1 | // $Id: spcurve.lib,v 1.2 1999-07-06 11:33:14 obachman Exp $ |
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2 | // (anne, last modified 20.7.98) |
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3 | ///////////////////////////////////////////////////////////////////////////// |
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4 | |
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5 | version="$Id: spcurve.lib,v 1.2 1999-07-06 11:33:14 obachman Exp $"; |
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6 | info=" |
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7 | LIBRARY: spcurve.lib PROCEDURES FOR CM CODIMENSION 2 SINGULARITIES |
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8 | |
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9 | PROCEDURES: |
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10 | isCMcod2(i); presentation matrix of the ideal i |
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11 | CMtype(i); Cohen-Macaulay type of the ideal i |
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12 | matrixT1(M,n); T1 in matrix description |
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13 | semi(M,t1); semiuniversal deformation |
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14 | discr(sem,n); discriminant given the semiuniversal |
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15 | deformation (not a powerful algorithm) |
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16 | qhmatrix(M); weights if M is quasihomogeneous |
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17 | relweight(N,W,a); relative matrix weight |
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18 | posweight(M,t1,i); perturbation of non-negative weight |
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19 | kod_ker(M,a); kernel of the Kodaira-Spencer map using eliminate |
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20 | kod_noelim(M); kernel of the Kodaira-Spencer map (weighted |
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21 | elimination) |
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22 | "; |
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23 | |
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24 | LIB "elim.lib"; |
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25 | LIB "homolog.lib"; |
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26 | LIB "inout.lib"; |
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27 | LIB "poly.lib"; |
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28 | ///////////////////////////////////////////////////////////////////////////// |
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29 | |
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30 | proc isCMcod2(ideal kurve) |
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31 | USAGE: isCMcod2(i); |
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32 | where i is an ideal |
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33 | RETURN: presentation matrix of i, if i is Cohen-Macaulay of codimension2 |
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34 | a zero matrix otherwise |
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35 | EXAMPLE: example isCMcod2; shows an example |
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36 | { |
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37 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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38 | //--------------------------------------------------------------------------- |
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39 | // Compute a minimal free resolution of the ideal and check if the |
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40 | // resolution has the expected structure |
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41 | //--------------------------------------------------------------------------- |
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42 | list kurveres=mres(kurve,0); |
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43 | matrix M=kurveres[2]; |
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44 | if ((size(kurveres)>3) && |
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45 | ((size(kurveres[3])>1) || |
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46 | ((size(kurveres[3])<=1) && (kurveres[3][1,1]!=0)))) |
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47 | { |
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48 | dbprint(p,"//not Cohen-Macaulay, codim 2"); |
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49 | matrix ret=0; |
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50 | return(ret); |
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51 | } |
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52 | return(M); |
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53 | } |
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54 | example |
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55 | { "EXAMPLE:"; echo=2; |
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56 | ring r=32003,(x,y,z),ds; |
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57 | ideal i=xz,yz,x^3-y^4; |
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58 | print(isCMcod2(i)); |
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59 | } |
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60 | ///////////////////////////////////////////////////////////////////////////// |
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61 | |
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62 | proc CMtype(ideal kurve) |
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63 | USAGE: CMtype(i); |
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64 | where i is an ideal, Cohen-Macaulay of codimension 2 |
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65 | RETURN: Cohen-Macaulay type of i (integer) |
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66 | (-1, if i is not Cohen-Macaulay of codimension 2) |
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67 | EXAMPLE: example CMtype; shows an example |
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68 | { |
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69 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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70 | int gt = -1; |
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71 | //--------------------------------------------------------------------------- |
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72 | // Compute a minimal free resolution of the ideal and check if the |
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73 | // resolution has the expected structure |
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74 | //--------------------------------------------------------------------------- |
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75 | list kurveres; |
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76 | kurveres=mres(kurve,0); |
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77 | if (size(kurveres)>3) |
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78 | { |
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79 | dbprint(p,"//not Cohen-Macaulay, codim 2"); |
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80 | return(gt); |
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81 | } |
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82 | //--------------------------------------------------------------------------- |
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83 | // Return the Cohen-Macaulay type of i |
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84 | //--------------------------------------------------------------------------- |
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85 | matrix M = matrix(kurveres[2]); |
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86 | gt = ncols(M); |
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87 | return(gt); |
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88 | } |
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89 | example |
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90 | { "EXAMPLE:"; echo=2; |
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91 | ring r=32003,(x,y,z),ds; |
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92 | ideal i=xy,xz,yz; |
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93 | CMtype(i); |
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94 | } |
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95 | ///////////////////////////////////////////////////////////////////////////// |
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96 | |
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97 | proc matrixT1(matrix M ,int n) |
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98 | USAGE: matrixT1(M,n); |
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99 | where M is a presentation matrix of an ideal, |
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100 | Cohen-Macaulay of codimension 2 |
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101 | and n determines the number of variables used as variables, not as |
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102 | parameters ( variables are assumed to precede parameters in the |
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103 | definition of the basering ) |
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104 | RETURN: T1 of i as a quotient of the module of matrices of the same |
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105 | size as M with polynomial entries |
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106 | EXAMPLE: example matrixT1; shows an example |
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107 | { |
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108 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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109 | //-------------------------------------------------------------------------- |
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110 | // Initialization and sanity checks |
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111 | //-------------------------------------------------------------------------- |
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112 | int nr=nrows(M); |
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113 | int nc=ncols(M); |
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114 | if ( nr < nc ) |
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115 | { |
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116 | M=transpose(M); |
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117 | temp=nc; |
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118 | nc=nr; |
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119 | nr=temp; |
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120 | tra=1; |
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121 | } |
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122 | if ( nr != (nc+1) ) |
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123 | { |
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124 | dbprint(p,"//not a k x (k+1) matrix"); |
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125 | return("ERROR"); |
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126 | } |
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127 | //--------------------------------------------------------------------------- |
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128 | // Construct the denominator - step by step |
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129 | // step 1: initialization |
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130 | //--------------------------------------------------------------------------- |
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131 | int gt=nc; |
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132 | int i,j; |
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133 | ideal m = M; |
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134 | ideal dx; |
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135 | ideal rv; |
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136 | ideal lv; |
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137 | matrix R[gt][gt]=0; |
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138 | matrix L[gt+1][gt+1]=0; |
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139 | matrix T1[n+gt*gt+(gt+1)*(gt+1)][gt*(gt+1)] = 0; |
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140 | //--------------------------------------------------------------------------- |
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141 | // step 2: the derivatives of the matrix are generators of the denominator |
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142 | //--------------------------------------------------------------------------- |
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143 | for( i=1; i<= n; i++ ) |
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144 | { |
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145 | dx=diff(m,var(i)); |
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146 | T1[i,1..gt*(gt+1)] = dx; |
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147 | } |
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148 | //--------------------------------------------------------------------------- |
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149 | // step 3: M*R is a generator as well |
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150 | //--------------------------------------------------------------------------- |
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151 | for( i=1; i <= gt; i++ ) |
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152 | { |
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153 | for ( j=1 ; j <= gt ; j++ ) |
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154 | { |
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155 | R[i,j]=1; |
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156 | rv = M * R; |
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157 | T1[n+(i-1)*gt+j,1..gt*(gt+1)] = rv; |
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158 | R[i,j]=0; |
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159 | } |
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160 | } |
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161 | //--------------------------------------------------------------------------- |
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162 | // step 4: so is L*M |
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163 | //--------------------------------------------------------------------------- |
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164 | for( i=1; i <= (gt+1); i++) |
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165 | { |
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166 | for( j=1 ; j <= (gt+1);j++ ) |
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167 | { |
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168 | L[i,j]=1; |
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169 | lv = L * M; |
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170 | T1[n+gt*gt+(i-1)*(gt+1)+j,1..gt*(gt+1)] = lv; |
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171 | L[i,j]=0; |
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172 | } |
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173 | } |
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174 | //--------------------------------------------------------------------------- |
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175 | // Compute the vectorspace basis of T1 |
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176 | //--------------------------------------------------------------------------- |
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177 | module t1 = module(transpose(T1)); |
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178 | list result=M,t1; |
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179 | return(result); |
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180 | } |
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181 | example |
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182 | { "EXAMPLE:"; echo = 2; |
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183 | ring r=32003,(x(1),x(2),x(3)),dp; |
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184 | ideal curve=x(1)*x(2),x(1)*x(3),x(2)*x(3); |
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185 | matrix M=isCMcod2(curve); |
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186 | matrixT1(M,3); |
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187 | } |
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188 | ///////////////////////////////////////////////////////////////////////////// |
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189 | |
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190 | proc semi(matrix M, module t1) |
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191 | USAGE: semi(M,t1); |
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192 | where M is a presentation matrix of a Cohen-Macaulay codimension 2 |
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193 | ideal and t1 is its T1 space in matrix notation |
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194 | The current basering should not contain any variables named |
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195 | A(i) where i is some integer! |
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196 | RETURN: after a change to a new basering |
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197 | ring rneu=char(basering),(var(1),..,var(nvars(basering)), |
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198 | A(1),..,A(vdim(t1))),(dp nvars(basering),dp vdim(t1)); |
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199 | an ideal describing the semiuniversal deformation is returned |
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200 | EXAMPLE: example semi; shows an example |
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201 | { |
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202 | //--------------------------------------------------------------------------- |
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203 | // Initialization |
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204 | //--------------------------------------------------------------------------- |
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205 | module t1erz=kbase(t1); |
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206 | int tau=vdim(t1); |
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207 | int gt=ncols(M); |
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208 | int i; |
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209 | def r=basering; |
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210 | if(size(M)!=gt*(gt+1)) |
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211 | { |
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212 | gt=gt-1; |
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213 | } |
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214 | //--------------------------------------------------------------------------- |
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215 | // Definition of the new ring and the image of M and t1 in the new ring |
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216 | //--------------------------------------------------------------------------- |
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217 | ring rtemp=0,(A(1..tau)),dp; |
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218 | def rneu=r+rtemp; |
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219 | setring rneu; |
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220 | matrix M=imap(r,M); |
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221 | ideal m=M; |
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222 | module t1erz=imap(r,t1erz); |
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223 | //--------------------------------------------------------------------------- |
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224 | // Construction of the presentation matrix of the versal deformation |
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225 | //--------------------------------------------------------------------------- |
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226 | matrix N=matrix(m); |
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227 | matrix Mtemp[gt*(gt+1)][1]; |
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228 | for( i=1; i<=tau; i++) |
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229 | { |
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230 | Mtemp=t1erz[i]; |
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231 | N=N+A(i)*transpose(Mtemp); |
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232 | } |
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233 | ideal n=N; |
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234 | matrix O[gt+1][gt]=n; |
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235 | //--------------------------------------------------------------------------- |
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236 | // Construction of the return value |
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237 | //--------------------------------------------------------------------------- |
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238 | ideal result=minor(O,gt); |
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239 | export rneu; |
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240 | keepring rneu; |
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241 | return(result); |
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242 | } |
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243 | example |
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244 | { "EXAMPLE:"; echo=2; |
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245 | ring r=32003,(x(1),x(2),x(3)),dp; |
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246 | ideal curve=x(1)*x(2),x(1)*x(3),x(2)*x(3); |
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247 | matrix M=isCMcod2(curve); |
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248 | list l=matrixT1(M,3); |
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249 | semi(l[1],std(l[2])); |
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250 | } |
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251 | ///////////////////////////////////////////////////////////////////////////// |
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252 | |
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253 | proc discr(ideal kurve, int n) |
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254 | USAGE: discr(sem,n) |
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255 | where sem is the versal deformation of an ideal |
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256 | and the first n variables of the ring are treated as variables |
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257 | all the others as parameters |
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258 | RETURN: ideal describing the discriminant |
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259 | EXAMPLE: example discr; shows an example |
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260 | { |
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261 | //--------------------------------------------------------------------------- |
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262 | // some sanity checks and initialization |
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263 | //--------------------------------------------------------------------------- |
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264 | int i; |
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265 | ideal sem=std(kurve); |
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266 | ideal semdiff; |
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267 | ideal J2; |
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268 | int ncol=ncols(matrix(sem)); |
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269 | matrix Jacob[n][ncol]; |
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270 | //--------------------------------------------------------------------------- |
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271 | // compute the Jacobian |
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272 | //--------------------------------------------------------------------------- |
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273 | for ( i=1; i<=n; i++) |
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274 | { |
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275 | semdiff=diff(sem,var(i)); |
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276 | Jacob[i,1..ncol]=semdiff; |
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277 | } |
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278 | //--------------------------------------------------------------------------- |
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279 | // eliminate the first n variables in the ideal generated by |
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280 | // the versal deformation and the 2x2 minors of the Jacobian |
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281 | //--------------------------------------------------------------------------- |
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282 | semdiff=minor(Jacob,2); |
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283 | J2=sem,semdiff; |
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284 | J2=std(J2); |
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285 | poly eli=1; |
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286 | for( i=1; i<=n; i++) |
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287 | { |
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288 | eli=eli*var(i); |
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289 | } |
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290 | ideal dis=eliminate(J2,eli); |
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291 | return(dis); |
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292 | } |
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293 | example |
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294 | { "EXAMPLE:"; echo=2; |
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295 | ring r=32003,(x(1),x(2),x(3)),dp; |
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296 | ideal curve=x(1)*x(2),x(1)*x(3),x(2)*x(3); |
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297 | matrix M=isCMcod2(curve); |
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298 | list l=matrixT1(M,3); |
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299 | def sem=semi(l[1],std(l[2])); |
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300 | basering; |
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301 | discr(sem,3); |
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302 | } |
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303 | ///////////////////////////////////////////////////////////////////////////// |
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304 | |
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305 | proc qhmatrix(matrix M) |
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306 | USAGE: qhmatrix(M); |
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307 | where M is a matrix with polynomial entries of size k x (k+1) |
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308 | RETURN: list consisting of an integer vector containing the weights of |
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309 | the variables of the basering and an integer matrix giving the |
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310 | weights of the entries |
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311 | integer vector and integer matrix both contain only 0, if the |
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312 | matrix is not quasihomogeneous, i.e. does not allow row and |
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313 | column weights |
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314 | EXAMPLE: example qhmatrix; shows an example |
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315 | { |
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316 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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317 | //--------------------------------------------------------------------------- |
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318 | // Initialization and sanity checks |
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319 | //--------------------------------------------------------------------------- |
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320 | def r=basering; |
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321 | int i,j,temp; |
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322 | int tra=0; |
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323 | int nr=nrows(M); |
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324 | int nc=ncols(M); |
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325 | if ( nr > nc ) |
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326 | { |
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327 | M=transpose(M); |
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328 | temp=nc; |
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329 | nc=nr; |
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330 | nr=temp; |
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331 | tra=1; |
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332 | } |
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333 | if ( nc != (nr+1) ) |
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334 | { |
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335 | dbprint(p,"//not a k x (k+1) matrix"); |
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336 | return("ERROR"); |
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337 | } |
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338 | ideal m=minor(M,nr); |
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339 | //--------------------------------------------------------------------------- |
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340 | // get the weight using the fact that the matrix is quasihomogeneous, if |
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341 | // its maximal minors are, and check, whether M is really quasihomogeneous |
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342 | //--------------------------------------------------------------------------- |
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343 | intvec a=weight(m); |
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344 | string tempstr="ring rneu=" + charstr(r) + ",(" + varstr(r) + "),ws(" + string(a) + ");"; |
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345 | execute(tempstr); |
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346 | def M=imap(r,M); |
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347 | int difset=0; |
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348 | list l; |
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349 | int dif; |
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350 | int donttest=0; |
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351 | int comprow=0; |
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352 | intmat W[nr][nc]; |
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353 | //--------------------------------------------------------------------------- |
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354 | // find a row not containing a 0 |
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355 | //--------------------------------------------------------------------------- |
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356 | for(i=1; i<=nr; i++) |
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357 | { |
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358 | if(comprow==0) |
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359 | { |
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360 | comprow=i; |
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361 | for(j=1; j<=nc; j++) |
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362 | { |
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363 | if(M[i,j]==0) |
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364 | { |
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365 | comprow=0; |
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366 | break; |
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367 | } |
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368 | } |
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369 | } |
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370 | } |
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371 | //--------------------------------------------------------------------------- |
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372 | // get the weights of the comprow'th row or use emergency exit |
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373 | //--------------------------------------------------------------------------- |
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374 | if(comprow==0) |
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375 | { |
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376 | intvec v=0; |
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377 | intmat V=0 |
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378 | list ret=v,V; |
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379 | return(ret); |
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380 | } |
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381 | else |
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382 | { |
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383 | for(j=1; j<=nc; j++) |
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384 | { |
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385 | l[j]=deg(lead(M[comprow,j])); |
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386 | } |
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387 | } |
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388 | //--------------------------------------------------------------------------- |
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389 | // do the checks |
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390 | //--------------------------------------------------------------------------- |
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391 | for(i=1; i<=nr; i++) |
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392 | { |
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393 | if ( i==comprow ) |
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394 | { |
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395 | // this row should not be tested against itself |
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396 | donttest=1; |
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397 | } |
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398 | else |
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399 | { |
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400 | // initialize the difference of the rows, but ignore 0-entries |
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401 | if (M[i,1]!=0) |
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402 | { |
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403 | dif=deg(lead(M[i,1]))-l[1]; |
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404 | difset=1; |
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405 | } |
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406 | else |
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407 | { |
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408 | list memo; |
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409 | memo[1]=1; |
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410 | } |
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411 | } |
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412 | // check column by column |
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413 | for(j=1; j<=nc; j++) |
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414 | { |
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415 | if(M[i,j]==0) |
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416 | { |
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417 | if(defined(memo)!=0) |
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418 | { |
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419 | memo[size(memo)+1]=j; |
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420 | } |
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421 | else |
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422 | { |
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423 | list memo; |
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424 | memo[1]=j; |
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425 | } |
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426 | } |
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427 | temp=deg(lead(M[i,j])); |
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428 | if((difset!=1) && (donttest!=1) && (M[i,j]!=0)) |
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429 | { |
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430 | // initialize the difference of the rows, if necessary - still ignore 0s |
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431 | dif=deg(lead(M[i,j]))-l[j]; |
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432 | difset=1; |
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433 | } |
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434 | // is M[i,j] quasihomogeneous - else emergency exit |
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435 | if(M[i,j]!=jet(M[i,j],temp,a)-jet(M[i,j],temp-1,a)) |
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436 | { |
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437 | intvec v=0; |
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438 | intmat V=0; |
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439 | list ret=v,V; |
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440 | return(ret); |
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441 | } |
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442 | if(donttest!=1) |
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443 | { |
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444 | // check row and column weights - else emergency exit |
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445 | if(((temp-l[j])!=dif) && (M[i,j]!=0) && (difset==1)) |
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446 | { |
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447 | intvec v=0; |
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448 | intmat V=0; |
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449 | list ret=v,V; |
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450 | return(ret); |
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451 | } |
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452 | } |
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453 | // set the weight matrix entry |
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454 | W[i,j]=temp; |
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455 | } |
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456 | // clean up the 0's we left out |
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457 | if((difset==1) && (defined(memo)!=0)) |
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458 | { |
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459 | for(j=1; j<=size(memo); j++) |
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460 | { |
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461 | W[i,memo[j]]=dif+l[memo[j]]; |
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462 | } |
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463 | kill memo; |
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464 | } |
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465 | donttest=0; |
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466 | } |
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467 | //--------------------------------------------------------------------------- |
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468 | // transpose, if M was transposed during initialization, and return the list |
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469 | //--------------------------------------------------------------------------- |
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470 | if ( tra==1 ) |
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471 | { |
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472 | W=transpose(W); |
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473 | } |
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474 | setring r; |
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475 | list ret=a,W; |
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476 | return(ret); |
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477 | } |
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478 | example |
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479 | { "EXAMPLE:"; echo=2; |
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480 | ring r=0,(x,y,z),ds; |
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481 | matrix M[3][2]=z,0,y,x,x^3,y; |
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482 | qhmatrix(M); |
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483 | pmat(M); |
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484 | } |
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485 | ///////////////////////////////////////////////////////////////////////////// |
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486 | |
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487 | proc relweight(matrix N, intmat W, intvec a) |
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488 | USAGE: relweight(N,W); |
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489 | where N is a non-zero matrix with polynomial entries, W is a |
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490 | matrix with integer entries of the same size and the entries |
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491 | of a give the weight of the variables of the basering |
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492 | RETURN: maximum of weight_a(N_\{ij\})-W_\{ij\} |
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493 | string "ERROR" is sizes do not match |
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494 | EXAMPLE: example relweight; shows an example |
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495 | { |
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496 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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497 | //--------------------------------------------------------------------------- |
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498 | // Initialization and sanity checks |
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499 | //--------------------------------------------------------------------------- |
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500 | if ((size(N)!=size(W)) || (ncols(N)!=ncols(W))) |
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501 | { |
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502 | dbprint(p,"//matrix size does not match"); |
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503 | return("ERROR"); |
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504 | } |
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505 | if (size(a)!=nvars(basering)) |
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506 | { |
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507 | dbprint(p,"//length of weight vector != number of variables"); |
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508 | return("ERROR"); |
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509 | } |
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510 | int i,j,temp; |
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511 | def r=basering; |
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512 | //--------------------------------------------------------------------------- |
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513 | // Comparision entry by entry |
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514 | //--------------------------------------------------------------------------- |
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515 | for(i=1; i<=nrows(N); i++) |
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516 | { |
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517 | for(j=1; j<=ncols(N); j++) |
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518 | { |
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519 | if (N[i,j]!=0) |
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520 | { |
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521 | temp=mindeg1(N[i,j],a)-W[i,j]; |
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522 | if (defined(ret)) |
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523 | { |
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524 | if(temp > ret) |
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525 | { |
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526 | ret=temp; |
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527 | } |
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528 | } |
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529 | else |
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530 | { |
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531 | int ret=temp; |
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532 | } |
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533 | } |
---|
534 | } |
---|
535 | } |
---|
536 | return(ret); |
---|
537 | } |
---|
538 | example |
---|
539 | { "EXAMPLE:"; echo=2; |
---|
540 | ring r=32003,(x,y,z),ds; |
---|
541 | matrix N[2][3]=z,0,y,x,x^3,y; |
---|
542 | intmat W[2][3]=1,1,1,1,1,1; |
---|
543 | intvec a=1,1,1; |
---|
544 | relweight(N,W,a); |
---|
545 | } |
---|
546 | ///////////////////////////////////////////////////////////////////////////// |
---|
547 | |
---|
548 | proc posweight(matrix M, module t1,int choose) |
---|
549 | USAGE: posweight(M,t1,choose); |
---|
550 | where M is a presentation matrix of a Cohen-Macaulay codimension 2 |
---|
551 | ideal and t1 is its T1 space in matrix notation |
---|
552 | The current basering should not contain any variables named |
---|
553 | T(i) where i is some integer! |
---|
554 | choose==0 => all perturbations of non-negative weight |
---|
555 | choose==1 => only those which do not change the CM-type |
---|
556 | choose==2 => all perturbations of positive weight |
---|
557 | RETURN: after a change to a new basering |
---|
558 | ring rneu=char(basering),(var(1),..,var(nvars(basering)), |
---|
559 | T(1),..,T(k),(dp nvars(basering),dp k); |
---|
560 | a presentation matrix describing the deformation given by the |
---|
561 | generators of T1 of non-negative weight is returned as the first |
---|
562 | list element, followed by the weight vector for the new variables |
---|
563 | EXAMPLE: example posweight; shows an example |
---|
564 | { |
---|
565 | //--------------------------------------------------------------------------- |
---|
566 | // Initialization and sanity checks |
---|
567 | //--------------------------------------------------------------------------- |
---|
568 | if (attrib(t1,"isSB")) |
---|
569 | { |
---|
570 | module t1erz=kbase(t1); |
---|
571 | int tau=vdim(t1); |
---|
572 | } |
---|
573 | else |
---|
574 | { module t1erz=kbase(std(t1)); |
---|
575 | int tau=vdim(std(t1)); |
---|
576 | } |
---|
577 | for(int i=1; i<=size(t1erz); i++) |
---|
578 | { |
---|
579 | if(rvar(T(i))) |
---|
580 | { |
---|
581 | int jj=-1; |
---|
582 | } |
---|
583 | } |
---|
584 | kill i; |
---|
585 | if (defined(jj)) |
---|
586 | { |
---|
587 | if (jj==-1) |
---|
588 | { |
---|
589 | "Your ring contains a variable T(" +string(i)+")!"; |
---|
590 | return("ERROR"); |
---|
591 | } |
---|
592 | } |
---|
593 | int pw=0; |
---|
594 | int i; |
---|
595 | def r=basering; |
---|
596 | list l=qhmatrix(M); |
---|
597 | int gt=ncols(M); |
---|
598 | if(size(M)!=gt*(gt+1)) |
---|
599 | { |
---|
600 | gt=gt-1; |
---|
601 | } |
---|
602 | matrix erzmat[gt+1][gt]; |
---|
603 | list erz; |
---|
604 | if ((size(l[1])==1) && (l[1][1]==0) && (size(l[2])==1) && (l[2][1,1]==0)) |
---|
605 | { |
---|
606 | return("ERROR"); |
---|
607 | } |
---|
608 | //--------------------------------------------------------------------------- |
---|
609 | // Find the generators of T1 of non-negative weight |
---|
610 | //--------------------------------------------------------------------------- |
---|
611 | int relw; |
---|
612 | list rlw; |
---|
613 | for(i=1; i<=tau; i++) |
---|
614 | { |
---|
615 | erzmat=t1erz[i]; |
---|
616 | kill relw; |
---|
617 | def relw=relweight(erzmat,l[2],l[1]); |
---|
618 | if(typeof(relw)=="int") |
---|
619 | { |
---|
620 | if (((choose==0) && (relw>=0)) |
---|
621 | || ((choose==1) && (relw>=0) && (CMtype(minor(M+erzmat,gt))==gt)) |
---|
622 | || ((choose==2) && (relw > 0))) |
---|
623 | { |
---|
624 | pw++; |
---|
625 | rlw[pw]=relw; |
---|
626 | erz[pw]=erzmat; |
---|
627 | } |
---|
628 | } |
---|
629 | else |
---|
630 | { |
---|
631 | return("ERROR"); |
---|
632 | } |
---|
633 | } |
---|
634 | //--------------------------------------------------------------------------- |
---|
635 | // Definition of the new ring and the image of M and erz in the new ring |
---|
636 | //--------------------------------------------------------------------------- |
---|
637 | if(size(rlw)==0) |
---|
638 | { |
---|
639 | return("ERROR"); |
---|
640 | } |
---|
641 | intvec iv=rlw[1..size(rlw)]; |
---|
642 | ring rtemp=0,(T(1..pw)),dp; |
---|
643 | def rneu=r+rtemp; |
---|
644 | setring rneu; |
---|
645 | matrix M=imap(r,M); |
---|
646 | ideal m=M; |
---|
647 | // we cannot imap erz, if its size=0 |
---|
648 | if(pw==0) |
---|
649 | { |
---|
650 | list erz1; |
---|
651 | } |
---|
652 | else |
---|
653 | { |
---|
654 | list erz1=imap(r,erz); |
---|
655 | } |
---|
656 | //--------------------------------------------------------------------------- |
---|
657 | // Construction of the presentation matrix of the deformation |
---|
658 | //--------------------------------------------------------------------------- |
---|
659 | matrix N=matrix(m); |
---|
660 | ideal mtemp; |
---|
661 | matrix Mtemp[gt*(gt+1)][1]; |
---|
662 | for( i=1; i<=pw; i++) |
---|
663 | { |
---|
664 | mtemp=erz1[i]; |
---|
665 | Mtemp=mtemp; |
---|
666 | N=N+T(i)*transpose(Mtemp); |
---|
667 | } |
---|
668 | ideal n=N; |
---|
669 | matrix O[gt+1][gt]=n; |
---|
670 | //--------------------------------------------------------------------------- |
---|
671 | // Keep the ring and return the matrix |
---|
672 | //--------------------------------------------------------------------------- |
---|
673 | export rneu; |
---|
674 | keepring rneu; |
---|
675 | list ret=O,iv; |
---|
676 | return(ret); |
---|
677 | } |
---|
678 | example |
---|
679 | { "EXAMPLE:"; echo=2; |
---|
680 | ring r=32003,(x(1),x(2),x(3)),dp; |
---|
681 | ideal curve=(x(3)-x(1)^2)*x(3),(x(3)-x(1)^2)*x(2),x(2)^2-x(1)^7*x(3); |
---|
682 | matrix M=isCMcod2(curve); |
---|
683 | list l=matrixT1(M,3); |
---|
684 | list li=posweight(l[1],std(l[2]),0); |
---|
685 | pmat(li[1]); |
---|
686 | li[2]; |
---|
687 | } |
---|
688 | ///////////////////////////////////////////////////////////////////////////// |
---|
689 | |
---|
690 | proc kod_ker(matrix Mat, intvec a); |
---|
691 | USAGE: kod_ker(Mat,a); |
---|
692 | where Mat and a are the return values of posweight, |
---|
693 | i.e. Mat is a k x (k+1) matrix perturbed by all generators of T1 |
---|
694 | of relative weight >= 0 and -a is the weight vector of the |
---|
695 | corresponding parameters |
---|
696 | RETURN: module K, a presentation of the kernel of the Kodaira-Spencer |
---|
697 | mapping |
---|
698 | EXAMPLE: example kod_ker; shows examples |
---|
699 | { |
---|
700 | //--------------------------------------------------------------------------- |
---|
701 | // Initialization and sanity checks |
---|
702 | //--------------------------------------------------------------------------- |
---|
703 | intvec optvec=option(get); |
---|
704 | int i,j; |
---|
705 | int gt=ncols(Mat); |
---|
706 | int first=nvars(basering)-size(a); |
---|
707 | int last=size(a); |
---|
708 | intvec iv; |
---|
709 | poly p=1; |
---|
710 | def baser=basering; |
---|
711 | if (nrows(Mat)!=gt+1) |
---|
712 | { |
---|
713 | if (nrows(Mat)!=gt-1) |
---|
714 | { |
---|
715 | return("ERROR"); |
---|
716 | } |
---|
717 | else |
---|
718 | { |
---|
719 | gt=gt-1; |
---|
720 | |
---|
721 | } |
---|
722 | } |
---|
723 | list l=matrixT1(Mat,first); |
---|
724 | matrix t1=l[2]; |
---|
725 | matrix kod[gt*(gt+1)][last]; |
---|
726 | for(i=1;i<=last;i++) |
---|
727 | { |
---|
728 | kod[1..gt*(gt+1),i]=diff(l[1],var(first+i)); |
---|
729 | } |
---|
730 | |
---|
731 | module ker=std(modulo(kod,t1)); |
---|
732 | option(redTail); |
---|
733 | ker=std(ker); |
---|
734 | |
---|
735 | for(j=1;j<=first;j++) |
---|
736 | { |
---|
737 | p=p*var(j); |
---|
738 | } |
---|
739 | for(i=1;i<=size(ker);i++) |
---|
740 | { |
---|
741 | matrix testlead=coef(lead(ker[i]),p); |
---|
742 | if((size(testlead)==2) && (testlead[1,1]=1)) |
---|
743 | { |
---|
744 | if((ker[i]/lead(ker[i]))*lead(ker[i])==ker[i]) |
---|
745 | { |
---|
746 | ker[i]=lead(ker[i]); |
---|
747 | } |
---|
748 | } |
---|
749 | kill testlead; |
---|
750 | } |
---|
751 | ker=eliminate(interred(ker),p); |
---|
752 | option(set,optvec); |
---|
753 | return(ker); |
---|
754 | } |
---|
755 | example |
---|
756 | {"EXAMPLE"; // echo=2; |
---|
757 | ring r=0,(x,y,z),(c,ds); |
---|
758 | matrix M[3][2]=z-x^4,0,y,z,x^12,y; |
---|
759 | list l=matrixT1(M,3); |
---|
760 | list posw=posweight(l[1],std(l[2]),1); |
---|
761 | posw; |
---|
762 | kod_ker(posw[1],posw[2]); |
---|
763 | } |
---|
764 | ///////////////////////////////////////////////////////////////////////////// |
---|
765 | |
---|
766 | proc kod_noelim(matrix M) |
---|
767 | USAGE: kod_noelim(M); |
---|
768 | where M is a quasihomogeneous n x (n+1) matrix defining an |
---|
769 | isolated space curve singularity |
---|
770 | NOTE: the initial basering should not contain variables with name |
---|
771 | e(i) or T(i), since those variable names will be internally |
---|
772 | used by the script |
---|
773 | RETURN: coefficient matrix representing the kernel of the Kodaira- |
---|
774 | Spencer map of the family of non-negative perturbations |
---|
775 | having the given singularity as special fibre |
---|
776 | EXAMPLE: example kod_noelim; shows an example |
---|
777 | { |
---|
778 | //--------------------------------------------------------------------------- |
---|
779 | // Initialization and sanity checks |
---|
780 | //--------------------------------------------------------------------------- |
---|
781 | intvec optvec=option(get); |
---|
782 | option(redTail); |
---|
783 | if (nvars(basering) != 3 ) |
---|
784 | { |
---|
785 | "It should be a curve in 3 space"; |
---|
786 | return("ERROR"); |
---|
787 | } |
---|
788 | def rt=basering; |
---|
789 | list wl=qhmatrix(M); |
---|
790 | if ((size(wl)!=2) || ((wl[1]==0) && (wl[2]==0))) |
---|
791 | { |
---|
792 | "The matrix was not n x (n+1) or not quasihomogenous"; |
---|
793 | return("ERROR"); |
---|
794 | } |
---|
795 | string ringre=" ring r=" + charstr(rt) + ",(x,y,z), ws(" + string(wl[1]) + ");"; |
---|
796 | execute ringre; |
---|
797 | matrix M=imap(rt,M); |
---|
798 | int ne=size(M); |
---|
799 | intvec iv=1..ne; |
---|
800 | list l=matrixT1(M,3); |
---|
801 | if (dim(std(l[2])) != 0) |
---|
802 | { |
---|
803 | "The matrix does not define an isolated space curve singularity"; |
---|
804 | return("ERROR"); |
---|
805 | } |
---|
806 | module t1qh=l[2]; |
---|
807 | //-------------------------------------------------------------------------- |
---|
808 | // Passing to a new ring with extra variables e(i) corresponding to |
---|
809 | // the module generators gen(i) for weighted standard basis computation |
---|
810 | // accepting weights for the gen(i) |
---|
811 | //-------------------------------------------------------------------------- |
---|
812 | for(int i=1; i<=ne; i++) |
---|
813 | { |
---|
814 | if(rvar(e(i))) |
---|
815 | { |
---|
816 | int jj=-1; |
---|
817 | } |
---|
818 | } |
---|
819 | if (defined(jj)) |
---|
820 | { |
---|
821 | if (jj==-1) |
---|
822 | { |
---|
823 | "Your ring contains a variable e(" +string(i)+")!"; |
---|
824 | return("ERROR"); |
---|
825 | } |
---|
826 | } |
---|
827 | ringre="ring re=" + charstr(r) +",(e(1.." + string(ne) + "),"+ varstr(basering) |
---|
828 | + "),ws(" + string((-1)*intvec(wl[2])) + "," |
---|
829 | + string(wl[1]) + ");"; |
---|
830 | execute ringre; |
---|
831 | module temp=imap(r,t1qh); |
---|
832 | ideal t1qh=mod2id(temp,iv); |
---|
833 | if (defined(Debug)) |
---|
834 | { |
---|
835 | option(prot); |
---|
836 | } |
---|
837 | ideal t1qhs=std(t1qh); |
---|
838 | if (defined(Debug)) |
---|
839 | { |
---|
840 | option(noprot); |
---|
841 | } |
---|
842 | ideal t1qhsl=lead(t1qhs); |
---|
843 | module mo=id2mod(t1qhsl,iv); |
---|
844 | //-------------------------------------------------------------------------- |
---|
845 | // Return to the initial ring to compute the kbase and noether there |
---|
846 | // (in the new ring t1qh is of course not of dimension 0 but of dimension 3 |
---|
847 | // so we have to go back) |
---|
848 | //-------------------------------------------------------------------------- |
---|
849 | setring r; |
---|
850 | module mo=imap(re,mo); |
---|
851 | attrib(mo,"isSB",1); // mo is monomial ==> SB |
---|
852 | attrib(mo,"isHomog",intvec(wl[2])); // highcorner has to respect the weights |
---|
853 | vector noe=highcorner(mo); |
---|
854 | if (defined(Debug)) |
---|
855 | { |
---|
856 | wl; |
---|
857 | mo; |
---|
858 | noe; |
---|
859 | } |
---|
860 | //-------------------------------------------------------------------------- |
---|
861 | // Define the family of curves with the same quasihomogeneous initial |
---|
862 | // matrix M, compute T1 and pass again to the ring with the variables e(i) |
---|
863 | //-------------------------------------------------------------------------- |
---|
864 | list li=posweight(M,mo,0); |
---|
865 | if (size(li)<=1) |
---|
866 | { |
---|
867 | return("ERROR") |
---|
868 | } |
---|
869 | if (defined(Debug)) |
---|
870 | { |
---|
871 | li; |
---|
872 | } |
---|
873 | list li2=matrixT1(li[1],3); |
---|
874 | module Mpert=transpose(matrix(ideal(li2[1]))); |
---|
875 | module t1pert=li2[2]; |
---|
876 | int nv=nvars(rneu)-nvars(r); |
---|
877 | ring rtemp=0,(T(1..nv)),wp(li[2]); |
---|
878 | def reneu=re+rtemp; |
---|
879 | setring reneu; |
---|
880 | module noe=matrix(imap(r,noe)); |
---|
881 | ideal noet=mod2id(noe,iv); |
---|
882 | module temp=imap(rneu,t1pert); |
---|
883 | ideal t1pert=mod2id(temp,iv); |
---|
884 | //-------------------------------------------------------------------------- |
---|
885 | // Compute the standard basis and select those generators with leading term |
---|
886 | // divisible by some T(i) |
---|
887 | //-------------------------------------------------------------------------- |
---|
888 | noether=noet[size(noet)]; |
---|
889 | if (defined(Debug)) |
---|
890 | { |
---|
891 | noether; |
---|
892 | option(prot); |
---|
893 | } |
---|
894 | ideal t1perts=std(t1pert); |
---|
895 | noether=noet[size(noet)]; |
---|
896 | t1perts=interred(t1perts); |
---|
897 | if (defined(Debug)) |
---|
898 | { |
---|
899 | option(noprot); |
---|
900 | } |
---|
901 | ideal templ=lead(t1perts); |
---|
902 | for(int j=1;j<=nv;j++) |
---|
903 | { |
---|
904 | templ=subst(templ,T(j),0); |
---|
905 | } |
---|
906 | ideal mx; |
---|
907 | ideal mt; |
---|
908 | for(j=1;j<=size(t1perts);j++) |
---|
909 | { |
---|
910 | if(templ[j]!=0) |
---|
911 | { |
---|
912 | mx=mx,t1perts[j]; |
---|
913 | } |
---|
914 | else |
---|
915 | { |
---|
916 | mt=mt,t1perts[j]; |
---|
917 | } |
---|
918 | } |
---|
919 | //-------------------------------------------------------------------------- |
---|
920 | // multiply by the initial ring variables to shift the generators with |
---|
921 | // leading term divisible by some T(i) and reduce afterwards |
---|
922 | //-------------------------------------------------------------------------- |
---|
923 | attrib(mx,"isSB",1); // This is obviously no SB, but we have to reduce by |
---|
924 | // it and setting isSB suppresses error messages |
---|
925 | noether=noet[size(noet)]; |
---|
926 | ideal ker_gen=reduce(mt,mx); |
---|
927 | ideal ovar=var(ne+1),var(ne+2),var(ne+3); |
---|
928 | ideal tvar=ovar; |
---|
929 | j=1; |
---|
930 | noether=noet[size(noet)]; |
---|
931 | if (defined(Debug)) |
---|
932 | { |
---|
933 | mt; |
---|
934 | noether; |
---|
935 | } |
---|
936 | int zeros; |
---|
937 | templ=ker_gen; |
---|
938 | while(zeros==0) |
---|
939 | { |
---|
940 | zeros=1; |
---|
941 | templ=templ*ovar; |
---|
942 | templ=reduce(templ,mx); |
---|
943 | // for(i=1;i<=size(tvar);i++) |
---|
944 | // { |
---|
945 | // templ=tvar[i]*mt; |
---|
946 | // templ=reduce(templ,mx); |
---|
947 | if(defined(Debug)) |
---|
948 | { |
---|
949 | if(Debug > 2) |
---|
950 | { |
---|
951 | // tvar[i]; |
---|
952 | templ; |
---|
953 | // ~; |
---|
954 | } |
---|
955 | } |
---|
956 | if (size(templ)!= 0) |
---|
957 | { |
---|
958 | zeros=0; |
---|
959 | ker_gen=ker_gen,templ; |
---|
960 | } |
---|
961 | // } |
---|
962 | // tvar=tvar*ovar; |
---|
963 | } |
---|
964 | //------------------------------------------------------------------------- |
---|
965 | // kill zero entries, keep only one of identical entries and |
---|
966 | // return the result |
---|
967 | //------------------------------------------------------------------------- |
---|
968 | ovar=var(1); |
---|
969 | for(i=2;i<=ne;i++) |
---|
970 | { |
---|
971 | ovar=ovar,var(i); |
---|
972 | } |
---|
973 | ker_gen=ker_gen,ovar^2; |
---|
974 | noether=noet[size(noet)]; |
---|
975 | ker_gen=simplify(ker_gen,10); |
---|
976 | //------------------------------------------------------------------------- |
---|
977 | // interreduce ker_gen as a k[T]-module |
---|
978 | //------------------------------------------------------------------------- |
---|
979 | intvec mgen=1..(ne+3); |
---|
980 | ideal Mpert=mod2id(imap(rneu,Mpert),iv); |
---|
981 | templ=0; |
---|
982 | for(i=1;i<=nv;i++) |
---|
983 | { |
---|
984 | templ[i]=diff(Mpert[size(Mpert)],T(i)); |
---|
985 | } |
---|
986 | templ=templ,ovar^2; |
---|
987 | ker_gen=subrInterred(templ,ker_gen,mgen); |
---|
988 | module ret=id2mod(ker_gen,iv); |
---|
989 | option(set,optvec); |
---|
990 | export reneu; |
---|
991 | keepring reneu; |
---|
992 | return(ret); |
---|
993 | } |
---|
994 | example |
---|
995 | { "EXAMPLE:"; echo=2; |
---|
996 | ring r=0,(x,y,z),ds; |
---|
997 | matrix M[3][2]=z,0,y,z-x^3,x^14,y; |
---|
998 | kod_noelim(M); |
---|
999 | } |
---|
1000 | /////////////////////////////////////////////////////////////////////////// |
---|
1001 | static |
---|
1002 | proc mod2id(matrix M,intvec vpos) |
---|
1003 | USAGE: mod2id(M,vpos); |
---|
1004 | where M is a matrix and vpos is an integer vector such that |
---|
1005 | gen(i) will correspond to var(vpos[i]) |
---|
1006 | NOTE: this procedure should be used in the following situation: |
---|
1007 | one wants to pass to a ring with new variables which stand |
---|
1008 | for the generators of the module and consider the module as |
---|
1009 | an ideal |
---|
1010 | RETURN: ideal i in which each gen(i) from the module is replaced by |
---|
1011 | var(vpos[i]) and all monomials var(vpos[i])*var(vpos[j]) have |
---|
1012 | been added to the generating set of i |
---|
1013 | EXAMPLE: example mod2id; shows an example |
---|
1014 | { |
---|
1015 | //---------------------------------------------------------------------- |
---|
1016 | // define the ideal generated by the var(vpos[i]) and set up the matrix |
---|
1017 | // for the multiplication |
---|
1018 | //---------------------------------------------------------------------- |
---|
1019 | ideal vars=var(vpos[1]); |
---|
1020 | for(int i=2;i<=size(vpos);i++) |
---|
1021 | { |
---|
1022 | vars=vars,var(vpos[i]); |
---|
1023 | } |
---|
1024 | matrix varm[1][size(vpos)]=vars; |
---|
1025 | if (size(vpos) > nrows(M)) |
---|
1026 | { |
---|
1027 | matrix Mt[size(vpos)][ncols(M)]; |
---|
1028 | Mt[1..nrows(M),1..ncols(M)]=M; |
---|
1029 | kill M; |
---|
1030 | matrix M=Mt; |
---|
1031 | } |
---|
1032 | //---------------------------------------------------------------------- |
---|
1033 | // define the desired ideal |
---|
1034 | //---------------------------------------------------------------------- |
---|
1035 | ideal ret=vars^2,varm*M; |
---|
1036 | return(ret); |
---|
1037 | } |
---|
1038 | example |
---|
1039 | { "EXAMPLE:"; echo=2; |
---|
1040 | ring r=0,(e(1),e(2),x,y,z),(dp(2),ds(3)); |
---|
1041 | module mo=x*gen(1)+y*gen(2); |
---|
1042 | intvec iv=2,1; |
---|
1043 | mod2id(mo,iv); |
---|
1044 | } |
---|
1045 | //////////////////////////////////////////////////////////////////////// |
---|
1046 | static |
---|
1047 | proc id2mod(ideal i,intvec vpos) |
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1048 | USAGE: id2mod(I,vpos); |
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1049 | where I is an ideal and vpos is an integer vector such that |
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1050 | gen(i) will correspond to var(vpos[i]) |
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1051 | NOTE: * use this procedure only makes sense if the ideal contains |
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1052 | all var(vpos[i])*var(vpos[j]) as monomial generators and |
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1053 | all other generators are linear combinations of the |
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1054 | var(vpos[i]) over the ring in the other variables |
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1055 | * this is the inverse procedure to mod2id |
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1056 | RETURN: module corresponding to the ideal by replacing var(vpos[i]) by |
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1057 | gen(i) and omitting all generators var(vpos[i])*var(vpos[j]) |
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1058 | EXAMPLE: example id2mod; shows an example; |
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1059 | { |
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1060 | //--------------------------------------------------------------------- |
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1061 | // Initialization |
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1062 | //--------------------------------------------------------------------- |
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1063 | int n=size(i); |
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1064 | int v=size(vpos); |
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1065 | matrix tempmat; |
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1066 | matrix mm[v][n]; |
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1067 | //--------------------------------------------------------------------- |
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1068 | // Conversion |
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1069 | //--------------------------------------------------------------------- |
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1070 | for(int j=1;j<=v;j++) |
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1071 | { |
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1072 | tempmat=coeffs(i,var(vpos[j])); |
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1073 | mm[j,1..n]=tempmat[2,1..n]; |
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1074 | } |
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1075 | for(j=1;j<=v;j++) |
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1076 | { |
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1077 | mm=subst(mm,var(vpos[j]),0); |
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1078 | } |
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1079 | module ret=simplify(mm,10); |
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1080 | return(ret); |
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1081 | } |
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1082 | example |
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1083 | { "EXAMPLE:"; echo=2; |
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1084 | ring r=0,(e(1),e(2),x,y,z),(dp(2),ds(3)); |
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1085 | ideal i=e(2)^2,e(1)*e(2),e(1)^2,e(1)*y+e(2)*x; |
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1086 | intvec iv=2,1; |
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1087 | id2mod(i,iv); |
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1088 | } |
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1089 | /////////////////////////////////////////////////////////////////////// |
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1090 | static |
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1091 | proc subrInterred(ideal mon, ideal sm, intvec iv) |
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1092 | USAGE: subrInterred(mon,sm); |
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1093 | sm ideal in a ring r with n + s variables, |
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1094 | e.g. x_1,..,x_n and t_1,..,t_s |
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1095 | mon ideal with monomial generators (not divisible by |
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1096 | one of the t_i) such that sm is contained in the module |
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1097 | k[t_1,..,t_s]*mon |
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1098 | iv intvec listing the variables which are supposed to be used |
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1099 | as x_i |
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1100 | RETURN: interreduced system of generators of sm seen as a submodule |
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1101 | of k[t_1,..,t_s]*mon |
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1102 | EXAMPLE: example subrInterred; shows an example |
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1103 | { |
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1104 | //----------------------------------------------------------------------- |
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1105 | // check that mon is really generated by monomials |
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1106 | // and sort its generators with respect to the monomial ordering |
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1107 | //----------------------------------------------------------------------- |
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1108 | int err; |
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1109 | for(int i=1;i<=ncols(mon);i++) |
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1110 | { |
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1111 | if ( size(mon[i]) > 1 ) |
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1112 | { |
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1113 | err=1; |
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1114 | } |
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1115 | } |
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1116 | if (err==1) |
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1117 | { |
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1118 | "mon has to be generated by monomials"; |
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1119 | return("ERROR"); |
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1120 | } |
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1121 | intvec sv=sortvec(mon); |
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1122 | ideal mons; |
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1123 | for(i=1;i<=size(sv);i++) |
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1124 | { |
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1125 | mons[i]=mon[sv[i]]; |
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1126 | } |
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1127 | ideal itemp=maxideal(1); |
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1128 | for(i=1;i<=size(iv);i++) |
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1129 | { |
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1130 | itemp=subst(itemp,var(iv[i]),0); |
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1131 | } |
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1132 | itemp=simplify(itemp,10); |
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1133 | def r=basering; |
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1134 | string tempstr="ring rtemp=" + charstr(basering) + ",(" + string(itemp) |
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1135 | + "),(C,dp);"; |
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1136 | //----------------------------------------------------------------------- |
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1137 | // express m in terms of the generators of mon and check whether m |
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1138 | // can be considered as a submodule of k[t_1,..,t_n]*mon |
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1139 | //----------------------------------------------------------------------- |
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1140 | module motemp; |
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1141 | motemp[ncols(sm)]=0; |
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1142 | poly ptemp; |
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1143 | int j; |
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1144 | for(i=1;i<=ncols(mons);i++) |
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1145 | { |
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1146 | for(j=1;j<=ncols(sm);j++) |
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1147 | { |
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1148 | ptemp=sm[j]/mons[i]; |
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1149 | motemp[j]=motemp[j]+ptemp*gen(i); |
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1150 | } |
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1151 | } |
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1152 | for(i=1;i<=size(iv);i++) |
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1153 | { |
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1154 | motemp=subst(motemp,var(iv[i]),0); |
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1155 | } |
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1156 | matrix monmat[1][ncols(mons)]=mons; |
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1157 | ideal dummy=monmat*motemp; |
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1158 | for(i=1;i<=size(sm);i++) |
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1159 | { |
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1160 | if(sm[i]-dummy[i]!=0) |
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1161 | { |
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1162 | "the second argument is not a submodule of the assumed structure"; |
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1163 | return("ERROR"); |
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1164 | } |
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1165 | } |
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1166 | //---------------------------------------------------------------------- |
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1167 | // do the interreduction and convert back |
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1168 | //---------------------------------------------------------------------- |
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1169 | execute tempstr; |
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1170 | def motemp=imap(r,motemp); |
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1171 | motemp=interred(motemp); |
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1172 | setring r; |
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1173 | kill motemp; |
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1174 | def motemp=imap(rtemp,motemp); |
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1175 | ideal ret=monmat*motemp; |
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1176 | for(i=1;i<=ncols(ret);i++) |
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1177 | { |
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1178 | ret[i]=cleardenom(ret[i]); |
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1179 | } |
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1180 | return(ret); |
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1181 | } |
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1182 | example |
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1183 | { "EXAMPLE:"; echo=2; |
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1184 | ring r=0,(x,y,z),dp; |
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1185 | ideal i=x^2+x*y^2,x*y+x^2*y,z; |
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1186 | ideal j=x^2+x*y^2,x*y,z; |
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1187 | ideal mon=x^2,z,x*y; |
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1188 | intvec iv=1,3; |
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1189 | subrInterred(mon,i,iv); |
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1190 | subrInterred(mon,j,iv); |
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1191 | } |
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1192 | //////////////////////////////////////////////////////////////////////// |
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