | if(A[row, j]!=0) |
---|
1608 | { |
---|
1609 | gcdvec = gcdcomb(A[row,column],A[row,j]); |
---|
1610 | // gcdvec; |
---|
1611 | // typeof(A[1..rr,column]); |
---|
1612 | v1 = A[1..rr,column]; |
---|
1613 | v2 = A[1..rr,j]; |
---|
1614 | transform = unitMatrix(cc); |
---|
1615 | transform[j,j] = v1[row]/gcdvec[1]; |
---|
1616 | transform[column, column] = gcdvec[2]; |
---|
1617 | transform[column,j] = -v2[row]/gcdvec[1]; |
---|
1618 | transform[j,column] = gcdvec[3]; |
---|
1619 | q = q*transform; |
---|
1620 | A = A*transform; |
---|
1621 | // A; |
---|
1622 | } |
---|
1623 | } |
---|
1624 | if(A[row,column]<0) |
---|
1625 | { |
---|
1626 | transform = unitMatrix(cc); |
---|
1627 | transform[column,column] = -1; |
---|
1628 | q = q*transform; |
---|
1629 | A = A*transform; |
---|
1630 | } |
---|
1631 | for( j=1; j<column; j++){ |
---|
1632 | if(A[row, j]!=0){ |
---|
1633 | transform = unitMatrix(cc); |
---|
1634 | transform[column, j] = (-A[row,j]+A[row, j]%A[row, column])/A[row, column]; |
---|
1635 | if(A[row,j]<0){ |
---|
1636 | transform[column,j]=transform[column,j]+1;} |
---|
1637 | q = q*transform; |
---|
1638 | A = A*transform; |
---|
1639 | } |
---|
1640 | } |
---|
1641 | column++; |
---|
1642 | } |
---|
1643 | if(size(#) > 0){ |
---|
1644 | return(list(A, q)); |
---|
1645 | } |
---|
1646 | return(A); |
---|
1647 | } |
---|
1648 | example |
---|
1649 | { |
---|
1650 | "EXAMPLE:"; echo=2; |
---|
1651 | |
---|
1652 | intmat M[2][5] = |
---|
1653 | 1, 2, 3, 4, 0, |
---|
1654 | 0,10,20,30, 1; |
---|
1655 | |
---|
1656 | // Hermite Normal Form of M: |
---|
1657 | print(hermiteNormalForm(M)); |
---|
1658 | |
---|
1659 | intmat T[3][4] = |
---|
1660 | 3,3,3,3, |
---|
1661 | 2,1,3,0, |
---|
1662 | 1,2,0,3; |
---|
1663 | |
---|
1664 | // Hermite Normal Form of T: |
---|
1665 | print(hermiteNormalForm(T)); |
---|
1666 | |
---|
1667 | intmat A[4][5] = |
---|
1668 | 1,2,3,2,2, |
---|
1669 | 1,2,3,4,0, |
---|
1670 | 0,5,4,2,1, |
---|
1671 | 3,2,4,0,2; |
---|
1672 | |
---|
1673 | // Hermite Normal Form of A: |
---|
1674 | print(hermiteNormalForm(A)); |
---|
1675 | } |
---|
1676 | |
---|
1677 | proc areZeroElements(intmat m, list #) |
---|
1678 | "same as isZeroElement but for an integer matrix considered as a collection of columns" |
---|
1679 | { |
---|
1680 | int r = nrows(m); |
---|
1681 | int i = ncols(m); |
---|
1682 | |
---|
1683 | intvec v; |
---|
1684 | |
---|
1685 | for( ; i > 0; i-- ) |
---|
1686 | { |
---|
1687 | v = m[1..r, i]; |
---|
1688 | if( !isZeroElement(v, #) ) |
---|
1689 | { |
---|
1690 | return (0); |
---|
1691 | } |
---|
1692 | } |
---|
1693 | return(1); |
---|
1694 | } |
---|
1695 | |
---|
1696 | example |
---|
1697 | { |
---|
1698 | "EXAMPLE:"; echo=2; |
---|
1699 | |
---|
1700 | ring r = 0,(x,y,z),dp; |
---|
1701 | |
---|
1702 | intmat g[2][3]= |
---|
1703 | 1,0,1, |
---|
1704 | 0,1,1; |
---|
1705 | intmat t[2][1]= |
---|
1706 | -2, |
---|
1707 | 1; |
---|
1708 | |
---|
1709 | intmat tt[2][1]= |
---|
1710 | 1, |
---|
1711 | -1; |
---|
1712 | |
---|
1713 | setBaseMultigrading(g,t); |
---|
1714 | |
---|
1715 | poly a = x10yz; |
---|
1716 | poly b = x8y2z; |
---|
1717 | poly c = x4z2; |
---|
1718 | poly d = y5; |
---|
1719 | poly e = x2y2; |
---|
1720 | poly f = z2; |
---|
1721 | |
---|
1722 | intmat m[5][2]=multiDeg(a)-multiDeg(b),multiDeg(b)-multiDeg(c),multiDeg(c)-multiDeg(d),multiDeg(d)-multiDeg(e),multiDeg(e)-multiDeg(f); |
---|
1723 | m=transpose(m); |
---|
1724 | areZeroElements(m); |
---|
1725 | areZeroElements(m,tt); |
---|
1726 | } |
---|
1727 | |
---|
1728 | |
---|
1729 | /******************************************************/ |
---|
1730 | proc isZeroElement(intvec mdeg, list #) |
---|
1731 | "USAGE: isZeroElement(d, [T]); intvec d, group T |
---|
1732 | PURPOSE: For a integer vector mdeg representing the multidegree of some polynomial |
---|
1733 | or vector this method computes if the multidegree is contained in the grading group |
---|
1734 | group (either set globally or given as an optional argument), i.e. if it is zero in the multigrading. |
---|
1735 | EXAMPLE: example isZeroElement; shows an example |
---|
1736 | " |
---|
1737 | { |
---|
1738 | if( size(#) > 0 ) |
---|
1739 | { |
---|
1740 | if( typeof(#[1]) == "intmat" ) |
---|
1741 | { |
---|
1742 | intmat H = hermiteNormalForm(#[1]); |
---|
1743 | } else |
---|
1744 | { |
---|
1745 | if( typeof(#[1]) == "list" ) |
---|
1746 | { |
---|
1747 | list L = #[1]; |
---|
1748 | intmat H = attrib(L, "hermite"); // todo |
---|
1749 | } |
---|
1750 | } |
---|
1751 | |
---|
1752 | } |
---|
1753 | if( !defined(H) ) |
---|
1754 | { |
---|
1755 | intmat H = getLattice("hermite"); |
---|
1756 | } |
---|
1757 | |
---|
1758 | int x, k, i, row; |
---|
1759 | |
---|
1760 | int r = nrows(H); |
---|
1761 | int c = ncols(H); |
---|
1762 | |
---|
1763 | int rr = nrows(mdeg); |
---|
1764 | row = 1; |
---|
1765 | intvec v; |
---|
1766 | for(i=1; (i<=r)&&(row<=r)&&(i<=c); i++) |
---|
1767 | { |
---|
1768 | while((H[row,i]==0)&&(row<=r)) |
---|
1769 | { |
---|
1770 | row++; |
---|
1771 | if(row == (r+1)){ |
---|
1772 | break; |
---|
1773 | } |
---|
1774 | } |
---|
1775 | if(row<=r){ |
---|
1776 | if(H[row,i]!=0) |
---|
1777 | { |
---|
1778 | v = H[1..r,i]; |
---|
1779 | mdeg = mdeg-(mdeg[row]-mdeg[row]%v[row])/v[row]*v; |
---|
1780 | } |
---|
1781 | } |
---|
1782 | } |
---|
1783 | return( mdeg == 0 ); |
---|
1784 | |
---|
1785 | } |
---|
1786 | example |
---|
1787 | { |
---|
1788 | "EXAMPLE:"; echo=2; |
---|
1789 | |
---|
1790 | ring r = 0,(x,y,z),dp; |
---|
1791 | |
---|
1792 | intmat g[2][3]= |
---|
1793 | 1,0,1, |
---|
1794 | 0,1,1; |
---|
1795 | intmat t[2][1]= |
---|
1796 | -2, |
---|
1797 | 1; |
---|
1798 | |
---|
1799 | intmat tt[2][1]= |
---|
1800 | 1, |
---|
1801 | -1; |
---|
1802 | |
---|
1803 | setBaseMultigrading(g,t); |
---|
1804 | |
---|
1805 | poly a = x10yz; |
---|
1806 | poly b = x8y2z; |
---|
1807 | poly c = x4z2; |
---|
1808 | poly d = y5; |
---|
1809 | poly e = x2y2; |
---|
1810 | poly f = z2; |
---|
1811 | |
---|
1812 | intvec v1 = multiDeg(a) - multiDeg(b); |
---|
1813 | v1; |
---|
1814 | isZeroElement(v1); |
---|
1815 | isZeroElement(v1, tt); |
---|
1816 | |
---|
1817 | intvec v2 = multiDeg(a) - multiDeg(c); |
---|
1818 | v2; |
---|
1819 | isZeroElement(v2); |
---|
1820 | isZeroElement(v2, tt); |
---|
1821 | |
---|
1822 | intvec v3 = multiDeg(e) - multiDeg(f); |
---|
1823 | v3; |
---|
1824 | isZeroElement(v3); |
---|
1825 | isZeroElement(v3, tt); |
---|
1826 | |
---|
1827 | intvec v4 = multiDeg(c) - multiDeg(d); |
---|
1828 | v4; |
---|
1829 | isZeroElement(v4); |
---|
1830 | isZeroElement(v4, tt); |
---|
1831 | } |
---|
1832 | |
---|
1833 | |
---|
1834 | /******************************************************/ |
---|
1835 | proc defineHomogeneous(poly f, list #) |
---|
1836 | "USAGE: defineHomogeneous(f[, G]); polynomial f, integer matrix G |
---|
1837 | PURPOSE: Yields a matrix which has to be appended to the grading group matrix to make the |
---|
1838 | polynomial f homogeneous in the grading by grad. |
---|
1839 | EXAMPLE: example defineHomogeneous; shows an example |
---|
1840 | " |
---|
1841 | { |
---|
1842 | if( size(#) > 0 ) |
---|
1843 | { |
---|
1844 | if( typeof(#[1]) == "intmat" ) |
---|
1845 | { |
---|
1846 | intmat grad = #[1]; |
---|
1847 | } |
---|
1848 | } |
---|
1849 | |
---|
1850 | if( !defined(grad) ) |
---|
1851 | { |
---|
1852 | intmat grad = getVariableWeights(); |
---|
1853 | } |
---|
1854 | |
---|
1855 | intmat newgg[nrows(grad)][size(f)-1]; |
---|
1856 | int i,j; |
---|
1857 | intvec l = grad*leadexp(f); |
---|
1858 | intvec v; |
---|
1859 | for(i=2; i <= size(f); i++) |
---|
1860 | { |
---|
1861 | v = grad * leadexp(f[i]) - l; |
---|
1862 | for( j=1; j<=size(v); j++) |
---|
1863 | { |
---|
1864 | newgg[j,i-1] = v[j]; |
---|
1865 | } |
---|
1866 | } |
---|
1867 | return(newgg); |
---|
1868 | } |
---|
1869 | example |
---|
1870 | { |
---|
1871 | "EXAMPLE:"; echo=2; |
---|
1872 | |
---|
1873 | ring r =0,(x,y,z),dp; |
---|
1874 | intmat grad[2][3] = |
---|
1875 | 1,0,1, |
---|
1876 | 0,1,1; |
---|
1877 | |
---|
1878 | setBaseMultigrading(grad); |
---|
1879 | |
---|
1880 | poly f = x2y3-z5+x-3zx; |
---|
1881 | |
---|
1882 | intmat M = defineHomogeneous(f); |
---|
1883 | M; |
---|
1884 | defineHomogeneous(f, grad) == M; |
---|
1885 | |
---|
1886 | isHomogeneous(f); |
---|
1887 | setBaseMultigrading(grad, M); |
---|
1888 | isHomogeneous(f); |
---|
1889 | } |
---|
1890 | |
---|
1891 | |
---|
1892 | proc gradiator(def h) |
---|
1893 | PURPOSE: coarsens the grading of the basering until the polynom or ideal h becomes homogeneous. |
---|
1894 | |
---|
1895 | { |
---|
1896 | if(typeof(h)=="poly"){ |
---|
1897 | intmat W = getVariableWeights(); |
---|
1898 | intmat L = getLattice(); |
---|
1899 | intmat toadd = defineHomogeneous(h); |
---|
1900 | //h; |
---|
1901 | //toadd; |
---|
1902 | if(ncols(toadd) == 0) |
---|
1903 | { |
---|
1904 | return(1==1); |
---|
1905 | } |
---|
1906 | int rr = nrows(W); |
---|
1907 | intmat newL[rr][ncols(L)+ncols(toadd)]; |
---|
1908 | newL[1..rr,1..ncols(L)] = L[1..rr,1..ncols(L)]; |
---|
1909 | newL[1..rr,(ncols(L)+1)..(ncols(L)+ncols(toadd))] = toadd[1..rr,1..ncols(toadd)]; |
---|
1910 | setBaseMultigrading(W,newL); |
---|
1911 | return(1==1); |
---|
1912 | } |
---|
1913 | if(typeof(h)=="ideal"){ |
---|
1914 | int i; |
---|
1915 | def s = (1==1); |
---|
1916 | for(i=1;i<=size(h);i++){ |
---|
1917 | s = s && gradiator(h[i]); |
---|
1918 | } |
---|
1919 | return(s); |
---|
1920 | } |
---|
1921 | return(1==0); |
---|
1922 | } |
---|
1923 | example |
---|
1924 | { |
---|
1925 | "EXAMPLE:"; echo=2; |
---|
1926 | ring r = 0,(x,y,z),dp; |
---|
1927 | intmat g[2][3] = 1,0,1,0,1,1; |
---|
1928 | intmat l[2][1] = 3,0; |
---|
1929 | |
---|
1930 | setBaseMultigrading(g,l); |
---|
1931 | |
---|
1932 | getLattice(); |
---|
1933 | |
---|
1934 | ideal i = -y5+x4, |
---|
1935 | y6+xz, |
---|
1936 | x2y; |
---|
1937 | gradiator(i); |
---|
1938 | getLattice(); |
---|
1939 | isHomogeneous(i); |
---|
1940 | } |
---|
1941 | |
---|
1942 | |
---|
1943 | proc pushForward(map f) |
---|
1944 | "USAGE: pushForward(f); |
---|
1945 | PURPOSE: Computes the finest grading of the image ring which makes the map f |
---|
1946 | a map of graded rings. The group map between the two grading groups is given |
---|
1947 | by transpose( (Id, 0) ). Pay attention that the group spanned by the columns of |
---|
1948 | the grading group matrix may not be a subgroup of the grading group. Still all columns |
---|
1949 | are needed to find the correct image of the preimage gradings. |
---|
1950 | EXAMPLE: example pushForward; shows an example |
---|
1951 | " |
---|
1952 | { |
---|
1953 | |
---|
1954 | int k,i,j; |
---|
1955 | // f; |
---|
1956 | |
---|
1957 | // listvar(); |
---|
1958 | def pre = preimage(f); |
---|
1959 | |
---|
1960 | // "pre: "; pre; |
---|
1961 | |
---|
1962 | intmat oldgrad=getVariableWeights(pre); |
---|
1963 | intmat oldtor=getLattice(pre); |
---|
1964 | |
---|
1965 | int n=nvars(pre); |
---|
1966 | int np=nvars(basering); |
---|
1967 | int p=nrows(oldgrad); |
---|
1968 | int pp=p+np; |
---|
1969 | |
---|
1970 | intmat newgrad[pp][np]; |
---|
1971 | |
---|
1972 | for(i=1;i<=np;i++){ newgrad[p+i,i]=1;} |
---|
1973 | |
---|
1974 | //newgrad; |
---|
1975 | |
---|
1976 | |
---|
1977 | |
---|
1978 | list newtor; |
---|
1979 | intmat toadd; |
---|
1980 | int columns=0; |
---|
1981 | |
---|
1982 | intmat toadd1[pp][n]; |
---|
1983 | intvec v; |
---|
1984 | poly im; |
---|
1985 | |
---|
1986 | for(i=1;i<=p;i++){ |
---|
1987 | for(j=1;j<=n;j++){ toadd1[i,j]=oldgrad[i,j];} |
---|
1988 | } |
---|
1989 | |
---|
1990 | for(i=1;i<=n;i++){ |
---|
1991 | im=f[i]; |
---|
1992 | //im; |
---|
1993 | toadd = defineHomogeneous(im, newgrad); |
---|
1994 | newtor=insert(newtor,toadd); |
---|
1995 | columns=columns+ncols(toadd); |
---|
1996 | |
---|
1997 | v=leadexp(f[i]); |
---|
1998 | for(j=p+1;j<=p+np;j++){ toadd1[j,i]=-v[j-p];} |
---|
1999 | } |
---|
2000 | |
---|
2001 | newtor=insert(newtor,toadd1); |
---|
2002 | columns=columns+ncols(toadd1); |
---|
2003 | |
---|
2004 | |
---|
2005 | if(typeof(basering)=="qring"){ |
---|
2006 | //"Entering qring"; |
---|
2007 | ideal a=ideal(basering); |
---|
2008 | for(i=1;i<=size(a);i++){ |
---|
2009 | toadd = defineHomogeneous(a[i], newgrad); |
---|
2010 | //toadd; |
---|
2011 | columns=columns+ncols(toadd); |
---|
2012 | newtor=insert(newtor,toadd); |
---|
2013 | } |
---|
2014 | } |
---|
2015 | |
---|
2016 | //newtor; |
---|
2017 | intmat imofoldtor[pp][ncols(oldtor)]; |
---|
2018 | for(i=1; i<=nrows(oldtor);i++){ |
---|
2019 | for(j=1; j<=ncols(oldtor); j++){ |
---|
2020 | imofoldtor[i,j]=oldtor[i,j]; |
---|
2021 | } |
---|
2022 | } |
---|
2023 | |
---|
2024 | columns=columns+ncols(oldtor); |
---|
2025 | newtor=insert(newtor, imofoldtor); |
---|
2026 | |
---|
2027 | intmat gragr[pp][columns]; |
---|
2028 | columns=0; |
---|
2029 | for(k=1;k<=size(newtor);k++){ |
---|
2030 | for(i=1;i<=pp;i++){ |
---|
2031 | for(j=1;j<=ncols(newtor[k]);j++){gragr[i,j+columns]=newtor[k][i,j];} |
---|
2032 | } |
---|
2033 | columns=columns+ncols(newtor[k]); |
---|
2034 | } |
---|
2035 | |
---|
2036 | gragr=hermiteNormalForm(gragr); |
---|
2037 | intmat result[pp][pp]; |
---|
2038 | for(i=1;i<=pp;i++){ |
---|
2039 | for(j=1;j<=pp;j++){result[i,j]=gragr[i,j];} |
---|
2040 | } |
---|
2041 | |
---|
2042 | setBaseMultigrading(newgrad, result); |
---|
2043 | |
---|
2044 | } |
---|
2045 | example |
---|
2046 | { |
---|
2047 | "EXAMPLE:"; echo=2; |
---|
2048 | |
---|
2049 | ring r = 0,(x,y,z),dp; |
---|
2050 | |
---|
2051 | |
---|
2052 | |
---|
2053 | // Setting degrees for preimage ring.; |
---|
2054 | intmat grad[3][3] = |
---|
2055 | 1,0,0, |
---|
2056 | 0,1,0, |
---|
2057 | 0,0,1; |
---|
2058 | |
---|
2059 | setBaseMultigrading(grad); |
---|
2060 | |
---|
2061 | // grading on r: |
---|
2062 | getVariableWeights(); |
---|
2063 | getLattice(); |
---|
2064 | |
---|
2065 | // only for the purpose of this example |
---|
2066 | if( voice > 1 ){ /*keepring(r);*/ export(r); } |
---|
2067 | |
---|
2068 | ring R = 0,(a,b),dp; |
---|
2069 | ideal i = a2-b2+a6-b5+ab3,a7b+b15-ab6+a6b6; |
---|
2070 | |
---|
2071 | // The quotient ring by this ideal will become our image ring.; |
---|
2072 | qring Q = std(i); |
---|
2073 | |
---|
2074 | listvar(); |
---|
2075 | |
---|
2076 | map f = r,-a2b6+b5+a3b+a2+ab,-a2b7-3a2b5+b4+a,a6-b6-b3+a2; f; |
---|
2077 | |
---|
2078 | |
---|
2079 | // TODO: Unfortunately this is not a very spectacular example...: |
---|
2080 | // Pushing forward f: |
---|
2081 | pushForward(f); |
---|
2082 | |
---|
2083 | // due to pushForward we have got new grading on Q |
---|
2084 | getVariableWeights(); |
---|
2085 | getLattice(); |
---|
2086 | |
---|
2087 | |
---|
2088 | // only for the purpose of this example |
---|
2089 | if( voice > 1 ){ kill r; } |
---|
2090 | |
---|
2091 | } |
---|
2092 | |
---|
2093 | |
---|
2094 | /******************************************************/ |
---|
2095 | proc equalMultiDeg(intvec exp1, intvec exp2, list #) |
---|
2096 | "USAGE: equalMultiDeg(exp1, exp2[, V]); intvec exp1, exp2, intmat V |
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2097 | PURPOSE: Tests if the exponent vectors of two monomials (given by exp1 and exp2) |
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2098 | represent the same multidegree. |
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2099 | NOTE: the integer matrix V encodes multidegrees of module components, |
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2100 | if module component is present in exp1 and exp2 |
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2101 | EXAMPLE: example equalMultiDeg; shows an example |
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2102 | " |
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2103 | { |
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2104 | if( size(exp1) != size(exp2) ) |
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2105 | { |
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2106 | ERROR("Sorry: we cannot compare exponents comming from a polynomial and a vector yet!"); |
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2107 | } |
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2108 | |
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2109 | if( exp1 == exp2) |
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2110 | { |
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2111 | return (1==1); |
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2112 | } |
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2113 | |
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2114 | |
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2115 | |
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2116 | intmat M = getVariableWeights(); |
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2117 | |
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2118 | if( nrows(exp1) > ncols(M) ) // vectors => last exponent is the module component! |
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2119 | { |
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2120 | if( (size(#) == 0) or (typeof(#[1])!="intmat") ) |
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2121 | { |
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2122 | ERROR("Sorry: wrong or missing module-unit-weights-matrix V!"); |
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2123 | } |
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2124 | intmat V = #[1]; |
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2125 | |
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2126 | // typeof(V); print(V); |
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2127 | |
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2128 | int N = ncols(M); |
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2129 | int r = nrows(M); |
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2130 | |
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2131 | intvec d = intvec(exp1[1..N]) - intvec(exp2[1..N]); |
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2132 | intvec dm = intvec(V[1..r, exp1[N+1]]) - intvec(V[1..r, exp2[N+1]]); |
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2133 | |
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2134 | intvec difference = M * d + dm; |
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2135 | } |
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2136 | else |
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2137 | { |
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2138 | intvec d = (exp1 - exp2); |
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2139 | intvec difference = M * d; |
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2140 | } |
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2141 | |
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2142 | if (isFreeRepresented()) // no grading group!? |
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2143 | { |
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2144 | return ( difference == 0); |
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2145 | } |
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2146 | return ( isZeroElement( difference ) ); |
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2147 | } |
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2148 | example |
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2149 | { |
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2150 | "EXAMPLE:"; echo=2;printlevel=3; |
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2151 | |
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2152 | ring r = 0,(x,y,z),dp; |
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2153 | |
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2154 | intmat g[2][3]= |
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2155 | 1,0,1, |
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2156 | 0,1,1; |
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2157 | |
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2158 | intmat t[2][1]= |
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2159 | -2, |
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2160 | 1; |
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2161 | |
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2162 | setBaseMultigrading(g,t); |
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2163 | |
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2164 | poly a = x10yz; |
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2165 | poly b = x8y2z; |
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2166 | poly c = x4z2; |
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2167 | poly d = y5; |
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2168 | poly e = x2y2; |
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2169 | poly f = z2; |
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2170 | |
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2171 | |
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2172 | equalMultiDeg(leadexp(a), leadexp(b)); |
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2173 | equalMultiDeg(leadexp(a), leadexp(c)); |
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2174 | equalMultiDeg(leadexp(a), leadexp(d)); |
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2175 | equalMultiDeg(leadexp(a), leadexp(e)); |
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2176 | equalMultiDeg(leadexp(a), leadexp(f)); |
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2177 | |
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2178 | equalMultiDeg(leadexp(b), leadexp(c)); |
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2179 | equalMultiDeg(leadexp(b), leadexp(d)); |
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2180 | equalMultiDeg(leadexp(b), leadexp(e)); |
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2181 | equalMultiDeg(leadexp(b), leadexp(f)); |
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2182 | |
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2183 | equalMultiDeg(leadexp(c), leadexp(d)); |
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2184 | equalMultiDeg(leadexp(c), leadexp(e)); |
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2185 | equalMultiDeg(leadexp(c), leadexp(f)); |
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2186 | |
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2187 | equalMultiDeg(leadexp(d), leadexp(e)); |
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2188 | equalMultiDeg(leadexp(d), leadexp(f)); |
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2189 | |
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2190 | equalMultiDeg(leadexp(e), leadexp(f)); |
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2191 | |
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2192 | } |
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2193 | |
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2194 | |
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2195 | |
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2196 | /******************************************************/ |
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2197 | static proc isFreeRepresented() |
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2198 | "check whether the base muligrading is free (it is zero). |
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2199 | " |
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2200 | { |
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2201 | intmat T = getLattice(); |
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2202 | |
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2203 | intmat Z[nrows(T)][ncols(T)]; |
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2204 | |
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2205 | return (T == Z); // no grading group! |
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2206 | } |
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2207 | |
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2208 | |
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2209 | /******************************************************/ |
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2210 | proc isHomogeneous(def a, list #) |
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2211 | "USAGE: isHomogeneous(a[, f]); a polynomial/vector/ideal/module |
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2212 | RETURN: boolean, TRUE if a is (multi)homogeneous, and FALSE otherwise |
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2213 | EXAMPLE: example isHomogeneous; shows an example |
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2214 | " |
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2215 | { |
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2216 | if( (typeof(a) == "poly") or (typeof(a) == "vector") ) |
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2217 | { |
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2218 | return ( size(multiDegPartition(a)) <= 1 ) |
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2219 | } |
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2220 | |
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2221 | if( (typeof(a) == "ideal") or (typeof(a) == "module") ) |
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2222 | { |
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2223 | if(size(#) > 0) |
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2224 | { |
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2225 | if (#[1] == "checkGens") |
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2226 | { |
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2227 | def aa; |
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2228 | for( int i = ncols(a); i > 0; i-- ) |
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2229 | { |
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2230 | aa = getGradedGenerator(a, i); |
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2231 | |
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2232 | if(!isHomogeneous(aa)) |
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2233 | { |
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2234 | return(0==1); |
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2235 | } |
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2236 | } |
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2237 | return(1==1); |
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2238 | } |
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2239 | } |
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2240 | |
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2241 | def g = groebner(a); // !!!! |
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2242 | |
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2243 | def b, aa; int j; |
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2244 | for( int i = ncols(a); i > 0; i-- ) |
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2245 | { |
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2246 | aa = getGradedGenerator(a, i); |
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2247 | |
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2248 | b = multiDegPartition(aa); |
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2249 | for( j = ncols(b); j > 0; j-- ) |
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2250 | { |
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2251 | if(NF(b[j],g) != 0) |
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2252 | { |
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2253 | return(0==1); |
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2254 | } |
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2255 | } |
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2256 | } |
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2257 | return(1==1); |
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2258 | } |
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2259 | } |
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2260 | example |
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2261 | { |
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2262 | "EXAMPLE:"; echo=2; |
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2263 | |
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2264 | ring r = 0,(x,y,z),dp; |
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2265 | |
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2266 | //Grading and Torsion matrices: |
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2267 | intmat M[3][3] = |
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2268 | 1,0,0, |
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2269 | 0,1,0, |
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2270 | 0,0,1; |
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2271 | |
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2272 | intmat T[3][1] = |
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2273 | 1,2,3; |
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2274 | |
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2275 | setBaseMultigrading(M,T); |
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2276 | |
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2277 | attrib(r); |
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2278 | |
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2279 | poly f = x-yz; |
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2280 | |
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2281 | multiDegPartition(f); |
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2282 | print(multiDeg(_)); |
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2283 | |
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2284 | isHomogeneous(f); // f: is not homogeneous |
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2285 | |
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2286 | poly g = 1-xy2z3; |
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2287 | isHomogeneous(g); // g: is homogeneous |
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2288 | multiDegPartition(g); |
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2289 | |
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2290 | kill T; |
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2291 | ///////////////////////////////////////////////////////// |
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2292 | // new Torsion matrix: |
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2293 | intmat T[3][4] = |
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2294 | 3,3,3,3, |
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2295 | 2,1,3,0, |
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2296 | 1,2,0,3; |
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2297 | |
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2298 | setBaseMultigrading(M,T); |
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2299 | |
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2300 | f; |
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2301 | isHomogeneous(f); |
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2302 | multiDegPartition(f); |
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2303 | |
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2304 | // --------------------- |
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2305 | g; |
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2306 | isHomogeneous(g); |
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2307 | multiDegPartition(g); |
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2308 | |
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2309 | kill r, T, M; |
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2310 | |
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2311 | ring R = 0, (x,y,z), dp; |
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2312 | |
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2313 | intmat A[2][3] = |
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2314 | 0,0,1, |
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2315 | 3,2,1; |
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2316 | intmat T[2][1] = |
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2317 | -1, |
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2318 | 4; |
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2319 | setBaseMultigrading(A, T); |
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2320 | |
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2321 | isHomogeneous(ideal(x2 - y3 -xy +z, x*y-z, x^3 - y^2*z + x^2 -y^3)); // 1 |
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2322 | isHomogeneous(ideal(x2 - y3 -xy +z, x*y-z, x^3 - y^2*z + x^2 -y^3), "checkGens"); |
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2323 | isHomogeneous(ideal(x+y, x2 - y2)); // 0 |
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2324 | |
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2325 | // Degree partition: |
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2326 | multiDegPartition(x2 - y3 -xy +z); |
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2327 | multiDegPartition(x3 -y2z + x2 -y3 + z + 1); |
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2328 | |
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2329 | |
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2330 | module N = gen(1) + (x+y) * gen(2), z*gen(3); |
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2331 | |
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2332 | intmat V[2][3] = 0; // 1, 2, 3, 4, 5, 6; // column-wise weights of components!!?? |
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2333 | |
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2334 | vector v1, v2; |
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2335 | |
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2336 | v1 = setModuleGrading(N[1], V); v1; |
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2337 | multiDegPartition(v1); |
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2338 | print( multiDeg(_) ); |
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2339 | |
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2340 | v2 = setModuleGrading(N[2], V); v2; |
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2341 | multiDegPartition(v2); |
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2342 | print( multiDeg(_) ); |
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2343 | |
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2344 | N = setModuleGrading(N, V); |
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2345 | isHomogeneous(N); |
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2346 | print( multiDeg(N) ); |
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2347 | |
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2348 | /////////////////////////////////////// |
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2349 | |
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2350 | V = |
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2351 | 1, 2, 3, |
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2352 | 4, 5, 6; |
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2353 | |
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2354 | v1 = setModuleGrading(N[1], V); v1; |
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2355 | multiDegPartition(v1); |
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2356 | print( multiDeg(_) ); |
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2357 | |
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2358 | v2 = setModuleGrading(N[2], V); v2; |
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2359 | multiDegPartition(v2); |
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2360 | print( multiDeg(_) ); |
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2361 | |
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2362 | N = setModuleGrading(N, V); |
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2363 | isHomogeneous(N); |
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2364 | print( multiDeg(N) ); |
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2365 | |
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2366 | /////////////////////////////////////// |
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2367 | |
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2368 | V = |
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2369 | 0, 0, 0, |
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2370 | 4, 1, 0; |
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2371 | |
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2372 | N = gen(1) + x * gen(2), z*gen(3); |
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2373 | N = setModuleGrading(N, V); print(N); |
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2374 | isHomogeneous(N); |
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2375 | print( multiDeg(N) ); |
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2376 | v1 = getGradedGenerator(N,1); print(v1); |
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2377 | multiDegPartition(v1); |
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2378 | print( multiDeg(_) ); |
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2379 | N = setModuleGrading(N, V); print(N); |
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2380 | isHomogeneous(N); |
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2381 | print( multiDeg(N) ); |
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2382 | } |
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2383 | |
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2384 | /******************************************************/ |
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2385 | proc multiDeg(def A) |
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2386 | "USAGE: multiDeg(A); polynomial/vector/ideal/module A |
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2387 | PURPOSE: compute multidegree |
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2388 | EXAMPLE: example multiDeg; shows an example |
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2389 | " |
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2390 | { |
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2391 | if( defined(attrib(A, "grad")) > 0 ) |
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2392 | { |
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2393 | return (attrib(A, "grad")); |
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2394 | } |
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2395 | |
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2396 | intmat M = getVariableWeights(); |
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2397 | int N = nvars(basering); |
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2398 | |
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2399 | if( ncols(M) != N ) |
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2400 | { |
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2401 | ERROR("Sorry wrong mgrad-size of M: " + string(ncols(M))); |
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2402 | } |
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2403 | |
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2404 | int r = nrows(M); |
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2405 | |
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2406 | if( (typeof(A) == "vector") or (typeof(A) == "module") ) |
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2407 | { |
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2408 | intmat V = getModuleGrading(A); |
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2409 | |
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2410 | if( nrows(V) != r ) |
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2411 | { |
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2412 | ERROR("Sorry wrong mgrad-size of V: " + string(nrows(V))); |
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2413 | } |
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2414 | } |
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2415 | |
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2416 | if( A == 0 ) |
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2417 | { |
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2418 | intvec v; v[r] = 0; |
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2419 | return (v); |
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2420 | } |
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2421 | |
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2422 | intvec m; m[r] = 0; |
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2423 | |
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2424 | if( typeof(A) == "poly" ) |
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2425 | { |
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2426 | intvec v = leadexp(A); // v; |
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2427 | m = M * v; |
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2428 | |
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2429 | // We assume homogeneous input! |
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2430 | return(m); |
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2431 | |
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2432 | A = A - lead(A); |
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2433 | while( size(A) > 0 ) |
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2434 | { |
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2435 | v = leadexp(A); // v; |
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2436 | m = max( m, M * v, r ); // ???? |
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2437 | A = A - lead(A); |
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2438 | } |
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2439 | |
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2440 | return(m); |
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2441 | } |
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2442 | |
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2443 | |
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2444 | if( typeof(A) == "vector" ) |
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2445 | { |
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2446 | intvec v; |
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2447 | v = leadexp(A); // v; |
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2448 | m = intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]); |
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2449 | |
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2450 | // We assume homogeneous input! |
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2451 | return(m); |
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2452 | |
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2453 | A = A - lead(A); |
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2454 | while( size(A) > 0 ) |
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2455 | { |
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2456 | v = leadexp(A); // v; |
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2457 | |
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2458 | // intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]); |
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2459 | |
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2460 | m = max( m, intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]), r ); // ??? |
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2461 | |
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2462 | A = A - lead(A); |
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2463 | } |
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2464 | |
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2465 | return(m); |
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2466 | } |
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2467 | |
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2468 | int i, j; intvec d; |
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2469 | |
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2470 | if( typeof(A) == "ideal" ) |
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2471 | { |
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2472 | intmat G[ r ] [ ncols(A)]; |
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2473 | for( i = ncols(A); i > 0; i-- ) |
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2474 | { |
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2475 | d = multiDeg( A[i] ); |
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2476 | |
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2477 | for( j = 1; j <= r; j++ ) // see ticket: 253 |
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2478 | { |
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2479 | G[j, i] = d[j]; |
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2480 | } |
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2481 | } |
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2482 | return(G); |
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2483 | } |
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2484 | |
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2485 | if( typeof(A) == "module" ) |
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2486 | { |
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2487 | intmat G[ r ] [ ncols(A)]; |
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2488 | vector v; |
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2489 | |
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2490 | for( i = ncols(A); i > 0; i-- ) |
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2491 | { |
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2492 | v = getGradedGenerator(A, i); |
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2493 | |
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2494 | // G[1..r, i] |
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2495 | d = multiDeg(v); |
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2496 | |
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2497 | for( j = 1; j <= r; j++ ) // see ticket: 253 |
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2498 | { |
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2499 | G[j, i] = d[j]; |
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2500 | } |
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2501 | |
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2502 | } |
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2503 | |
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2504 | return(G); |
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2505 | } |
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2506 | |
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2507 | } |
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2508 | example |
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2509 | { |
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2510 | "EXAMPLE:"; echo=2; |
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2511 | |
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2512 | ring r = 0,(x, y), dp; |
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2513 | |
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2514 | intmat A[2][2] = 1, 0, 0, 1; |
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2515 | print(A); |
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2516 | |
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2517 | intmat Ta[2][1] = 0, 3; |
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2518 | print(Ta); |
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2519 | |
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2520 | // attrib(A, "gradingGroup", Ta); // to think about |
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2521 | |
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2522 | // "poly:"; |
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2523 | setBaseMultigrading(A); |
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2524 | |
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2525 | |
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2526 | multiDeg( x*x, A ); |
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2527 | multiDeg( y*y*y, A ); |
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2528 | |
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2529 | setBaseMultigrading(A, Ta); |
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2530 | |
---|
2531 | multiDeg( x*x*y ); |
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2532 | |
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2533 | multiDeg( y*y*y*x ); |
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2534 | |
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2535 | multiDeg( x*y + x + 1 ); |
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2536 | |
---|
2537 | multiDegPartition(x*y + x + 1); |
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2538 | |
---|
2539 | print ( multiDeg(0) ); |
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2540 | poly zero = 0; |
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2541 | print ( multiDeg(zero) ); |
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2542 | |
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2543 | // "ideal:"; |
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2544 | |
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2545 | ideal I = y*x*x, x*y*y*y; |
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2546 | print( multiDeg(I) ); |
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2547 | |
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2548 | print ( multiDeg(ideal(0)) ); |
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2549 | print ( multiDeg(ideal(0,0,0)) ); |
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2550 | |
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2551 | // "vectors:"; |
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2552 | |
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2553 | intmat B[2][2] = 0, 1, 1, 0; |
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2554 | print(B); |
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2555 | |
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2556 | multiDeg( setModuleGrading(y*y*y*gen(2), B )); |
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2557 | multiDeg( setModuleGrading(x*x*gen(1), B )); |
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2558 | |
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2559 | |
---|
2560 | vector V = x*gen(1) + y*gen(2); |
---|
2561 | V = setModuleGrading(V, B); |
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2562 | multiDeg( V ); |
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2563 | |
---|
2564 | vector v1 = setModuleGrading([0, 0, 0], B); |
---|
2565 | print( multiDeg( v1 ) ); |
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2566 | |
---|
2567 | vector v2 = setModuleGrading([0], B); |
---|
2568 | print( multiDeg( v2 ) ); |
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2569 | |
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2570 | // "module:"; |
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2571 | |
---|
2572 | module D = x*gen(1), y*gen(2); |
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2573 | D; |
---|
2574 | D = setModuleGrading(D, B); |
---|
2575 | print( multiDeg( D ) ); |
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2576 | |
---|
2577 | |
---|
2578 | module DD = [0, 0],[0, 0, 0]; |
---|
2579 | DD = setModuleGrading(DD, B); |
---|
2580 | print( multiDeg( DD ) ); |
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2581 | |
---|
2582 | module DDD = [0, 0]; |
---|
2583 | DDD = setModuleGrading(DDD, B); |
---|
2584 | print( multiDeg( DDD ) ); |
---|
2585 | |
---|
2586 | }; |
---|
2587 | |
---|
2588 | |
---|
2589 | |
---|
2590 | |
---|
2591 | |
---|
2592 | /******************************************************/ |
---|
2593 | proc multiDegPartition(def p) |
---|
2594 | "USAGE: multiDegPartition(def p), p polynomial/vector |
---|
2595 | RETURNS: an ideal/module consisting of multigraded-homogeneous parts of p |
---|
2596 | EXAMPLE: example multiDegPartition; shows an example |
---|
2597 | " |
---|
2598 | { // TODO: What about an ideal or module??? |
---|
2599 | |
---|
2600 | if( typeof(p) == "poly" ) |
---|
2601 | { |
---|
2602 | ideal I; |
---|
2603 | poly mp, t, tt; |
---|
2604 | } |
---|
2605 | else |
---|
2606 | { |
---|
2607 | if( typeof(p) == "vector" ) |
---|
2608 | { |
---|
2609 | module I; |
---|
2610 | vector mp, t, tt; |
---|
2611 | } |
---|
2612 | else |
---|
2613 | { |
---|
2614 | ERROR("Wrong ARGUMENT type!"); |
---|
2615 | } |
---|
2616 | } |
---|
2617 | |
---|
2618 | if( typeof(p) == "vector" ) |
---|
2619 | { |
---|
2620 | intmat V = getModuleGrading(p); |
---|
2621 | } |
---|
2622 | else |
---|
2623 | { |
---|
2624 | intmat V; |
---|
2625 | } |
---|
2626 | |
---|
2627 | if( size(p) > 1) |
---|
2628 | { |
---|
2629 | intvec m; |
---|
2630 | |
---|
2631 | while( p != 0 ) |
---|
2632 | { |
---|
2633 | m = leadexp(p); |
---|
2634 | mp = lead(p); |
---|
2635 | p = p - lead(p); |
---|
2636 | tt = p; t = 0; |
---|
2637 | |
---|
2638 | while( size(tt) > 0 ) |
---|
2639 | { |
---|
2640 | // TODO: we do not cache matrices (M,T,H,V), which remain the same :( |
---|
2641 | // TODO: we need some low-level procedure with all these arguments...! |
---|
2642 | if( equalMultiDeg( leadexp(tt), m, V ) ) |
---|
2643 | { |
---|
2644 | mp = mp + lead(tt); // "mp", mp; |
---|
2645 | } |
---|
2646 | else |
---|
2647 | { |
---|
2648 | t = t + lead(tt); // "t", t; |
---|
2649 | } |
---|
2650 | |
---|
2651 | tt = tt - lead(tt); |
---|
2652 | } |
---|
2653 | |
---|
2654 | I[size(I)+1] = mp; |
---|
2655 | |
---|
2656 | p = t; |
---|
2657 | } |
---|
2658 | } |
---|
2659 | else |
---|
2660 | { |
---|
2661 | I[1] = p; // single monom |
---|
2662 | } |
---|
2663 | |
---|
2664 | if( typeof(I) == "module" ) |
---|
2665 | { |
---|
2666 | I = setModuleGrading(I, V); |
---|
2667 | } |
---|
2668 | |
---|
2669 | return (I); |
---|
2670 | } |
---|
2671 | example |
---|
2672 | { |
---|
2673 | "EXAMPLE:"; echo=2; |
---|
2674 | |
---|
2675 | ring r = 0,(x,y,z),dp; |
---|
2676 | |
---|
2677 | intmat g[2][3]= |
---|
2678 | 1,0,1, |
---|
2679 | 0,1,1; |
---|
2680 | intmat t[2][1]= |
---|
2681 | -2, |
---|
2682 | 1; |
---|
2683 | |
---|
2684 | setBaseMultigrading(g,t); |
---|
2685 | |
---|
2686 | poly f = x10yz+x8y2z-x4z2+y5+x2y2-z2+x17z3-y6; |
---|
2687 | |
---|
2688 | multiDegPartition(f); |
---|
2689 | |
---|
2690 | vector v = xy*gen(1)-x3y2*gen(2)+x4y*gen(3); |
---|
2691 | intmat B[2][3]=1,-1,-2,0,0,1; |
---|
2692 | v = setModuleGrading(v,B); |
---|
2693 | getModuleGrading(v); |
---|
2694 | |
---|
2695 | multiDegPartition(v, B); |
---|
2696 | } |
---|
2697 | |
---|
2698 | |
---|
2699 | |
---|
2700 | /******************************************************/ |
---|
2701 | static proc unitMatrix(int n) |
---|
2702 | { |
---|
2703 | intmat A[n][n]; |
---|
2704 | |
---|
2705 | for( int i = n; i > 0; i-- ) |
---|
2706 | { |
---|
2707 | A[i,i] = 1; |
---|
2708 | } |
---|
2709 | |
---|
2710 | return (A); |
---|
2711 | } |
---|
2712 | |
---|
2713 | |
---|
2714 | |
---|
2715 | /******************************************************/ |
---|
2716 | static proc finestMDeg(def r) |
---|
2717 | " |
---|
2718 | USAGE: finestMDeg(r); ring r |
---|
2719 | RETURN: ring, r endowed with the finest multigrading |
---|
2720 | TODO: not yet... |
---|
2721 | " |
---|
2722 | { |
---|
2723 | def save = basering; |
---|
2724 | setring (r); |
---|
2725 | |
---|
2726 | // in basering |
---|
2727 | ideal I = ideal(basering); |
---|
2728 | |
---|
2729 | int n = 0; int i; poly p; |
---|
2730 | for( i = ncols(I); i > 0; i-- ) |
---|
2731 | { |
---|
2732 | p = I[i]; |
---|
2733 | if( size(p) > 1 ) |
---|
2734 | { |
---|
2735 | n = n + (size(p) - 1); |
---|
2736 | } |
---|
2737 | else |
---|
2738 | { |
---|
2739 | I[i] = 0; |
---|
2740 | } |
---|
2741 | } |
---|
2742 | |
---|
2743 | int N = nvars(basering); |
---|
2744 | intmat A = unitMatrix(N); |
---|
2745 | |
---|
2746 | |
---|
2747 | |
---|
2748 | if( n > 0) |
---|
2749 | { |
---|
2750 | |
---|
2751 | intmat L[N][n]; |
---|
2752 | // list L; |
---|
2753 | int j = n; |
---|
2754 | |
---|
2755 | for( i = ncols(I); i > 0; i-- ) |
---|
2756 | { |
---|
2757 | p = I[i]; |
---|
2758 | |
---|
2759 | if( size(p) > 1 ) |
---|
2760 | { |
---|
2761 | intvec m0 = leadexp(p); |
---|
2762 | p = p - lead(p); |
---|
2763 | |
---|
2764 | while( size(p) > 0 ) |
---|
2765 | { |
---|
2766 | L[ 1..N, j ] = leadexp(p) - m0; |
---|
2767 | p = p - lead(p); |
---|
2768 | j--; |
---|
2769 | } |
---|
2770 | } |
---|
2771 | } |
---|
2772 | |
---|
2773 | print(L); |
---|
2774 | setBaseMultigrading(A, L); |
---|
2775 | } |
---|
2776 | else |
---|
2777 | { |
---|
2778 | setBaseMultigrading(A); |
---|
2779 | } |
---|
2780 | |
---|
2781 | // ERROR("nope"); |
---|
2782 | |
---|
2783 | // ring T = integer, (x), (C, dp); |
---|
2784 | |
---|
2785 | setring(save); |
---|
2786 | return (r); |
---|
2787 | } |
---|
2788 | example |
---|
2789 | { |
---|
2790 | "EXAMPLE:"; echo=2; |
---|
2791 | |
---|
2792 | ring r = 0,(x, y), dp; |
---|
2793 | qring q = std(x^2 - y); |
---|
2794 | |
---|
2795 | finestMDeg(q); |
---|
2796 | |
---|
2797 | } |
---|
2798 | |
---|
2799 | |
---|
2800 | |
---|
2801 | |
---|
2802 | /******************************************************/ |
---|
2803 | static proc newMap(map F, intmat Q, list #) |
---|
2804 | " |
---|
2805 | USAGE: newMap(F, Q[, P]); map F, intmat Q[, intmat P] |
---|
2806 | PURPOSE: endowe the map F with the integer matrices P [and Q] |
---|
2807 | " |
---|
2808 | { |
---|
2809 | attrib(F, "Q", Q); |
---|
2810 | |
---|
2811 | if( size(#) > 0 and typeof(#[1]) == "intmat" ) |
---|
2812 | { |
---|
2813 | attrib(F, "P", #[1]); |
---|
2814 | } |
---|
2815 | return (F); |
---|
2816 | } |
---|
2817 | |
---|
2818 | /******************************************************/ |
---|
2819 | static proc matrix2intmat( matrix M ) |
---|
2820 | { |
---|
2821 | execute( "intmat A[ "+ string(nrows(M)) + "]["+ string(ncols(M)) + "] = " + string(M) + ";" ); |
---|
2822 | return (A); |
---|
2823 | } |
---|
2824 | |
---|
2825 | |
---|
2826 | /******************************************************/ |
---|
2827 | static proc leftKernelZ(intmat M) |
---|
2828 | "USAGE: leftKernel(M); M a matrix |
---|
2829 | RETURN: module |
---|
2830 | PURPOSE: computes left kernel of matrix M (a module of all elements v such that vM=0) |
---|
2831 | EXAMPLE: example leftKernel; shows an example |
---|
2832 | " |
---|
2833 | { |
---|
2834 | if( nameof(basering) != "basering" ) |
---|
2835 | { |
---|
2836 | def save = basering; |
---|
2837 | } |
---|
2838 | |
---|
2839 | ring r = integer, (x), dp; |
---|
2840 | |
---|
2841 | |
---|
2842 | // basering; |
---|
2843 | module N = matrix((M)); // transpose |
---|
2844 | // print(N); |
---|
2845 | |
---|
2846 | def MM = modulo( N, std(0) ) ; |
---|
2847 | // print(MM); |
---|
2848 | |
---|
2849 | intmat R = ( matrix2intmat( MM ) ); // transpose |
---|
2850 | |
---|
2851 | if( defined(save) > 0 ) |
---|
2852 | { |
---|
2853 | setring save; |
---|
2854 | } |
---|
2855 | |
---|
2856 | kill r; |
---|
2857 | return( R ); |
---|
2858 | } |
---|
2859 | example |
---|
2860 | { |
---|
2861 | "EXAMPLE:"; echo=2; |
---|
2862 | |
---|
2863 | ring r= 0,(x,y,z),dp; |
---|
2864 | matrix M[3][1] = x,y,z; |
---|
2865 | print(M); |
---|
2866 | matrix L = leftKernel(M); |
---|
2867 | print(L); |
---|
2868 | // check: |
---|
2869 | print(L*M); |
---|
2870 | }; |
---|
2871 | |
---|
2872 | |
---|
2873 | |
---|
2874 | /******************************************************/ |
---|
2875 | // the following is taken from "sing4ti2.lib" as we need 'hilbert' from 4ti2 |
---|
2876 | |
---|
2877 | static proc hilbert4ti2intmat(intmat A, list #) |
---|
2878 | "USAGE: hilbert4ti2(A[,i]); |
---|
2879 | @* A=intmat |
---|
2880 | @* i=int |
---|
2881 | ASSUME: - A is a matrix with integer entries which describes the lattice |
---|
2882 | @* as ker(A), if second argument is not present, and |
---|
2883 | @* as the left image Im(A) = {zA : z \in ZZ^k}, if second argument is a positive integer |
---|
2884 | @* - number of variables of basering equals number of columns of A |
---|
2885 | @* (for ker(A)) resp. of rows of A (for Im(A)) |
---|
2886 | CREATE: temporary files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar |
---|
2887 | @* in the current directory (I/O files for communication with 4ti2) |
---|
2888 | NOTE: input rules for 4ti2 also apply to input to this procedure |
---|
2889 | @* hence ker(A)={x|Ax=0} and Im(A)={xA} |
---|
2890 | RETURN: toric ideal specified by Hilbert basis thereof |
---|
2891 | EXAMPLE: example graver4ti2; shows an example |
---|
2892 | " |
---|
2893 | { |
---|
2894 | if( system("sh","which hilbert 2> /dev/null 1> /dev/null") != 0 ) |
---|
2895 | { |
---|
2896 | ERROR("Sorry: cannot find 'hilbert' command from 4ti2. Please install 4ti2!"); |
---|
2897 | } |
---|
2898 | |
---|
2899 | //-------------------------------------------------------------------------- |
---|
2900 | // Initialization and Sanity Checks |
---|
2901 | //-------------------------------------------------------------------------- |
---|
2902 | int i,j; |
---|
2903 | int nr=nrows(A); |
---|
2904 | int nc=ncols(A); |
---|
2905 | string fileending="mat"; |
---|
2906 | if (size(#)!=0) |
---|
2907 | { |
---|
2908 | //--- default behaviour: use ker(A) as lattice |
---|
2909 | //--- if #[1]!=0 use Im(A) as lattice |
---|
2910 | if(typeof(#[1])!="int") |
---|
2911 | { |
---|
2912 | ERROR("optional parameter needs to be integer value"); |
---|
2913 | } |
---|
2914 | if(#[1]!=0) |
---|
2915 | { |
---|
2916 | fileending="lat"; |
---|
2917 | } |
---|
2918 | } |
---|
2919 | //--- we should also be checking whether all entries are indeed integers |
---|
2920 | //--- or whether there are fractions, but in this case the error message |
---|
2921 | //--- of 4ti2 is printed directly |
---|
2922 | |
---|
2923 | //-------------------------------------------------------------------------- |
---|
2924 | // preparing input file for 4ti2 |
---|
2925 | //-------------------------------------------------------------------------- |
---|
2926 | link eing=":w sing4ti2."+fileending; |
---|
2927 | string eingstring=string(nr)+" "+string(nc); |
---|
2928 | write(eing,eingstring); |
---|
2929 | for(i=1;i<=nr;i++) |
---|
2930 | { |
---|
2931 | kill eingstring; |
---|
2932 | string eingstring; |
---|
2933 | for(j=1;j<=nc;j++) |
---|
2934 | { |
---|
2935 | // if(g(A[i,j])>0)||(char(basering)!=0)||(npars(basering)>0)) |
---|
2936 | // { |
---|
2937 | // ERROR("Input to hilbert4ti2 needs to be a matrix with integer entries"); |
---|
2938 | // } |
---|
2939 | eingstring=eingstring+string(A[i,j])+" "; |
---|
2940 | } |
---|
2941 | write(eing, eingstring); |
---|
2942 | } |
---|
2943 | close(eing); |
---|
2944 | |
---|
2945 | //---------------------------------------------------------------------- |
---|
2946 | // calling 4ti2 and converting output |
---|
2947 | // Singular's string is too clumsy for this, hence we first prepare |
---|
2948 | // using standard unix commands |
---|
2949 | //---------------------------------------------------------------------- |
---|
2950 | |
---|
2951 | |
---|
2952 | j=system("sh","hilbert -q -n sing4ti2"); ////////// be quiet + no loggin!!! |
---|
2953 | |
---|
2954 | j=system("sh", "awk \'BEGIN{ORS=\",\";}{print $0;}\' sing4ti2.hil " + |
---|
2955 | "| sed s/[\\\ \\\t\\\v\\\f]/,/g " + |
---|
2956 | "| sed s/,+/,/g|sed s/,,/,/g " + |
---|
2957 | "| sed s/,,/,/g " + |
---|
2958 | "> sing4ti2.converted" ); |
---|
2959 | |
---|
2960 | |
---|
2961 | //---------------------------------------------------------------------- |
---|
2962 | // reading output of 4ti2 |
---|
2963 | //---------------------------------------------------------------------- |
---|
2964 | link ausg=":r sing4ti2.converted"; |
---|
2965 | //--- last entry ideal(0) is used to tie the list to the basering |
---|
2966 | //--- it will not be processed any further |
---|
2967 | |
---|
2968 | string s = read(ausg); |
---|
2969 | |
---|
2970 | if( defined(keepfiles) <= 0) |
---|
2971 | { |
---|
2972 | j=system("sh",("rm -f sing4ti2.hil sing4ti2.converted sing4ti2."+fileending)); |
---|
2973 | } |
---|
2974 | |
---|
2975 | string ergstr = "intvec erglist = " + s + "0;"; |
---|
2976 | execute(ergstr); |
---|
2977 | |
---|
2978 | // print(erglist); |
---|
2979 | |
---|
2980 | int Rnc = erglist[1]; |
---|
2981 | int Rnr = erglist[2]; |
---|
2982 | |
---|
2983 | intmat R[Rnr][Rnc]; |
---|
2984 | |
---|
2985 | int k = 3; |
---|
2986 | |
---|
2987 | for(i=1;i<=Rnc;i++) |
---|
2988 | { |
---|
2989 | for(j=1;j<=Rnr;j++) |
---|
2990 | { |
---|
2991 | // "i: ", i, ", j: ", j, ", v: ", erglist[k]; |
---|
2992 | R[j, i] = erglist[k]; |
---|
2993 | k = k + 1; |
---|
2994 | } |
---|
2995 | } |
---|
2996 | |
---|
2997 | |
---|
2998 | |
---|
2999 | return (R); |
---|
3000 | //--- get rid of leading entry 0; |
---|
3001 | // toric=toric[2..ncols(toric)]; |
---|
3002 | // return(toric); |
---|
3003 | } |
---|
3004 | // A nice example here is the 3x3 Magic Squares |
---|
3005 | example |
---|
3006 | { |
---|
3007 | "EXAMPLE:"; echo=2; |
---|
3008 | |
---|
3009 | ring r=0,(x1,x2,x3,x4,x5,x6,x7,x8,x9),dp; |
---|
3010 | intmat M[7][9]= |
---|
3011 | 1, 1, 1, -1, -1, -1, 0, 0, 0, |
---|
3012 | 1, 1, 1, 0, 0, 0,-1,-1,-1, |
---|
3013 | 0, 1, 1, -1, 0, 0,-1, 0, 0, |
---|
3014 | 1, 0, 1, 0, -1, 0, 0,-1, 0, |
---|
3015 | 1, 1, 0, 0, 0, -1, 0, 0,-1, |
---|
3016 | 0, 1, 1, 0, -1, 0, 0, 0,-1, |
---|
3017 | 1, 1, 0, 0, -1, 0,-1, 0, 0; |
---|
3018 | hilbert4ti2intmat(M); |
---|
3019 | hermiteNormalForm(M); |
---|
3020 | } |
---|
3021 | |
---|
3022 | ///////////////////////////////////////////////////////////////////////////// |
---|
3023 | static proc getMonomByExponent(intvec exp) |
---|
3024 | { |
---|
3025 | int n = nvars(basering); |
---|
3026 | |
---|
3027 | if( nrows(exp) < n ) |
---|
3028 | { |
---|
3029 | n = nrows(exp); |
---|
3030 | } |
---|
3031 | |
---|
3032 | poly m = 1; int e; |
---|
3033 | |
---|
3034 | for( int i = 1; i <= n; i++ ) |
---|
3035 | { |
---|
3036 | e = exp[i]; |
---|
3037 | if( e < 0 ) |
---|
3038 | { |
---|
3039 | ERROR("Negative exponent!!!"); |
---|
3040 | } |
---|
3041 | |
---|
3042 | m = m * (var(i)^e); |
---|
3043 | } |
---|
3044 | |
---|
3045 | return (m); |
---|
3046 | |
---|
3047 | } |
---|
3048 | |
---|
3049 | /******************************************************/ |
---|
3050 | proc multiDegBasis(intvec d) |
---|
3051 | " |
---|
3052 | USAGE: multidegree d |
---|
3053 | ASSUME: current ring is multigraded, monomial ordering is global |
---|
3054 | PURPOSE: compute all monomials of multidegree d |
---|
3055 | EXAMPLE: example multiDegBasis; shows an example |
---|
3056 | " |
---|
3057 | { |
---|
3058 | def R = basering; // setring R; |
---|
3059 | |
---|
3060 | intmat M = getVariableWeights(R); |
---|
3061 | |
---|
3062 | // print(M); |
---|
3063 | |
---|
3064 | int nr = nrows(M); |
---|
3065 | int nc = ncols(M); |
---|
3066 | |
---|
3067 | intmat A[nr][nc+1]; |
---|
3068 | A[1..nr, 1..nc] = M[1..nr, 1..nc]; |
---|
3069 | //typeof(A[1..nr, nc+1]); |
---|
3070 | if( nr==1) |
---|
3071 | { |
---|
3072 | A[1..nr, nc+1]=-d[1]; |
---|
3073 | } |
---|
3074 | else |
---|
3075 | { |
---|
3076 | A[1..nr, nc+1] = -d; |
---|
3077 | } |
---|
3078 | |
---|
3079 | intmat T = getLattice(R); |
---|
3080 | |
---|
3081 | if( isFreeRepresented() ) |
---|
3082 | { |
---|
3083 | intmat B = hilbert4ti2intmat(A); |
---|
3084 | |
---|
3085 | // matrix B = unitMatrix(nrows(T)); |
---|
3086 | } |
---|
3087 | else |
---|
3088 | { |
---|
3089 | int n = ncols(T); |
---|
3090 | |
---|
3091 | nc = ncols(A); |
---|
3092 | |
---|
3093 | intmat AA[nr][nc + 2 * n]; |
---|
3094 | AA[1..nr, 1.. nc] = A[1..nr, 1.. nc]; |
---|
3095 | AA[1..nr, nc + (1.. n)] = T[1..nr, 1.. n]; |
---|
3096 | AA[1..nr, nc + n + (1.. n)] = -T[1..nr, 1.. n]; |
---|
3097 | |
---|
3098 | |
---|
3099 | // print ( AA ); |
---|
3100 | |
---|
3101 | intmat K = leftKernelZ(( AA ) ); // |
---|
3102 | |
---|
3103 | // print(K); |
---|
3104 | |
---|
3105 | intmat KK[nc][ncols(K)] = K[ 1.. nc, 1.. ncols(K) ]; |
---|
3106 | |
---|
3107 | // print(KK); |
---|
3108 | // "!"; |
---|
3109 | |
---|
3110 | intmat B = hilbert4ti2intmat(transpose(KK), 1); |
---|
3111 | |
---|
3112 | // "!"; print(B); |
---|
3113 | |
---|
3114 | } |
---|
3115 | |
---|
3116 | |
---|
3117 | // print(A); |
---|
3118 | |
---|
3119 | |
---|
3120 | |
---|
3121 | int i; |
---|
3122 | int nnr = nrows(B); |
---|
3123 | int nnc = ncols(B); |
---|
3124 | ideal I, J; |
---|
3125 | if(nnc==0){ |
---|
3126 | I=0; |
---|
3127 | return(I); |
---|
3128 | } |
---|
3129 | I[nnc] = 0; |
---|
3130 | J[nnc] = 0; |
---|
3131 | |
---|
3132 | for( i = 1; i <= nnc; i++ ) |
---|
3133 | { |
---|
3134 | // "i: ", i; B[nnr, i]; |
---|
3135 | |
---|
3136 | if( B[nnr, i] == 1) |
---|
3137 | { |
---|
3138 | // intvec(B[1..nnr-1, i]); |
---|
3139 | I[i] = getMonomByExponent(intvec(B[1..nnr-1, i])); |
---|
3140 | } |
---|
3141 | else |
---|
3142 | { |
---|
3143 | if( B[nnr, i] == 0) |
---|
3144 | { |
---|
3145 | // intvec(B[1..nnr-1, i]); |
---|
3146 | J[i] = getMonomByExponent(intvec(B[1..nnr-1, i])); |
---|
3147 | } |
---|
3148 | } |
---|
3149 | // I[i]; |
---|
3150 | } |
---|
3151 | |
---|
3152 | ideal Q = (ideal(basering)); |
---|
3153 | |
---|
3154 | if ( size(Q) > 0 ) |
---|
3155 | { |
---|
3156 | I = NF( I, lead(Q) ); |
---|
3157 | J = NF( J, lead(Q) ); // Global ordering!!! |
---|
3158 | } |
---|
3159 | |
---|
3160 | I = simplify(I, 2); // d |
---|
3161 | J = simplify(J, 2); // d |
---|
3162 | |
---|
3163 | attrib(I, "ZeroPart", J); |
---|
3164 | |
---|
3165 | return (I); |
---|
3166 | |
---|
3167 | // setring ; |
---|
3168 | } |
---|
3169 | example |
---|
3170 | { |
---|
3171 | "EXAMPLE:"; echo=2; |
---|
3172 | |
---|
3173 | ring R = 0, (x, y), dp; |
---|
3174 | |
---|
3175 | intmat g1[2][2]=1,0,0,1; |
---|
3176 | intmat l[2][1]=2,0; |
---|
3177 | intmat g2[2][2]=1,1,1,1; |
---|
3178 | intvec v1=4,0; |
---|
3179 | intvec v2=4,4; |
---|
3180 | |
---|
3181 | intmat g3[1][2]=1,1; |
---|
3182 | setBaseMultigrading(g3); |
---|
3183 | intvec v3=4:1; |
---|
3184 | v3; |
---|
3185 | multiDegBasis(v3); |
---|
3186 | |
---|
3187 | setBaseMultigrading(g1,l); |
---|
3188 | multiDegBasis(v1); |
---|
3189 | setBaseMultigrading(g2); |
---|
3190 | multiDegBasis(v2); |
---|
3191 | |
---|
3192 | intmat M[2][2] = 1, -1, -1, 1; |
---|
3193 | intvec d = -2, 2; |
---|
3194 | |
---|
3195 | setBaseMultigrading(M); |
---|
3196 | |
---|
3197 | multiDegBasis(d); |
---|
3198 | attrib(_, "ZeroPart"); |
---|
3199 | |
---|
3200 | kill R, M, d; |
---|
3201 | ring R = 0, (x, y, z), dp; |
---|
3202 | |
---|
3203 | intmat M[2][3] = 1, -2, 1, 1, 1, 0; |
---|
3204 | |
---|
3205 | intmat L[2][1] = 0, 2; |
---|
3206 | |
---|
3207 | intvec d = 4, 1; |
---|
3208 | |
---|
3209 | setBaseMultigrading(M, L); |
---|
3210 | |
---|
3211 | multiDegBasis(d); |
---|
3212 | attrib(_, "ZeroPart"); |
---|
3213 | |
---|
3214 | |
---|
3215 | kill R, M, d; |
---|
3216 | |
---|
3217 | ring R = 0, (x, y, z), dp; |
---|
3218 | qring Q = std(ideal( y^6+ x*y^3*z-x^2*z^2 )); |
---|
3219 | |
---|
3220 | |
---|
3221 | intmat M[2][3] = 1, 1, 2, 2, 1, 1; |
---|
3222 | // intmat T[2][1] = 0, 2; |
---|
3223 | |
---|
3224 | setBaseMultigrading(M); // BUG???? |
---|
3225 | |
---|
3226 | intvec d = 6, 6; |
---|
3227 | multiDegBasis(d); |
---|
3228 | attrib(_, "ZeroPart"); |
---|
3229 | |
---|
3230 | |
---|
3231 | |
---|
3232 | kill R, Q, M, d; |
---|
3233 | ring R = 0, (x, y, z), dp; |
---|
3234 | qring Q = std(ideal( x*z^3 - y *z^6, x*y*z - x^4*y^2 )); |
---|
3235 | |
---|
3236 | |
---|
3237 | intmat M[2][3] = 1, -2, 1, 1, 1, 0; |
---|
3238 | intmat T[2][1] = 0, 2; |
---|
3239 | |
---|
3240 | intvec d = 4, 1; |
---|
3241 | |
---|
3242 | setBaseMultigrading(M, T); // BUG???? |
---|
3243 | |
---|
3244 | multiDegBasis(d); |
---|
3245 | attrib(_, "ZeroPart"); |
---|
3246 | } |
---|
3247 | |
---|
3248 | |
---|
3249 | proc multiDegSyzygy(def I) |
---|
3250 | "USAGE: multiDegSyzygy(I); I is a ideal or a module |
---|
3251 | PURPOSE: computes the multigraded syzygy module of I |
---|
3252 | RETURNS: module, the syzygy module of I |
---|
3253 | NOTE: generators of I must be multigraded homogeneous |
---|
3254 | EXAMPLE: example multiDegSyzygy; shows an example |
---|
3255 | " |
---|
3256 | { |
---|
3257 | if( isHomogeneous(I, "checkGens") == 0) |
---|
3258 | { |
---|
3259 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3260 | } |
---|
3261 | module S = syz(I); |
---|
3262 | S = setModuleGrading(S, multiDeg(I)); |
---|
3263 | return (S); |
---|
3264 | } |
---|
3265 | example |
---|
3266 | { |
---|
3267 | "EXAMPLE:"; echo=2; |
---|
3268 | |
---|
3269 | |
---|
3270 | ring r = 0,(x,y,z,w),dp; |
---|
3271 | intmat MM[2][4]= |
---|
3272 | 1,1,1,1, |
---|
3273 | 0,1,3,4; |
---|
3274 | setBaseMultigrading(MM); |
---|
3275 | module M = ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3276 | |
---|
3277 | |
---|
3278 | intmat v[2][nrows(M)]= |
---|
3279 | 1, |
---|
3280 | 0; |
---|
3281 | |
---|
3282 | M = setModuleGrading(M, v); |
---|
3283 | |
---|
3284 | isHomogeneous(M); |
---|
3285 | "Multidegrees: "; print(multiDeg(M)); |
---|
3286 | // Let's compute syzygies! |
---|
3287 | def S = multiDegSyzygy(M); S; |
---|
3288 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3289 | "Multidegrees: "; print(multiDeg(S)); |
---|
3290 | |
---|
3291 | isHomogeneous(S); |
---|
3292 | } |
---|
3293 | |
---|
3294 | |
---|
3295 | |
---|
3296 | proc multiDegModulo(def I, def J) |
---|
3297 | "USAGE: multiDegModulo(I); I, J are ideals or modules |
---|
3298 | PURPOSE: computes the multigraded 'modulo' module of I and J |
---|
3299 | RETURNS: module, see 'modulo' command |
---|
3300 | NOTE: I and J should have the same multigrading, and their |
---|
3301 | generators must be multigraded homogeneous |
---|
3302 | EXAMPLE: example multiDegModulo; shows an example |
---|
3303 | " |
---|
3304 | { |
---|
3305 | if( (isHomogeneous(I, "checkGens") == 0) or (isHomogeneous(J, "checkGens") == 0) ) |
---|
3306 | { |
---|
3307 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3308 | } |
---|
3309 | module K = modulo(I, J); |
---|
3310 | K = setModuleGrading(K, multiDeg(I)); |
---|
3311 | return (K); |
---|
3312 | } |
---|
3313 | example |
---|
3314 | { |
---|
3315 | "EXAMPLE:"; echo=2; |
---|
3316 | |
---|
3317 | ring r = 0,(x,y,z),dp; |
---|
3318 | intmat MM[2][3]= |
---|
3319 | -1,1,1, |
---|
3320 | 0,1,3; |
---|
3321 | setBaseMultigrading(MM); |
---|
3322 | |
---|
3323 | ideal h1 = x, y, z; |
---|
3324 | ideal h2 = x; |
---|
3325 | |
---|
3326 | "Multidegrees: "; print(multiDeg(h1)); |
---|
3327 | |
---|
3328 | // Let's compute modulo(h1, h2): |
---|
3329 | def K = multiDegModulo(h1, h2); K; |
---|
3330 | |
---|
3331 | "Module Units Multigrading: "; print( getModuleGrading(K) ); |
---|
3332 | "Multidegrees: "; print(multiDeg(K)); |
---|
3333 | |
---|
3334 | isHomogeneous(K); |
---|
3335 | } |
---|
3336 | |
---|
3337 | |
---|
3338 | proc multiDegGroebner(def I) |
---|
3339 | "USAGE: multiDegGroebner(I); I is a poly/vector/ideal/module |
---|
3340 | PURPOSE: computes the multigraded standard/groebner basis of I |
---|
3341 | NOTE: I must be multigraded homogeneous |
---|
3342 | RETURNS: ideal/module, the computed basis |
---|
3343 | EXAMPLE: example multiDegGroebner; shows an example |
---|
3344 | " |
---|
3345 | { |
---|
3346 | if( isHomogeneous(I) == 0) |
---|
3347 | { |
---|
3348 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3349 | } |
---|
3350 | |
---|
3351 | def S = groebner(I); |
---|
3352 | |
---|
3353 | if( typeof(I) == "module" or typeof(I) == "vector" ) |
---|
3354 | { |
---|
3355 | S = setModuleGrading(S, getModuleGrading(I)); |
---|
3356 | } |
---|
3357 | |
---|
3358 | return(S); |
---|
3359 | } |
---|
3360 | example |
---|
3361 | { |
---|
3362 | "EXAMPLE:"; echo=2; |
---|
3363 | |
---|
3364 | ring r = 0,(x,y,z,w),dp; |
---|
3365 | |
---|
3366 | intmat MM[2][4]= |
---|
3367 | 1,1,1,1, |
---|
3368 | 0,1,3,4; |
---|
3369 | |
---|
3370 | setBaseMultigrading(MM); |
---|
3371 | |
---|
3372 | |
---|
3373 | module M = ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3374 | |
---|
3375 | |
---|
3376 | intmat v[2][nrows(M)]= |
---|
3377 | 1, |
---|
3378 | 0; |
---|
3379 | |
---|
3380 | M = setModuleGrading(M, v); |
---|
3381 | |
---|
3382 | |
---|
3383 | ///////////////////////////////////////////////////////////////////////////// |
---|
3384 | // GB: |
---|
3385 | M = multiDegGroebner(M); M; |
---|
3386 | "Module Units Multigrading: "; print( getModuleGrading(M) ); |
---|
3387 | "Multidegrees: "; print(multiDeg(M)); |
---|
3388 | |
---|
3389 | isHomogeneous(M); |
---|
3390 | |
---|
3391 | ///////////////////////////////////////////////////////////////////////////// |
---|
3392 | // Let's compute Syzygy! |
---|
3393 | def S = multiDegSyzygy(M); S; |
---|
3394 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3395 | "Multidegrees: "; print(multiDeg(S)); |
---|
3396 | |
---|
3397 | isHomogeneous(S); |
---|
3398 | |
---|
3399 | ///////////////////////////////////////////////////////////////////////////// |
---|
3400 | // GB: |
---|
3401 | S = multiDegGroebner(S); S; |
---|
3402 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3403 | "Multidegrees: "; print(multiDeg(S)); |
---|
3404 | |
---|
3405 | isHomogeneous(S); |
---|
3406 | } |
---|
3407 | |
---|
3408 | |
---|
3409 | /******************************************************/ |
---|
3410 | proc multiDegResolution(def I, int ll, list #) |
---|
3411 | "USAGE: multiDegResolution(I,l,[f]); I is poly/vector/ideal/module; l,f are integers |
---|
3412 | PURPOSE: computes the multigraded resolution of I of the length l, |
---|
3413 | or the whole resolution if l is zero. Returns minimal resolution if an optional |
---|
3414 | argument 1 is supplied |
---|
3415 | NOTE: input must have multigraded-homogeneous generators. |
---|
3416 | The returned list is truncated beginning with the first zero differential. |
---|
3417 | RETURNS: list, the computed resolution |
---|
3418 | EXAMPLE: example multiDegResolution; shows an example |
---|
3419 | " |
---|
3420 | { |
---|
3421 | if( isHomogeneous(I, "checkGens") == 0) |
---|
3422 | { |
---|
3423 | ERROR ("Sorry: inhomogeneous input!"); |
---|
3424 | } |
---|
3425 | |
---|
3426 | def R = res(I, ll, #); list L = R; int l = size(L); |
---|
3427 | def V = getModuleGrading(I); |
---|
3428 | if( (typeof(I) == "module") or (typeof(I) == "vector") ) |
---|
3429 | { |
---|
3430 | L[1] = setModuleGrading(L[1], V); |
---|
3431 | } |
---|
3432 | |
---|
3433 | int i; |
---|
3434 | for( i = 2; i <= l; i++ ) |
---|
3435 | { |
---|
3436 | if( size(L[i]) > 0 ) |
---|
3437 | { |
---|
3438 | L[i] = setModuleGrading( L[i], multiDeg(L[i-1]) ); |
---|
3439 | } else |
---|
3440 | { |
---|
3441 | return (L[1..(i-1)]); |
---|
3442 | } |
---|
3443 | } |
---|
3444 | |
---|
3445 | return (L); |
---|
3446 | |
---|
3447 | |
---|
3448 | } |
---|
3449 | example |
---|
3450 | { |
---|
3451 | "EXAMPLE:"; echo=2; |
---|
3452 | |
---|
3453 | ring r = 0,(x,y,z,w),dp; |
---|
3454 | |
---|
3455 | intmat M[2][4]= |
---|
3456 | 1,1,1,1, |
---|
3457 | 0,1,3,4; |
---|
3458 | |
---|
3459 | setBaseMultigrading(M); |
---|
3460 | |
---|
3461 | |
---|
3462 | module m= ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3463 | |
---|
3464 | isHomogeneous(ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3), "checkGens"); |
---|
3465 | |
---|
3466 | ideal A = xw-yz, x2z-y3, xz2-y2w, yw2-z3; |
---|
3467 | |
---|
3468 | int j; |
---|
3469 | |
---|
3470 | for(j=1; j<=ncols(A); j++) |
---|
3471 | { |
---|
3472 | multiDegPartition(A[j]); |
---|
3473 | } |
---|
3474 | |
---|
3475 | intmat v[2][1]= |
---|
3476 | 1, |
---|
3477 | 0; |
---|
3478 | |
---|
3479 | m = setModuleGrading(m, v); |
---|
3480 | |
---|
3481 | // Let's compute Syzygy! |
---|
3482 | def S = multiDegSyzygy(m); S; |
---|
3483 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3484 | "Multidegrees: "; print(multiDeg(S)); |
---|
3485 | |
---|
3486 | ///////////////////////////////////////////////////////////////////////////// |
---|
3487 | |
---|
3488 | S = multiDegGroebner(S); S; |
---|
3489 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
3490 | "Multidegrees: "; print(multiDeg(S)); |
---|
3491 | |
---|
3492 | ///////////////////////////////////////////////////////////////////////////// |
---|
3493 | |
---|
3494 | def L = multiDegResolution(m, 0, 1); |
---|
3495 | |
---|
3496 | for( j =1; j<=size(L); j++) |
---|
3497 | { |
---|
3498 | "----------------------------------- ", j, " -----------------------------"; |
---|
3499 | L[j]; |
---|
3500 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
3501 | "Multigrading: "; print(multiDeg(L[j])); |
---|
3502 | } |
---|
3503 | |
---|
3504 | ///////////////////////////////////////////////////////////////////////////// |
---|
3505 | |
---|
3506 | L = multiDegResolution(maxideal(1), 0, 1); |
---|
3507 | |
---|
3508 | for( j =1; j<=size(L); j++) |
---|
3509 | { |
---|
3510 | "----------------------------------- ", j, " -----------------------------"; |
---|
3511 | L[j]; |
---|
3512 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
3513 | "Multigrading: "; print(multiDeg(L[j])); |
---|
3514 | } |
---|
3515 | |
---|
3516 | kill v; |
---|
3517 | |
---|
3518 | |
---|
3519 | def h = hilbertSeries(m); |
---|
3520 | setring h; |
---|
3521 | |
---|
3522 | numerator1; |
---|
3523 | factorize(numerator1); |
---|
3524 | |
---|
3525 | denominator1; |
---|
3526 | factorize(denominator1); |
---|
3527 | |
---|
3528 | numerator2; |
---|
3529 | factorize(numerator2); |
---|
3530 | |
---|
3531 | denominator2; |
---|
3532 | factorize(denominator2); |
---|
3533 | } |
---|
3534 | |
---|
3535 | /******************************************************/ |
---|
3536 | proc hilbertSeries(def I) |
---|
3537 | "USAGE: hilbertSeries(I); I is poly/vector/ideal/module |
---|
3538 | PURPOSE: computes the multigraded Hilbert Series of I |
---|
3539 | NOTE: input must have multigraded-homogeneous generators. |
---|
3540 | Multigrading should be positive. |
---|
3541 | RETURNS: a ring in variables t_(i), s_(i), with polynomials |
---|
3542 | numerator1 and denominator1 and mutually prime numerator2 |
---|
3543 | and denominator2, quotients of which give the series. |
---|
3544 | EXAMPLE: example hilbertSeries; shows an example |
---|
3545 | " |
---|
3546 | { |
---|
3547 | |
---|
3548 | if( !isFreeRepresented() ) |
---|
3549 | { |
---|
3550 | "Things might happen, since we are not free."; |
---|
3551 | //ERROR("SORRY: ONLY TORSION-FREE CASE (POSITIVE GRADING)"); |
---|
3552 | } |
---|
3553 | |
---|
3554 | int i, j, k, v; |
---|
3555 | |
---|
3556 | intmat M = getVariableWeights(); |
---|
3557 | |
---|
3558 | int cc = ncols(M); |
---|
3559 | int n = nrows(M); |
---|
3560 | |
---|
3561 | if( n == 0 ) |
---|
3562 | { |
---|
3563 | ERROR("Error: wrong Variable Weights?"); |
---|
3564 | } |
---|
3565 | |
---|
3566 | list RES = multiDegResolution(I,0,1); |
---|
3567 | |
---|
3568 | int l = size(RES); |
---|
3569 | |
---|
3570 | list L; L[l + 1] = 0; |
---|
3571 | |
---|
3572 | if(typeof(I) == "ideal") |
---|
3573 | { |
---|
3574 | intmat zeros[n][1]; |
---|
3575 | L[1] = zeros; |
---|
3576 | } |
---|
3577 | else |
---|
3578 | { |
---|
3579 | L[1] = getModuleGrading(RES[1]); |
---|
3580 | } |
---|
3581 | |
---|
3582 | for( j = 1; j <= l; j++) |
---|
3583 | { |
---|
3584 | L[j + 1] = multiDeg(RES[j]); |
---|
3585 | } |
---|
3586 | |
---|
3587 | l++; |
---|
3588 | |
---|
3589 | ring R = 0,(t_(1..n),s_(1..n)),dp; |
---|
3590 | |
---|
3591 | ideal units; |
---|
3592 | for( i=n; i>=1; i--) |
---|
3593 | { |
---|
3594 | units[i] = (var(i) * var(n + i) - 1); |
---|
3595 | } |
---|
3596 | |
---|
3597 | qring Q = std(units); |
---|
3598 | |
---|
3599 | // TODO: should not it be a quotient ring depending on Torsion??? |
---|
3600 | // I am not sure about what to do in the torsion case, but since |
---|
3601 | // we want to evaluate the polynomial at certain points to get |
---|
3602 | // a dimension we need uniqueness for this. I think we would lose |
---|
3603 | // this uniqueness if switching to this torsion ring. |
---|
3604 | |
---|
3605 | poly monom, summand, numerator; |
---|
3606 | poly denominator = 1; |
---|
3607 | |
---|
3608 | for( i = 1; i <= cc; i++) |
---|
3609 | { |
---|
3610 | monom = 1; |
---|
3611 | for( k = 1; k <= n; k++) |
---|
3612 | { |
---|
3613 | v = M[k,i]; |
---|
3614 | |
---|
3615 | if(v >= 0) |
---|
3616 | { |
---|
3617 | monom = monom * (var(k)^(v)); |
---|
3618 | } |
---|
3619 | else |
---|
3620 | { |
---|
3621 | monom = monom * (var(n+k)^(-v)); |
---|
3622 | } |
---|
3623 | } |
---|
3624 | |
---|
3625 | if( monom == 1) |
---|
3626 | { |
---|
3627 | ERROR("Multigrading not positive."); |
---|
3628 | } |
---|
3629 | |
---|
3630 | denominator = denominator * (1 - monom); |
---|
3631 | } |
---|
3632 | |
---|
3633 | for( j = 1; j<= l; j++) |
---|
3634 | { |
---|
3635 | summand = 0; |
---|
3636 | M = L[j]; |
---|
3637 | |
---|
3638 | for( i = 1; i <= ncols(M); i++) |
---|
3639 | { |
---|
3640 | monom = 1; |
---|
3641 | for( k = 1; k <= n; k++) |
---|
3642 | { |
---|
3643 | v = M[k,i]; |
---|
3644 | if( v > 0 ) |
---|
3645 | { |
---|
3646 | monom = monom * (var(k)^v); |
---|
3647 | } |
---|
3648 | else |
---|
3649 | { |
---|
3650 | monom = monom * (var(n+k)^(-v)); |
---|
3651 | } |
---|
3652 | } |
---|
3653 | summand = summand + monom; |
---|
3654 | } |
---|
3655 | numerator = numerator - (-1)^j * summand; |
---|
3656 | } |
---|
3657 | |
---|
3658 | if( denominator == 0 ) |
---|
3659 | { |
---|
3660 | ERROR("Multigrading not positive."); |
---|
3661 | } |
---|
3662 | |
---|
3663 | poly denominator1 = denominator; |
---|
3664 | poly numerator1 = numerator; |
---|
3665 | |
---|
3666 | export denominator1; |
---|
3667 | export numerator1; |
---|
3668 | |
---|
3669 | if( numerator != 0 ) |
---|
3670 | { |
---|
3671 | poly d = gcd(denominator, numerator); |
---|
3672 | |
---|
3673 | poly denominator2 = denominator/d; |
---|
3674 | poly numerator2 = numerator/d; |
---|
3675 | |
---|
3676 | if( gcd(denominator2, numerator2) != 1 ) |
---|
3677 | { |
---|
3678 | ERROR("Sorry: gcd should be 1 (after dividing out gcd)! Something went wrong!"); |
---|
3679 | } |
---|
3680 | } |
---|
3681 | else |
---|
3682 | { |
---|
3683 | poly denominator2 = denominator; |
---|
3684 | poly numerator2 = numerator; |
---|
3685 | } |
---|
3686 | |
---|
3687 | |
---|
3688 | export denominator2; |
---|
3689 | export numerator2; |
---|
3690 | |
---|
3691 | " ------------ "; |
---|
3692 | "This proc returns a ring with polynomials called 'numerator1/2' and 'denominator1/2'!"; |
---|
3693 | "They represent the first and the second Hilbert Series."; |
---|
3694 | "The s_(i)-variables are defined to be the inverse of the t_(i)-variables."; |
---|
3695 | " ------------ "; |
---|
3696 | |
---|
3697 | return(Q); |
---|
3698 | } |
---|
3699 | example |
---|
3700 | { |
---|
3701 | "EXAMPLE:"; echo=2; |
---|
3702 | |
---|
3703 | ring r = 0,(x,y,z,w),dp; |
---|
3704 | intmat g[2][4]= |
---|
3705 | 1,1,1,1, |
---|
3706 | 0,1,3,4; |
---|
3707 | setBaseMultigrading(g); |
---|
3708 | |
---|
3709 | module M = ideal(xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
3710 | intmat V[2][1]= |
---|
3711 | 1, |
---|
3712 | 0; |
---|
3713 | |
---|
3714 | M = setModuleGrading(M, V); |
---|
3715 | |
---|
3716 | def h = hilbertSeries(M); setring h; |
---|
3717 | |
---|
3718 | factorize(numerator2); |
---|
3719 | factorize(denominator2); |
---|
3720 | |
---|
3721 | kill g, h; setring r; |
---|
3722 | |
---|
3723 | intmat g[2][4]= |
---|
3724 | 1,2,3,4, |
---|
3725 | 0,0,5,8; |
---|
3726 | |
---|
3727 | setBaseMultigrading(g); |
---|
3728 | |
---|
3729 | ideal I = x^2, y, z^3; |
---|
3730 | I = std(I); |
---|
3731 | def L = multiDegResolution(I, 0, 1); |
---|
3732 | |
---|
3733 | for( int j = 1; j<=size(L); j++) |
---|
3734 | { |
---|
3735 | "----------------------------------- ", j, " -----------------------------"; |
---|
3736 | L[j]; |
---|
3737 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
3738 | "Multigrading: "; print(multiDeg(L[j])); |
---|
3739 | } |
---|
3740 | |
---|
3741 | multiDeg(I); |
---|
3742 | def h = hilbertSeries(I); setring h; |
---|
3743 | |
---|
3744 | factorize(numerator2); |
---|
3745 | factorize(denominator2); |
---|
3746 | |
---|
3747 | kill r, h, g, V; |
---|
3748 | //////////////////////////////////////////////// |
---|
3749 | ring R = 0,(x,y,z),dp; |
---|
3750 | intmat W[2][3] = |
---|
3751 | 1,1, 1, |
---|
3752 | 0,0,-1; |
---|
3753 | setBaseMultigrading(W); |
---|
3754 | ideal I = x3y,yz2,y2z,z4; |
---|
3755 | |
---|
3756 | def h = hilbertSeries(I); setring h; |
---|
3757 | |
---|
3758 | factorize(numerator2); |
---|
3759 | factorize(denominator2); |
---|
3760 | |
---|
3761 | kill R, W, h; |
---|
3762 | //////////////////////////////////////////////// |
---|
3763 | ring R = 0,(x,y,z,a,b,c),dp; |
---|
3764 | intmat W[2][6] = |
---|
3765 | 1,1, 1,1,1,1, |
---|
3766 | 0,0,-1,0,0,0; |
---|
3767 | setBaseMultigrading(W); |
---|
3768 | ideal I = x3y,yz2,y2z,z4; |
---|
3769 | |
---|
3770 | def h = hilbertSeries(I); setring h; |
---|
3771 | |
---|
3772 | factorize(numerator2); |
---|
3773 | factorize(denominator2); |
---|
3774 | |
---|
3775 | kill R, W, h; |
---|
3776 | //////////////////////////////////////////////// |
---|
3777 | // This is example 5.3.9. from Robbianos book. |
---|
3778 | |
---|
3779 | ring R = 0,(x,y,z,w),dp; |
---|
3780 | intmat W[1][4] = |
---|
3781 | 1,1, 1,1; |
---|
3782 | setBaseMultigrading(W); |
---|
3783 | ideal I = z3,y3zw2,x2y4w2xyz2; |
---|
3784 | |
---|
3785 | hilb(std(I)); |
---|
3786 | |
---|
3787 | def h = hilbertSeries(I); setring h; |
---|
3788 | |
---|
3789 | numerator1; |
---|
3790 | denominator1; |
---|
3791 | |
---|
3792 | factorize(numerator2); |
---|
3793 | factorize(denominator2); |
---|
3794 | |
---|
3795 | |
---|
3796 | kill h; |
---|
3797 | //////////////////////////////////////////////// |
---|
3798 | setring R; |
---|
3799 | |
---|
3800 | ideal I2 = x2,y2,z2; I2; |
---|
3801 | |
---|
3802 | hilb(std(I2)); |
---|
3803 | |
---|
3804 | def h = hilbertSeries(I2); setring h; |
---|
3805 | |
---|
3806 | numerator1; |
---|
3807 | denominator1; |
---|
3808 | |
---|
3809 | |
---|
3810 | kill h; |
---|
3811 | //////////////////////////////////////////////// |
---|
3812 | setring R; |
---|
3813 | |
---|
3814 | W = 2,2,2,2; |
---|
3815 | |
---|
3816 | setBaseMultigrading(W); |
---|
3817 | |
---|
3818 | getVariableWeights(); |
---|
3819 | |
---|
3820 | intvec w = 2,2,2,2; |
---|
3821 | |
---|
3822 | hilb(std(I2), 1, w); |
---|
3823 | |
---|
3824 | kill w; |
---|
3825 | |
---|
3826 | |
---|
3827 | def h = hilbertSeries(I2); setring h; |
---|
3828 | |
---|
3829 | |
---|
3830 | numerator1; denominator1; |
---|
3831 | kill h; |
---|
3832 | |
---|
3833 | |
---|
3834 | kill R, W; |
---|
3835 | |
---|
3836 | //////////////////////////////////////////////// |
---|
3837 | ring R = 0,(x),dp; |
---|
3838 | intmat W[1][1] = |
---|
3839 | 1; |
---|
3840 | setBaseMultigrading(W); |
---|
3841 | |
---|
3842 | ideal I; |
---|
3843 | |
---|
3844 | I = 1; I; |
---|
3845 | |
---|
3846 | hilb(std(I)); |
---|
3847 | |
---|
3848 | def h = hilbertSeries(I); setring h; |
---|
3849 | |
---|
3850 | numerator1; denominator1; |
---|
3851 | |
---|
3852 | kill h; |
---|
3853 | //////////////////////////////////////////////// |
---|
3854 | setring R; |
---|
3855 | |
---|
3856 | I = x; I; |
---|
3857 | |
---|
3858 | hilb(std(I)); |
---|
3859 | |
---|
3860 | def h = hilbertSeries(I); setring h; |
---|
3861 | |
---|
3862 | numerator1; denominator1; |
---|
3863 | |
---|
3864 | kill h; |
---|
3865 | //////////////////////////////////////////////// |
---|
3866 | setring R; |
---|
3867 | |
---|
3868 | I = x^5; I; |
---|
3869 | |
---|
3870 | hilb(std(I)); |
---|
3871 | hilb(std(I), 1); |
---|
3872 | |
---|
3873 | def h = hilbertSeries(I); setring h; |
---|
3874 | |
---|
3875 | numerator1; denominator1; |
---|
3876 | |
---|
3877 | |
---|
3878 | kill h; |
---|
3879 | //////////////////////////////////////////////// |
---|
3880 | setring R; |
---|
3881 | |
---|
3882 | I = x^10; I; |
---|
3883 | |
---|
3884 | hilb(std(I)); |
---|
3885 | |
---|
3886 | def h = hilbertSeries(I); setring h; |
---|
3887 | |
---|
3888 | numerator1; denominator1; |
---|
3889 | |
---|
3890 | kill h; |
---|
3891 | //////////////////////////////////////////////// |
---|
3892 | setring R; |
---|
3893 | |
---|
3894 | module M = 1; |
---|
3895 | |
---|
3896 | M = setModuleGrading(M, W); |
---|
3897 | |
---|
3898 | |
---|
3899 | hilb(std(M)); |
---|
3900 | |
---|
3901 | def h = hilbertSeries(M); setring h; |
---|
3902 | |
---|
3903 | numerator1; denominator1; |
---|
3904 | |
---|
3905 | kill h; |
---|
3906 | //////////////////////////////////////////////// |
---|
3907 | setring R; |
---|
3908 | |
---|
3909 | kill M; module M = x^5*gen(1); |
---|
3910 | // intmat V[1][3] = 0; // TODO: this would lead to a wrong result!!!? |
---|
3911 | intmat V[1][1] = 0; // all gen(i) of degree 0! |
---|
3912 | |
---|
3913 | M = setModuleGrading(M, V); |
---|
3914 | |
---|
3915 | hilb(std(M)); |
---|
3916 | |
---|
3917 | def h = hilbertSeries(M); setring h; |
---|
3918 | |
---|
3919 | numerator1; denominator1; |
---|
3920 | |
---|
3921 | kill h; |
---|
3922 | //////////////////////////////////////////////// |
---|
3923 | setring R; |
---|
3924 | |
---|
3925 | module N = x^5*gen(3); |
---|
3926 | |
---|
3927 | kill V; |
---|
3928 | |
---|
3929 | intmat V[1][3] = 0; // all gen(i) of degree 0! |
---|
3930 | |
---|
3931 | N = setModuleGrading(N, V); |
---|
3932 | |
---|
3933 | hilb(std(N)); |
---|
3934 | |
---|
3935 | def h = hilbertSeries(N); setring h; |
---|
3936 | |
---|
3937 | numerator1; denominator1; |
---|
3938 | |
---|
3939 | kill h; |
---|
3940 | //////////////////////////////////////////////// |
---|
3941 | setring R; |
---|
3942 | |
---|
3943 | |
---|
3944 | module S = M + N; |
---|
3945 | |
---|
3946 | S = setModuleGrading(S, V); |
---|
3947 | |
---|
3948 | hilb(std(S)); |
---|
3949 | |
---|
3950 | def h = hilbertSeries(S); setring h; |
---|
3951 | |
---|
3952 | numerator1; denominator1; |
---|
3953 | |
---|
3954 | kill h; |
---|
3955 | |
---|
3956 | kill V; |
---|
3957 | kill R, W; |
---|
3958 | |
---|
3959 | } |
---|
3960 | |
---|
3961 | proc evalHilbertSeries(def h, intvec v) |
---|
3962 | " |
---|
3963 | evaluate hilbert series h by substibuting v[i] for t_(i) (1/v[i] for s_(i)) |
---|
3964 | return: int (h(v)) |
---|
3965 | " |
---|
3966 | { |
---|
3967 | if( 2*size(v) != nvars(h) ) |
---|
3968 | { |
---|
3969 | ERROR("Wrong input/size!"); |
---|
3970 | } |
---|
3971 | |
---|
3972 | setring h; |
---|
3973 | |
---|
3974 | if( defined(numerator2) and defined(denominator2) ) |
---|
3975 | { |
---|
3976 | poly n = numerator2; poly d = denominator2; |
---|
3977 | } else |
---|
3978 | { |
---|
3979 | poly n = numerator1; poly d = denominator1; |
---|
3980 | } |
---|
3981 | |
---|
3982 | int N = size(v); |
---|
3983 | int i; number k; |
---|
3984 | ideal V; |
---|
3985 | |
---|
3986 | for( i = N; i > 0; i -- ) |
---|
3987 | { |
---|
3988 | k = v[i]; |
---|
3989 | V[i] = var(i) - k; |
---|
3990 | } |
---|
3991 | |
---|
3992 | V = groebner(V); |
---|
3993 | |
---|
3994 | n = NF(n, V); |
---|
3995 | d = NF(d, V); |
---|
3996 | |
---|
3997 | n; |
---|
3998 | d; |
---|
3999 | |
---|
4000 | if( d == 0 ) |
---|
4001 | { |
---|
4002 | ERROR("Sorry: denominator is zero!"); |
---|
4003 | } |
---|
4004 | |
---|
4005 | if( n == 0 ) |
---|
4006 | { |
---|
4007 | return (0); |
---|
4008 | } |
---|
4009 | |
---|
4010 | poly g = gcd(n, d); |
---|
4011 | |
---|
4012 | if( g != leadcoef(g) ) |
---|
4013 | { |
---|
4014 | n = n / g; |
---|
4015 | d = d / g; |
---|
4016 | } |
---|
4017 | |
---|
4018 | n; |
---|
4019 | d; |
---|
4020 | |
---|
4021 | |
---|
4022 | for( i = N; i > 0; i -- ) |
---|
4023 | { |
---|
4024 | "i: ", i; |
---|
4025 | n; |
---|
4026 | d; |
---|
4027 | |
---|
4028 | k = v[i]; |
---|
4029 | k; |
---|
4030 | |
---|
4031 | n = subst(n, var(i), k); |
---|
4032 | d = subst(d, var(i), k); |
---|
4033 | |
---|
4034 | if( k != 0 ) |
---|
4035 | { |
---|
4036 | k = 1/k; |
---|
4037 | n = subst(n, var(N+i), k); |
---|
4038 | d = subst(d, var(N+i), k); |
---|
4039 | } |
---|
4040 | } |
---|
4041 | |
---|
4042 | n; |
---|
4043 | d; |
---|
4044 | |
---|
4045 | if( d == 0 ) |
---|
4046 | { |
---|
4047 | ERROR("Sorry: denominator is zero!"); |
---|
4048 | } |
---|
4049 | |
---|
4050 | if( n == 0 ) |
---|
4051 | { |
---|
4052 | return (0); |
---|
4053 | } |
---|
4054 | |
---|
4055 | poly g = gcd(n, d); |
---|
4056 | |
---|
4057 | if( g != leadcoef(g) ) |
---|
4058 | { |
---|
4059 | n = n / g; |
---|
4060 | d = d / g; |
---|
4061 | } |
---|
4062 | |
---|
4063 | n; |
---|
4064 | d; |
---|
4065 | |
---|
4066 | if( n != leadcoef(n) || d != leadcoef(d) ) |
---|
4067 | { |
---|
4068 | ERROR("Sorry cannot completely evaluate. Partial result: (" + string(n) + ")/(" + string(d) + ")"); |
---|
4069 | } |
---|
4070 | |
---|
4071 | n; |
---|
4072 | d; |
---|
4073 | |
---|
4074 | return (leadcoef(n)/leadcoef(d)); |
---|
4075 | } |
---|
4076 | example |
---|
4077 | { |
---|
4078 | "EXAMPLE:"; echo=2; |
---|
4079 | |
---|
4080 | // TODO! |
---|
4081 | |
---|
4082 | } |
---|
4083 | |
---|
4084 | |
---|
4085 | proc isPositive() |
---|
4086 | "USAGE: isPositive() |
---|
4087 | PURPOSE: Computes whether the multigrading of the ring is positive. |
---|
4088 | For computation theorem 8.6 of the Miller/Sturmfels book is used. |
---|
4089 | RETURNS: true if the multigrading is positive |
---|
4090 | EXAMPLE: example isPositive; shows an example |
---|
4091 | " |
---|
4092 | { |
---|
4093 | ideal I = multiDegBasis(0); |
---|
4094 | ideal J = attrib(I,"ZeroPart"); |
---|
4095 | /* |
---|
4096 | I am not quite sure what this ZeroPart is anymore. I thought it |
---|
4097 | should contain all monomials of degree 0, but then apparently 1 should |
---|
4098 | be contained. It makes sense to exclude 1, but was this also the intention? |
---|
4099 | */ |
---|
4100 | return(J==0); |
---|
4101 | } |
---|
4102 | example |
---|
4103 | { |
---|
4104 | echo = 2; printlevel = 3; |
---|
4105 | ring r = 0,(x,y),dp; |
---|
4106 | intmat A[1][2]=-1,1; |
---|
4107 | setBaseMultigrading(A); |
---|
4108 | isPositive(); |
---|
4109 | |
---|
4110 | intmat B[1][2]=1,1; |
---|
4111 | setBaseMultigrading(B); |
---|
4112 | isPositive(B); |
---|
4113 | } |
---|
4114 | |
---|
4115 | /////////////////////////////////////////////////////////////////////////////// |
---|
4116 | // testing for consistency of the library: |
---|
4117 | proc testMultigradingLib () |
---|
4118 | { |
---|
4119 | example setBaseMultigrading; |
---|
4120 | example setModuleGrading; |
---|
4121 | |
---|
4122 | example getVariableWeights; |
---|
4123 | example getLattice; |
---|
4124 | example getGradingGroup; |
---|
4125 | example getModuleGrading; |
---|
4126 | |
---|
4127 | |
---|
4128 | example multiDeg; |
---|
4129 | example multiDegPartition; |
---|
4130 | |
---|
4131 | |
---|
4132 | example hermiteNormalForm; |
---|
4133 | example isHomogeneous; |
---|
4134 | example isTorsionFree; |
---|
4135 | example pushForward; |
---|
4136 | example defineHomogeneous; |
---|
4137 | |
---|
4138 | example equalMultiDeg; |
---|
4139 | example isZeroElement; |
---|
4140 | |
---|
4141 | example multiDegResolution; |
---|
4142 | |
---|
4143 | "// ******************* example hilbertSeries ************************//"; |
---|
4144 | example hilbertSeries; |
---|
4145 | |
---|
4146 | |
---|
4147 | // example multiDegBasis; // needs 4ti2! |
---|
4148 | |
---|
4149 | "The End!"; |
---|
4150 | } |
---|
4151 | |
---|
4152 | |
---|
4153 | static proc multiDegTruncate(def M, intvec md) |
---|
4154 | { |
---|
4155 | "d: "; |
---|
4156 | print(md); |
---|
4157 | |
---|
4158 | "M: "; |
---|
4159 | module LL = M; // + L for d+1 |
---|
4160 | LL; |
---|
4161 | print(multiDeg(LL)); |
---|
4162 | |
---|
4163 | |
---|
4164 | intmat V = getModuleGrading(M); |
---|
4165 | intvec vi; |
---|
4166 | int s = nrows(M); |
---|
4167 | int r = nrows(V); |
---|
4168 | int i; |
---|
4169 | module L; def B; |
---|
4170 | for (i=s; i>0; i--) |
---|
4171 | { |
---|
4172 | "comp: ", i; |
---|
4173 | vi = V[1..r, i]; |
---|
4174 | "v[i]: "; vi; |
---|
4175 | |
---|
4176 | B = multiDegBasis(md - vi); // ZeroPart is always the same... |
---|
4177 | "B: "; B; |
---|
4178 | |
---|
4179 | L = L, B*gen(i); |
---|
4180 | } |
---|
4181 | L = simplify(L, 2); |
---|
4182 | L = setModuleGrading(L,V); |
---|
4183 | |
---|
4184 | "L: "; L; |
---|
4185 | print(multiDeg(L)); |
---|
4186 | |
---|
4187 | L = multiDegModulo(L, LL); |
---|
4188 | L = multiDegGroebner(L); |
---|
4189 | // L = minbase(prune(L)); |
---|
4190 | |
---|
4191 | "??????????"; |
---|
4192 | print(L); |
---|
4193 | print(multiDeg(L)); |
---|
4194 | |
---|
4195 | V = getModuleGrading(L); |
---|
4196 | |
---|
4197 | // take out other degrees |
---|
4198 | for(i = ncols(L); i > 0; i-- ) |
---|
4199 | { |
---|
4200 | if( !equalMultiDeg( multiDeg(getGradedGenerator(L, i)), md ) ) |
---|
4201 | { |
---|
4202 | L[i] = 0; |
---|
4203 | } |
---|
4204 | } |
---|
4205 | |
---|
4206 | L = simplify(L, 2); |
---|
4207 | L = setModuleGrading(L, V); |
---|
4208 | print(L); |
---|
4209 | print(multiDeg(L)); |
---|
4210 | |
---|
4211 | return(L); |
---|
4212 | } |
---|
4213 | example |
---|
4214 | { |
---|
4215 | "EXAMPLE:"; echo=2; |
---|
4216 | |
---|
4217 | // TODO! |
---|
4218 | ring r = 32003, (x,y), dp; |
---|
4219 | |
---|
4220 | intmat M[2][2] = |
---|
4221 | 1, 0, |
---|
4222 | 0, 1; |
---|
4223 | |
---|
4224 | setBaseMultigrading(M); |
---|
4225 | |
---|
4226 | intmat V[2][1] = |
---|
4227 | 0, |
---|
4228 | 0; |
---|
4229 | |
---|
4230 | "X:"; |
---|
4231 | module h1 = x; |
---|
4232 | h1 = setModuleGrading(h1, V); |
---|
4233 | multiDegTruncate(h1, multiDeg(x)); |
---|
4234 | multiDegTruncate(h1, multiDeg(y)); |
---|
4235 | |
---|
4236 | "XY:"; |
---|
4237 | module h2 = ideal(x, y); |
---|
4238 | h2 = setModuleGrading(h2, V); |
---|
4239 | multiDegTruncate(h2, multiDeg(x)); |
---|
4240 | multiDegTruncate(h2, multiDeg(y)); |
---|
4241 | multiDegTruncate(h2, multiDeg(xy)); |
---|
4242 | } |
---|
4243 | |
---|
4244 | |
---|
4245 | /******************************************************/ |
---|
4246 | /* Some functions on lattices. |
---|
4247 | TODO Tuebingen: - add functionality (see wiki) and |
---|
4248 | - adjust them to work for groups as well.*/ |
---|
4249 | /******************************************************/ |
---|
4250 | |
---|
4251 | |
---|
4252 | |
---|
4253 | /******************************************************/ |
---|
4254 | proc imageLattice(intmat Q, intmat L) |
---|
4255 | "USAGE: imageLattice(Q,L); Q and L are of type intmat |
---|
4256 | PURPOSE: compute an integral basis for the image of the |
---|
4257 | lattice L under the homomorphism of lattices Q. |
---|
4258 | RETURN: intmat |
---|
4259 | EXAMPLE: example imageLattice; shows an example |
---|
4260 | " |
---|
4261 | { |
---|
4262 | intmat Mul = Q*L; |
---|
4263 | intmat LL = latticeBasis(Mul); |
---|
4264 | |
---|
4265 | return(LL); |
---|
4266 | } |
---|
4267 | example |
---|
4268 | { |
---|
4269 | "EXAMPLE:"; echo=2; |
---|
4270 | |
---|
4271 | intmat Q[2][3] = |
---|
4272 | 1,2,3, |
---|
4273 | 3,2,1; |
---|
4274 | |
---|
4275 | intmat L[3][2] = |
---|
4276 | 1,4, |
---|
4277 | 2,5, |
---|
4278 | 3,6; |
---|
4279 | |
---|
4280 | // should be a 2x2 matrix with columns |
---|
4281 | // [2,-14], [0,36] |
---|
4282 | imageLattice(Q,L); |
---|
4283 | |
---|
4284 | } |
---|
4285 | |
---|
4286 | /******************************************************/ |
---|
4287 | proc intRank(intmat A) |
---|
4288 | " |
---|
4289 | USAGE: intRank(A); intmat A |
---|
4290 | PURPOSE: compute the rank of the integral matrix A |
---|
4291 | by computing a hermite normalform. |
---|
4292 | RETURNS: int |
---|
4293 | EXAMPLE: example intRank; shows an example |
---|
4294 | " |
---|
4295 | { |
---|
4296 | intmat B = hermiteNormalForm(A); |
---|
4297 | |
---|
4298 | // get number of zero columns |
---|
4299 | int nzerocols = 0; |
---|
4300 | int j; |
---|
4301 | int i; |
---|
4302 | int iszero; |
---|
4303 | for ( j = 1; j <= ncols(B); j++ ) |
---|
4304 | { |
---|
4305 | iszero = 1; |
---|
4306 | |
---|
4307 | for ( i = 1; i <= nrows(B); i++ ) |
---|
4308 | { |
---|
4309 | if ( B[i,j] != 0 ) |
---|
4310 | { |
---|
4311 | iszero = 0; |
---|
4312 | break; |
---|
4313 | } |
---|
4314 | } |
---|
4315 | |
---|
4316 | if ( iszero == 1 ) |
---|
4317 | { |
---|
4318 | nzerocols++; |
---|
4319 | } |
---|
4320 | } |
---|
4321 | |
---|
4322 | // get number of zero rows |
---|
4323 | int nzerorows = 0; |
---|
4324 | |
---|
4325 | for ( i = 1; i <= nrows(B); i++ ) |
---|
4326 | { |
---|
4327 | iszero = 1; |
---|
4328 | |
---|
4329 | for ( j = 1; j <= ncols(B); j++ ) |
---|
4330 | { |
---|
4331 | if ( B[i,j] != 0 ) |
---|
4332 | { |
---|
4333 | iszero = 0; |
---|
4334 | break; |
---|
4335 | } |
---|
4336 | } |
---|
4337 | |
---|
4338 | if ( iszero == 1 ) |
---|
4339 | { |
---|
4340 | nzerorows++; |
---|
4341 | } |
---|
4342 | } |
---|
4343 | |
---|
4344 | int r = nrows(B) - nzerorows; |
---|
4345 | |
---|
4346 | if ( ncols(B) - nzerocols < r ) |
---|
4347 | { |
---|
4348 | r = ncols(B) - nzerocols; |
---|
4349 | } |
---|
4350 | |
---|
4351 | return(r); |
---|
4352 | } |
---|
4353 | example |
---|
4354 | { |
---|
4355 | |
---|
4356 | intmat A[3][4] = |
---|
4357 | 1,0,1,0, |
---|
4358 | 1,2,0,0, |
---|
4359 | 0,0,0,0; |
---|
4360 | |
---|
4361 | int r = intRank(A); |
---|
4362 | |
---|
4363 | print(A); |
---|
4364 | print(r); // Should be 2 |
---|
4365 | |
---|
4366 | kill A; |
---|
4367 | |
---|
4368 | } |
---|
4369 | |
---|
4370 | /*****************************************************/ |
---|
4371 | |
---|
4372 | proc isSublattice(intmat L, intmat S) |
---|
4373 | "USAGE: isSublattice(L, S); L, S are of tpye intmat |
---|
4374 | PURPOSE: checks whether the lattice created by L is a |
---|
4375 | sublattice of the lattice created by S. |
---|
4376 | The procedure checks whether each generator of L is |
---|
4377 | contained in S. |
---|
4378 | RETURN: 0 if false, 1 if true |
---|
4379 | EXAMPLE: example isSublattice; shows an example |
---|
4380 | " |
---|
4381 | { |
---|
4382 | int a,b,g,i,j,k; |
---|
4383 | intmat Ker; |
---|
4384 | |
---|
4385 | // check whether each column v of L is contained in |
---|
4386 | // the lattice generated by S |
---|
4387 | for ( i = 1; i <= ncols(L); i++ ) |
---|
4388 | { |
---|
4389 | |
---|
4390 | // v is the i-th column of L |
---|
4391 | intvec v; |
---|
4392 | for ( j = 1; j <= nrows(L); j++ ) |
---|
4393 | { |
---|
4394 | v[j] = L[j,i]; |
---|
4395 | } |
---|
4396 | |
---|
4397 | // concatenate B = [S,v] |
---|
4398 | intmat B[nrows(L)][ncols(S) + 1]; |
---|
4399 | |
---|
4400 | for ( a = 1; a <= nrows(S); a++ ) |
---|
4401 | { |
---|
4402 | for ( b = 1; b <= ncols(S); b++ ) |
---|
4403 | { |
---|
4404 | B[a,b] = S[a,b]; |
---|
4405 | } |
---|
4406 | } |
---|
4407 | |
---|
4408 | for ( a = 1; a <= size(v); a++ ) |
---|
4409 | { |
---|
4410 | B[a,ncols(B)] = v[a]; |
---|
4411 | } |
---|
4412 | |
---|
4413 | |
---|
4414 | // check gcd |
---|
4415 | Ker = kernelLattice(B); |
---|
4416 | k = nrows(Ker); |
---|
4417 | list R; // R is the last row |
---|
4418 | |
---|
4419 | for ( j = 1; j <= ncols(Ker); j++ ) |
---|
4420 | { |
---|
4421 | R[j] = Ker[k,j]; |
---|
4422 | } |
---|
4423 | |
---|
4424 | g = R[1]; |
---|
4425 | |
---|
4426 | for ( j = 2; j <= size(R); j++ ) |
---|
4427 | { |
---|
4428 | g = gcd(g,R[j]); |
---|
4429 | } |
---|
4430 | |
---|
4431 | if ( g != 1 and g != -1 ) |
---|
4432 | { |
---|
4433 | return(0); |
---|
4434 | } |
---|
4435 | |
---|
4436 | kill B, v, R; |
---|
4437 | |
---|
4438 | } |
---|
4439 | |
---|
4440 | return(1); |
---|
4441 | } |
---|
4442 | example |
---|
4443 | { |
---|
4444 | "EXAMPLE:"; echo=2; |
---|
4445 | |
---|
4446 | //ring R = 0,(x,y),dp; |
---|
4447 | intmat S2[3][3]= |
---|
4448 | 0, 2, 3, |
---|
4449 | 0, 1, 1, |
---|
4450 | 3, 0, 2; |
---|
4451 | |
---|
4452 | intmat S1[3][2]= |
---|
4453 | 0, 6, |
---|
4454 | 0, 2, |
---|
4455 | 3, 4; |
---|
4456 | |
---|
4457 | isSublattice(S1,S2); // Yes! |
---|
4458 | |
---|
4459 | intmat S3[3][1] = |
---|
4460 | 0, |
---|
4461 | 0, |
---|
4462 | 1; |
---|
4463 | |
---|
4464 | not(isSublattice(S3,S2)); // Yes! |
---|
4465 | |
---|
4466 | } |
---|
4467 | |
---|
4468 | /******************************************************/ |
---|
4469 | |
---|
4470 | proc latticeBasis(intmat B) |
---|
4471 | "USAGE: latticeBasis(B); intmat B |
---|
4472 | PURPOSE: compute an integral basis for the lattice defined by |
---|
4473 | the columns of B. |
---|
4474 | RETURNS: intmat |
---|
4475 | EXAMPLE: example latticeBasis; shows an example |
---|
4476 | " |
---|
4477 | { |
---|
4478 | int n = ncols(B); |
---|
4479 | int r = intRank(B); |
---|
4480 | |
---|
4481 | if ( r == 0 ) |
---|
4482 | { |
---|
4483 | intmat H[nrows(B)][1]; |
---|
4484 | int j; |
---|
4485 | |
---|
4486 | for ( j = 1; j <= nrows(B); j++ ) |
---|
4487 | { |
---|
4488 | H[j,1] = 0; |
---|
4489 | } |
---|
4490 | } |
---|
4491 | else |
---|
4492 | { |
---|
4493 | intmat H = hermiteNormalForm(B);; |
---|
4494 | |
---|
4495 | if (r < n) |
---|
4496 | { |
---|
4497 | // delete columns r+1 to n |
---|
4498 | // should be identical with the function |
---|
4499 | // H = submat(H,1..nrows(H),1..r); |
---|
4500 | // for matrices |
---|
4501 | intmat Hdel[nrows(H)][r]; |
---|
4502 | int k; |
---|
4503 | int m; |
---|
4504 | |
---|
4505 | for ( k = 1; k <= nrows(H); k++ ) |
---|
4506 | { |
---|
4507 | for ( m = 1; m <= r; m++ ) |
---|
4508 | { |
---|
4509 | Hdel[k,m] = H[k,m]; |
---|
4510 | } |
---|
4511 | } |
---|
4512 | |
---|
4513 | H = Hdel; |
---|
4514 | } |
---|
4515 | } |
---|
4516 | |
---|
4517 | return(H); |
---|
4518 | } |
---|
4519 | example |
---|
4520 | { |
---|
4521 | "EXAMPLE:"; echo=2; |
---|
4522 | |
---|
4523 | intmat L[3][3] = |
---|
4524 | 1,4,8, |
---|
4525 | 2,5,10, |
---|
4526 | 3,6,12; |
---|
4527 | |
---|
4528 | intmat B = latticeBasis(L); |
---|
4529 | print(B); // should be a matrix with columns [1,2,3] and [0,3,6] |
---|
4530 | |
---|
4531 | kill B,L; |
---|
4532 | } |
---|
4533 | |
---|
4534 | /******************************************************/ |
---|
4535 | |
---|
4536 | proc kernelLattice(def P) |
---|
4537 | " |
---|
4538 | USAGE: kernelLattice(P); intmat P |
---|
4539 | PURPOSE: compute a integral basis for the kernel of the |
---|
4540 | homomorphism of lattices defined by the intmat P. |
---|
4541 | RETURNS: intmat |
---|
4542 | EXAMPLE: example kernelLattice; shows an example |
---|
4543 | " |
---|
4544 | { |
---|
4545 | int n = ncols(P); |
---|
4546 | int r = intRank(P); |
---|
4547 | |
---|
4548 | if ( r == 0 ) |
---|
4549 | { |
---|
4550 | intmat U = unitMatrix(n); |
---|
4551 | } |
---|
4552 | else |
---|
4553 | { |
---|
4554 | if ( r == n ) |
---|
4555 | { |
---|
4556 | intmat U[n][1]; // now all entries are zero. |
---|
4557 | } |
---|
4558 | else |
---|
4559 | { |
---|
4560 | list L = hermiteNormalForm(P, "transform"); //hermite(P, "transform"); // now, Hermite = L[1] = A*L[2] |
---|
4561 | intmat U = L[2]; |
---|
4562 | |
---|
4563 | // delete columns 1 to r |
---|
4564 | // should be identical with the function |
---|
4565 | // U = submat(U,1..nrows(U),r+1..); |
---|
4566 | // for matrices |
---|
4567 | intmat Udel[nrows(U)][ncols(U) - r]; |
---|
4568 | int k; |
---|
4569 | int m; |
---|
4570 | |
---|
4571 | for ( k = 1; k <= nrows(U); k++ ) |
---|
4572 | { |
---|
4573 | for ( m = r + 1; m <= ncols(U); m++ ) |
---|
4574 | { |
---|
4575 | Udel[k,m - r] = U[k,m]; |
---|
4576 | } |
---|
4577 | } |
---|
4578 | |
---|
4579 | U = Udel; |
---|
4580 | |
---|
4581 | } |
---|
4582 | } |
---|
4583 | |
---|
4584 | return(U); |
---|
4585 | } |
---|
4586 | example |
---|
4587 | { |
---|
4588 | "EXAMPLE"; echo = 2; |
---|
4589 | |
---|
4590 | intmat LL[3][4] = |
---|
4591 | 1,4,7,10, |
---|
4592 | 2,5,8,11, |
---|
4593 | 3,6,9,12; |
---|
4594 | |
---|
4595 | // should be a 4x2 matrix with colums |
---|
4596 | // [-1,2,-1,0],[2,-3,0,1] |
---|
4597 | intmat B = kernelLattice(LL); |
---|
4598 | |
---|
4599 | print(B); |
---|
4600 | |
---|
4601 | kill B; |
---|
4602 | |
---|
4603 | } |
---|
4604 | |
---|
4605 | /*****************************************************/ |
---|
4606 | |
---|
4607 | proc preimageLattice(def P, def B) |
---|
4608 | " |
---|
4609 | USAGE: preimageLattice(P, B); intmat P, intmat B |
---|
4610 | PURPOSE: compute an integral basis for the preimage of B under |
---|
4611 | the homomorphism of lattices defined by the intmat P. |
---|
4612 | RETURNS: intmat |
---|
4613 | EXAMPLE: example preimageLattice; shows an example |
---|
4614 | " |
---|
4615 | { |
---|
4616 | // concatenate matrices: Con = [P,-B] |
---|
4617 | intmat Con[nrows(P)][ncols(P) + ncols(B)]; |
---|
4618 | int i; |
---|
4619 | int j; |
---|
4620 | |
---|
4621 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4622 | { |
---|
4623 | for ( j = 1; j <= ncols(P); j++ ) // P first |
---|
4624 | { |
---|
4625 | Con[i,j] = P[i,j]; |
---|
4626 | } |
---|
4627 | } |
---|
4628 | |
---|
4629 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4630 | { |
---|
4631 | for ( j = 1; j <= ncols(B); j++ ) // now -B |
---|
4632 | { |
---|
4633 | Con[i,ncols(P) + j] = - B[i,j]; |
---|
4634 | } |
---|
4635 | } |
---|
4636 | |
---|
4637 | |
---|
4638 | // print(Con); |
---|
4639 | |
---|
4640 | intmat L = kernelLattice(Con); |
---|
4641 | /* |
---|
4642 | print(L); |
---|
4643 | print(ncols(P)); |
---|
4644 | print(ncols(L)); |
---|
4645 | */ |
---|
4646 | // delete rows ncols(P)+1 to nrows(L) out of L |
---|
4647 | intmat Del[ncols(P)][ncols(L)]; |
---|
4648 | int k; |
---|
4649 | int m; |
---|
4650 | |
---|
4651 | for ( k = 1; k <= nrows(Del); k++ ) |
---|
4652 | { |
---|
4653 | for ( m = 1; m <= ncols(Del); m++ ) |
---|
4654 | { |
---|
4655 | Del[k,m] = L[k,m]; |
---|
4656 | } |
---|
4657 | } |
---|
4658 | |
---|
4659 | L = latticeBasis(Del); |
---|
4660 | |
---|
4661 | return(L); |
---|
4662 | |
---|
4663 | } |
---|
4664 | example |
---|
4665 | { |
---|
4666 | "EXAMPLE"; echo = 2; |
---|
4667 | |
---|
4668 | intmat P[2][3] = |
---|
4669 | 2,6,10, |
---|
4670 | 4,8,12; |
---|
4671 | |
---|
4672 | intmat B[2][1] = |
---|
4673 | 1, |
---|
4674 | 0; |
---|
4675 | |
---|
4676 | intmat L = preimageLattice(P,B); |
---|
4677 | |
---|
4678 | print(L); // should be a (3x2)-matrix with columns e.g. [1,1,-1] and [0,3,-2] (the generated lattice should be identical) |
---|
4679 | |
---|
4680 | kill B, P, L; |
---|
4681 | |
---|
4682 | } |
---|
4683 | |
---|
4684 | /******************************************************/ |
---|
4685 | proc isPrimitiveSublattice(intmat A); |
---|
4686 | "USAGE: isPrimitiveSublattice(A); intmat A |
---|
4687 | PURPOSE: check whether the given set of integral vectors in ZZ^m, |
---|
4688 | i.e. the columns of A, generate a primitive sublattice in ZZ^m |
---|
4689 | (a direct summand of ZZ^m). |
---|
4690 | RETURNS: int, where 0 is false and 1 is true. |
---|
4691 | EXAMPLE: example isPrimitiveSublattice; shows an example |
---|
4692 | " |
---|
4693 | { |
---|
4694 | intmat B = smithNormalForm(A); |
---|
4695 | int r = intRank(B); |
---|
4696 | |
---|
4697 | if ( r == 0 ) |
---|
4698 | { |
---|
4699 | return(1); |
---|
4700 | } |
---|
4701 | |
---|
4702 | if ( 1 < B[r,r] ) |
---|
4703 | { |
---|
4704 | return(0); |
---|
4705 | } |
---|
4706 | |
---|
4707 | return(1); |
---|
4708 | } |
---|
4709 | example |
---|
4710 | { |
---|
4711 | "EXAMPLE"; echo = 2; |
---|
4712 | |
---|
4713 | intmat A[3][2] = |
---|
4714 | 1,4, |
---|
4715 | 2,5, |
---|
4716 | 3,6; |
---|
4717 | |
---|
4718 | // should be 0 |
---|
4719 | int b = isPrimitiveSublattice(A); |
---|
4720 | b; |
---|
4721 | |
---|
4722 | if( b != 0 ){ ERROR("Sorry, something went wrong..."); } |
---|
4723 | |
---|
4724 | kill A,b; |
---|
4725 | } |
---|
4726 | |
---|
4727 | /******************************************************/ |
---|
4728 | proc isIntegralSurjective(intmat P); |
---|
4729 | "USAGE: isIntegralSurjective(P); intmat P |
---|
4730 | PURPOSE: test whether the given linear map P of lattices is |
---|
4731 | surjective. |
---|
4732 | RETURNS: int, where 0 is false and 1 is true. |
---|
4733 | EXAMPLE: example isIntegralSurjective; shows an example |
---|
4734 | " |
---|
4735 | { |
---|
4736 | int r = intRank(P); |
---|
4737 | |
---|
4738 | if ( r < nrows(P) ) |
---|
4739 | { |
---|
4740 | return(0); |
---|
4741 | } |
---|
4742 | |
---|
4743 | if ( isPrimitiveSublattice(A) == 1 ) |
---|
4744 | { |
---|
4745 | return(1); |
---|
4746 | } |
---|
4747 | |
---|
4748 | return(0); |
---|
4749 | } |
---|
4750 | example |
---|
4751 | { |
---|
4752 | "EXAMPLE"; echo = 2; |
---|
4753 | |
---|
4754 | intmat A[3][2] = |
---|
4755 | 1,3,5, |
---|
4756 | 2,4,6; |
---|
4757 | |
---|
4758 | // should be 0 |
---|
4759 | int b = isIntegralSurjective(A); |
---|
4760 | print(b); |
---|
4761 | |
---|
4762 | kill A,b; |
---|
4763 | } |
---|
4764 | |
---|
4765 | /******************************************************/ |
---|
4766 | proc projectLattice(intmat B) |
---|
4767 | "USAGE: projectLattice(B); intmat B |
---|
4768 | PURPOSE: A set of vectors in ZZ^m is given as the columns of B. |
---|
4769 | Then this function provides a linear map ZZ^m --> ZZ^n |
---|
4770 | having the primitive span of B its kernel. |
---|
4771 | RETURNS: intmat |
---|
4772 | EXAMPLE: example projectLattice; shows an example |
---|
4773 | " |
---|
4774 | { |
---|
4775 | int n = nrows(B); |
---|
4776 | int r = intRank(B); |
---|
4777 | |
---|
4778 | if ( r == 0 ) |
---|
4779 | { |
---|
4780 | intmat U = unitMatrix(n); |
---|
4781 | } |
---|
4782 | else |
---|
4783 | { |
---|
4784 | if ( r == n ) |
---|
4785 | { |
---|
4786 | intmat U[n][1]; // U now is the n-dim zero-vector |
---|
4787 | } |
---|
4788 | else |
---|
4789 | { |
---|
4790 | // we want a matrix with column operations so we transpose |
---|
4791 | list L = hermiteNormalForm(B, "transform"); //hermite(transpose(B), "transform"); |
---|
4792 | intmat U = transpose(L[2]); |
---|
4793 | |
---|
4794 | // delete rows 1 to r |
---|
4795 | intmat Udel[nrows(U) - r][ncols(U)]; |
---|
4796 | int k; |
---|
4797 | int m; |
---|
4798 | |
---|
4799 | for ( k = 1; k <= nrows(U) - r ; k++ ) |
---|
4800 | { |
---|
4801 | for ( m = 1; m <= ncols(U); m++ ) |
---|
4802 | { |
---|
4803 | Udel[k,m] = U[k + r,m]; |
---|
4804 | } |
---|
4805 | } |
---|
4806 | |
---|
4807 | U = Udel; |
---|
4808 | |
---|
4809 | } |
---|
4810 | } |
---|
4811 | |
---|
4812 | return(U); |
---|
4813 | } |
---|
4814 | example |
---|
4815 | { |
---|
4816 | "EXAMPLE"; echo = 2; |
---|
4817 | |
---|
4818 | intmat B[4][2] = |
---|
4819 | 1,5, |
---|
4820 | 2,6, |
---|
4821 | 3,7, |
---|
4822 | 4,8; |
---|
4823 | |
---|
4824 | // should result in a (2x4)-matrix with columns |
---|
4825 | // [-1, 2], [2, -3], [-1, 0] and [0, 1] |
---|
4826 | intmat U = projectLattice(B); |
---|
4827 | |
---|
4828 | } |
---|
4829 | |
---|
4830 | /******************************************************/ |
---|
4831 | proc intersectLattices(intmat A, intmat B) |
---|
4832 | "USAGE: intersectLattices(A, B); intmat A, intmat B |
---|
4833 | PURPOSE: compute an integral basis for the intersection of the |
---|
4834 | lattices A and B. |
---|
4835 | RETURNS: intmat |
---|
4836 | EXAMPLE: example intersectLattices; shows an example |
---|
4837 | " |
---|
4838 | { |
---|
4839 | // concatenate matrices: Con = [A,-B] |
---|
4840 | intmat Con[nrows(A)][ncols(A) + ncols(B)]; |
---|
4841 | int i; |
---|
4842 | int j; |
---|
4843 | |
---|
4844 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4845 | { |
---|
4846 | for ( j = 1; j <= ncols(A); j++ ) // A first |
---|
4847 | { |
---|
4848 | Con[i,j] = A[i,j]; |
---|
4849 | } |
---|
4850 | } |
---|
4851 | |
---|
4852 | for ( i = 1; i <= nrows(Con); i++ ) |
---|
4853 | { |
---|
4854 | for ( j = 1; j <= ncols(B); j++ ) // now -B |
---|
4855 | { |
---|
4856 | Con[i,ncols(A) + j] = - B[i,j]; |
---|
4857 | } |
---|
4858 | } |
---|
4859 | |
---|
4860 | intmat K = kernelLattice(Con); |
---|
4861 | |
---|
4862 | // delete all rows in K from ncols(A)+1 onwards |
---|
4863 | intmat Bas[ncols(A)][ncols(K)]; |
---|
4864 | |
---|
4865 | for ( i = 1; i <= nrows(Bas); i++ ) |
---|
4866 | { |
---|
4867 | for ( j = 1; j <= ncols(Bas); j++ ) |
---|
4868 | { |
---|
4869 | Bas[i,j] = K[i,j]; |
---|
4870 | } |
---|
4871 | } |
---|
4872 | |
---|
4873 | // take product in order to obtain the intersection |
---|
4874 | intmat S = A * Bas; |
---|
4875 | intmat Cut = hermiteNormalForm(S); //hermite(S); |
---|
4876 | int r = intRank(Cut); |
---|
4877 | |
---|
4878 | if ( r == 0 ) |
---|
4879 | { |
---|
4880 | intmat Cutdel[nrows(Cut)][1]; // is now the zero-vector |
---|
4881 | |
---|
4882 | Cut = Cutdel; |
---|
4883 | } |
---|
4884 | else |
---|
4885 | { |
---|
4886 | // delte columns from r+1 onwards |
---|
4887 | intmat Cutdel[nrows(Cut)][r]; |
---|
4888 | |
---|
4889 | for ( i = 1; i <= nrows(Cutdel); i++ ) |
---|
4890 | { |
---|
4891 | for ( j = 1; j <= r; j++ ) |
---|
4892 | { |
---|
4893 | Cutdel[i,j] = Cut[i,j]; |
---|
4894 | } |
---|
4895 | } |
---|
4896 | |
---|
4897 | Cut = Cutdel; |
---|
4898 | } |
---|
4899 | |
---|
4900 | return(Cut); |
---|
4901 | } |
---|
4902 | example |
---|
4903 | { |
---|
4904 | "EXAMPLE"; echo = 2; |
---|
4905 | |
---|
4906 | intmat A[3][2] = |
---|
4907 | 1,4, |
---|
4908 | 2,5, |
---|
4909 | 3,6; |
---|
4910 | |
---|
4911 | intmat B[3][2] = |
---|
4912 | 6,9, |
---|
4913 | 7,10, |
---|
4914 | 8,11; |
---|
4915 | |
---|
4916 | intmat U = intersectLattices(A,B); |
---|
4917 | |
---|
4918 | // should result in a (2x4)-matrix with columns |
---|
4919 | // e.g. [3, 3, 3], [-3, 0, 3] (the lattice should be the same) |
---|
4920 | print(U); |
---|
4921 | } |
---|
4922 | |
---|
4923 | proc intInverse(intmat A); |
---|
4924 | "USAGE: intInverse(A); intmat A |
---|
4925 | PURPOSE: compute the integral inverse of the intmat A. |
---|
4926 | If det(A) is neither 1 nor -1 an error is returned. |
---|
4927 | RETURNS: intmat |
---|
4928 | EXAMPLE: example intInverse; shows an example |
---|
4929 | " |
---|
4930 | { |
---|
4931 | int d = det(A); |
---|
4932 | |
---|
4933 | if ( d * d != 1 ) // is d = 1 or -1? Else: error |
---|
4934 | { |
---|
4935 | ERROR("determinant of the given intmat has to be 1 or -1."); |
---|
4936 | } |
---|
4937 | |
---|
4938 | int c; |
---|
4939 | int i,j; |
---|
4940 | intmat C[nrows(A)][ncols(A)]; |
---|
4941 | intmat Ad; |
---|
4942 | int s; |
---|
4943 | |
---|
4944 | for ( i = 1; i <= nrows(C); i++ ) |
---|
4945 | { |
---|
4946 | for ( j = 1; j <= ncols(C); j++ ) |
---|
4947 | { |
---|
4948 | Ad = intAdjoint(A,i,j); |
---|
4949 | s = 1; |
---|
4950 | |
---|
4951 | if ( ((i + j) % 2) > 0 ) |
---|
4952 | { |
---|
4953 | s = -1; |
---|
4954 | } |
---|
4955 | |
---|
4956 | C[i,j] = d * s * det(Ad); // mult by d is equal to div by det |
---|
4957 | } |
---|
4958 | } |
---|
4959 | |
---|
4960 | C = transpose(C); |
---|
4961 | |
---|
4962 | return(C); |
---|
4963 | } |
---|
4964 | example |
---|
4965 | { |
---|
4966 | "EXAMPLE"; echo = 2; |
---|
4967 | |
---|
4968 | intmat A[3][3] = |
---|
4969 | 1,1,3, |
---|
4970 | 3,2,0, |
---|
4971 | 0,0,1; |
---|
4972 | |
---|
4973 | intmat B = intInverse(A); |
---|
4974 | |
---|
4975 | // should be the unit matrix |
---|
4976 | print(A * B); |
---|
4977 | |
---|
4978 | kill A,B; |
---|
4979 | } |
---|
4980 | |
---|
4981 | |
---|
4982 | /******************************************************/ |
---|
4983 | proc intAdjoint(intmat A, int indrow, int indcol) |
---|
4984 | "USAGE: intAdjoint(A); intmat A |
---|
4985 | PURPOSE: return the matrix where the given row and column are deleted. |
---|
4986 | RETURNS: intmat |
---|
4987 | EXAMPLE: example intAdjoint; shows an example |
---|
4988 | " |
---|
4989 | { |
---|
4990 | int n = nrows(A); |
---|
4991 | int m = ncols(A); |
---|
4992 | int i, j; |
---|
4993 | intmat B[n - 1][m - 1]; |
---|
4994 | int a, b; |
---|
4995 | |
---|
4996 | for ( i = 1; i < indrow; i++ ) |
---|
4997 | { |
---|
4998 | for ( j = 1; j < indcol; j++ ) |
---|
4999 | { |
---|
5000 | B[i,j] = A[i,j]; |
---|
5001 | } |
---|
5002 | for ( j = indcol + 1; j <= ncols(A); j++ ) |
---|
5003 | { |
---|
5004 | B[i,j - 1] = A[i,j]; |
---|
5005 | } |
---|
5006 | } |
---|
5007 | |
---|
5008 | for ( i = indrow + 1; i <= nrows(A); i++ ) |
---|
5009 | { |
---|
5010 | for ( j = 1; j < indcol; j++ ) |
---|
5011 | { |
---|
5012 | B[i - 1,j] = A[i,j]; |
---|
5013 | } |
---|
5014 | for ( j = indcol+1; j <= ncols(A); j++ ) |
---|
5015 | { |
---|
5016 | B[i - 1,j - 1] = A[i,j]; |
---|
5017 | } |
---|
5018 | } |
---|
5019 | |
---|
5020 | return(B); |
---|
5021 | } |
---|
5022 | example |
---|
5023 | { |
---|
5024 | "EXAMPLE"; echo = 2; |
---|
5025 | |
---|
5026 | intmat A[2][3] = |
---|
5027 | 1,3,5, |
---|
5028 | 2,4,6; |
---|
5029 | |
---|
5030 | intmat B = intAdjoint(A,2,2); |
---|
5031 | print(B); |
---|
5032 | |
---|
5033 | kill A,B; |
---|
5034 | } |
---|
5035 | |
---|
5036 | /******************************************************/ |
---|
5037 | proc integralSection(intmat P); |
---|
5038 | "USAGE: integralSection(P); intmat P |
---|
5039 | PURPOSE: for a given linear surjective map P of lattices |
---|
5040 | this procedure returns an integral section of P. |
---|
5041 | RETURNS: intmat |
---|
5042 | EXAMPLE: example integralSection; shows an example |
---|
5043 | " |
---|
5044 | { |
---|
5045 | int m = nrows(P); |
---|
5046 | int n = ncols(P); |
---|
5047 | |
---|
5048 | if ( m == n ) |
---|
5049 | { |
---|
5050 | intmat U = intInverse(P); |
---|
5051 | } |
---|
5052 | else |
---|
5053 | { |
---|
5054 | intmat U = (hermiteNormalForm(P, "transform"))[2]; |
---|
5055 | |
---|
5056 | // delete columns m+1 to n |
---|
5057 | intmat Udel[nrows(U)][ncols(U) - (n - m)]; |
---|
5058 | int k; |
---|
5059 | int z; |
---|
5060 | |
---|
5061 | for ( k = 1; k <= nrows(U); k++ ) |
---|
5062 | { |
---|
5063 | for ( z = 1; z <= m; z++ ) |
---|
5064 | { |
---|
5065 | Udel[k,z] = U[k,z]; |
---|
5066 | } |
---|
5067 | } |
---|
5068 | |
---|
5069 | U = Udel; |
---|
5070 | } |
---|
5071 | |
---|
5072 | return(U); |
---|
5073 | } |
---|
5074 | example |
---|
5075 | { |
---|
5076 | "EXAMPLE"; echo = 2; |
---|
5077 | |
---|
5078 | intmat P[2][4] = |
---|
5079 | 1,3,4,6, |
---|
5080 | 2,4,5,7; |
---|
5081 | |
---|
5082 | // should be a matrix with two columns |
---|
5083 | // for example: [-2, 1, 0, 0], [3, -3, 0, 1] |
---|
5084 | intmat U = integralSection(P); |
---|
5085 | |
---|
5086 | print(U); |
---|
5087 | print(P * U); |
---|
5088 | |
---|
5089 | kill U; |
---|
5090 | } |
---|
5091 | |
---|
5092 | |
---|
5093 | |
---|
5094 | /******************************************************/ |
---|
5095 | proc factorgroup(G,H) |
---|
5096 | "USAGE: factorgroup(G,H); list G, list H |
---|
5097 | PURPOSE: returns a representation of the factor group G mod H using the first isomorphism thm |
---|
5098 | RETURNS: list |
---|
5099 | EXAMPLE: example factorgroup(G,H); shows an example |
---|
5100 | " |
---|
5101 | { |
---|
5102 | intmat S1 = G[1]; |
---|
5103 | intmat L1 = G[2]; |
---|
5104 | intmat S2 = H[1]; |
---|
5105 | intmat L2 = H[2]; |
---|
5106 | |
---|
5107 | // check whether G,H are subgroups of a common group, i.e. whether L1 and L2 span the same lattice |
---|
5108 | if ( !isSublattice(L1,L2) || !isSublattice(L2,L1)) |
---|
5109 | { |
---|
5110 | ERROR("G and H are not subgroups of a common group."); |
---|
5111 | } |
---|
5112 | |
---|
5113 | // check whether H is a subgroup of G, i.e. whether S2 is a sublattice of S1+L1 |
---|
5114 | intmat B = concatintmat(S1,L1); // check whether this gives the concatinated matrix |
---|
5115 | if ( !isSublattice(S2,B) ) |
---|
5116 | { |
---|
5117 | ERROR("H is not a subgroup of G"); |
---|
5118 | } |
---|
5119 | // use first isomorphism thm to get the factor group |
---|
5120 | intmat L = concatintmat(L1,S2); // check whether this gives the concatinated matrix |
---|
5121 | list GmodH; |
---|
5122 | GmodH[1]=S1; |
---|
5123 | GmodH[2]=L; |
---|
5124 | return(GmodH); |
---|
5125 | } |
---|
5126 | example |
---|
5127 | { |
---|
5128 | "EXAMPLE"; echo = 2; |
---|
5129 | |
---|
5130 | intmat S1[2][2] = |
---|
5131 | 1,0, |
---|
5132 | 0,1; |
---|
5133 | intmat L1[2][1] = |
---|
5134 | 2, |
---|
5135 | 0; |
---|
5136 | |
---|
5137 | intmat S2[2][1] = |
---|
5138 | 1, |
---|
5139 | 0; |
---|
5140 | intmat L2[2][1] = |
---|
5141 | 2, |
---|
5142 | 0; |
---|
5143 | |
---|
5144 | list G = createGroup(S1,L1); |
---|
5145 | list H = createGroup(S2,L2); |
---|
5146 | |
---|
5147 | list N = factorgroup(G,H); |
---|
5148 | print(N); |
---|
5149 | |
---|
5150 | kill G,H,N,S1,L1,S2,L2; |
---|
5151 | |
---|
5152 | } |
---|
5153 | |
---|
5154 | /******************************************************/ |
---|
5155 | proc productgroup(G,H) |
---|
5156 | "USAGE: productgroup(G,H); list G, list H |
---|
5157 | PURPOSE: returns a representation of the G x H |
---|
5158 | RETURNS: list |
---|
5159 | EXAMPLE: example productgroup(G,H); shows an example |
---|
5160 | " |
---|
5161 | { |
---|
5162 | intmat S1 = G[1]; |
---|
5163 | intmat L1 = G[2]; |
---|
5164 | intmat S2 = H[1]; |
---|
5165 | intmat L2 = H[2]; |
---|
5166 | intmat OS1[nrows(S1)][ncols(S2)]; |
---|
5167 | intmat OS2[nrows(S2)][ncols(S1)]; |
---|
5168 | intmat OL1[nrows(L1)][ncols(L2)]; |
---|
5169 | intmat OL2[nrows(L2)][ncols(L1)]; |
---|
5170 | |
---|
5171 | // concatinate matrices to get S |
---|
5172 | intmat A = concatintmat(S1,OS1); |
---|
5173 | intmat B = concatintmat(OS2,S2); |
---|
5174 | intmat At = transpose(A); |
---|
5175 | intmat Bt = transpose(B); |
---|
5176 | intmat St = concatintmat(At,Bt); |
---|
5177 | intmat S = transpose(St); |
---|
5178 | |
---|
5179 | // concatinate matrices to get L |
---|
5180 | intmat C = concatintmat(L1,OL1); |
---|
5181 | intmat D = concatintmat(OL2,L2); |
---|
5182 | intmat Ct = transpose(C); |
---|
5183 | intmat Dt = transpose(D); |
---|
5184 | intmat Lt = concatintmat(Ct,Dt); |
---|
5185 | intmat L = transpose(Lt); |
---|
5186 | |
---|
5187 | list GxH; |
---|
5188 | GxH[1]=S; |
---|
5189 | GxH[2]=L; |
---|
5190 | return(GxH); |
---|
5191 | } |
---|
5192 | example |
---|
5193 | { |
---|
5194 | "EXAMPLE"; echo = 2; |
---|
5195 | |
---|
5196 | intmat S1[2][2] = |
---|
5197 | 1,0, |
---|
5198 | 0,1; |
---|
5199 | intmat L1[2][1] = |
---|
5200 | 2, |
---|
5201 | 0; |
---|
5202 | |
---|
5203 | intmat S2[2][2] = |
---|
5204 | 1,0, |
---|
5205 | 0,2; |
---|
5206 | intmat L2[2][1] = |
---|
5207 | 0, |
---|
5208 | 3; |
---|
5209 | |
---|
5210 | list G = createGroup(S1,L1); |
---|
5211 | list H = createGroup(S2,L2); |
---|
5212 | |
---|
5213 | list N = productgroup(G,H); |
---|
5214 | print(N); |
---|
5215 | |
---|
5216 | kill G,H,N,S1,L1,S2,L2; |
---|
5217 | |
---|
5218 | } |
---|
5219 | |
---|
5220 | /******************************************************/ |
---|
5221 | proc primitiveSpan(intmat V); |
---|
5222 | "USAGE: isIntegralSurjective(V); intmat V |
---|
5223 | PURPOSE: compute an integral basis for the minimal primitive |
---|
5224 | sublattice that contains the given vectors, i.e. the columns of V. |
---|
5225 | RETURNS: int, where 0 is false and 1 is true. |
---|
5226 | EXAMPLE: example isIntegralSurjective; shows an example |
---|
5227 | " |
---|
5228 | { |
---|
5229 | int n = ncols(V); |
---|
5230 | int m = nrows(V); |
---|
5231 | int r = intRank(V); |
---|
5232 | |
---|
5233 | |
---|
5234 | if ( r == 0 ) |
---|
5235 | { |
---|
5236 | intmat P[m][1]; // this is the m-zero-vector now |
---|
5237 | } |
---|
5238 | else |
---|
5239 | { |
---|
5240 | list L = smithNormalForm(V, "transform"); // L = [A,S,B] where S is the smith-NF and S = A*S*B |
---|
5241 | intmat P = intInverse(L[1]); |
---|
5242 | |
---|
5243 | // print(L); |
---|
5244 | |
---|
5245 | if ( r < m ) |
---|
5246 | { |
---|
5247 | // delete columns r+1 to m in P: |
---|
5248 | intmat Pdel[nrows(P)][r]; |
---|
5249 | int i,j; |
---|
5250 | |
---|
5251 | for ( i = 1; i <= nrows(Pdel); i++ ) |
---|
5252 | { |
---|
5253 | for ( j = 1; j <= ncols(Pdel); j++ ) |
---|
5254 | { |
---|
5255 | Pdel[i,j] = P[i,j]; |
---|
5256 | } |
---|
5257 | } |
---|
5258 | |
---|
5259 | P = Pdel; |
---|
5260 | } |
---|
5261 | } |
---|
5262 | |
---|
5263 | return(P); |
---|
5264 | } |
---|
5265 | example |
---|
5266 | { |
---|
5267 | "EXAMPLE"; echo = 2; |
---|
5268 | |
---|
5269 | intmat V[2][4] = |
---|
5270 | 1,4, |
---|
5271 | 2,5, |
---|
5272 | 3,6; |
---|
5273 | |
---|
5274 | // should return a (4x2)-matrix with columns |
---|
5275 | // [1, 2, 3] and [1, 1, 1] (or similar) |
---|
5276 | intmat R = primitiveSpan(V); |
---|
5277 | print(R); |
---|
5278 | |
---|
5279 | kill V,R; |
---|
5280 | } |
---|
5281 | |
---|
5282 | /***********************************************************/ |
---|