1 | /////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Combinatorial Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: multigrading.lib Multigraded Rings |
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6 | |
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7 | AUTHORS: Rene Birkner, rbirkner@math.fu-berlin.de |
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8 | @* Lars Kastner, lkastner@math.fu-berlin.de |
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9 | @* Oleksandr Motsak, U@D, where U={motsak}, D={mathematik.uni-kl.de} |
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10 | |
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11 | |
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12 | OVERVIEW: using this library allows one can virtually add multigrading to Singular. |
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13 | For more see http://code.google.com/p/convex-singular/wiki/Multigrading |
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14 | For theoretical references see: |
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15 | E. Miller, B. Sturmfels: 'Combinatorial Commutative Algebra' and |
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16 | M. Kreuzer, L. Robbiano: 'Computational Commutative Algebra'. |
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17 | |
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18 | NOTE: 'mDegBasis' relies on 4ti2 for computing Hilbert Bases. |
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19 | |
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20 | PROCEDURES: |
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21 | setBaseMultigrading(M,T); attach multiweights/torsion matrices to the basering |
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22 | getVariableWeights([R]); get matrix of multidegrees of vars attached to a ring |
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23 | getTorsion([R]); get torsion matrix attached to a ring |
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24 | |
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25 | setModuleGrading(M,v); attach multiweights of units to a module and return it |
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26 | getModuleGrading(M); get multiweights of module units (attached to M) |
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27 | |
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28 | mDeg(A); compute the multidegree of A |
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29 | mDegBasis(d); compute all monomials of multidegree d |
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30 | mDegPartition(p); compute the multigraded-homogenous components of p |
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31 | |
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32 | isTorsionFree(); test whether the current multigrading is torsion free |
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33 | isTorsionElement(p); test whether p has zero multidegree |
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34 | isHomogenous(a); test whether 'a' is multigraded-homogenous |
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35 | |
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36 | equalMDeg(e1,e2[,V]); test whether e1==e2 in the current multigrading |
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37 | |
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38 | mDegGroebner(M); compute the multigraded GB/SB of M |
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39 | mDegSyzygy(M); compute the multigraded syzygies of M |
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40 | mDegResolution(M,l[,m]); compute the multigraded resolution of M |
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41 | |
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42 | defineHomogenous(p); get a torsion matrix wrt which p becomes homogenous |
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43 | pushForward(f); find the finest grading on the image ring, homogenizing f |
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44 | |
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45 | hermite(A); compute the Hermite Normal Form of a matrix |
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46 | |
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47 | hilbertSeries(M); compute the multigraded Hilbert Series of M |
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48 | |
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49 | (parameters in square brackets [] are optional) |
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50 | |
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51 | KEYWORDS: multigradeding, multidegree, multiweights, multigraded-homogenous |
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52 | "; |
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53 | |
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54 | // finestMDeg(def r) |
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55 | // newMap(map F, intmat Q, list #) |
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56 | |
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57 | LIB "standard.lib"; // for groebner |
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58 | |
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59 | /******************************************************/ |
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60 | proc setBaseMultigrading(intmat M, list #) |
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61 | "USAGE: setBaseMultigrading(M[, T]); M, T are integer matrices |
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62 | PURPOSE: attaches weights of variables and torsion to the basering. |
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63 | NOTE: M encodes the weights of variables column-wise. |
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64 | The torsion is given by the lattice spanned by the columns of the integer |
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65 | matrix T in Z^nrows(M) over Z. |
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66 | RETURN: nothing |
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67 | EXAMPLE: example setBaseMultigrading; shows an example |
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68 | " |
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69 | { |
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70 | string attrMgrad = "mgrad"; |
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71 | string attrTorsion = "torsion"; |
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72 | string attrTorsionHNF = "torsion_hermite"; |
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73 | |
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74 | |
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75 | attrib(basering, attrMgrad, M); |
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76 | |
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77 | if( size(#) > 0 and typeof(#[1]) == "intmat" ) |
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78 | { |
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79 | def T = #[1]; |
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80 | } else |
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81 | { |
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82 | intmat T[nrows(M)][1]; |
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83 | } |
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84 | |
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85 | if( nrows(T) == nrows(M) ) |
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86 | { |
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87 | attrib(basering, attrTorsion, T); |
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88 | def H; |
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89 | attrib(basering, attrTorsionHNF, H); |
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90 | } |
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91 | else |
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92 | { |
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93 | ERROR("Incompatible matrix sizes!"); |
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94 | } |
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95 | } |
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96 | example |
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97 | { |
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98 | "EXAMPLE:"; echo=2; |
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99 | |
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100 | ring R = 0, (x, y, z), dp; |
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101 | |
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102 | // Weights of variables |
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103 | intmat M[3][3] = |
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104 | 1, 0, 0, |
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105 | 0, 1, 0, |
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106 | 0, 0, 1; |
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107 | |
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108 | // Torsion: |
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109 | intmat L[3][2] = |
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110 | 1, 1, |
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111 | 1, 3, |
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112 | 1, 5; |
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113 | |
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114 | // attaches M & L to R (==basering): |
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115 | setBaseMultigrading(M, L); // Grading: Z^3/L |
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116 | |
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117 | // Weights are accessible via "getVariableWeights()": |
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118 | getVariableWeights() == M; |
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119 | getVariableWeights(R) == M; |
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120 | getVariableWeights(basering) == M; |
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121 | |
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122 | // Torsion is accessible via "getTorsion()": |
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123 | getTorsion() == L; |
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124 | getTorsion(R) == L; |
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125 | getTorsion(basering) == L; |
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126 | |
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127 | // And its hermite NF via getTorsion("hermite"): |
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128 | intmat H = hermite(L); |
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129 | getTorsion("hermite") == H; |
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130 | getTorsion(R, "hermite") == H; |
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131 | getTorsion(basering, "hermite") == H; |
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132 | |
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133 | |
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134 | |
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135 | kill L, M; |
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136 | |
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137 | // ----------- isomorphic multigrading -------- // |
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138 | |
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139 | // Weights of variables |
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140 | intmat M[2][3] = |
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141 | 1, -2, 1, |
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142 | 1, 1, 0; |
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143 | |
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144 | // Torsion: |
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145 | intmat L[2][1] = |
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146 | 0, |
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147 | 2; |
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148 | |
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149 | // attaches M & L to R (==basering): |
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150 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
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151 | |
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152 | // Weights are accessible via "getVariableWeights()": |
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153 | getVariableWeights() == M; |
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154 | |
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155 | // Torsion is accessible via "getTorsion()": |
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156 | getTorsion() == L; |
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157 | |
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158 | kill L, M; |
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159 | // ----------- extreme case ------------ // |
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160 | |
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161 | // Weights of variables |
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162 | intmat M[1][3] = |
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163 | 1, -1, 10; |
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164 | |
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165 | // Torsion: |
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166 | intmat L[1][1] = |
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167 | 0; |
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168 | |
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169 | // attaches M & L to R (==basering): |
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170 | setBaseMultigrading(M); // Grading: Z^3 |
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171 | |
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172 | // Weights are accessible via "getVariableWeights()": |
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173 | getVariableWeights() == M; |
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174 | |
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175 | // Torsion is accessible via "getTorsion()": |
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176 | getTorsion() == L; |
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177 | } |
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178 | |
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179 | |
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180 | /******************************************************/ |
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181 | proc getVariableWeights(list #) |
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182 | "USAGE: getVariableWeights([R]) |
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183 | PURPOSE: get associated multigrading matrix for the basering [or R] |
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184 | RETURN: intmat, matrix of multidegrees of variables |
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185 | EXAMPLE: example getVariableWeights; shows an example |
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186 | " |
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187 | { |
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188 | string attrMgrad = "mgrad"; |
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189 | |
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190 | |
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191 | if( size(#) > 0 ) |
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192 | { |
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193 | if(( typeof(#[1]) == "ring" ) || ( typeof(#[1]) == "qring" )) |
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194 | { |
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195 | def R = #[1]; |
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196 | } |
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197 | else |
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198 | { |
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199 | ERROR("Optional argument must be a ring!"); |
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200 | } |
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201 | } |
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202 | else |
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203 | { |
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204 | def R = basering; |
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205 | } |
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206 | |
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207 | def M = attrib(R, attrMgrad); |
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208 | if( typeof(M) == "intmat"){ return (M); } |
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209 | ERROR( "Sorry no multigrading matrix!" ); |
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210 | } |
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211 | example |
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212 | { |
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213 | "EXAMPLE:"; echo=2; |
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214 | |
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215 | ring R = 0, (x, y, z), dp; |
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216 | |
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217 | // Weights of variables |
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218 | intmat M[3][3] = |
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219 | 1, 0, 0, |
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220 | 0, 1, 0, |
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221 | 0, 0, 1; |
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222 | |
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223 | // Torsion: |
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224 | intmat L[3][2] = |
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225 | 1, 1, |
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226 | 1, 3, |
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227 | 1, 5; |
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228 | |
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229 | // attaches M & L to R (==basering): |
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230 | setBaseMultigrading(M, L); // Grading: Z^3/L |
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231 | |
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232 | // Weights are accessible via "getVariableWeights()": |
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233 | getVariableWeights() == M; |
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234 | |
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235 | kill L, M; |
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236 | |
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237 | // ----------- isomorphic multigrading -------- // |
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238 | |
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239 | // Weights of variables |
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240 | intmat M[2][3] = |
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241 | 1, -2, 1, |
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242 | 1, 1, 0; |
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243 | |
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244 | // Torsion: |
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245 | intmat L[2][1] = |
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246 | 0, |
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247 | 2; |
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248 | |
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249 | // attaches M & L to R (==basering): |
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250 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
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251 | |
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252 | // Weights are accessible via "getVariableWeights()": |
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253 | getVariableWeights() == M; |
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254 | |
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255 | kill L, M; |
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256 | |
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257 | // ----------- extreme case ------------ // |
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258 | |
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259 | // Weights of variables |
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260 | intmat M[1][3] = |
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261 | 1, -1, 10; |
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262 | |
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263 | // Torsion: |
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264 | intmat L[1][1] = |
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265 | 0; |
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266 | |
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267 | // attaches M & L to R (==basering): |
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268 | setBaseMultigrading(M); // Grading: Z^3 |
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269 | |
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270 | // Weights are accessible via "getVariableWeights()": |
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271 | getVariableWeights() == M; |
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272 | } |
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273 | |
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274 | /******************************************************/ |
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275 | proc getTorsion(list #) |
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276 | "USAGE: getTorsion([R[,opt]]) |
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277 | PURPOSE: get associated torsion matrix, i.e. generators (cols) of the Torsion group |
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278 | RETURN: intmat, the torsion matrix, or its hermite normal form |
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279 | if an optional argument (\"hermite\") is given |
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280 | " |
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281 | { |
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282 | string attrTorsion = "torsion"; |
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283 | string attrTorsionHNF = "torsion_hermite"; |
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284 | |
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285 | int i = 1; |
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286 | |
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287 | if( size(#) >= i ) |
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288 | { |
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289 | if( ( typeof(#[i]) == "ring" ) or ( typeof(#[i]) == "qring" ) ) |
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290 | { |
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291 | def R = #[i]; |
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292 | i++; |
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293 | } |
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294 | } |
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295 | |
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296 | if( !defined(R) ) |
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297 | { |
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298 | def R = basering; |
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299 | } |
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300 | |
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301 | if( size(#) >= i ) |
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302 | { |
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303 | if( #[i] == "hermite" ) |
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304 | { |
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305 | if( typeof(attrib(R, attrTorsionHNF)) != "intmat" ) |
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306 | { |
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307 | def M = getTorsion(R); |
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308 | if( typeof(M) != "intmat") |
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309 | { |
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310 | ERROR( "Sorry no torsion matrix!" ); |
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311 | } |
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312 | attrib(R, attrTorsionHNF, hermite(M)); // this might not work with R != basering... |
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313 | } |
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314 | return (attrib(R, attrTorsionHNF)); |
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315 | } |
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316 | } |
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317 | |
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318 | def M = attrib(R, attrTorsion); |
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319 | if( typeof(M) != "intmat") |
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320 | { |
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321 | ERROR( "Sorry no torsion matrix!" ); |
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322 | } |
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323 | return (M); |
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324 | } |
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325 | example |
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326 | { |
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327 | "EXAMPLE:"; echo=2; |
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328 | |
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329 | ring R = 0, (x, y, z), dp; |
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330 | |
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331 | // Weights of variables |
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332 | intmat M[3][3] = |
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333 | 1, 0, 0, |
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334 | 0, 1, 0, |
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335 | 0, 0, 1; |
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336 | |
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337 | // Torsion: |
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338 | intmat L[3][2] = |
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339 | 1, 1, |
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340 | 1, 3, |
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341 | 1, 5; |
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342 | |
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343 | // attaches M & L to R (==basering): |
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344 | setBaseMultigrading(M, L); // Grading: Z^3/L |
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345 | |
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346 | // Torsion is accessible via "getTorsion()": |
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347 | getTorsion() == L; |
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348 | |
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349 | // its hermite NF: |
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350 | print(getTorsion("hermite")); |
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351 | |
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352 | kill L, M; |
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353 | |
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354 | // ----------- isomorphic multigrading -------- // |
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355 | |
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356 | // Weights of variables |
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357 | intmat M[2][3] = |
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358 | 1, -2, 1, |
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359 | 1, 1, 0; |
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360 | |
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361 | // Torsion: |
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362 | intmat L[2][1] = |
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363 | 0, |
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364 | 2; |
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365 | |
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366 | // attaches M & L to R (==basering): |
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367 | setBaseMultigrading(M, L); // Grading: Z + (Z/2Z) |
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368 | |
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369 | // Torsion is accessible via "getTorsion()": |
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370 | getTorsion() == L; |
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371 | |
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372 | // its hermite NF: |
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373 | print(getTorsion("hermite")); |
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374 | |
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375 | kill L, M; |
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376 | |
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377 | // ----------- extreme case ------------ // |
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378 | |
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379 | // Weights of variables |
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380 | intmat M[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 | // attaches M & L to R (==basering): |
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388 | setBaseMultigrading(M); // Grading: Z^3 |
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389 | |
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390 | // Torsion is accessible via "getTorsion()": |
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391 | getTorsion() == L; |
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392 | |
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393 | // its hermite NF: |
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394 | print(getTorsion("hermite")); |
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395 | } |
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396 | |
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397 | /******************************************************/ |
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398 | proc getModuleGrading(def m) |
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399 | "USAGE: getModuleGrading(m), 'm' module/vector |
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400 | RETURN: integer matrix of the multiweights of free module generators attached to 'm' |
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401 | " |
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402 | { |
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403 | string attrModuleGrading = "genWeights"; |
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404 | |
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405 | // print(m); typeof(m); attrib(m); |
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406 | |
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407 | def V = attrib(m, attrModuleGrading); |
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408 | |
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409 | if( typeof(V) != "intmat" ) |
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410 | { |
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411 | if( (typeof(m) == "ideal") or (typeof(m) == "poly") ) |
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412 | { |
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413 | intmat M = getVariableWeights(); |
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414 | intmat VV[nrows(M)][1]; |
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415 | return (VV); |
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416 | } |
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417 | |
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418 | ERROR("Sorry: vector or module need module-grading-matrix! See 'getModuleGrading'."); |
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419 | } |
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420 | |
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421 | if( nrows(V) != nrows(getVariableWeights()) ) |
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422 | { |
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423 | ERROR("Sorry wrong height of V: " + string(nrows(V))); |
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424 | } |
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425 | |
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426 | if( ncols(V) < nrows(m) ) |
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427 | { |
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428 | ERROR("Sorry wrong width of V: " + string(ncols(V))); |
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429 | } |
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430 | |
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431 | return (V); |
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432 | } |
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433 | example |
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434 | { |
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435 | "EXAMPLE:"; echo=2; |
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436 | |
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437 | ring R = 0, (x,y), dp; |
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438 | intmat M[2][2]= |
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439 | 1, 1, |
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440 | 0, 2; |
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441 | intmat T[2][5]= |
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442 | 1, 2, 3, 4, 0, |
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443 | 0, 10, 20, 30, 1; |
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444 | |
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445 | setBaseMultigrading(M, T); |
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446 | |
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447 | ideal I = x, y, xy^5; |
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448 | isHomogenous(I); |
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449 | |
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450 | intmat V = mDeg(I); print(V); |
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451 | |
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452 | module S = syz(I); print(S); |
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453 | |
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454 | S = setModuleGrading(S, V); |
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455 | |
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456 | getModuleGrading(S) == V; |
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457 | |
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458 | vector v = setModuleGrading(S[1], V); |
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459 | getModuleGrading(v) == V; |
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460 | isHomogenous(v); |
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461 | print( mDeg(v) ); |
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462 | |
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463 | isHomogenous(S); |
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464 | print( mDeg(S) ); |
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465 | } |
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466 | |
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467 | /******************************************************/ |
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468 | proc setModuleGrading(def m, intmat G) |
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469 | "USAGE: setModuleGrading(m, G), m module/vector, G intmat |
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470 | PURPOSE: attaches the multiweights of free module generators to 'm' |
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471 | WARNING: The method does not verify that the multigrading makes the |
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472 | module/vector homogenous. One can do that using isHomogenous(m). |
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473 | " |
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474 | { |
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475 | string attrModuleGrading = "genWeights"; |
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476 | |
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477 | intmat R = getVariableWeights(); |
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478 | |
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479 | if(nrows(G) != nrows(R)){ ERROR("Incompatible gradings.");} |
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480 | if(ncols(G) < nrows(m)){ ERROR("Multigrading does not fit to module.");} |
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481 | |
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482 | attrib(m, attrModuleGrading, G); |
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483 | return(m); |
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484 | } |
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485 | example |
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486 | { |
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487 | "EXAMPLE:"; echo=2; |
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488 | |
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489 | ring R = 0, (x,y), dp; |
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490 | intmat M[2][2]= |
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491 | 1, 1, |
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492 | 0, 2; |
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493 | intmat T[2][5]= |
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494 | 1, 2, 3, 4, 0, |
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495 | 0, 10, 20, 30, 1; |
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496 | |
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497 | setBaseMultigrading(M, T); |
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498 | |
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499 | ideal I = x, y, xy^5; |
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500 | intmat V = mDeg(I); |
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501 | |
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502 | // V == M; modulo T |
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503 | print(V); |
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504 | |
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505 | module S = syz(I); |
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506 | |
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507 | S = setModuleGrading(S, V); |
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508 | getModuleGrading(S) == V; |
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509 | |
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510 | print(S); |
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511 | |
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512 | vector v = S[1]; v = setModuleGrading(v, V); |
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513 | getModuleGrading(v) == V; |
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514 | |
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515 | print( mDeg(v) ); |
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516 | |
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517 | isHomogenous(S); |
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518 | |
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519 | print( mDeg(S) ); |
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520 | } |
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521 | |
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522 | |
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523 | |
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524 | |
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525 | /******************************************************/ |
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526 | proc isTorsionFree() |
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527 | "USAGE: isTorsionFree() |
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528 | PURPOSE: Determines whether the multigrading attached to the current ring is torsion-free. |
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529 | RETURN: boolean, the result of the test |
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530 | " |
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531 | { |
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532 | |
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533 | intmat H = hermite(transpose(getTorsion("hermite"))); // TODO: ?cache it? |
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534 | |
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535 | int i, j; |
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536 | int r = nrows(H); |
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537 | int c = ncols(H); |
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538 | int d = 1; |
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539 | for( i = 1; (i <= c) && (i <= r); i++ ) |
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540 | { |
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541 | for( j = i; (H[j, i] == 0)&&(j < r); j++ ) |
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542 | { |
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543 | } |
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544 | |
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545 | if(H[j, i]!=0) |
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546 | { |
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547 | d=d*H[j, i]; |
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548 | } |
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549 | } |
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550 | |
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551 | if( (d*d)==1 ) |
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552 | { |
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553 | return(1==1); |
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554 | } |
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555 | return(0==1); |
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556 | } |
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557 | example |
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558 | { |
---|
559 | "EXAMPLE:"; echo=2; |
---|
560 | |
---|
561 | ring R = 0,(x,y),dp; |
---|
562 | intmat M[2][2]= |
---|
563 | 1,0, |
---|
564 | 0,1; |
---|
565 | intmat T[2][5]= |
---|
566 | 1, 2, 3, 4, 0, |
---|
567 | 0,10,20,30, 1; |
---|
568 | |
---|
569 | setBaseMultigrading(M,T); |
---|
570 | |
---|
571 | // Is the resulting group torsion free? |
---|
572 | isTorsionFree(); |
---|
573 | |
---|
574 | kill R, M, T; |
---|
575 | /////////////////////////////////////////// |
---|
576 | |
---|
577 | ring R=0,(x,y,z),dp; |
---|
578 | intmat A[3][3] = |
---|
579 | 1,0,0, |
---|
580 | 0,1,0, |
---|
581 | 0,0,1; |
---|
582 | intmat B[3][4]= |
---|
583 | 3,3,3,3, |
---|
584 | 2,1,3,0, |
---|
585 | 1,2,0,3; |
---|
586 | setBaseMultigrading(A,B); |
---|
587 | // Is the resulting group torsion free? |
---|
588 | isTorsionFree(); |
---|
589 | |
---|
590 | kill R, A, B; |
---|
591 | } |
---|
592 | |
---|
593 | |
---|
594 | |
---|
595 | /******************************************************/ |
---|
596 | proc hermite(intmat A) |
---|
597 | "USAGE: hermite( A ); |
---|
598 | PROCEDURE: Computes the Hermite Normal Form of the matrix A by column operations. |
---|
599 | RETURN: intmat, the Hermite Normal Form of A |
---|
600 | " |
---|
601 | { |
---|
602 | intmat g; |
---|
603 | int i,j,k, save, d, a1, a2, c, l; |
---|
604 | c=0; |
---|
605 | l=0; |
---|
606 | for(i=1; ((i+l)<=nrows(A))&&((i+l)<=ncols(A)); i++) |
---|
607 | { |
---|
608 | //i; |
---|
609 | if(A[i+l, i]==0) |
---|
610 | { |
---|
611 | for(j=i;j<=ncols(A); j++) |
---|
612 | { |
---|
613 | if(A[i+l, j]!=0) |
---|
614 | { |
---|
615 | for(k=1;k<=nrows(A);k++) |
---|
616 | { |
---|
617 | save=A[k, i]; |
---|
618 | A[k, i]=A[k, j]; |
---|
619 | A[k, j]=save; |
---|
620 | } |
---|
621 | break; |
---|
622 | } |
---|
623 | if(j==ncols(A)){ c=1; l=l+1; } |
---|
624 | } |
---|
625 | } |
---|
626 | |
---|
627 | if((i+l)>nrows(A)){ break; } |
---|
628 | |
---|
629 | if(A[i+l, i]<0) |
---|
630 | { |
---|
631 | for(k=1;k<=nrows(A);k++){ A[k, i]=A[k, i]*(-1); } |
---|
632 | } |
---|
633 | |
---|
634 | if(c==0) |
---|
635 | { |
---|
636 | for(j=i+1;j<=ncols(A);j++) |
---|
637 | { |
---|
638 | //A; |
---|
639 | if(A[i+l, j]<0) |
---|
640 | { |
---|
641 | for(k=1;k<=nrows(A);k++){ A[k, j]=(-1)*A[k, j];} |
---|
642 | } |
---|
643 | |
---|
644 | if(A[i+l, i]==1) |
---|
645 | { |
---|
646 | d=A[i+l, j]; |
---|
647 | for(k=1;k<=nrows(A);k++) |
---|
648 | { |
---|
649 | A[k, j]=A[k, j]-d*A[k, i]; |
---|
650 | } |
---|
651 | } |
---|
652 | else |
---|
653 | { |
---|
654 | while((A[i+l, i] * A[i+l, j])!=0) |
---|
655 | { |
---|
656 | if(A[i+l, i]> A[i+l, j]) |
---|
657 | { |
---|
658 | |
---|
659 | for(k=1;k<=nrows(A);k++) |
---|
660 | { |
---|
661 | save=A[k, i]; |
---|
662 | A[k, i]=A[k, j]; |
---|
663 | A[k, j]=save; |
---|
664 | } |
---|
665 | } |
---|
666 | a1=A[i+l, j]%A[i+l,i]; |
---|
667 | a2=A[i+l, j]-a1; |
---|
668 | d=a2/A[i+l, i]; |
---|
669 | for(k=1;k<=nrows(A);k++) |
---|
670 | { |
---|
671 | A[k, j]=A[k, j]- d*A[k, i]; |
---|
672 | } |
---|
673 | } |
---|
674 | } |
---|
675 | } |
---|
676 | for(j=1;j<i;j++) |
---|
677 | { |
---|
678 | a1=A[i+l, j]%A[i+l,i]; |
---|
679 | d=(A[i+l, j]-a1)/A[i+l, i]; |
---|
680 | for(k=1;k<=nrows(A);k++){ A[k, j]=A[k, j]-d*A[k, i];} |
---|
681 | } |
---|
682 | } |
---|
683 | c=0; |
---|
684 | } |
---|
685 | return( A); |
---|
686 | } |
---|
687 | example |
---|
688 | { |
---|
689 | "EXAMPLE:"; echo=2; |
---|
690 | |
---|
691 | intmat M[2][5] = |
---|
692 | 1, 2, 3, 4, 0, |
---|
693 | 0,10,20,30, 1; |
---|
694 | |
---|
695 | // Hermite Normal Form of M: |
---|
696 | print(hermite(M)); |
---|
697 | |
---|
698 | intmat T[3][4] = |
---|
699 | 3,3,3,3, |
---|
700 | 2,1,3,0, |
---|
701 | 1,2,0,3; |
---|
702 | |
---|
703 | // Hermite Normal Form of T: |
---|
704 | print(hermite(T)); |
---|
705 | |
---|
706 | intmat A[4][5] = |
---|
707 | 1,2,3,2,2, |
---|
708 | 1,2,3,4,0, |
---|
709 | 0,5,4,2,1, |
---|
710 | 3,2,4,0,2; |
---|
711 | |
---|
712 | // Hermite Normal Form of A: |
---|
713 | print(hermite(A)); |
---|
714 | } |
---|
715 | |
---|
716 | |
---|
717 | /******************************************************/ |
---|
718 | proc isTorsionElement(intvec mdeg) |
---|
719 | "USAGE: isTorsionElement(intvec mdeg); |
---|
720 | PROCEDURE: For a integer vector mdeg representing the multidegree of some polynomial |
---|
721 | or vector this method computes if the multidegree is contained in the torsion |
---|
722 | group, i.e. if it is zero in the multigrading. |
---|
723 | " |
---|
724 | { |
---|
725 | intmat H = getTorsion("hermite"); |
---|
726 | int x, k, i; |
---|
727 | |
---|
728 | int r = nrows(H); |
---|
729 | int c = ncols(H); |
---|
730 | |
---|
731 | int rr = nrows(mdeg); |
---|
732 | |
---|
733 | for( i = 1; (i<=r) && (i<=c); i++) |
---|
734 | { |
---|
735 | if(H[i, i]!=0) |
---|
736 | { |
---|
737 | x = mdeg[i]%H[i, i]; |
---|
738 | |
---|
739 | if(x!=0) |
---|
740 | { |
---|
741 | return(1==0); |
---|
742 | } |
---|
743 | |
---|
744 | x = mdeg[i] / H[i,i]; |
---|
745 | |
---|
746 | for( k=1; k <= rr; k++) |
---|
747 | { |
---|
748 | mdeg[k] = mdeg[k] - x*H[k,i]; |
---|
749 | } |
---|
750 | } |
---|
751 | } |
---|
752 | |
---|
753 | return( mdeg == 0 ); |
---|
754 | |
---|
755 | } |
---|
756 | example |
---|
757 | { |
---|
758 | "EXAMPLE:"; echo=2; |
---|
759 | |
---|
760 | ring r = 0,(x,y,z),dp; |
---|
761 | |
---|
762 | intmat g[2][3]= |
---|
763 | 1,0,1, |
---|
764 | 0,1,1; |
---|
765 | intmat t[2][1]= |
---|
766 | -2, |
---|
767 | 1; |
---|
768 | |
---|
769 | setBaseMultigrading(g,t); |
---|
770 | |
---|
771 | poly a = x10yz; |
---|
772 | poly b = x8y2z; |
---|
773 | poly c = x4z2; |
---|
774 | poly d = y5; |
---|
775 | poly e = x2y2; |
---|
776 | poly f = z2; |
---|
777 | |
---|
778 | intvec v1 = mDeg(a) - mDeg(b); |
---|
779 | v1; |
---|
780 | isTorsionElement(v1); |
---|
781 | |
---|
782 | intvec v2 = mDeg(a) - mDeg(c); |
---|
783 | v2; |
---|
784 | isTorsionElement(v2); |
---|
785 | |
---|
786 | intvec v3 = mDeg(e) - mDeg(f); |
---|
787 | v3; |
---|
788 | isTorsionElement(v3); |
---|
789 | |
---|
790 | intvec v4 = mDeg(c) - mDeg(d); |
---|
791 | v4; |
---|
792 | isTorsionElement(v4); |
---|
793 | } |
---|
794 | |
---|
795 | |
---|
796 | /******************************************************/ |
---|
797 | proc defineHomogenous(poly f, list #) |
---|
798 | "USAGE: defineHomogenous(f[, G]); polynomial f, integer matrix G |
---|
799 | PURPOSE: Yields a matrix which has to be appended to the torsion matrix to make the |
---|
800 | polynomial f homogenous in the grading by grad. |
---|
801 | " |
---|
802 | { |
---|
803 | if( size(#) > 0 ) |
---|
804 | { |
---|
805 | if( typeof(#[1]) == "intmat" ) |
---|
806 | { |
---|
807 | intmat grad = #[1]; |
---|
808 | } |
---|
809 | } |
---|
810 | |
---|
811 | if( !defined(grad) ) |
---|
812 | { |
---|
813 | intmat grad = getVariableWeights(); |
---|
814 | } |
---|
815 | |
---|
816 | intmat newtor[nrows(grad)][size(f)-1]; |
---|
817 | int i,j; |
---|
818 | intvec l = grad*leadexp(f); |
---|
819 | intvec v; |
---|
820 | for(i=2; i <= size(f); i++) |
---|
821 | { |
---|
822 | v = grad * leadexp(f[i]) - l; |
---|
823 | for( j=1; j<=size(v); j++) |
---|
824 | { |
---|
825 | newtor[j,i-1] = v[j]; |
---|
826 | } |
---|
827 | } |
---|
828 | return(newtor); |
---|
829 | } |
---|
830 | example |
---|
831 | { |
---|
832 | "EXAMPLE:"; echo=2; |
---|
833 | |
---|
834 | ring r =0,(x,y,z),dp; |
---|
835 | intmat grad[2][3] = |
---|
836 | 1,0,1, |
---|
837 | 0,1,1; |
---|
838 | |
---|
839 | setBaseMultigrading(grad); |
---|
840 | |
---|
841 | poly f = x2y3-z5+x-3zx; |
---|
842 | |
---|
843 | intmat M = defineHomogenous(f); |
---|
844 | M; |
---|
845 | defineHomogenous(f, grad) == M; |
---|
846 | |
---|
847 | isHomogenous(f); |
---|
848 | setBaseMultigrading(grad, M); |
---|
849 | isHomogenous(f); |
---|
850 | } |
---|
851 | |
---|
852 | /******************************************************/ |
---|
853 | proc pushForward(map f) |
---|
854 | "USAGE: pushForward(f); |
---|
855 | PURPOSE: Computes the finest grading of the image ring which makes the map f |
---|
856 | a map of graded rings. The group map between the two grading groups is given |
---|
857 | by transpose( (Id, 0) ). Pay attention that the group spanned by the columns of |
---|
858 | the torsion matrix may not be a subgroup of the grading group. Still all columns |
---|
859 | are needed to find the correct image of the preimage gradings. |
---|
860 | " |
---|
861 | { |
---|
862 | |
---|
863 | int k,i,j; |
---|
864 | f; |
---|
865 | |
---|
866 | intmat oldgrad=getVariableWeights(preimage(f)); |
---|
867 | intmat oldtor=getTorsion(preimage(f)); |
---|
868 | |
---|
869 | int n=nvars(preimage(f)); |
---|
870 | int np=nvars(basering); |
---|
871 | int p=nrows(oldgrad); |
---|
872 | int pp=p+np; |
---|
873 | |
---|
874 | intmat newgrad[pp][np]; |
---|
875 | |
---|
876 | for(i=1;i<=np;i++){ newgrad[p+i,i]=1;} |
---|
877 | |
---|
878 | //newgrad; |
---|
879 | |
---|
880 | |
---|
881 | |
---|
882 | list newtor; |
---|
883 | intmat toadd; |
---|
884 | int columns=0; |
---|
885 | |
---|
886 | intmat toadd1[pp][n]; |
---|
887 | intvec v; |
---|
888 | poly im; |
---|
889 | |
---|
890 | for(i=1;i<=p;i++){ |
---|
891 | for(j=1;j<=n;j++){ toadd1[i,j]=oldgrad[i,j];} |
---|
892 | } |
---|
893 | |
---|
894 | for(i=1;i<=n;i++){ |
---|
895 | im=f[i]; |
---|
896 | //im; |
---|
897 | toadd = defineHomogenous(im, newgrad); |
---|
898 | newtor=insert(newtor,toadd); |
---|
899 | columns=columns+ncols(toadd); |
---|
900 | |
---|
901 | v=leadexp(f[i]); |
---|
902 | for(j=p+1;j<=p+np;j++){ toadd1[j,i]=-v[j-p];} |
---|
903 | } |
---|
904 | |
---|
905 | newtor=insert(newtor,toadd1); |
---|
906 | columns=columns+ncols(toadd1); |
---|
907 | |
---|
908 | |
---|
909 | if(typeof(basering)=="qring"){ |
---|
910 | //"Entering qring"; |
---|
911 | ideal a=ideal(basering); |
---|
912 | for(i=1;i<=size(a);i++){ |
---|
913 | toadd = defineHomogenous(a[i], newgrad); |
---|
914 | //toadd; |
---|
915 | columns=columns+ncols(toadd); |
---|
916 | newtor=insert(newtor,toadd); |
---|
917 | } |
---|
918 | } |
---|
919 | |
---|
920 | //newtor; |
---|
921 | intmat imofoldtor[pp][ncols(oldtor)]; |
---|
922 | for(i=1; i<=nrows(oldtor);i++){ |
---|
923 | for(j=1; j<=ncols(oldtor); j++){ |
---|
924 | imofoldtor[i,j]=oldtor[i,j]; |
---|
925 | } |
---|
926 | } |
---|
927 | |
---|
928 | columns=columns+ncols(oldtor); |
---|
929 | newtor=insert(newtor, imofoldtor); |
---|
930 | |
---|
931 | intmat torsion[pp][columns]; |
---|
932 | columns=0; |
---|
933 | for(k=1;k<=size(newtor);k++){ |
---|
934 | for(i=1;i<=pp;i++){ |
---|
935 | for(j=1;j<=ncols(newtor[k]);j++){torsion[i,j+columns]=newtor[k][i,j];} |
---|
936 | } |
---|
937 | columns=columns+ncols(newtor[k]); |
---|
938 | } |
---|
939 | |
---|
940 | torsion=hermite(torsion); |
---|
941 | intmat result[pp][pp]; |
---|
942 | for(i=1;i<=pp;i++){ |
---|
943 | for(j=1;j<=pp;j++){result[i,j]=torsion[i,j];} |
---|
944 | } |
---|
945 | |
---|
946 | setBaseMultigrading(newgrad, result); |
---|
947 | |
---|
948 | } |
---|
949 | example |
---|
950 | { |
---|
951 | "EXAMPLE:"; echo=2; |
---|
952 | |
---|
953 | // Setting degrees for preimage ring.; |
---|
954 | intmat grad[3][3] = |
---|
955 | 1,0,0, |
---|
956 | 0,1,0, |
---|
957 | 0,0,1; |
---|
958 | |
---|
959 | ring r = 0,(x,y,z),dp; |
---|
960 | keepring(r); |
---|
961 | setBaseMultigrading(grad); |
---|
962 | |
---|
963 | // grading on r: |
---|
964 | getVariableWeights(); |
---|
965 | getTorsion(); |
---|
966 | |
---|
967 | ring R = 0,(a,b),dp; |
---|
968 | ideal i=a2-b2+a6-b5+ab3,a7b+b15-ab6+a6b6; |
---|
969 | |
---|
970 | // The quotient ring by this ideal will become our image ring.; |
---|
971 | qring Q = std(i); |
---|
972 | map f = r,-a2b6+b5+a3b+a2+ab,-a2b7-3a2b5+b4+a,a6-b6-b3+a2; |
---|
973 | f; |
---|
974 | |
---|
975 | // Pushing forward f: |
---|
976 | pushForward(f); |
---|
977 | |
---|
978 | // due to pushForward we have got new grading on Q |
---|
979 | getVariableWeights(); |
---|
980 | getTorsion(); |
---|
981 | |
---|
982 | // TODO: Unfortunately this is not a very spectacular example.; |
---|
983 | |
---|
984 | } |
---|
985 | |
---|
986 | |
---|
987 | /******************************************************/ |
---|
988 | proc equalMDeg(intvec exp1, intvec exp2, list #) |
---|
989 | "USAGE: equalMDeg(exp1, exp2[, V]); intvec exp1, exp2, intmat V |
---|
990 | PURPOSE: Tests if the exponent vectors of two monomials (given by exp1 and exp2) |
---|
991 | represent the same multidegree. |
---|
992 | NOTE: the integer matrix V encodes multidegrees of module components, |
---|
993 | if module component is present in exp1 and exp2 |
---|
994 | " |
---|
995 | { |
---|
996 | if( size(exp1) != size(exp2) ) |
---|
997 | { |
---|
998 | ERROR("Sorry: we cannot compare exponents comming from a polynomial and a vector yet!"); |
---|
999 | } |
---|
1000 | |
---|
1001 | if( exp1 == exp2) |
---|
1002 | { |
---|
1003 | return (1==1); |
---|
1004 | } |
---|
1005 | |
---|
1006 | |
---|
1007 | |
---|
1008 | intmat M = getVariableWeights(); |
---|
1009 | |
---|
1010 | if( nrows(exp1) > ncols(M) ) // vectors => last exponent is the module component! |
---|
1011 | { |
---|
1012 | if( (size(#) == 0) or (typeof(#[1])!="intmat") ) |
---|
1013 | { |
---|
1014 | ERROR("Sorry: wrong or missing module-unit-weights-matrix V!"); |
---|
1015 | } |
---|
1016 | intmat V = #[1]; |
---|
1017 | |
---|
1018 | // typeof(V); print(V); |
---|
1019 | |
---|
1020 | int N = ncols(M); |
---|
1021 | int r = nrows(M); |
---|
1022 | |
---|
1023 | intvec d = intvec(exp1[1..N]) - intvec(exp2[1..N]); |
---|
1024 | intvec dm = intvec(V[1..r, exp1[N+1]]) - intvec(V[1..r, exp2[N+1]]); |
---|
1025 | |
---|
1026 | intvec difference = M * d + dm; |
---|
1027 | } |
---|
1028 | else |
---|
1029 | { |
---|
1030 | intvec d = (exp1 - exp2); |
---|
1031 | intvec difference = M * d; |
---|
1032 | } |
---|
1033 | |
---|
1034 | if (isFreeRepresented()) // no torsion!? |
---|
1035 | { |
---|
1036 | return ( difference == 0); |
---|
1037 | } |
---|
1038 | return ( isTorsionElement( difference ) ); |
---|
1039 | } |
---|
1040 | example |
---|
1041 | { |
---|
1042 | "EXAMPLE:"; echo=2; |
---|
1043 | |
---|
1044 | ring r = 0,(x,y,z),dp; |
---|
1045 | |
---|
1046 | intmat g[2][3]= |
---|
1047 | 1,0,1, |
---|
1048 | 0,1,1; |
---|
1049 | |
---|
1050 | intmat t[2][1]= |
---|
1051 | -2, |
---|
1052 | 1; |
---|
1053 | |
---|
1054 | setBaseMultigrading(g,t); |
---|
1055 | |
---|
1056 | poly a = x10yz; |
---|
1057 | poly b = x8y2z; |
---|
1058 | poly c = x4z2; |
---|
1059 | poly d = y5; |
---|
1060 | poly e = x2y2; |
---|
1061 | poly f = z2; |
---|
1062 | |
---|
1063 | |
---|
1064 | equalMDeg(leadexp(a), leadexp(b)); |
---|
1065 | equalMDeg(leadexp(a), leadexp(c)); |
---|
1066 | equalMDeg(leadexp(a), leadexp(d)); |
---|
1067 | equalMDeg(leadexp(a), leadexp(e)); |
---|
1068 | equalMDeg(leadexp(a), leadexp(f)); |
---|
1069 | |
---|
1070 | equalMDeg(leadexp(b), leadexp(c)); |
---|
1071 | equalMDeg(leadexp(b), leadexp(d)); |
---|
1072 | equalMDeg(leadexp(b), leadexp(e)); |
---|
1073 | equalMDeg(leadexp(b), leadexp(f)); |
---|
1074 | |
---|
1075 | equalMDeg(leadexp(c), leadexp(d)); |
---|
1076 | equalMDeg(leadexp(c), leadexp(e)); |
---|
1077 | equalMDeg(leadexp(c), leadexp(f)); |
---|
1078 | |
---|
1079 | equalMDeg(leadexp(d), leadexp(e)); |
---|
1080 | equalMDeg(leadexp(d), leadexp(f)); |
---|
1081 | |
---|
1082 | equalMDeg(leadexp(e), leadexp(f)); |
---|
1083 | |
---|
1084 | } |
---|
1085 | |
---|
1086 | |
---|
1087 | |
---|
1088 | /******************************************************/ |
---|
1089 | static proc isFreeRepresented() |
---|
1090 | "check whether the base muligrading is torsion-free (it is zero). |
---|
1091 | " |
---|
1092 | { |
---|
1093 | intmat T = getTorsion(); |
---|
1094 | |
---|
1095 | intmat Z[nrows(T)][ncols(T)]; |
---|
1096 | |
---|
1097 | return (T == Z); // no torsion! |
---|
1098 | } |
---|
1099 | |
---|
1100 | |
---|
1101 | /******************************************************/ |
---|
1102 | proc isHomogenous(def a, list #) |
---|
1103 | "USAGE: isHomogenous(a[, f]); a polynomial/vector/ideal/module |
---|
1104 | RETURN: boolean, TRUE if a is (multi)homogenous, and FALSE otherwise |
---|
1105 | " |
---|
1106 | { |
---|
1107 | if( (typeof(a) == "poly") or (typeof(a) == "vector") ) |
---|
1108 | { |
---|
1109 | return ( size(mDegPartition(a)) <= 1 ) |
---|
1110 | } |
---|
1111 | |
---|
1112 | if( typeof(a) == "vector" or typeof(a) == "module" ) |
---|
1113 | { |
---|
1114 | intmat V = getModuleGrading(a); |
---|
1115 | } |
---|
1116 | |
---|
1117 | if( (typeof(a) == "ideal") or (typeof(a) == "module") ) |
---|
1118 | { |
---|
1119 | if(size(#) > 0) |
---|
1120 | { |
---|
1121 | if (#[1] == "checkGens") |
---|
1122 | { |
---|
1123 | def aa; |
---|
1124 | for( int i = ncols(a); i > 0; i-- ) |
---|
1125 | { |
---|
1126 | aa = a[i]; |
---|
1127 | |
---|
1128 | if(defined(V)) |
---|
1129 | { |
---|
1130 | aa = setModuleGrading(aa, V); |
---|
1131 | } |
---|
1132 | |
---|
1133 | if(!isHomogenous(aa)) |
---|
1134 | { |
---|
1135 | return(0==1); |
---|
1136 | } |
---|
1137 | } |
---|
1138 | return(1==1); |
---|
1139 | } |
---|
1140 | } |
---|
1141 | |
---|
1142 | def g = groebner(a); // !!!! |
---|
1143 | |
---|
1144 | def b, aa; int j; |
---|
1145 | for( int i = ncols(a); i > 0; i-- ) |
---|
1146 | { |
---|
1147 | aa = a[i]; |
---|
1148 | |
---|
1149 | if(defined(V)) |
---|
1150 | { |
---|
1151 | aa = setModuleGrading(aa, V); |
---|
1152 | } |
---|
1153 | |
---|
1154 | b = mDegPartition(aa); |
---|
1155 | for( j = ncols(b); j > 0; j-- ) |
---|
1156 | { |
---|
1157 | if(NF(b[j],g) != 0) |
---|
1158 | { |
---|
1159 | return(0==1); |
---|
1160 | } |
---|
1161 | } |
---|
1162 | } |
---|
1163 | return(1==1); |
---|
1164 | } |
---|
1165 | } |
---|
1166 | example |
---|
1167 | { |
---|
1168 | "EXAMPLE:"; echo=2; |
---|
1169 | |
---|
1170 | ring r = 0,(x,y,z),dp; |
---|
1171 | |
---|
1172 | //Grading and Torsion matrices: |
---|
1173 | intmat M[3][3] = |
---|
1174 | 1,0,0, |
---|
1175 | 0,1,0, |
---|
1176 | 0,0,1; |
---|
1177 | |
---|
1178 | intmat T[3][1] = |
---|
1179 | 1,2,3; |
---|
1180 | |
---|
1181 | setBaseMultigrading(M,T); |
---|
1182 | |
---|
1183 | attrib(r); |
---|
1184 | |
---|
1185 | poly f = x-yz; |
---|
1186 | |
---|
1187 | mDegPartition(f); |
---|
1188 | print(mDeg(_)); |
---|
1189 | |
---|
1190 | isHomogenous(f); // f: is not homogenous |
---|
1191 | |
---|
1192 | poly g = 1-xy2z3; |
---|
1193 | isHomogenous(g); // g: is homogenous |
---|
1194 | mDegPartition(g); |
---|
1195 | |
---|
1196 | kill T; |
---|
1197 | ///////////////////////////////////////////////////////// |
---|
1198 | // new Torsion matrix: |
---|
1199 | intmat T[3][4] = |
---|
1200 | 3,3,3,3, |
---|
1201 | 2,1,3,0, |
---|
1202 | 1,2,0,3; |
---|
1203 | |
---|
1204 | setBaseMultigrading(M,T); |
---|
1205 | |
---|
1206 | f; |
---|
1207 | isHomogenous(f); |
---|
1208 | mDegPartition(f); |
---|
1209 | |
---|
1210 | // --------------------- |
---|
1211 | g; |
---|
1212 | isHomogenous(g); |
---|
1213 | mDegPartition(g); |
---|
1214 | |
---|
1215 | kill r, T, M; |
---|
1216 | |
---|
1217 | ring R = 0, (x,y,z), dp; |
---|
1218 | |
---|
1219 | intmat A[2][3] = |
---|
1220 | 0,0,1, |
---|
1221 | 3,2,1; |
---|
1222 | intmat T[2][1] = |
---|
1223 | -1, |
---|
1224 | 4; |
---|
1225 | setBaseMultigrading(A, T); |
---|
1226 | |
---|
1227 | isHomogenous(ideal(x2 - y3 -xy +z, x*y-z, x^3 - y^2*z + x^2 -y^3)); // 1 |
---|
1228 | isHomogenous(ideal(x2 - y3 -xy +z, x*y-z, x^3 - y^2*z + x^2 -y^3), "checkGens"); |
---|
1229 | isHomogenous(ideal(x+y, x2 - y2)); // 0 |
---|
1230 | |
---|
1231 | // Degree partition: |
---|
1232 | mDegPartition(x2 - y3 -xy +z); |
---|
1233 | mDegPartition(x3 -y2z + x2 -y3 + z + 1); |
---|
1234 | |
---|
1235 | |
---|
1236 | module N = gen(1) + (x+y) * gen(2), z*gen(3); |
---|
1237 | |
---|
1238 | intmat V[2][3] = 0; // 1, 2, 3, 4, 5, 6; // column-wise weights of components!!?? |
---|
1239 | |
---|
1240 | vector v1, v2; |
---|
1241 | |
---|
1242 | v1 = setModuleGrading(N[1], V); v1; |
---|
1243 | mDegPartition(v1); |
---|
1244 | print( mDeg(_) ); |
---|
1245 | |
---|
1246 | v2 = setModuleGrading(N[2], V); v2; |
---|
1247 | mDegPartition(v2); |
---|
1248 | print( mDeg(_) ); |
---|
1249 | |
---|
1250 | N = setModuleGrading(N, V); |
---|
1251 | isHomogenous(N); |
---|
1252 | print( mDeg(N) ); |
---|
1253 | |
---|
1254 | /////////////////////////////////////// |
---|
1255 | |
---|
1256 | V = |
---|
1257 | 1, 2, 3, |
---|
1258 | 4, 5, 6; |
---|
1259 | |
---|
1260 | v1 = setModuleGrading(N[1], V); v1; |
---|
1261 | mDegPartition(v1); |
---|
1262 | print( mDeg(_) ); |
---|
1263 | |
---|
1264 | v2 = setModuleGrading(N[2], V); v2; |
---|
1265 | mDegPartition(v2); |
---|
1266 | print( mDeg(_) ); |
---|
1267 | |
---|
1268 | N = setModuleGrading(N, V); |
---|
1269 | isHomogenous(N); |
---|
1270 | print( mDeg(N) ); |
---|
1271 | |
---|
1272 | /////////////////////////////////////// |
---|
1273 | |
---|
1274 | V = |
---|
1275 | 0, 0, 0, |
---|
1276 | 4, 1, 0; |
---|
1277 | |
---|
1278 | N = gen(1) + x * gen(2), z*gen(3); |
---|
1279 | |
---|
1280 | N = setModuleGrading(N, V); N; |
---|
1281 | isHomogenous(N); |
---|
1282 | print( mDeg(N) ); |
---|
1283 | |
---|
1284 | v1 = setModuleGrading(N[1], V); v1; |
---|
1285 | mDegPartition(v1); |
---|
1286 | print( mDeg(_) ); |
---|
1287 | |
---|
1288 | N = setModuleGrading(N, V); N; |
---|
1289 | isHomogenous(N); |
---|
1290 | print( mDeg(N) ); |
---|
1291 | } |
---|
1292 | |
---|
1293 | /******************************************************/ |
---|
1294 | proc mDeg(def A) |
---|
1295 | "USAGE: mDeg(A); any A |
---|
1296 | PURPOSE: compute multidegree |
---|
1297 | " |
---|
1298 | { |
---|
1299 | if( defined(attrib(A, "grad")) > 0 ) |
---|
1300 | { |
---|
1301 | return (attrib(A, "grad")); |
---|
1302 | } |
---|
1303 | |
---|
1304 | intmat M = getVariableWeights(); |
---|
1305 | int N = nvars(basering); |
---|
1306 | |
---|
1307 | if( ncols(M) != N ) |
---|
1308 | { |
---|
1309 | ERROR("Sorry wrong mgrad-size of M: " + string(ncols(M))); |
---|
1310 | } |
---|
1311 | |
---|
1312 | int r = nrows(M); |
---|
1313 | |
---|
1314 | if( A == 0 ) |
---|
1315 | { |
---|
1316 | intvec v; v[r] = 0; |
---|
1317 | return (v); |
---|
1318 | } |
---|
1319 | |
---|
1320 | if( (typeof(A) == "vector") or (typeof(A) == "module") ) |
---|
1321 | { |
---|
1322 | intmat V = getModuleGrading(A); |
---|
1323 | |
---|
1324 | if( nrows(V) != r ) |
---|
1325 | { |
---|
1326 | ERROR("Sorry wrong mgrad-size of V: " + string(nrows(V))); |
---|
1327 | } |
---|
1328 | } |
---|
1329 | |
---|
1330 | intvec m; m[r] = 0; |
---|
1331 | |
---|
1332 | if( typeof(A) == "poly" ) |
---|
1333 | { |
---|
1334 | intvec v = leadexp(A); // v; |
---|
1335 | m = M * v; |
---|
1336 | |
---|
1337 | // We assume homogeneous input! |
---|
1338 | return(m); |
---|
1339 | |
---|
1340 | A = A - lead(A); |
---|
1341 | while( size(A) > 0 ) |
---|
1342 | { |
---|
1343 | v = leadexp(A); // v; |
---|
1344 | m = max( m, M * v, r ); // ???? |
---|
1345 | A = A - lead(A); |
---|
1346 | } |
---|
1347 | |
---|
1348 | return(m); |
---|
1349 | } |
---|
1350 | |
---|
1351 | |
---|
1352 | if( typeof(A) == "vector" ) |
---|
1353 | { |
---|
1354 | intvec v; |
---|
1355 | v = leadexp(A); // v; |
---|
1356 | m = intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]); |
---|
1357 | |
---|
1358 | // We assume homogeneous input! |
---|
1359 | return(m); |
---|
1360 | |
---|
1361 | A = A - lead(A); |
---|
1362 | while( size(A) > 0 ) |
---|
1363 | { |
---|
1364 | v = leadexp(A); // v; |
---|
1365 | |
---|
1366 | // intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]); |
---|
1367 | |
---|
1368 | m = max( m, intvec(M * intvec(v[1..N])) + intvec(V[1..r, v[N+1]]), r ); // ??? |
---|
1369 | |
---|
1370 | A = A - lead(A); |
---|
1371 | } |
---|
1372 | |
---|
1373 | return(m); |
---|
1374 | } |
---|
1375 | |
---|
1376 | int i, j; intvec d; |
---|
1377 | |
---|
1378 | if( typeof(A) == "ideal" ) |
---|
1379 | { |
---|
1380 | intmat G[ r ] [ ncols(A)]; |
---|
1381 | for( i = ncols(A); i > 0; i-- ) |
---|
1382 | { |
---|
1383 | d = mDeg( A[i] ); |
---|
1384 | |
---|
1385 | for( j = 1; j <= r; j++ ) // see ticket: 253 |
---|
1386 | { |
---|
1387 | G[j, i] = d[j]; |
---|
1388 | } |
---|
1389 | } |
---|
1390 | return(G); |
---|
1391 | } |
---|
1392 | |
---|
1393 | if( typeof(A) == "module" ) |
---|
1394 | { |
---|
1395 | intmat G[ r ] [ ncols(A)]; |
---|
1396 | vector v; |
---|
1397 | |
---|
1398 | for( i = ncols(A); i > 0; i-- ) |
---|
1399 | { |
---|
1400 | v = setModuleGrading(A[i], V); |
---|
1401 | |
---|
1402 | // G[1..r, i] |
---|
1403 | d = mDeg(v); |
---|
1404 | |
---|
1405 | for( j = 1; j <= r; j++ ) // see ticket: 253 |
---|
1406 | { |
---|
1407 | G[j, i] = d[j]; |
---|
1408 | } |
---|
1409 | |
---|
1410 | } |
---|
1411 | |
---|
1412 | return(G); |
---|
1413 | } |
---|
1414 | |
---|
1415 | } |
---|
1416 | example |
---|
1417 | { |
---|
1418 | "EXAMPLE:"; echo=2; |
---|
1419 | |
---|
1420 | ring r = 0,(x, y), dp; |
---|
1421 | |
---|
1422 | intmat A[2][2] = 1, 0, 0, 1; |
---|
1423 | print(A); |
---|
1424 | |
---|
1425 | intmat Ta[2][1] = 0, 3; |
---|
1426 | print(Ta); |
---|
1427 | |
---|
1428 | // attrib(A, "torsion", Ta); // to think about |
---|
1429 | |
---|
1430 | // "poly:"; |
---|
1431 | setBaseMultigrading(A); |
---|
1432 | |
---|
1433 | |
---|
1434 | mDeg( x*x, A ); |
---|
1435 | mDeg( y*y*y, A ); |
---|
1436 | |
---|
1437 | setBaseMultigrading(A, Ta); |
---|
1438 | |
---|
1439 | mDeg( x*x*y ); |
---|
1440 | |
---|
1441 | mDeg( y*y*y*x ); |
---|
1442 | |
---|
1443 | mDeg( x*y + x + 1 ); |
---|
1444 | |
---|
1445 | mDegPartition(x*y + x + 1); |
---|
1446 | |
---|
1447 | print ( mDeg(0) ); |
---|
1448 | poly zero = 0; |
---|
1449 | print ( mDeg(zero) ); |
---|
1450 | |
---|
1451 | // "ideal:"; |
---|
1452 | |
---|
1453 | ideal I = y*x*x, x*y*y*y; |
---|
1454 | print( mDeg(I) ); |
---|
1455 | |
---|
1456 | print ( mDeg(ideal(0)) ); |
---|
1457 | print ( mDeg(ideal(0,0,0)) ); |
---|
1458 | |
---|
1459 | // "vectors:"; |
---|
1460 | |
---|
1461 | intmat B[2][2] = 0, 1, 1, 0; |
---|
1462 | print(B); |
---|
1463 | |
---|
1464 | mDeg( setModuleGrading(y*y*y*gen(2), B )); |
---|
1465 | mDeg( setModuleGrading(x*x*gen(1), B )); |
---|
1466 | |
---|
1467 | |
---|
1468 | vector V = x*gen(1) + y*gen(2); |
---|
1469 | V = setModuleGrading(V, B); |
---|
1470 | mDeg( V ); |
---|
1471 | |
---|
1472 | vector v1 = setModuleGrading([0, 0, 0], B); |
---|
1473 | print( mDeg( v1 ) ); |
---|
1474 | |
---|
1475 | vector v2 = setModuleGrading([0], B); |
---|
1476 | print( mDeg( v2 ) ); |
---|
1477 | |
---|
1478 | // "module:"; |
---|
1479 | |
---|
1480 | module D = x*gen(1), y*gen(2); |
---|
1481 | D; |
---|
1482 | D = setModuleGrading(D, B); |
---|
1483 | print( mDeg( D ) ); |
---|
1484 | |
---|
1485 | |
---|
1486 | module DD = [0, 0],[0, 0, 0]; |
---|
1487 | DD = setModuleGrading(DD, B); |
---|
1488 | print( mDeg( DD ) ); |
---|
1489 | |
---|
1490 | module DDD = [0, 0]; |
---|
1491 | DDD = setModuleGrading(DDD, B); |
---|
1492 | print( mDeg( DDD ) ); |
---|
1493 | |
---|
1494 | }; |
---|
1495 | |
---|
1496 | |
---|
1497 | |
---|
1498 | |
---|
1499 | |
---|
1500 | /******************************************************/ |
---|
1501 | proc mDegPartition(def p) |
---|
1502 | "USAGE: mDegPartition(def p), p polynomial/vector |
---|
1503 | RETURNS: an ideal/module consisting of multigraded-homogeneous parts of p |
---|
1504 | " |
---|
1505 | { |
---|
1506 | if( typeof(p) == "poly" ) |
---|
1507 | { |
---|
1508 | ideal I; |
---|
1509 | poly mp, t, tt; |
---|
1510 | } |
---|
1511 | else |
---|
1512 | { |
---|
1513 | if( typeof(p) == "vector" ) |
---|
1514 | { |
---|
1515 | module I; |
---|
1516 | vector mp, t, tt; |
---|
1517 | } |
---|
1518 | else |
---|
1519 | { |
---|
1520 | ERROR("Wrong ARGUMENT type!"); |
---|
1521 | } |
---|
1522 | } |
---|
1523 | |
---|
1524 | if( typeof(p) == "vector" ) |
---|
1525 | { |
---|
1526 | intmat V = getModuleGrading(p); |
---|
1527 | } |
---|
1528 | else |
---|
1529 | { |
---|
1530 | intmat V; |
---|
1531 | } |
---|
1532 | |
---|
1533 | if( size(p) > 1) |
---|
1534 | { |
---|
1535 | intvec m; |
---|
1536 | |
---|
1537 | while( p != 0 ) |
---|
1538 | { |
---|
1539 | m = leadexp(p); |
---|
1540 | mp = lead(p); |
---|
1541 | p = p - lead(p); |
---|
1542 | tt = p; t = 0; |
---|
1543 | |
---|
1544 | while( size(tt) > 0 ) |
---|
1545 | { |
---|
1546 | // TODO: we make no caching of matrices (M,T,H,V), which remain the same! |
---|
1547 | if( equalMDeg( leadexp(tt), m, V ) ) |
---|
1548 | { |
---|
1549 | mp = mp + lead(tt); // "mp", mp; |
---|
1550 | } |
---|
1551 | else |
---|
1552 | { |
---|
1553 | t = t + lead(tt); // "t", t; |
---|
1554 | } |
---|
1555 | |
---|
1556 | tt = tt - lead(tt); |
---|
1557 | } |
---|
1558 | |
---|
1559 | I[size(I)+1] = mp; |
---|
1560 | |
---|
1561 | p = t; |
---|
1562 | } |
---|
1563 | } |
---|
1564 | else |
---|
1565 | { |
---|
1566 | I[1] = p; // single monom |
---|
1567 | } |
---|
1568 | |
---|
1569 | if( typeof(I) == "module" ) |
---|
1570 | { |
---|
1571 | I = setModuleGrading(I, V); |
---|
1572 | } |
---|
1573 | |
---|
1574 | return (I); |
---|
1575 | } |
---|
1576 | example |
---|
1577 | { |
---|
1578 | "EXAMPLE:"; echo=2; |
---|
1579 | |
---|
1580 | ring r = 0,(x,y,z),dp; |
---|
1581 | |
---|
1582 | intmat g[2][3]= |
---|
1583 | 1,0,1, |
---|
1584 | 0,1,1; |
---|
1585 | intmat t[2][1]= |
---|
1586 | -2, |
---|
1587 | 1; |
---|
1588 | |
---|
1589 | setBaseMultigrading(g,t); |
---|
1590 | |
---|
1591 | poly f = x10yz+x8y2z-x4z2+y5+x2y2-z2+x17z3-y6; |
---|
1592 | |
---|
1593 | mDegPartition(f); |
---|
1594 | |
---|
1595 | vector v = xy*gen(1)-x3y2*gen(2)+x4y*gen(3); |
---|
1596 | intmat B[2][3]=1,-1,-2,0,0,1; |
---|
1597 | v = setModuleGrading(v,B); |
---|
1598 | getModuleGrading(v); |
---|
1599 | |
---|
1600 | mDegPartition(v, B); |
---|
1601 | } |
---|
1602 | |
---|
1603 | |
---|
1604 | |
---|
1605 | /******************************************************/ |
---|
1606 | static proc unitMatrix(int n) |
---|
1607 | { |
---|
1608 | intmat A[n][n]; |
---|
1609 | |
---|
1610 | for( int i = n; i > 0; i-- ) |
---|
1611 | { |
---|
1612 | A[i,i] = 1; |
---|
1613 | } |
---|
1614 | |
---|
1615 | return (A); |
---|
1616 | } |
---|
1617 | |
---|
1618 | |
---|
1619 | |
---|
1620 | /******************************************************/ |
---|
1621 | static proc finestMDeg(def r) |
---|
1622 | " |
---|
1623 | USAGE: |
---|
1624 | PURPOSE: finest multigrading |
---|
1625 | " |
---|
1626 | { |
---|
1627 | def save = basering; |
---|
1628 | setring (r); |
---|
1629 | |
---|
1630 | // in basering |
---|
1631 | ideal I = ideal(basering); |
---|
1632 | |
---|
1633 | int n = 0; int i; poly p; |
---|
1634 | for( i = ncols(I); i > 0; i-- ) |
---|
1635 | { |
---|
1636 | p = I[i]; |
---|
1637 | if( size(p) > 1 ) |
---|
1638 | { |
---|
1639 | n = n + (size(p) - 1); |
---|
1640 | } |
---|
1641 | else |
---|
1642 | { |
---|
1643 | I[i] = 0; |
---|
1644 | } |
---|
1645 | } |
---|
1646 | |
---|
1647 | int N = nvars(basering); |
---|
1648 | intmat A = unitMatrix(N); |
---|
1649 | |
---|
1650 | |
---|
1651 | |
---|
1652 | if( n > 0) |
---|
1653 | { |
---|
1654 | |
---|
1655 | intmat L[N][n]; |
---|
1656 | // list L; |
---|
1657 | int j = n; |
---|
1658 | |
---|
1659 | for( i = ncols(I); i > 0; i-- ) |
---|
1660 | { |
---|
1661 | p = I[i]; |
---|
1662 | |
---|
1663 | if( size(p) > 1 ) |
---|
1664 | { |
---|
1665 | intvec m0 = leadexp(p); |
---|
1666 | p = p - lead(p); |
---|
1667 | |
---|
1668 | while( size(p) > 0 ) |
---|
1669 | { |
---|
1670 | L[ 1..N, j ] = leadexp(p) - m0; |
---|
1671 | p = p - lead(p); |
---|
1672 | j--; |
---|
1673 | } |
---|
1674 | } |
---|
1675 | } |
---|
1676 | |
---|
1677 | print(L); |
---|
1678 | setBaseMultigrading(A, L); |
---|
1679 | } |
---|
1680 | else |
---|
1681 | { |
---|
1682 | setBaseMultigrading(A); |
---|
1683 | } |
---|
1684 | |
---|
1685 | // ERROR("nope"); |
---|
1686 | |
---|
1687 | // ring T = integer, (x), (C, dp); |
---|
1688 | |
---|
1689 | setring(save); |
---|
1690 | return (r); |
---|
1691 | } |
---|
1692 | example |
---|
1693 | { |
---|
1694 | "EXAMPLE:"; echo=2; |
---|
1695 | |
---|
1696 | ring r = 0,(x, y), dp; |
---|
1697 | qring q = std(x^2 - y); |
---|
1698 | |
---|
1699 | finestMDeg(q); |
---|
1700 | |
---|
1701 | } |
---|
1702 | |
---|
1703 | |
---|
1704 | |
---|
1705 | |
---|
1706 | /******************************************************/ |
---|
1707 | static proc newMap(map F, intmat Q, list #) |
---|
1708 | " |
---|
1709 | USAGE: ?? no use now... |
---|
1710 | PURPOSE: ?? |
---|
1711 | " |
---|
1712 | { |
---|
1713 | attrib(F, "Q", Q); |
---|
1714 | |
---|
1715 | if( size(#) > 0 and typeof(#[1]) == "intmat" ) |
---|
1716 | { |
---|
1717 | attrib(F, "P", #[1]); |
---|
1718 | } |
---|
1719 | return (F); |
---|
1720 | } |
---|
1721 | |
---|
1722 | |
---|
1723 | /******************************************************/ |
---|
1724 | static proc matrix2intmat( matrix M ) |
---|
1725 | { |
---|
1726 | execute( "intmat A[ "+ string(nrows(M)) + "]["+ string(ncols(M)) + "] = " + string(M) + ";" ); |
---|
1727 | return (A); |
---|
1728 | } |
---|
1729 | |
---|
1730 | |
---|
1731 | /******************************************************/ |
---|
1732 | static proc leftKernelZ(intmat M) |
---|
1733 | "USAGE: leftKernel(M); M a matrix |
---|
1734 | RETURN: module |
---|
1735 | PURPOSE: computes left kernel of matrix M (a module of all elements v such that vM=0) |
---|
1736 | EXAMPLE: example leftKernel; shows an example |
---|
1737 | " |
---|
1738 | { |
---|
1739 | if( nameof(basering) != "basering" ) |
---|
1740 | { |
---|
1741 | def save = basering; |
---|
1742 | } |
---|
1743 | |
---|
1744 | ring r = integer, (x), dp; |
---|
1745 | |
---|
1746 | |
---|
1747 | // basering; |
---|
1748 | module N = matrix((M)); // transpose |
---|
1749 | // print(N); |
---|
1750 | |
---|
1751 | def MM = modulo( N, std(0) ) ; |
---|
1752 | // print(MM); |
---|
1753 | |
---|
1754 | intmat R = ( matrix2intmat( MM ) ); // transpose |
---|
1755 | |
---|
1756 | if( defined(save) > 0 ) |
---|
1757 | { |
---|
1758 | setring save; |
---|
1759 | } |
---|
1760 | |
---|
1761 | kill r; |
---|
1762 | return( R ); |
---|
1763 | } |
---|
1764 | example |
---|
1765 | { |
---|
1766 | "EXAMPLE:"; echo=2; |
---|
1767 | |
---|
1768 | ring r= 0,(x,y,z),dp; |
---|
1769 | matrix M[3][1] = x,y,z; |
---|
1770 | print(M); |
---|
1771 | matrix L = leftKernel(M); |
---|
1772 | print(L); |
---|
1773 | // check: |
---|
1774 | print(L*M); |
---|
1775 | }; |
---|
1776 | |
---|
1777 | |
---|
1778 | |
---|
1779 | /******************************************************/ |
---|
1780 | // the following is taken from "sing4ti2.lib" as we need 'hilbert' from 4ti2 |
---|
1781 | |
---|
1782 | static proc hilbert4ti2intmat(intmat A, list #) |
---|
1783 | "USAGE: hilbert4ti2(A[,i]); |
---|
1784 | @* A=intmat |
---|
1785 | @* i=int |
---|
1786 | ASSUME: - A is a matrix with integer entries which describes the lattice |
---|
1787 | @* as ker(A), if second argument is not present, |
---|
1788 | @* as the left image Im(A) = {zA : z \in ZZ^k}, if second argument is a positive integer |
---|
1789 | @* - number of variables of basering equals number of columns of A |
---|
1790 | @* (for ker(A)) resp. of rows of A (for Im(A)) |
---|
1791 | CREATE: temporary files sing4ti2.mat, sing4ti2.lat, sing4ti2.mar |
---|
1792 | @* in the current directory (I/O files for communication with 4ti2) |
---|
1793 | NOTE: input rules for 4ti2 also apply to input to this procedure |
---|
1794 | @* hence ker(A)={x|Ax=0} and Im(A)={xA} |
---|
1795 | RETURN: toric ideal specified by Hilbert basis thereof |
---|
1796 | EXAMPLE: example graver4ti2; shows an example |
---|
1797 | " |
---|
1798 | { |
---|
1799 | //-------------------------------------------------------------------------- |
---|
1800 | // Initialization and Sanity Checks |
---|
1801 | //-------------------------------------------------------------------------- |
---|
1802 | int i,j; |
---|
1803 | int nr=nrows(A); |
---|
1804 | int nc=ncols(A); |
---|
1805 | string fileending="mat"; |
---|
1806 | if (size(#)!=0) |
---|
1807 | { |
---|
1808 | //--- default behaviour: use ker(A) as lattice |
---|
1809 | //--- if #[1]!=0 use Im(A) as lattice |
---|
1810 | if(typeof(#[1])!="int") |
---|
1811 | { |
---|
1812 | ERROR("optional parameter needs to be integer value"); |
---|
1813 | } |
---|
1814 | if(#[1]!=0) |
---|
1815 | { |
---|
1816 | fileending="lat"; |
---|
1817 | } |
---|
1818 | } |
---|
1819 | //--- we should also be checking whether all entries are indeed integers |
---|
1820 | //--- or whether there are fractions, but in this case the error message |
---|
1821 | //--- of 4ti2 is printed directly |
---|
1822 | |
---|
1823 | //-------------------------------------------------------------------------- |
---|
1824 | // preparing input file for 4ti2 |
---|
1825 | //-------------------------------------------------------------------------- |
---|
1826 | link eing=":w sing4ti2."+fileending; |
---|
1827 | string eingstring=string(nr)+" "+string(nc); |
---|
1828 | write(eing,eingstring); |
---|
1829 | for(i=1;i<=nr;i++) |
---|
1830 | { |
---|
1831 | kill eingstring; |
---|
1832 | string eingstring; |
---|
1833 | for(j=1;j<=nc;j++) |
---|
1834 | { |
---|
1835 | // if(g(A[i,j])>0)||(char(basering)!=0)||(npars(basering)>0)) |
---|
1836 | // { |
---|
1837 | // ERROR("Input to hilbert4ti2 needs to be a matrix with integer entries"); |
---|
1838 | // } |
---|
1839 | eingstring=eingstring+string(A[i,j])+" "; |
---|
1840 | } |
---|
1841 | write(eing, eingstring); |
---|
1842 | } |
---|
1843 | close(eing); |
---|
1844 | |
---|
1845 | //---------------------------------------------------------------------- |
---|
1846 | // calling 4ti2 and converting output |
---|
1847 | // Singular's string is too clumsy for this, hence we first prepare |
---|
1848 | // using standard unix commands |
---|
1849 | //---------------------------------------------------------------------- |
---|
1850 | j=system("sh","hilbert -q -n sing4ti2"); ////////// be quiet + no loggin!!! |
---|
1851 | |
---|
1852 | j=system("sh", "awk \'BEGIN{ORS=\",\";}{print $0;}\' sing4ti2.hil " + |
---|
1853 | "| sed s/[\\\ \\\t\\\v\\\f]/,/g " + |
---|
1854 | "| sed s/,+/,/g|sed s/,,/,/g " + |
---|
1855 | "| sed s/,,/,/g " + |
---|
1856 | "> sing4ti2.converted" ); |
---|
1857 | if( defined(keepfiles) <= 0) |
---|
1858 | { |
---|
1859 | j=system("sh",("rm -f sing4ti2.hil sing4ti2."+fileending)); |
---|
1860 | } |
---|
1861 | //---------------------------------------------------------------------- |
---|
1862 | // reading output of 4ti2 |
---|
1863 | //---------------------------------------------------------------------- |
---|
1864 | link ausg=":r sing4ti2.converted"; |
---|
1865 | //--- last entry ideal(0) is used to tie the list to the basering |
---|
1866 | //--- it will not be processed any further |
---|
1867 | |
---|
1868 | string s = read(ausg); |
---|
1869 | string ergstr = "intvec erglist = " + s + "0;"; |
---|
1870 | execute(ergstr); |
---|
1871 | |
---|
1872 | // print(erglist); |
---|
1873 | |
---|
1874 | int Rnc = erglist[1]; |
---|
1875 | int Rnr = erglist[2]; |
---|
1876 | |
---|
1877 | intmat R[Rnr][Rnc]; |
---|
1878 | |
---|
1879 | int k = 3; |
---|
1880 | |
---|
1881 | for(i=1;i<=Rnc;i++) |
---|
1882 | { |
---|
1883 | for(j=1;j<=Rnr;j++) |
---|
1884 | { |
---|
1885 | // "i: ", i, ", j: ", j, ", v: ", erglist[k]; |
---|
1886 | R[j, i] = erglist[k]; |
---|
1887 | k = k + 1; |
---|
1888 | } |
---|
1889 | } |
---|
1890 | |
---|
1891 | return (R); |
---|
1892 | //--- get rid of leading entry 0; |
---|
1893 | // toric=toric[2..ncols(toric)]; |
---|
1894 | // return(toric); |
---|
1895 | } |
---|
1896 | // A nice example here is the 3x3 Magic Squares |
---|
1897 | example |
---|
1898 | { |
---|
1899 | "EXAMPLE:"; echo=2; |
---|
1900 | |
---|
1901 | ring r=0,(x1,x2,x3,x4,x5,x6,x7,x8,x9),dp; |
---|
1902 | intmat M[7][9]= |
---|
1903 | 1, 1, 1, -1, -1, -1, 0, 0, 0, |
---|
1904 | 1, 1, 1, 0, 0, 0,-1,-1,-1, |
---|
1905 | 0, 1, 1, -1, 0, 0,-1, 0, 0, |
---|
1906 | 1, 0, 1, 0, -1, 0, 0,-1, 0, |
---|
1907 | 1, 1, 0, 0, 0, -1, 0, 0,-1, |
---|
1908 | 0, 1, 1, 0, -1, 0, 0, 0,-1, |
---|
1909 | 1, 1, 0, 0, -1, 0,-1, 0, 0; |
---|
1910 | hilbert4ti2intmat(M); |
---|
1911 | hermite(M); |
---|
1912 | } |
---|
1913 | |
---|
1914 | ///////////////////////////////////////////////////////////////////////////// |
---|
1915 | static proc getMonomByExponent(intvec exp) |
---|
1916 | { |
---|
1917 | int n = nvars(basering); |
---|
1918 | |
---|
1919 | if( nrows(exp) < n ) |
---|
1920 | { |
---|
1921 | n = nrows(exp); |
---|
1922 | } |
---|
1923 | |
---|
1924 | poly m = 1; int e; |
---|
1925 | |
---|
1926 | for( int i = 1; i <= n; i++ ) |
---|
1927 | { |
---|
1928 | e = exp[i]; |
---|
1929 | if( e < 0 ) |
---|
1930 | { |
---|
1931 | ERROR("Negative exponent!!!"); |
---|
1932 | } |
---|
1933 | |
---|
1934 | m = m * (var(i)^e); |
---|
1935 | } |
---|
1936 | |
---|
1937 | return (m); |
---|
1938 | |
---|
1939 | } |
---|
1940 | |
---|
1941 | /******************************************************/ |
---|
1942 | proc mDegBasis(intvec d) |
---|
1943 | " |
---|
1944 | USAGE: multidegree d |
---|
1945 | ASSUME: current ring is multigraded, monomial ordering is global |
---|
1946 | PURPOSE: compute all monomials of multidegree d |
---|
1947 | " |
---|
1948 | { |
---|
1949 | def R = basering; // setring R; |
---|
1950 | |
---|
1951 | intmat M = getVariableWeights(R); |
---|
1952 | |
---|
1953 | // print(M); |
---|
1954 | |
---|
1955 | int nr = nrows(M); |
---|
1956 | int nc = ncols(M); |
---|
1957 | |
---|
1958 | intmat A[nr][nc+1]; |
---|
1959 | A[1..nr, 1..nc] = M[1..nr, 1..nc]; |
---|
1960 | //typeof(A[1..nr, nc+1]); |
---|
1961 | if( nr==1) |
---|
1962 | { |
---|
1963 | A[1..nr, nc+1]=-d[1]; |
---|
1964 | } |
---|
1965 | else |
---|
1966 | { |
---|
1967 | A[1..nr, nc+1] = -d; |
---|
1968 | } |
---|
1969 | |
---|
1970 | intmat T = getTorsion(R); |
---|
1971 | |
---|
1972 | if( isFreeRepresented() ) |
---|
1973 | { |
---|
1974 | intmat B = hilbert4ti2intmat(A); |
---|
1975 | |
---|
1976 | // matrix B = unitMatrix(nrows(T)); |
---|
1977 | } |
---|
1978 | else |
---|
1979 | { |
---|
1980 | int n = ncols(T); |
---|
1981 | |
---|
1982 | nc = ncols(A); |
---|
1983 | |
---|
1984 | intmat AA[nr][nc + 2 * n]; |
---|
1985 | AA[1..nr, 1.. nc] = A[1..nr, 1.. nc]; |
---|
1986 | AA[1..nr, nc + (1.. n)] = T[1..nr, 1.. n]; |
---|
1987 | AA[1..nr, nc + n + (1.. n)] = -T[1..nr, 1.. n]; |
---|
1988 | |
---|
1989 | |
---|
1990 | // print ( AA ); |
---|
1991 | |
---|
1992 | intmat K = leftKernelZ(( AA ) ); // |
---|
1993 | |
---|
1994 | // print(K); |
---|
1995 | |
---|
1996 | intmat KK[nc][ncols(K)] = K[ 1.. nc, 1.. ncols(K) ]; |
---|
1997 | |
---|
1998 | // print(KK); |
---|
1999 | // "!"; |
---|
2000 | |
---|
2001 | intmat B = hilbert4ti2intmat(transpose(KK), 1); |
---|
2002 | |
---|
2003 | // "!"; print(B); |
---|
2004 | |
---|
2005 | } |
---|
2006 | |
---|
2007 | |
---|
2008 | // print(A); |
---|
2009 | |
---|
2010 | |
---|
2011 | |
---|
2012 | int i; |
---|
2013 | int nnr = nrows(B); |
---|
2014 | int nnc = ncols(B); |
---|
2015 | ideal I, J; |
---|
2016 | if(nnc==0){ |
---|
2017 | I=0; |
---|
2018 | return(I); |
---|
2019 | } |
---|
2020 | I[nnc] = 0; |
---|
2021 | J[nnc] = 0; |
---|
2022 | |
---|
2023 | for( i = 1; i <= nnc; i++ ) |
---|
2024 | { |
---|
2025 | // "i: ", i; B[nnr, i]; |
---|
2026 | |
---|
2027 | if( B[nnr, i] == 1) |
---|
2028 | { |
---|
2029 | // intvec(B[1..nnr-1, i]); |
---|
2030 | I[i] = getMonomByExponent(intvec(B[1..nnr-1, i])); |
---|
2031 | } |
---|
2032 | else |
---|
2033 | { |
---|
2034 | if( B[nnr, i] == 0) |
---|
2035 | { |
---|
2036 | // intvec(B[1..nnr-1, i]); |
---|
2037 | J[i] = getMonomByExponent(intvec(B[1..nnr-1, i])); |
---|
2038 | } |
---|
2039 | } |
---|
2040 | // I[i]; |
---|
2041 | } |
---|
2042 | |
---|
2043 | ideal Q = (ideal(basering)); |
---|
2044 | |
---|
2045 | if ( size(Q) > 0 ) |
---|
2046 | { |
---|
2047 | I = NF( I, lead(Q) ); |
---|
2048 | J = NF( J, lead(Q) ); // Global ordering!!! |
---|
2049 | } |
---|
2050 | |
---|
2051 | I = simplify(I, 2); // d |
---|
2052 | J = simplify(J, 2); // d |
---|
2053 | |
---|
2054 | attrib(I, "ZeroPart", J); |
---|
2055 | |
---|
2056 | return (I); |
---|
2057 | |
---|
2058 | // setring ; |
---|
2059 | } |
---|
2060 | example |
---|
2061 | { |
---|
2062 | "EXAMPLE:"; echo=2; |
---|
2063 | |
---|
2064 | ring R = 0, (x, y), dp; |
---|
2065 | |
---|
2066 | intmat g1[2][2]=1,0,0,1; |
---|
2067 | intmat t[2][1]=2,0; |
---|
2068 | intmat g2[2][2]=1,1,1,1; |
---|
2069 | intvec v1=4,0; |
---|
2070 | intvec v2=4,4; |
---|
2071 | |
---|
2072 | intmat g3[1][2]=1,1; |
---|
2073 | setBaseMultigrading(g3); |
---|
2074 | intvec v3=4:1; |
---|
2075 | v3; |
---|
2076 | mDegBasis(v3); |
---|
2077 | |
---|
2078 | setBaseMultigrading(g1,t); |
---|
2079 | mDegBasis(v1); |
---|
2080 | setBaseMultigrading(g2); |
---|
2081 | mDegBasis(v2); |
---|
2082 | |
---|
2083 | intmat M[2][2] = 1, -1, -1, 1; |
---|
2084 | intvec d = -2, 2; |
---|
2085 | |
---|
2086 | setBaseMultigrading(M); |
---|
2087 | |
---|
2088 | mDegBasis(d); |
---|
2089 | attrib(_, "ZeroPart"); |
---|
2090 | |
---|
2091 | kill R; |
---|
2092 | ring R = 0, (x, y, z), dp; |
---|
2093 | |
---|
2094 | intmat M[2][3] = 1, -2, 1, 1, 1, 0; |
---|
2095 | |
---|
2096 | intmat T[2][1] = 0, 2; |
---|
2097 | |
---|
2098 | intvec d = 4, 1; |
---|
2099 | |
---|
2100 | setBaseMultigrading(M, T); |
---|
2101 | |
---|
2102 | mDegBasis(d); |
---|
2103 | attrib(_, "ZeroPart"); |
---|
2104 | |
---|
2105 | |
---|
2106 | kill R; |
---|
2107 | |
---|
2108 | ring R = 0, (x, y, z), dp; |
---|
2109 | qring Q = std(ideal( y^6+ x*y^3*z-x^2*z^2 )); |
---|
2110 | |
---|
2111 | |
---|
2112 | intmat M[2][3] = 1, 1, 2, 2, 1, 1; |
---|
2113 | // intmat T[2][1] = 0, 2; |
---|
2114 | |
---|
2115 | setBaseMultigrading(M); |
---|
2116 | |
---|
2117 | intvec d = 6, 6; |
---|
2118 | mDegBasis(d); |
---|
2119 | attrib(_, "ZeroPart"); |
---|
2120 | |
---|
2121 | |
---|
2122 | |
---|
2123 | kill R; |
---|
2124 | ring R = 0, (x, y, z), dp; |
---|
2125 | qring Q = std(ideal( x*z^3 - y *z^6, x*y*z - x^4*y^2 )); |
---|
2126 | |
---|
2127 | |
---|
2128 | intmat M[2][3] = 1, -2, 1, 1, 1, 0; |
---|
2129 | intmat T[2][1] = 0, 2; |
---|
2130 | |
---|
2131 | intvec d = 4, 1; |
---|
2132 | |
---|
2133 | setBaseMultigrading(M, T); |
---|
2134 | |
---|
2135 | mDegBasis(d); |
---|
2136 | attrib(_, "ZeroPart"); |
---|
2137 | } |
---|
2138 | |
---|
2139 | |
---|
2140 | proc mDegSyzygy(def I) |
---|
2141 | "USAGE: mDegSyzygy(I); I is a poly/vector/ideal/module |
---|
2142 | PURPOSE: computes the multigraded syzygy of I |
---|
2143 | RETURNS: module, the syzygy of I |
---|
2144 | NOTE: generators of I must be multigraded homogeneous |
---|
2145 | " |
---|
2146 | { |
---|
2147 | if( isHomogenous(I, "checkGens") == 0) |
---|
2148 | { |
---|
2149 | ERROR ("Sorry: inhomogenous input!"); |
---|
2150 | } |
---|
2151 | module S = syz(I); |
---|
2152 | S = setModuleGrading(S, mDeg(I)); |
---|
2153 | return (S); |
---|
2154 | } |
---|
2155 | example |
---|
2156 | { |
---|
2157 | "EXAMPLE:"; echo=2; |
---|
2158 | |
---|
2159 | |
---|
2160 | ring r = 0,(x,y,z,w),dp; |
---|
2161 | |
---|
2162 | intmat M[2][4]= |
---|
2163 | 1,1,1,1, |
---|
2164 | 0,1,3,4; |
---|
2165 | |
---|
2166 | setBaseMultigrading(M); |
---|
2167 | |
---|
2168 | |
---|
2169 | module M = ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
2170 | |
---|
2171 | |
---|
2172 | intmat v[2][nrows(M)]= |
---|
2173 | 1, |
---|
2174 | 0; |
---|
2175 | |
---|
2176 | M = setModuleGrading(M, v); |
---|
2177 | |
---|
2178 | isHomogenous(M); |
---|
2179 | "Multidegrees: "; print(mDeg(M)); |
---|
2180 | |
---|
2181 | // Let's compute Syzygy! |
---|
2182 | def S = mDegSyzygy(M); S; |
---|
2183 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
2184 | "Multidegrees: "; print(mDeg(S)); |
---|
2185 | |
---|
2186 | isHomogenous(S); |
---|
2187 | } |
---|
2188 | |
---|
2189 | |
---|
2190 | proc mDegGroebner(def I) |
---|
2191 | "USAGE: mDegGroebner(I); I is a poly/vector/ideal/module |
---|
2192 | PURPOSE: computes the multigraded standard/groebner basis of I |
---|
2193 | NOTE: I must be multigraded homogeneous |
---|
2194 | RETURNS: ideal/module, the computed basis |
---|
2195 | " |
---|
2196 | { |
---|
2197 | if( isHomogenous(I) == 0) |
---|
2198 | { |
---|
2199 | ERROR ("Sorry: inhomogenous input!"); |
---|
2200 | } |
---|
2201 | |
---|
2202 | def S = groebner(I); |
---|
2203 | |
---|
2204 | if( typeof(I) == "module" or typeof(I) == "vector" ) |
---|
2205 | { |
---|
2206 | S = setModuleGrading(S, getModuleGrading(I)); |
---|
2207 | } |
---|
2208 | |
---|
2209 | return(S); |
---|
2210 | } |
---|
2211 | example |
---|
2212 | { |
---|
2213 | "EXAMPLE:"; echo=2; |
---|
2214 | |
---|
2215 | ring r = 0,(x,y,z,w),dp; |
---|
2216 | |
---|
2217 | intmat M[2][4]= |
---|
2218 | 1,1,1,1, |
---|
2219 | 0,1,3,4; |
---|
2220 | |
---|
2221 | setBaseMultigrading(M); |
---|
2222 | |
---|
2223 | |
---|
2224 | module M = ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
2225 | |
---|
2226 | |
---|
2227 | intmat v[2][nrows(M)]= |
---|
2228 | 1, |
---|
2229 | 0; |
---|
2230 | |
---|
2231 | M = setModuleGrading(M, v); |
---|
2232 | |
---|
2233 | |
---|
2234 | ///////////////////////////////////////////////////////////////////////////// |
---|
2235 | // GB: |
---|
2236 | M = mDegGroebner(M); M; |
---|
2237 | "Module Units Multigrading: "; print( getModuleGrading(M) ); |
---|
2238 | "Multidegrees: "; print(mDeg(M)); |
---|
2239 | |
---|
2240 | isHomogenous(M); |
---|
2241 | |
---|
2242 | ///////////////////////////////////////////////////////////////////////////// |
---|
2243 | // Let's compute Syzygy! |
---|
2244 | def S = mDegSyzygy(M); S; |
---|
2245 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
2246 | "Multidegrees: "; print(mDeg(S)); |
---|
2247 | |
---|
2248 | isHomogenous(S); |
---|
2249 | |
---|
2250 | ///////////////////////////////////////////////////////////////////////////// |
---|
2251 | // GB: |
---|
2252 | S = mDegGroebner(S); S; |
---|
2253 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
2254 | "Multidegrees: "; print(mDeg(S)); |
---|
2255 | |
---|
2256 | isHomogenous(S); |
---|
2257 | } |
---|
2258 | |
---|
2259 | |
---|
2260 | /******************************************************/ |
---|
2261 | proc mDegResolution(def I, int ll, list #) |
---|
2262 | "USAGE: mDegResolution(I,l,[f]); I is poly/vector/ideal/module; l,f are integers |
---|
2263 | PURPOSE: computes the multigraded resolution of I of the length l, |
---|
2264 | or the whole resolution if l is zero. Returns minimal resolution if an optional |
---|
2265 | argument 1 is supplied |
---|
2266 | NOTE: input must have multigraded-homogeneous generators. |
---|
2267 | The returned list is trunkated at the first zero entry. |
---|
2268 | RETURNS: list, the computed resolution |
---|
2269 | " |
---|
2270 | { |
---|
2271 | if( isHomogenous(I, "checkGens") == 0) |
---|
2272 | { |
---|
2273 | ERROR ("Sorry: inhomogenous input!"); |
---|
2274 | } |
---|
2275 | |
---|
2276 | def R = res(I, ll, #); list L = R; int l = size(L); |
---|
2277 | |
---|
2278 | if( (typeof(I) == "module") or (typeof(I) == "vector") ) |
---|
2279 | { |
---|
2280 | L[1] = setModuleGrading(L[1], getModuleGrading(I)); |
---|
2281 | } |
---|
2282 | |
---|
2283 | int i; |
---|
2284 | for( i = 2; i <= l; i++ ) |
---|
2285 | { |
---|
2286 | if( size(L[i]) > 0 ) |
---|
2287 | { |
---|
2288 | L[i] = setModuleGrading( L[i], mDeg(L[i-1]) ); |
---|
2289 | } else |
---|
2290 | { |
---|
2291 | return (L[1..(i-1)]); |
---|
2292 | } |
---|
2293 | } |
---|
2294 | |
---|
2295 | return (L); |
---|
2296 | |
---|
2297 | |
---|
2298 | } |
---|
2299 | example |
---|
2300 | { |
---|
2301 | "EXAMPLE:"; echo=2; |
---|
2302 | |
---|
2303 | ring r = 0,(x,y,z,w),dp; |
---|
2304 | |
---|
2305 | intmat M[2][4]= |
---|
2306 | 1,1,1,1, |
---|
2307 | 0,1,3,4; |
---|
2308 | |
---|
2309 | setBaseMultigrading(M); |
---|
2310 | |
---|
2311 | |
---|
2312 | module m= ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
2313 | |
---|
2314 | isHomogenous(ideal( xw-yz, x2z-y3, xz2-y2w, yw2-z3), "checkGens"); |
---|
2315 | |
---|
2316 | ideal A = xw-yz, x2z-y3, xz2-y2w, yw2-z3; |
---|
2317 | |
---|
2318 | int j; |
---|
2319 | |
---|
2320 | for(j=1; j<=ncols(A); j++) |
---|
2321 | { |
---|
2322 | mDegPartition(A[j]); |
---|
2323 | } |
---|
2324 | |
---|
2325 | intmat v[2][1]= |
---|
2326 | 1, |
---|
2327 | 0; |
---|
2328 | |
---|
2329 | m = setModuleGrading(m, v); |
---|
2330 | |
---|
2331 | // Let's compute Syzygy! |
---|
2332 | def S = mDegSyzygy(m); S; |
---|
2333 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
2334 | "Multidegrees: "; print(mDeg(S)); |
---|
2335 | |
---|
2336 | ///////////////////////////////////////////////////////////////////////////// |
---|
2337 | |
---|
2338 | S = mDegGroebner(S); S; |
---|
2339 | "Module Units Multigrading: "; print( getModuleGrading(S) ); |
---|
2340 | "Multidegrees: "; print(mDeg(S)); |
---|
2341 | |
---|
2342 | ///////////////////////////////////////////////////////////////////////////// |
---|
2343 | |
---|
2344 | def L = mDegResolution(m, 0, 1); |
---|
2345 | |
---|
2346 | for( j =1; j<=size(L); j++) |
---|
2347 | { |
---|
2348 | "----------------------------------- ", j, " -----------------------------"; |
---|
2349 | L[j]; |
---|
2350 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
2351 | "Multigrading: "; print(mDeg(L[j])); |
---|
2352 | } |
---|
2353 | |
---|
2354 | ///////////////////////////////////////////////////////////////////////////// |
---|
2355 | |
---|
2356 | L = mDegResolution(maxideal(1), 0, 1); |
---|
2357 | |
---|
2358 | for( j =1; j<=size(L); j++) |
---|
2359 | { |
---|
2360 | "----------------------------------- ", j, " -----------------------------"; |
---|
2361 | L[j]; |
---|
2362 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
2363 | "Multigrading: "; print(mDeg(L[j])); |
---|
2364 | } |
---|
2365 | |
---|
2366 | kill v; |
---|
2367 | |
---|
2368 | |
---|
2369 | def h = hilbertSeries(m); |
---|
2370 | setring h; |
---|
2371 | |
---|
2372 | numerator1; |
---|
2373 | factorize(numerator1); |
---|
2374 | |
---|
2375 | denominator1; |
---|
2376 | factorize(denominator1); |
---|
2377 | |
---|
2378 | numerator2; |
---|
2379 | factorize(numerator2); |
---|
2380 | |
---|
2381 | denominator2; |
---|
2382 | factorize(denominator2); |
---|
2383 | } |
---|
2384 | |
---|
2385 | /******************************************************/ |
---|
2386 | proc hilbertSeries(def I) |
---|
2387 | "USAGE: hilbertSeries(I); I is poly/vector/ideal/module |
---|
2388 | PURPOSE: computes the multigraded Hilbert Series of M |
---|
2389 | NOTE: input must have multigraded-homogeneous generators. |
---|
2390 | Multigrading should be positive. |
---|
2391 | RETURNS: a ring in variables t_(i), s_(i), with polynomials |
---|
2392 | numerator1 and denominator1 and muturally prime numerator2 |
---|
2393 | and denominator2, quotients of which give the series. |
---|
2394 | " |
---|
2395 | { |
---|
2396 | |
---|
2397 | if( !isFreeRepresented() ) |
---|
2398 | { |
---|
2399 | ERROR("SORRY: ONLY TORSION-FREE CASE (POSITIVE GRADING)"); |
---|
2400 | } |
---|
2401 | |
---|
2402 | int i, j, k, v; |
---|
2403 | |
---|
2404 | intmat M = getVariableWeights(); |
---|
2405 | |
---|
2406 | int cc = ncols(M); |
---|
2407 | int n = nrows(M); |
---|
2408 | |
---|
2409 | if( n == 0 ) |
---|
2410 | { |
---|
2411 | ERROR("Error: wrong Variable Weights?"); |
---|
2412 | } |
---|
2413 | |
---|
2414 | list RES = mDegResolution(I,0,1); |
---|
2415 | |
---|
2416 | int l = size(RES); |
---|
2417 | |
---|
2418 | list L; L[l + 1] = 0; |
---|
2419 | |
---|
2420 | if(typeof(I) == "ideal") |
---|
2421 | { |
---|
2422 | intmat zeros[n][1]; |
---|
2423 | L[1] = zeros; |
---|
2424 | } |
---|
2425 | else |
---|
2426 | { |
---|
2427 | L[1] = getModuleGrading(RES[1]); |
---|
2428 | } |
---|
2429 | |
---|
2430 | for( j = 1; j <= l; j++) |
---|
2431 | { |
---|
2432 | L[j + 1] = mDeg(RES[j]); |
---|
2433 | } |
---|
2434 | |
---|
2435 | l++; |
---|
2436 | |
---|
2437 | ring R = 0,(t_(1..n),s_(1..n)),dp; |
---|
2438 | |
---|
2439 | ideal units; |
---|
2440 | for( i=n; i>=1; i--) |
---|
2441 | { |
---|
2442 | units[i] = (var(i) * var(n + i) - 1); |
---|
2443 | } |
---|
2444 | |
---|
2445 | qring Q = std(units); |
---|
2446 | |
---|
2447 | // TODO: should not it be a quotient ring depending on Torsion??? |
---|
2448 | // I am not sure about what to do in the torsion case, but since |
---|
2449 | // we want to evaluate the polynomial at certain points to get |
---|
2450 | // a dimension we need uniqueness for this. I think we would lose |
---|
2451 | // this uniqueness if switching to this torsion ring. |
---|
2452 | |
---|
2453 | poly monom, summand, numerator; |
---|
2454 | poly denominator = 1; |
---|
2455 | |
---|
2456 | for( i = 1; i <= cc; i++) |
---|
2457 | { |
---|
2458 | monom = 1; |
---|
2459 | |
---|
2460 | for( k = 1; k <= n; k++) |
---|
2461 | { |
---|
2462 | v = M[k,i]; |
---|
2463 | |
---|
2464 | if(v >= 0) |
---|
2465 | { |
---|
2466 | monom = monom * (var(k)^(v)); |
---|
2467 | } |
---|
2468 | else |
---|
2469 | { |
---|
2470 | monom = monom * (var(n+k)^(-v)); |
---|
2471 | } |
---|
2472 | } |
---|
2473 | |
---|
2474 | if( monom == 1) |
---|
2475 | { |
---|
2476 | ERROR("Multigrading not positive."); |
---|
2477 | } |
---|
2478 | |
---|
2479 | denominator = denominator * (1 - monom); |
---|
2480 | } |
---|
2481 | |
---|
2482 | for( j = 1; j<= l; j++) |
---|
2483 | { |
---|
2484 | summand = 0; |
---|
2485 | M = L[j]; |
---|
2486 | |
---|
2487 | for( i = 1; i <= ncols(M); i++) |
---|
2488 | { |
---|
2489 | monom = 1; |
---|
2490 | for( k = 1; k <= n; k++) |
---|
2491 | { |
---|
2492 | v = M[k,i]; |
---|
2493 | if( v > 0 ) |
---|
2494 | { |
---|
2495 | monom = monom * (var(k)^v); |
---|
2496 | } |
---|
2497 | else |
---|
2498 | { |
---|
2499 | monom = monom * (var(n+k)^(-v)); |
---|
2500 | } |
---|
2501 | } |
---|
2502 | summand = summand + monom; |
---|
2503 | } |
---|
2504 | numerator = numerator - (-1)^j * summand; |
---|
2505 | } |
---|
2506 | |
---|
2507 | if( denominator == 0 ) |
---|
2508 | { |
---|
2509 | ERROR("Multigrading not positive."); |
---|
2510 | } |
---|
2511 | |
---|
2512 | poly denominator1 = denominator; |
---|
2513 | poly numerator1 = numerator; |
---|
2514 | |
---|
2515 | export denominator1; |
---|
2516 | export numerator1; |
---|
2517 | |
---|
2518 | if( numerator != 0 ) |
---|
2519 | { |
---|
2520 | poly d = gcd(denominator, numerator); |
---|
2521 | |
---|
2522 | poly denominator2 = denominator/d; |
---|
2523 | poly numerator2 = numerator/d; |
---|
2524 | |
---|
2525 | if( gcd(denominator2, numerator2) != 1 ) |
---|
2526 | { |
---|
2527 | ERROR("Sorry: gcd should be 1 (after dividing out gcd)! Something went wrong!"); |
---|
2528 | } |
---|
2529 | } |
---|
2530 | else |
---|
2531 | { |
---|
2532 | poly denominator2 = denominator; |
---|
2533 | poly numerator2 = numerator; |
---|
2534 | } |
---|
2535 | |
---|
2536 | |
---|
2537 | export denominator2; |
---|
2538 | export numerator2; |
---|
2539 | |
---|
2540 | " ------------ "; |
---|
2541 | "This proc returns a ring with polynomials called 'numerator1/2' and 'denominator1/2'!"; |
---|
2542 | "They represent the first and the second Hilbert Series."; |
---|
2543 | "The s_(i)-variables are defined to be the inverse of the t_(i)-variables."; |
---|
2544 | " ------------ "; |
---|
2545 | |
---|
2546 | return(Q); |
---|
2547 | } |
---|
2548 | example |
---|
2549 | { |
---|
2550 | "EXAMPLE:"; echo=2; |
---|
2551 | |
---|
2552 | ring r = 0,(x,y,z,w),dp; |
---|
2553 | intmat g[2][4]= |
---|
2554 | 1,1,1,1, |
---|
2555 | 0,1,3,4; |
---|
2556 | setBaseMultigrading(g); |
---|
2557 | |
---|
2558 | module M = ideal(xw-yz, x2z-y3, xz2-y2w, yw2-z3); |
---|
2559 | intmat V[2][1]= |
---|
2560 | 1, |
---|
2561 | 0; |
---|
2562 | |
---|
2563 | M = setModuleGrading(M, V); |
---|
2564 | |
---|
2565 | def h = hilbertSeries(M); setring h; |
---|
2566 | |
---|
2567 | factorize(numerator2); |
---|
2568 | factorize(denominator2); |
---|
2569 | |
---|
2570 | kill g, h; setring r; |
---|
2571 | |
---|
2572 | intmat g[2][4]= |
---|
2573 | 1,2,3,4, |
---|
2574 | 0,0,5,8; |
---|
2575 | |
---|
2576 | setBaseMultigrading(g); |
---|
2577 | |
---|
2578 | ideal I = x^2, y, z^3; |
---|
2579 | I = std(I); |
---|
2580 | def L = mDegResolution(I, 0, 1); |
---|
2581 | |
---|
2582 | for( int j = 1; j<=size(L); j++) |
---|
2583 | { |
---|
2584 | "----------------------------------- ", j, " -----------------------------"; |
---|
2585 | L[j]; |
---|
2586 | "Module Multigrading: "; print( getModuleGrading(L[j]) ); |
---|
2587 | "Multigrading: "; print(mDeg(L[j])); |
---|
2588 | } |
---|
2589 | |
---|
2590 | mDeg(I); |
---|
2591 | def h = hilbertSeries(I); setring h; |
---|
2592 | |
---|
2593 | factorize(numerator2); |
---|
2594 | factorize(denominator2); |
---|
2595 | |
---|
2596 | kill r, h, g, V; |
---|
2597 | //////////////////////////////////////////////// |
---|
2598 | //////////////////////////////////////////////// |
---|
2599 | |
---|
2600 | ring R = 0,(x,y,z),dp; |
---|
2601 | intmat W[2][3] = |
---|
2602 | 1,1, 1, |
---|
2603 | 0,0,-1; |
---|
2604 | setBaseMultigrading(W); |
---|
2605 | ideal I = x3y,yz2,y2z,z4; |
---|
2606 | |
---|
2607 | def h = hilbertSeries(I); setring h; |
---|
2608 | |
---|
2609 | factorize(numerator2); |
---|
2610 | factorize(denominator2); |
---|
2611 | |
---|
2612 | kill R, W, h; |
---|
2613 | //////////////////////////////////////////////// |
---|
2614 | //////////////////////////////////////////////// |
---|
2615 | |
---|
2616 | ring R = 0,(x,y,z,a,b,c),dp; |
---|
2617 | intmat W[2][6] = |
---|
2618 | 1,1, 1,1,1,1, |
---|
2619 | 0,0,-1,0,0,0; |
---|
2620 | setBaseMultigrading(W); |
---|
2621 | ideal I = x3y,yz2,y2z,z4; |
---|
2622 | |
---|
2623 | def h = hilbertSeries(I); setring h; |
---|
2624 | |
---|
2625 | factorize(numerator2); |
---|
2626 | factorize(denominator2); |
---|
2627 | |
---|
2628 | kill R, W, h; |
---|
2629 | |
---|
2630 | //////////////////////////////////////////////// |
---|
2631 | //////////////////////////////////////////////// |
---|
2632 | //////////////////////////////////////////////// |
---|
2633 | // This is example 5.3.9. from Robbianos book. |
---|
2634 | |
---|
2635 | ring R = 0,(x,y,z,w),dp; |
---|
2636 | intmat W[1][4] = |
---|
2637 | 1,1, 1,1; |
---|
2638 | setBaseMultigrading(W); |
---|
2639 | ideal I = z3,y3zw2,x2y4w2xyz2; |
---|
2640 | |
---|
2641 | hilb(std(I)); |
---|
2642 | |
---|
2643 | def h = hilbertSeries(I); setring h; |
---|
2644 | |
---|
2645 | numerator1; |
---|
2646 | denominator1; |
---|
2647 | |
---|
2648 | factorize(numerator2); |
---|
2649 | factorize(denominator2); |
---|
2650 | |
---|
2651 | |
---|
2652 | kill h; |
---|
2653 | //////////////////////////////////////////////// |
---|
2654 | setring R; |
---|
2655 | |
---|
2656 | ideal I2 = x2,y2,z2; I2; |
---|
2657 | |
---|
2658 | hilb(std(I2)); |
---|
2659 | |
---|
2660 | def h = hilbertSeries(I2); setring h; |
---|
2661 | |
---|
2662 | numerator1; |
---|
2663 | denominator1; |
---|
2664 | |
---|
2665 | |
---|
2666 | kill h; |
---|
2667 | //////////////////////////////////////////////// |
---|
2668 | setring R; |
---|
2669 | |
---|
2670 | W = 2,2,2,2; |
---|
2671 | |
---|
2672 | setBaseMultigrading(W); |
---|
2673 | |
---|
2674 | getVariableWeights(); |
---|
2675 | |
---|
2676 | intvec w = 2,2,2,2; |
---|
2677 | |
---|
2678 | hilb(std(I2), 1, w); |
---|
2679 | |
---|
2680 | kill w; |
---|
2681 | |
---|
2682 | |
---|
2683 | def h = hilbertSeries(I2); setring h; |
---|
2684 | |
---|
2685 | |
---|
2686 | numerator1; denominator1; |
---|
2687 | kill h; |
---|
2688 | |
---|
2689 | |
---|
2690 | kill R, W; |
---|
2691 | |
---|
2692 | //////////////////////////////////////////////// |
---|
2693 | //////////////////////////////////////////////// |
---|
2694 | //////////////////////////////////////////////// |
---|
2695 | |
---|
2696 | ring R = 0,(x),dp; |
---|
2697 | intmat W[1][1] = |
---|
2698 | 1; |
---|
2699 | setBaseMultigrading(W); |
---|
2700 | |
---|
2701 | ideal I; |
---|
2702 | |
---|
2703 | I = 1; I; |
---|
2704 | |
---|
2705 | hilb(std(I)); |
---|
2706 | |
---|
2707 | def h = hilbertSeries(I); setring h; |
---|
2708 | |
---|
2709 | numerator1; denominator1; |
---|
2710 | |
---|
2711 | kill h; |
---|
2712 | //////////////////////////////////////////////// |
---|
2713 | setring R; |
---|
2714 | |
---|
2715 | I = x; I; |
---|
2716 | |
---|
2717 | hilb(std(I)); |
---|
2718 | |
---|
2719 | def h = hilbertSeries(I); setring h; |
---|
2720 | |
---|
2721 | numerator1; denominator1; |
---|
2722 | |
---|
2723 | kill h; |
---|
2724 | //////////////////////////////////////////////// |
---|
2725 | setring R; |
---|
2726 | |
---|
2727 | I = x^5; I; |
---|
2728 | |
---|
2729 | hilb(std(I)); |
---|
2730 | hilb(std(I), 1); |
---|
2731 | |
---|
2732 | def h = hilbertSeries(I); setring h; |
---|
2733 | |
---|
2734 | numerator1; denominator1; |
---|
2735 | |
---|
2736 | |
---|
2737 | kill h; |
---|
2738 | //////////////////////////////////////////////// |
---|
2739 | setring R; |
---|
2740 | |
---|
2741 | I = x^10; I; |
---|
2742 | |
---|
2743 | hilb(std(I)); |
---|
2744 | |
---|
2745 | def h = hilbertSeries(I); setring h; |
---|
2746 | |
---|
2747 | numerator1; denominator1; |
---|
2748 | |
---|
2749 | kill h; |
---|
2750 | //////////////////////////////////////////////// |
---|
2751 | setring R; |
---|
2752 | |
---|
2753 | module M = 1; |
---|
2754 | |
---|
2755 | M = setModuleGrading(M, W); |
---|
2756 | |
---|
2757 | |
---|
2758 | hilb(std(M)); |
---|
2759 | |
---|
2760 | def h = hilbertSeries(M); setring h; |
---|
2761 | |
---|
2762 | numerator1; denominator1; |
---|
2763 | |
---|
2764 | kill h; |
---|
2765 | //////////////////////////////////////////////// |
---|
2766 | setring R; |
---|
2767 | |
---|
2768 | kill M; module M = x^5*gen(1); |
---|
2769 | |
---|
2770 | // intmat V[1][3] = 0; // TODO: this would lead to a wrong result!!!? |
---|
2771 | |
---|
2772 | intmat V[1][1] = 0; // all gen(i) of degree 0! |
---|
2773 | |
---|
2774 | M = setModuleGrading(M, V); |
---|
2775 | |
---|
2776 | hilb(std(M)); |
---|
2777 | |
---|
2778 | def h = hilbertSeries(M); setring h; |
---|
2779 | |
---|
2780 | numerator1; denominator1; |
---|
2781 | |
---|
2782 | kill h; |
---|
2783 | //////////////////////////////////////////////// |
---|
2784 | setring R; |
---|
2785 | |
---|
2786 | module N = x^5*gen(3); |
---|
2787 | |
---|
2788 | kill V; |
---|
2789 | |
---|
2790 | intmat V[1][3] = 0; // all gen(i) of degree 0! |
---|
2791 | |
---|
2792 | N = setModuleGrading(N, V); |
---|
2793 | |
---|
2794 | hilb(std(N)); |
---|
2795 | |
---|
2796 | def h = hilbertSeries(N); setring h; |
---|
2797 | |
---|
2798 | numerator1; denominator1; |
---|
2799 | |
---|
2800 | kill h; |
---|
2801 | //////////////////////////////////////////////// |
---|
2802 | setring R; |
---|
2803 | |
---|
2804 | |
---|
2805 | module S = M + N; |
---|
2806 | |
---|
2807 | S = setModuleGrading(S, V); |
---|
2808 | |
---|
2809 | hilb(std(S)); |
---|
2810 | |
---|
2811 | def h = hilbertSeries(S); setring h; |
---|
2812 | |
---|
2813 | numerator1; denominator1; |
---|
2814 | |
---|
2815 | kill h; |
---|
2816 | |
---|
2817 | kill V; |
---|
2818 | kill R, W; |
---|
2819 | |
---|
2820 | } |
---|
2821 | |
---|
2822 | |
---|
2823 | |
---|
2824 | /////////////////////////////////////////////////////////////////////////////// |
---|
2825 | // testing for consistency of the library: |
---|
2826 | proc testMultigradingLib () |
---|
2827 | { |
---|
2828 | example setBaseMultigrading; |
---|
2829 | example setModuleGrading; |
---|
2830 | |
---|
2831 | example getVariableWeights; |
---|
2832 | example getTorsion; |
---|
2833 | example getModuleGrading; |
---|
2834 | |
---|
2835 | |
---|
2836 | example mDeg; |
---|
2837 | example mDegPartition; |
---|
2838 | |
---|
2839 | |
---|
2840 | example hermite; |
---|
2841 | example isHomogenous; |
---|
2842 | example isTorsionFree; |
---|
2843 | example pushForward; |
---|
2844 | example defineHomogenous; |
---|
2845 | |
---|
2846 | example equalMDeg; |
---|
2847 | example isTorsionElement; |
---|
2848 | |
---|
2849 | example mDegResolution; |
---|
2850 | example hilbertSeries; |
---|
2851 | |
---|
2852 | |
---|
2853 | // example mDegBasis; // needs 4ti2! |
---|
2854 | } |
---|