1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="version modwalk.lib 4.1.1.0 Dec_2017 "; // $Id$ |
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3 | category = "Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: modwalk.lib Groebner basis convertion |
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6 | |
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7 | AUTHORS: S. Oberfranz oberfran@mathematik.uni-kl.de |
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8 | |
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9 | OVERVIEW: |
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10 | |
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11 | A library for converting Groebner bases of an ideal in the polynomial |
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12 | ring over the rational numbers using modular methods. The procedures are |
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13 | inspired by the following paper: |
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14 | Elizabeth A. Arnold: Modular algorithms for computing Groebner bases. |
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15 | Journal of Symbolic Computation 35, 403-419 (2003). |
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16 | |
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17 | PROCEDURES: |
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18 | |
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19 | modWalk(I,#); standard basis conversion of I by Groebner Walk using modular methods |
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20 | modrWalk(I,radius,#); standard basis conversion of I by Random Walk using modular methods |
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21 | modfWalk(I,#); standard basis conversion of I by Fractal Walk using modular methods |
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22 | modfrWalk(I,radius,#); standard basis conversion of I by Random Fractal Walk using modular methods |
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23 | |
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24 | KEYWORDS: walk, groebner;Groebnerwalk |
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25 | SEE ALSO: grwalk_lib;swalk_lib;rwalk_lib |
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26 | "; |
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27 | |
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28 | LIB "rwalk.lib"; |
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29 | LIB "grwalk.lib"; |
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30 | LIB "modular.lib"; |
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31 | |
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32 | proc modWalk(ideal I, list #) |
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33 | "USAGE: modWalk(I, [, v, w]); I ideal, v intvec or string, w intvec |
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34 | If v intvec, then I is assumed to be a standard basis with respect to (a(v),lp). |
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35 | If v string, then either v="dp" or v="Dp" and I is assumed to be a standard basis with |
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36 | respect to dp or Dp, respectively. |
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37 | If w is given, then a standard basis with respect to (a(w),lp) will be computed. Otherwise, |
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38 | the output will be a standard basis with respect to lp. |
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39 | If no optional argument is given, I is assumed to be a standard basis with respect to dp |
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40 | and a standard basis with respect to lp will be computed. |
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41 | RETURN: a standard basis of I |
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42 | NOTE: The procedure computes a standard basis of I (over the rational |
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43 | numbers) by using modular methods. |
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44 | SEE ALSO: modular |
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45 | EXAMPLE: example modWalk; shows an example" |
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46 | { |
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47 | /* save options */ |
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48 | intvec opt = option(get); |
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49 | option(redSB); |
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50 | |
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51 | /* call modular() */ |
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52 | if (size(#) > 0) { |
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53 | I = modular("gwalk", list(I,#), primeTest_std, |
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54 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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55 | } |
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56 | else { |
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57 | I = modular("gwalk", list(I), primeTest_std, |
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58 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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59 | } |
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60 | |
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61 | /* return the result */ |
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62 | attrib(I, "isSB", 1); |
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63 | option(set, opt); |
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64 | return(I); |
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65 | } |
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66 | example |
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67 | { |
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68 | "EXAMPLE:"; |
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69 | echo = 2; |
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70 | ring R1 = 0, (x,y,z,t), dp; |
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71 | ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4; |
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72 | I = std(I); |
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73 | ring R2 = 0, (x,y,z,t), lp; |
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74 | ideal I = fetch(R1, I); |
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75 | ideal J = modWalk(I); |
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76 | J; |
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77 | ring S1 = 0, (a,b,c,d), Dp; |
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78 | ideal I = 5b2, ac2+9d3+3a2+5b, 2a2c+7abd+bcd+4a2, 2ad2+6b2d+7c3+8ad+4c; |
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79 | I = std(I); |
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80 | ring S2 = 0, (c,d,b,a), lp; |
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81 | ideal I = fetch(S1,I); |
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82 | // I is assumed to be a Dp-Groebner basis. |
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83 | // We compute a lp-Groebner basis. |
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84 | ideal J = modWalk(I,"Dp"); |
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85 | J; |
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86 | intvec w = 3,2,1,2; |
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87 | ring S3 = 0, (c,d,b,a), (a(w),lp); |
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88 | ideal I = fetch(S1,I); |
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89 | // I is assumed to be a Dp-Groebner basis. |
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90 | // We compute a (a(w),lp)-Groebner basis. |
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91 | ideal J = modWalk(I,"Dp",w); |
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92 | J; |
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93 | } |
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94 | |
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95 | proc modrWalk(ideal I, int radius, list #) |
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96 | "USAGE: modrWalk(I, radius[, v, w]); |
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97 | I ideal, radius int, pertdeg int, v intvec or string, w intvec |
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98 | If v intvec, then I is assumed to be a standard basis with respect to (a(v),lp). |
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99 | If v string, then either v="dp" or v="Dp" and I is assumed to be a standard basis with |
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100 | respect to dp or Dp, respectively. |
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101 | If w is given, then a standard basis with respect to (a(w),lp) will be computed. Otherwise, |
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102 | the output will be a standard basis with respect to lp. |
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103 | If no optional argument is given, I is assumed to be a standard basis with respect to dp |
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104 | and a standard basis with respect to lp will be computed. |
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105 | RETURN: a standard basis of I |
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106 | NOTE: The procedure computes a standard basis of I (over the rational |
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107 | numbers) by using modular methods. |
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108 | SEE ALSO: modular |
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109 | EXAMPLE: example modrWalk; shows an example" |
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110 | { |
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111 | /* save options */ |
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112 | intvec opt = option(get); |
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113 | option(redSB); |
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114 | |
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115 | /* call modular() */ |
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116 | if (size(#) > 0) { |
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117 | I = modular("rwalk", list(I,radius,1,#), primeTest_std, |
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118 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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119 | } |
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120 | else { |
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121 | I = modular("rwalk", list(I,radius,1), primeTest_std, deleteUnluckyPrimes_std, |
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122 | pTest_std,finalTest_std); |
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123 | } |
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124 | |
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125 | /* return the result */ |
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126 | attrib(I, "isSB", 1); |
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127 | option(set, opt); |
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128 | return(I); |
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129 | } |
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130 | example |
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131 | { |
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132 | "EXAMPLE:"; |
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133 | echo = 2; |
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134 | ring R1 = 0, (x,y,z,t), dp; |
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135 | ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4; |
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136 | I = std(I); |
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137 | ring R2 = 0, (x,y,z,t), lp; |
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138 | ideal I = fetch(R1, I); |
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139 | int radius = 2; |
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140 | ideal J = modrWalk(I,radius); |
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141 | J; |
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142 | ring S1 = 0, (a,b,c,d), Dp; |
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143 | ideal I = 5b2, ac2+9d3+3a2+5b, 2a2c+7abd+bcd+4a2, 2ad2+6b2d+7c3+8ad+4c; |
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144 | I = std(I); |
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145 | ring S2 = 0, (c,d,b,a), lp; |
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146 | ideal I = fetch(S1,I); |
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147 | // I is assumed to be a Dp-Groebner basis. |
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148 | // We compute a lp-Groebner basis. |
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149 | ideal J = modrWalk(I,radius,"Dp"); |
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150 | J; |
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151 | intvec w = 3,2,1,2; |
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152 | ring S3 = 0, (c,d,b,a), (a(w),lp); |
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153 | ideal I = fetch(S1,I); |
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154 | // I is assumed to be a Dp-Groebner basis. |
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155 | // We compute a (a(w),lp)-Groebner basis. |
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156 | ideal J = modrWalk(I,radius,"Dp",w); |
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157 | J; |
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158 | } |
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159 | |
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160 | proc modfWalk(ideal I, list #) |
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161 | "USAGE: modfWalk(I, [, v, w]); I ideal, v intvec or string, w intvec |
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162 | If v intvec, then I is assumed to be a standard basis with respect to (a(v),lp). |
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163 | If v string, then either v="dp" or v="Dp" and I is assumed to be a standard basis with |
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164 | respect to dp or Dp, respectively. |
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165 | If w is given, then a standard basis with respect to (a(w),lp) will be computed. Otherwise, |
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166 | the output will be a standard basis with respect to lp. |
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167 | If no optional argument is given, I is assumed to be a standard basis with respect to dp |
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168 | and a standard basis with respect to lp will be computed. |
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169 | RETURN: a standard basis of I |
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170 | NOTE: The procedure computes a standard basis of I (over the rational |
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171 | numbers) by using modular methods. |
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172 | SEE ALSO: modular |
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173 | EXAMPLE: example modfWalk; shows an example" |
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174 | { |
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175 | /* save options */ |
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176 | intvec opt = option(get); |
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177 | option(redSB); |
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178 | |
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179 | /* call modular() */ |
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180 | if (size(#) > 0) { |
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181 | I = modular("fwalk", list(I,#), primeTest_std, |
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182 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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183 | } |
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184 | else { |
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185 | I = modular("fwalk", list(I), primeTest_std, |
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186 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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187 | } |
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188 | |
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189 | /* return the result */ |
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190 | attrib(I, "isSB", 1); |
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191 | option(set, opt); |
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192 | return(I); |
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193 | } |
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194 | example |
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195 | { |
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196 | "EXAMPLE:"; |
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197 | echo = 2; |
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198 | ring R1 = 0, (x,y,z,t), dp; |
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199 | ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4; |
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200 | I = std(I); |
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201 | ring R2 = 0, (x,y,z,t), lp; |
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202 | ideal I = fetch(R1, I); |
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203 | ideal J = modfWalk(I); |
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204 | J; |
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205 | ring S1 = 0, (a,b,c,d), Dp; |
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206 | ideal I = 5b2, ac2+9d3+3a2+5b, 2a2c+7abd+bcd+4a2, 2ad2+6b2d+7c3+8ad+4c; |
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207 | I = std(I); |
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208 | ring S2 = 0, (c,d,b,a), lp; |
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209 | ideal I = fetch(S1,I); |
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210 | // I is assumed to be a Dp-Groebner basis. |
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211 | // We compute a lp-Groebner basis. |
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212 | ideal J = modfWalk(I,"Dp"); |
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213 | J; |
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214 | intvec w = 3,2,1,2; |
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215 | ring S3 = 0, (c,d,b,a), (a(w),lp); |
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216 | ideal I = fetch(S1,I); |
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217 | // I is assumed to be a Dp-Groebner basis. |
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218 | // We compute a (a(w),lp)-Groebner basis. |
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219 | ideal J = modfWalk(I,"Dp",w); |
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220 | J; |
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221 | } |
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222 | |
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223 | proc modfrWalk(ideal I, int radius, list #) |
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224 | "USAGE: modfrWalk(I, radius [, v, w]); I ideal, radius int, v intvec or string, w intvec |
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225 | If v intvec, then I is assumed to be a standard basis with respect to (a(v),lp). |
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226 | If v string, then either v="dp" or v="Dp" and I is assumed to be a standard basis with |
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227 | respect to dp or Dp, respectively. |
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228 | If w is given, then a standard basis with respect to (a(w),lp) will be computed. Otherwise, |
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229 | the output will be a standard basis with respect to lp. |
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230 | If no optional argument is given, I is assumed to be a standard basis with respect to dp |
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231 | and a standard basis with respect to lp will be computed. |
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232 | RETURN: a standard basis of I |
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233 | NOTE: The procedure computes a standard basis of I (over the rational |
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234 | numbers) by using modular methods. |
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235 | SEE ALSO: modular |
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236 | EXAMPLE: example modfrWalk; shows an example" |
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237 | { |
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238 | /* save options */ |
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239 | intvec opt = option(get); |
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240 | option(redSB); |
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241 | |
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242 | /* call modular() */ |
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243 | if (size(#) > 0) { |
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244 | I = modular("frandwalk", list(I,radius,#), primeTest_std, |
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245 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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246 | } |
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247 | else { |
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248 | I = modular("frandwalk", list(I,radius), primeTest_std, |
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249 | deleteUnluckyPrimes_std, pTest_std, finalTest_std); |
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250 | } |
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251 | |
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252 | /* return the result */ |
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253 | attrib(I, "isSB", 1); |
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254 | option(set, opt); |
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255 | return(I); |
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256 | } |
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257 | example |
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258 | { |
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259 | "EXAMPLE:"; |
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260 | echo = 2; |
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261 | ring R1 = 0, (x,y,z,t), dp; |
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262 | ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4; |
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263 | I = std(I); |
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264 | ring R2 = 0, (x,y,z,t), lp; |
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265 | ideal I = fetch(R1, I); |
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266 | int radius = 2; |
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267 | ideal J = modfrWalk(I,radius); |
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268 | J; |
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269 | ring S1 = 0, (a,b,c,d), Dp; |
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270 | ideal I = 5b2, ac2+9d3+3a2+5b, 2a2c+7abd+bcd+4a2, 2ad2+6b2d+7c3+8ad+4c; |
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271 | I = std(I); |
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272 | ring S2 = 0, (c,d,b,a), lp; |
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273 | ideal I = fetch(S1,I); |
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274 | // I is assumed to be a Dp-Groebner basis. |
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275 | // We compute a lp-Groebner basis. |
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276 | ideal J = modfrWalk(I,radius,"Dp"); |
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277 | J; |
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278 | intvec w = 3,2,1,2; |
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279 | ring S3 = 0, (c,d,b,a), (a(w),lp); |
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280 | ideal I = fetch(S1,I); |
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281 | // I is assumed to be a Dp-Groebner basis. |
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282 | // We compute a (a(w),lp)-Groebner basis. |
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283 | ideal J = modfrWalk(I,radius,"Dp",w); |
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284 | J; |
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285 | } |
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286 | |
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287 | /* test if the prime p is suitable for the input, i.e. it does not divide |
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288 | * the numerator or denominator of any of the coefficients */ |
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289 | static proc primeTest_std(int p, alias list args) |
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290 | { |
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291 | /* erase zero generators */ |
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292 | ideal I = simplify(args[1], 2); |
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293 | |
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294 | /* clear denominators and count the terms */ |
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295 | ideal J; |
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296 | ideal K; |
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297 | int n = ncols(I); |
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298 | intvec sizes; |
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299 | number cnt; |
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300 | int i; |
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301 | for(i = n; i > 0; i--) { |
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302 | J[i] = cleardenom(I[i]); |
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303 | cnt = leadcoef(J[i])/leadcoef(I[i]); |
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304 | K[i] = numerator(cnt)*var(1)+denominator(cnt); |
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305 | } |
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306 | sizes = size(J[1..n]); |
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307 | |
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308 | /* change to characteristic p */ |
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309 | def br = basering; |
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310 | list lbr = ringlist(br); |
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311 | if (typeof(lbr[1]) == "int") { |
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312 | lbr[1] = p; |
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313 | } |
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314 | else { |
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315 | lbr[1][1] = p; |
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316 | } |
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317 | def rp = ring(lbr); |
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318 | setring(rp); |
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319 | ideal Jp = fetch(br, J); |
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320 | ideal Kp = fetch(br, K); |
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321 | |
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322 | /* test if any coefficient is missing */ |
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323 | if (intvec(size(Kp[1..n])) != 2:n) { |
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324 | setring(br); |
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325 | return(0); |
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326 | } |
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327 | if (intvec(size(Jp[1..n])) != sizes) { |
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328 | setring(br); |
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329 | return(0); |
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330 | } |
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331 | setring(br); |
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332 | return(1); |
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333 | } |
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334 | |
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335 | /* find entries in modresults which come from unlucky primes. |
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336 | * For this, sort the entries into categories depending on their leading |
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337 | * ideal and return the indices in all but the biggest category. */ |
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338 | static proc deleteUnluckyPrimes_std(alias list modresults) |
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339 | { |
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340 | int size_modresults = size(modresults); |
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341 | |
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342 | /* sort results into categories. |
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343 | * each category is represented by three entries: |
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344 | * - the corresponding leading ideal |
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345 | * - the number of elements |
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346 | * - the indices of the elements |
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347 | */ |
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348 | list cat; |
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349 | int size_cat; |
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350 | ideal L; |
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351 | int i; |
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352 | int j; |
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353 | for (i = 1; i <= size_modresults; i++) { |
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354 | L = lead(modresults[i]); |
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355 | attrib(L, "isSB", 1); |
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356 | for (j = 1; j <= size_cat; j++) { |
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357 | if (size(L) == size(cat[j][1]) |
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358 | && size(reduce(L, cat[j][1],5)) == 0 |
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359 | && size(reduce(cat[j][1], L,5)) == 0) |
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360 | { |
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361 | cat[j][2] = cat[j][2]+1; |
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362 | cat[j][3][cat[j][2]] = i; |
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363 | break; |
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364 | } |
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365 | } |
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366 | if (j > size_cat) { |
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367 | size_cat++; |
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368 | cat[size_cat] = list(); |
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369 | cat[size_cat][1] = L; |
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370 | cat[size_cat][2] = 1; |
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371 | cat[size_cat][3] = list(i); |
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372 | } |
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373 | } |
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374 | |
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375 | /* find the biggest categories */ |
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376 | int cat_max = 1; |
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377 | int max = cat[1][2]; |
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378 | for (i = 2; i <= size_cat; i++) { |
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379 | if (cat[i][2] > max) { |
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380 | cat_max = i; |
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381 | max = cat[i][2]; |
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382 | } |
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383 | } |
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384 | |
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385 | /* return all other indices */ |
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386 | list unluckyIndices; |
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387 | for (i = 1; i <= size_cat; i++) { |
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388 | if (i != cat_max) { |
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389 | unluckyIndices = unluckyIndices + cat[i][3]; |
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390 | } |
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391 | } |
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392 | return(unluckyIndices); |
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393 | } |
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394 | |
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395 | /* test if 'command' applied to 'args' in characteristic p is the same as |
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396 | 'result' mapped to characteristic p */ |
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397 | static proc pTest_std(string command, alias list args, alias ideal result, |
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398 | int p) |
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399 | { |
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400 | /* change to characteristic p */ |
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401 | def br = basering; |
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402 | list lbr = ringlist(br); |
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403 | if (typeof(lbr[1]) == "int") { |
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404 | lbr[1] = p; |
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405 | } |
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406 | else { |
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407 | lbr[1][1] = p; |
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408 | } |
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409 | def rp = ring(lbr); |
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410 | setring(rp); |
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411 | ideal Ip = fetch(br, args)[1]; |
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412 | list Arg = fetch(br, args); |
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413 | string exstr; |
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414 | ideal Gp = fetch(br, result); |
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415 | attrib(Gp, "isSB", 1); |
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416 | |
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417 | /* test if Ip is in Gp */ |
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418 | int i; |
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419 | for (i = ncols(Ip); i > 0; i--) { |
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420 | if (reduce(Ip[i], Gp, 1) != 0) { |
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421 | setring(br); |
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422 | return(0); |
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423 | } |
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424 | } |
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425 | |
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426 | /* compute command(args) */ |
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427 | exstr = "Ip = "+command+" (Ip"; |
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428 | |
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429 | for(i=2; i<=size(Arg); i++) { |
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430 | exstr = exstr+",Arg["+string(eval(i))+"]"; |
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431 | } |
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432 | exstr = exstr+");"; |
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433 | |
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434 | execute(exstr); |
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435 | |
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436 | /* test if Gp is in Ip */ |
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437 | for (i = ncols(Gp); i > 0; i--) { |
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438 | if (reduce(Gp[i], Ip, 1) != 0) { |
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439 | setring(br); |
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440 | return(0); |
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441 | } |
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442 | } |
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443 | |
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444 | setring(br); |
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445 | return(1); |
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446 | } |
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447 | |
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448 | /* test if 'result' is a GB of the input ideal */ |
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449 | static proc finalTest_std(string command, alias list args, ideal result) |
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450 | { |
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451 | /* test if args[1] is in result */ |
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452 | attrib(result, "isSB", 1); |
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453 | int i; |
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454 | for (i = ncols(args[1]); i > 0; i--) { |
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455 | if (reduce(args[1][i], result, 1) != 0) { |
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456 | return(0); |
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457 | } |
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458 | } |
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459 | |
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460 | /* test if result is in args[1]. */ |
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461 | /* args[1] is given by a Groebner basis. Thus we may */ |
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462 | /* reduce the result with respect to args[1]. */ |
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463 | int n=nvars(basering); |
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464 | string ord_str = "dp"; |
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465 | |
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466 | for(i=2; i<=size(args); i++) |
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467 | { |
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468 | if(typeof(args[i]) == "list") { |
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469 | if(typeof(args[i][1]) == "intvec") { |
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470 | ord_str = "(a("+string(args[i][1])+"),lp("+string(n) + "),C)"; |
---|
471 | break; |
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472 | } |
---|
473 | if(typeof(args[i][1]) == "string") { |
---|
474 | if(args[i][1] == "Dp") { |
---|
475 | ord_str = "Dp"; |
---|
476 | } |
---|
477 | break; |
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478 | } |
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479 | } |
---|
480 | } |
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481 | ideal xI = args[1]; |
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482 | ring xR = basering; |
---|
483 | execute("ring yR = ("+charstr(xR)+"),("+varstr(xR)+"),"+ord_str+";"); |
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484 | ideal yI = fetch(xR,xI); |
---|
485 | ideal yresult = fetch(xR,result); |
---|
486 | attrib(yI, "isSB", 1); |
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487 | for(i=size(yresult); i>0; i--) |
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488 | { |
---|
489 | if(reduce(yresult[i],yI) != 0) |
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490 | { |
---|
491 | return(0); |
---|
492 | } |
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493 | } |
---|
494 | setring xR; |
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495 | kill yR; |
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496 | |
---|
497 | /* test if result is a Groebner basis */ |
---|
498 | link l1="ssi:fork"; |
---|
499 | open(l1); |
---|
500 | link l2="ssi:fork"; |
---|
501 | open(l2); |
---|
502 | list l=list(l1,l2); |
---|
503 | write(l1,quote(TestSBred(result))); |
---|
504 | write(l2,quote(TestSBstd(result))); |
---|
505 | i=waitfirst(l); |
---|
506 | if(i==1) { |
---|
507 | i=read(l1); |
---|
508 | } |
---|
509 | else { |
---|
510 | i=read(l2); |
---|
511 | } |
---|
512 | close(l1); |
---|
513 | close(l2); |
---|
514 | return(i); |
---|
515 | } |
---|
516 | |
---|
517 | /* return 1, if I_reduce is _not_ in G_reduce, |
---|
518 | * 0, otherwise |
---|
519 | * (same as size(reduce(I_reduce, G_reduce))). |
---|
520 | * Uses parallelization. */ |
---|
521 | static proc reduce_parallel(def I_reduce, def G_reduce) |
---|
522 | { |
---|
523 | exportto(Modwalk, I_reduce); |
---|
524 | exportto(Modwalk, G_reduce); |
---|
525 | int size_I = ncols(I_reduce); |
---|
526 | int chunks = Modular::par_range(size_I); |
---|
527 | intvec range; |
---|
528 | int i; |
---|
529 | for (i = chunks; i > 0; i--) { |
---|
530 | range = Modular::par_range(size_I, i); |
---|
531 | task t(i) = "Modwalk::reduce_task", list(range); |
---|
532 | } |
---|
533 | startTasks(t(1..chunks)); |
---|
534 | waitAllTasks(t(1..chunks)); |
---|
535 | int result = 0; |
---|
536 | for (i = chunks; i > 0; i--) { |
---|
537 | if (getResult(t(i))) { |
---|
538 | result = 1; |
---|
539 | break; |
---|
540 | } |
---|
541 | } |
---|
542 | kill I_reduce; |
---|
543 | kill G_reduce; |
---|
544 | return(result); |
---|
545 | } |
---|
546 | |
---|
547 | /* compute a chunk of reductions for reduce_parallel */ |
---|
548 | static proc reduce_task(intvec range) |
---|
549 | { |
---|
550 | int result = 0; |
---|
551 | int i; |
---|
552 | for (i = range[1]; i <= range[2]; i++) { |
---|
553 | if (reduce(I_reduce[i], G_reduce, 1) != 0) { |
---|
554 | result = 1; |
---|
555 | break; |
---|
556 | } |
---|
557 | } |
---|
558 | return(result); |
---|
559 | } |
---|
560 | |
---|
561 | /* test if result is a GB with std*/ |
---|
562 | static proc TestSBstd(ideal result) |
---|
563 | { |
---|
564 | ideal G = std(result); |
---|
565 | if(reduce_parallel(G,result)) { |
---|
566 | return(0); |
---|
567 | } |
---|
568 | return(1); |
---|
569 | } |
---|
570 | |
---|
571 | /* test if result is a GB by reducing s-polynomials*/ |
---|
572 | static proc TestSBred(ideal result) |
---|
573 | { |
---|
574 | int i,j; |
---|
575 | for(i=1; i<=size(result); i++) { |
---|
576 | for(j=i; j<=size(result); j++) { |
---|
577 | if(reduce(sPolynomial(result[i],result[j]),result)!=0) { |
---|
578 | return(0); |
---|
579 | } |
---|
580 | } |
---|
581 | } |
---|
582 | return(1); |
---|
583 | } |
---|
584 | |
---|
585 | /* compute s-polynomial of f and g */ |
---|
586 | static proc sPolynomial(poly f,poly g) |
---|
587 | { |
---|
588 | int i; |
---|
589 | poly lcmp = 1; |
---|
590 | |
---|
591 | intvec lexpf = leadexp(f); |
---|
592 | intvec lexpg = leadexp(g); |
---|
593 | |
---|
594 | for(i=1; i<=nvars(basering); i++) { |
---|
595 | if(lexpf[i]>=lexpg[i]) { |
---|
596 | lcmp=lcmp*var(i)**lexpf[i]; |
---|
597 | } |
---|
598 | else { |
---|
599 | lcmp=lcmp*var(i)**lexpg[i]; |
---|
600 | } |
---|
601 | } |
---|
602 | |
---|
603 | poly fmult=lcmp/leadmonom(f); |
---|
604 | poly gmult=lcmp/leadmonom(g); |
---|
605 | poly result=leadcoef(g)*fmult*f-leadcoef(f)*gmult*g; |
---|
606 | |
---|
607 | return(result); |
---|
608 | } |
---|