1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="version modnormal.lib 4.0.0.0 Dec_2013 "; // $Id$ |
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3 | category = "Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: modnormal.lib Normalization of affine domains using modular methods |
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6 | |
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7 | AUTHORS: J. Boehm boehm@mathematik.uni-kl.de |
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8 | W. Decker decker@mathematik.uni-kl.de |
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9 | S. Laplagne slaplagn@dm.uba.ar |
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10 | G. Pfister pfister@mathematik.uni-kl.de |
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11 | A. Steenpass steenpass@mathematik.uni-kl.de |
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12 | S. Steidel steidel@mathematik.uni-kl.de |
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13 | @* |
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14 | |
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15 | OVERVIEW: |
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16 | Suppose A is an affine domain over a perfect field.@* |
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17 | This library implements a modular strategy for finding the normalization of A. |
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18 | Following [1], the idea is to apply the normalization algorithm given in [2] |
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19 | over finite fields and lift the results via Chinese remaindering and rational |
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20 | reconstruction as described in [3]. This approch is inherently parallel.@* |
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21 | The strategy is available both as a randomized and as a verified algorithm. |
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22 | |
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23 | REFERENCES: |
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24 | |
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25 | [1] Janko Boehm, Wolfram Decker, Santiago Laplagne, Gerhard Pfister, Stefan Steidel, |
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26 | Andreas Steenpass: Parallel algorithms for normalization, preprint, 2011. |
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27 | |
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28 | [2] Gert-Martin Greuel, Santiago Laplagne, Frank Seelisch: Normalization of Rings, |
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29 | Journal of Symbolic Computation 9 (2010), p. 887-901 |
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30 | |
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31 | [3] Janko Boehm, Wolfram Decker, Claus Fieker, Gerhard Pfister: |
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32 | The use of Bad Primes in Rational Reconstruction, preprint, 2012. |
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33 | |
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34 | KEYWORDS: |
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35 | normalization; modular methods |
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36 | |
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37 | SEE ALSO: normal_lib, locnormal_lib |
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38 | |
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39 | PROCEDURES: |
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40 | modNormal(I); normalization of R/I using modular methods |
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41 | |
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42 | "; |
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43 | |
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44 | LIB "poly.lib"; |
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45 | LIB "ring.lib"; |
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46 | LIB "normal.lib"; |
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47 | LIB "modstd.lib"; |
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48 | LIB "parallel.lib"; |
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49 | |
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50 | //////////////////////////////////////////////////////////////////////////////// |
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51 | // Verify the char 0 result L of normalization of I modulo a prime p |
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52 | |
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53 | static proc pTestNormal(ideal I, list L, int p, ideal normalIP) |
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54 | { |
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55 | // We change the characteristic of the ring to p. |
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56 | def R0 = basering; |
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57 | ideal U = L[1]; |
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58 | poly condu=L[2]; |
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59 | list rl = ringlist(R0); |
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60 | rl[1] = p; |
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61 | def @r = ring(rl); |
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62 | setring @r; |
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63 | ideal IP = fetch(R0,I); |
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64 | ideal UP = fetch(R0,U); |
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65 | poly conduP = fetch(R0, condu); |
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66 | ideal outP = fetch(R0,normalIP); |
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67 | poly denOutP = outP[1]; |
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68 | |
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69 | // Check if the universal denominator is valid |
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70 | ideal cOut = conduP*outP; |
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71 | ideal dI = ideal(denOutP) + IP; |
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72 | int inc = size(reduce(cOut, groebner(dI))); |
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73 | if(inc > 0) |
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74 | { |
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75 | "Inclusion is not satisfied. Unlucky prime?"; |
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76 | return(ideal(0)); |
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77 | } |
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78 | return(outComp(UP, outP, conduP, denOutP, IP)) |
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79 | } |
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80 | |
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81 | //////////////////////////////////////////////////////////////////////////////// |
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82 | |
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83 | |
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84 | // Computes the normalization of I in characterisitic p. |
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85 | // Returns an ideal Out such that the normalization mod p is the |
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86 | // module 1/condu * Out |
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87 | static proc modpNormal(ideal I, int p, poly condu,printTimings,list #) |
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88 | { |
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89 | int tt = timer; |
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90 | int liftRelations; |
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91 | // We change the characteristic of the ring to p. |
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92 | def R0 = basering; |
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93 | list rl = ringlist(R0); |
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94 | rl[1] = p; |
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95 | def @r = ring(rl); |
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96 | int loc; |
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97 | int i; |
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98 | for ( i=1; i <= size(#); i++ ) |
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99 | { |
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100 | if ( typeof(#[i]) == "string" ) |
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101 | { |
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102 | if (#[i]=="inputJ") { loc = 1;ideal J=#[i][2];} |
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103 | } |
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104 | } |
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105 | setring @r; |
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106 | if (loc==1) {ideal JP = fetch(R0,J)}; |
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107 | //int t=timer; |
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108 | ideal IP = groebner(fetch(R0,I)); |
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109 | //"Time for groebner mod p "+string(timer -t); |
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110 | poly conduP = fetch(R0, condu); |
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111 | |
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112 | option(redSB); |
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113 | |
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114 | int t = timer; |
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115 | // We compute the normalization mod p |
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116 | if (loc==0) |
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117 | { |
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118 | //global |
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119 | list l = normal(IP); |
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120 | } else { |
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121 | //local |
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122 | list l = normal(IP,list(list("inputJ", JP))); |
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123 | } |
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124 | if (printTimings==1) {"Time for modular normal: "+string(timer - t);} |
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125 | |
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126 | t = timer; |
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127 | |
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128 | // Currently, the algorithm only works if no splitting occurs during the |
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129 | // normalization process. (For example, if I is prime.) |
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130 | if(size(l[2]) > 1){ |
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131 | ERRROR("Original ideal is not prime (Not implemented.) or unlucky prime"); |
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132 | } |
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133 | |
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134 | ideal outP = l[2][1]; |
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135 | poly denOutP = outP[size(outP)]; |
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136 | |
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137 | // Check if the universal denominator is valid |
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138 | ideal cOut = conduP*outP; |
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139 | ideal dI = ideal(denOutP) + IP; |
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140 | int inc = size(reduce(cOut, groebner(dI))); |
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141 | if(inc > 0) |
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142 | { |
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143 | ERROR("Inclusion is not satisfied. Unlucky prime?"); |
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144 | } |
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145 | |
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146 | // We change the denominator to the universal denominator |
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147 | outP = changeDenominator(outP, denOutP, conduP, IP); |
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148 | if(size(outP) > 1) |
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149 | { |
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150 | ideal JP = conduP, outP[1..size(outP)-1]; |
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151 | } else |
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152 | { |
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153 | ERROR("Normal ring - Special case not fully implemented."); |
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154 | ideal JP = conduP; |
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155 | ideal norid = 0; |
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156 | export norid; |
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157 | def RP = @r; |
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158 | } |
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159 | |
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160 | setring R0; |
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161 | ideal out = fetch(@r, JP); |
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162 | |
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163 | if (printTimings==1) {"Prime: "+string(p);} |
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164 | tt = timer-tt; |
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165 | return(list(out, p, tt)); |
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166 | } |
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167 | |
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168 | // Computes the normalization using modular methods. |
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169 | // Basic algorithm based on modstd. |
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170 | proc modNormal(ideal I, int nPrimes, list #) |
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171 | "USAGE: modNormal(I, n [,options]); I = prime ideal, n = positive integer, options = list of options. @* |
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172 | Optional parameters in list options (can be entered in any order):@* |
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173 | noVerificication: do not verify the result.@* |
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174 | printTimings: print timings.@* |
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175 | int ncores: number of cores to be used (default = 1). |
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176 | ASSUME: I is a prime ideal (the algorithm will also work for radical ideals as long as the |
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177 | normal command does not detect that the ideal under consideration is not prime). |
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178 | RETURN: a list of an ideal U and a universal denominator d such that U/d is the normalization. |
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179 | REMARKS: We use the algorithm given in [1] to compute the normalization of A = R/I where R is the |
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180 | basering. We apply the algorithm for n primes at a time until the result lifted to the |
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181 | rationals is correct modulo one additional prime. Depending on whether the option |
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182 | noVerificication is used or not, the result is returned as a probabilistic result |
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183 | or verified over the rationals.@* |
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184 | The normalization of A is represented as an R-module by returning a list of U and d, |
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185 | where U is an ideal of A and d is an element of A such that U/d is the normalization of A. |
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186 | In fact, U and d are returned as an ideal and a polynomial of the base ring R. |
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187 | KEYWORDS: normalization; modular techniques. |
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188 | SEE ALSO: normal_lib, locnormal_lib. |
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189 | EXAMPLE: example modNormal; shows an example |
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190 | " |
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191 | { |
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192 | |
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193 | int i,noVerif,printTimings; |
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194 | int liftRelations; |
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195 | int ncores = 1; |
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196 | for ( i=1; i <= size(#); i++ ) |
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197 | { |
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198 | if ( typeof(#[i]) == "string" ) |
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199 | { |
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200 | if (#[i]=="noVerification") { noVerif = 1;} |
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201 | if (#[i]=="printTimings") { printTimings = 1;} |
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202 | } |
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203 | if ( typeof(#[i]) == "int" ) |
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204 | { |
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205 | ncores = #[i]; |
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206 | } |
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207 | } |
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208 | |
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209 | |
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210 | int totalTime = timer; |
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211 | |
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212 | intvec LTimer; |
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213 | int t; |
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214 | |
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215 | def R = basering; |
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216 | int j; |
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217 | |
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218 | //-------------------- Initialize the list of primes ------------------------- |
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219 | int n2 = nPrimes; |
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220 | |
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221 | |
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222 | |
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223 | //---Computation of the jacobian ideal and the universal denominator |
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224 | |
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225 | list IM = mstd(I); |
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226 | I = IM[1]; |
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227 | int d = dim(I); |
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228 | ideal IMin = IM[2]; |
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229 | qring Q = I; // We work in the quotient by the groebner base of the ideal I |
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230 | option("redSB"); |
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231 | option("returnSB"); |
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232 | ideal I = fetch(R, I); |
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233 | attrib(I, "isSB", 1); |
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234 | ideal IMin = fetch(R, IMin); |
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235 | dbprint(dbg, "Computing the jacobian ideal..."); |
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236 | ideal J = minor(jacob(IMin), nvars(basering) - d, I); |
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237 | t=timer; |
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238 | J = modStd(J); |
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239 | if (printTimings==1) {"Time for modStd Jacobian "+string(timer-t);} |
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240 | |
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241 | setring R; |
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242 | ideal J = fetch(Q, J); |
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243 | |
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244 | //------------------ We check if the singular locus is empty ------------- |
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245 | if(J[1] == 1) |
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246 | { |
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247 | // The original ring R/I was normal. Nothing to do. |
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248 | return(ideal(1)); |
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249 | } |
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250 | |
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251 | //--- Universal denominator--- |
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252 | poly condu = getSmallest(J); // Choses the polynomial of smallest degree |
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253 | // of J as universal denominator. |
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254 | if (printTimings==1) {"conductor: ", condu;} |
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255 | |
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256 | //-------------- Main standard basis computations in positive ---------------- |
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257 | //---------------------- characteristic start here --------------------------- |
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258 | list resultNormal,currentPrimes; |
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259 | list resultNormalX,currentPrimesX; |
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260 | list LL; |
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261 | |
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262 | ideal ChremLift; |
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263 | ideal Out; |
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264 | list OutCondu; |
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265 | |
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266 | int ptn; |
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267 | int k = 1; |
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268 | int sh; |
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269 | int p; |
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270 | int h; |
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271 | |
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272 | intvec L; |
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273 | bigint N; |
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274 | |
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275 | int totalModularTime; |
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276 | int maxModularTime; |
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277 | int sumMaxModularTime; |
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278 | int sumTotalModularTime; |
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279 | |
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280 | ideal normalIP; |
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281 | |
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282 | I = groebner(I); |
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283 | // Largest prime: 2147483647 |
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284 | // Max prime for gcd: 536870909 |
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285 | |
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286 | // loop increasing the number of primes by n2 until pTest is true |
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287 | list modarguments; |
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288 | list modresults; |
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289 | int lastPrime; |
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290 | while (ptn==0) |
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291 | { |
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292 | L = primeList(I,k*n2+1,intvec(536870627),1); |
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293 | maxModularTime=0; |
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294 | totalModularTime = timer; |
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295 | if (k==1) {sh=0;} else {sh=1;} |
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296 | if (ncores == 1) |
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297 | { |
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298 | for(j = (k-1)*n2+1+sh; j <= k*n2+1; j++) |
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299 | { |
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300 | t = timer; |
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301 | normalIP = modpNormal(I, L[j], condu,printTimings,#)[1]; |
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302 | if(timer - t > maxModularTime) { maxModularTime = timer - t; } |
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303 | LTimer[j] = timer - t; |
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304 | setring R; |
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305 | resultNormalX[j] = normalIP; |
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306 | currentPrimesX[j] = bigint(L[j]); |
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307 | } |
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308 | lastPrime = L[k*n2+1]; |
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309 | } |
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310 | else |
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311 | { |
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312 | for(j = (k-1)*n2+1+sh; j <= k*n2+1; j++) |
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313 | { |
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314 | modarguments[j-(k-1)*n2-sh] = list(I, L[j], condu, printTimings, #); |
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315 | } |
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316 | modresults = parallelWaitAll("modpNormal", modarguments, 0, ncores); |
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317 | for(j = (k-1)*n2+1+sh; j <= k*n2+1; j++) |
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318 | { |
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319 | resultNormalX[j] = modresults[j-(k-1)*n2-sh][1]; |
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320 | currentPrimesX[j] = bigint(modresults[j-(k-1)*n2-sh][2]); |
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321 | LTimer[j] = modresults[j-(k-1)*n2-sh][3]; |
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322 | if(LTimer[j] > maxModularTime) { maxModularTime = LTimer[j]; } |
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323 | } |
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324 | normalIP = resultNormalX[k*n2+1]; |
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325 | lastPrime = modresults[n2-sh+1][2]; |
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326 | } |
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327 | |
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328 | if (printTimings==1) {"List of times for all modular computations so far: "+string(LTimer);} |
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329 | if (printTimings==1) {"Maximal modular time of current step: "+string(maxModularTime);} |
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330 | sumMaxModularTime=sumMaxModularTime+maxModularTime; |
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331 | totalModularTime = timer - totalModularTime; |
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332 | sumTotalModularTime=sumTotalModularTime+totalModularTime; |
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333 | if (printTimings==1) {"Total modular time of current step: "+string(totalModularTime);} |
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334 | resultNormal=delete(resultNormalX,size(resultNormalX)); |
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335 | currentPrimes=delete(currentPrimesX,size(currentPrimesX)); |
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336 | //------------------------ Delete unlucky primes ----------------------------- |
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337 | //------------- unlucky if and only if the leading ideal is wrong ------------ |
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338 | // Polynomials are not homogeneous: h = 0 |
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339 | LL = deleteUnluckyPrimes(resultNormal,currentPrimes,h); |
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340 | resultNormal = LL[1]; |
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341 | currentPrimes = LL[2]; |
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342 | if (printTimings==1) {"Number of lucky primes: ", size(currentPrimes);} |
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343 | |
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344 | //------------------- Now all leading ideals are the same -------------------- |
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345 | //------------------- Lift results to basering via farey --------------------- |
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346 | N = currentPrimes[1]; |
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347 | for(i = 2; i <= size(currentPrimes); i++) |
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348 | { |
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349 | N = N*currentPrimes[i]; |
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350 | } |
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351 | // Chinese remainder |
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352 | ChremLift = chinrem(resultNormal,currentPrimes); |
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353 | // Farey lifting |
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354 | Out = farey(ChremLift,N); |
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355 | |
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356 | OutCondu=Out,condu; |
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357 | // pTest |
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358 | if (pTestNormal(I,OutCondu,lastPrime,normalIP)==0) |
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359 | { |
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360 | if (printTimings==1) {"pTestNormal has failed, increasing the number of primes by "+string(n2);} |
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361 | k=k+1; |
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362 | } else |
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363 | { |
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364 | ptn=1; |
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365 | } |
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366 | } |
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367 | if (printTimings==1) |
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368 | { |
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369 | "Time for all modular computations: "+string(sumTotalModularTime); |
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370 | "Parallel time for all modular computations: "+string(sumMaxModularTime); |
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371 | "Time for randomized normal: "+string(timer - totalTime); |
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372 | "Simulated parallel time for randomized normal: "+string(timer - totalTime + sumMaxModularTime - sumTotalModularTime); |
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373 | } |
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374 | // return the result if no verification |
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375 | if (noVerif==1) { |
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376 | Out[size(Out) + 1] = Out[1]; |
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377 | Out = Out[2..size(Out)]; |
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378 | OutCondu=modStd(Out),condu; |
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379 | return(OutCondu); |
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380 | }; |
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381 | |
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382 | //------------------- Optional tests to ensure correctness -------------------- |
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383 | // Check for finiteness. We do this by checking if the reconstruction of |
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384 | // the ring structure is still valid |
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385 | |
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386 | t = timer; |
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387 | int tVerif=timer; |
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388 | if (printTimings==1) {"Verification:";} |
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389 | setring R; |
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390 | |
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391 | int isNormal = normalCheck(Out, I,printTimings); |
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392 | |
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393 | |
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394 | if(isNormal == 0) |
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395 | { |
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396 | ERROR("Not normal!"); |
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397 | } else { |
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398 | if (printTimings==1) {"Normal!";} |
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399 | } |
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400 | |
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401 | |
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402 | if (printTimings==1) |
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403 | { |
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404 | "Time for verifying normal: "+string(timer - t); |
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405 | "Time for all verification tests: "+string(timer - tVerif); |
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406 | "Simulated parallel time including verfications: "+string(timer - totalTime + sumMaxModularTime - sumTotalModularTime); |
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407 | "Total time: "+string(timer - totalTime); |
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408 | } |
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409 | // We put the denominator at the end |
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410 | // however we return condu anyway |
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411 | Out[size(Out) + 1] = Out[1]; |
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412 | Out = Out[2..size(Out)]; |
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413 | OutCondu=modStd(Out),condu; |
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414 | return(OutCondu); |
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415 | } |
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416 | |
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417 | example |
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418 | { "EXAMPLE:"; |
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419 | ring R = 0,(x,y,z),dp; |
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420 | int k = 4; |
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421 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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422 | f = subst(f,z,3x-2y+1); |
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423 | ring S = 0,(x,y),dp; |
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424 | poly f = imap(R,f); |
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425 | ideal i = f; |
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426 | list L = modNormal(i,1,"noVerification"); |
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427 | } |
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428 | |
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429 | |
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430 | // Computes the Jacobian ideal |
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431 | // I is assumed to be a groebner base |
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432 | static proc jacobIdOne(ideal I,int printTimings) |
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433 | { |
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434 | def R = basering; |
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435 | |
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436 | int d = dim(I); |
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437 | |
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438 | if (printTimings==1) {"Computing the ideal of minors...";} |
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439 | ideal J = minor(jacob(I), nvars(basering) - d, I); |
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440 | if (printTimings==1) {"Computing the modstd of the ideal of minors...";} |
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441 | J = modStd(J); |
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442 | if (printTimings==1) |
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443 | { |
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444 | "Groebner base computed."; |
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445 | "ideal of minors: "; J; |
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446 | } |
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447 | return(J); |
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448 | } |
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449 | |
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450 | // Procedure for comparing timings and outputs between the modular approach |
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451 | // and the classical approach. Remove static to be used. |
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452 | static proc norComp(ideal I, int nPrimes) |
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453 | { |
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454 | // nPrimes is the number of primes to use. |
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455 | |
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456 | int t = timer; |
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457 | list Out2 = modNormal(I, nPrimes,"noVerification"); |
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458 | "Time modNormal: ", timer - t; |
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459 | t = timer; |
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460 | ideal Out1 = normal(I)[2][1]; |
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461 | "Time normal: ", timer - t; |
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462 | "Same output?"; |
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463 | outComp(Out1, Out2[1], Out1[size(Out1)], Out2[2], I); |
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464 | } |
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465 | |
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466 | static proc outComp(ideal Out1, ideal Out2, poly den1, poly den2, ideal I) |
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467 | { |
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468 | I = groebner(I); |
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469 | Out1 = changeDenominator(Out1, den1, den1, I); |
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470 | Out2 = changeDenominator(Out2, den2, den1, I); |
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471 | Out1 = groebner(I+Out1); |
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472 | Out2 = groebner(I+Out2); |
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473 | return((size(reduce(Out1, Out2)) == 0) * (size(reduce(Out2, Out1)) == 0)); |
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474 | } |
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475 | |
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476 | |
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477 | // Make p homogeneous of degree d taking h as the aux variable of deg 1. |
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478 | static proc polyHomogenize(poly p, int d, intvec degs, poly h) |
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479 | { |
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480 | int i; |
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481 | poly q; |
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482 | for(i = 1; i <= size(p); i++) |
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483 | { |
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484 | q = q + p[i]*h^(d-deg(p[i], degs)); |
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485 | } |
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486 | return(q); |
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487 | } |
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488 | |
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489 | |
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490 | // verification procedure |
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491 | static proc normalCheck(ideal U, ideal I,int printTimings) |
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492 | // U / U[1] = output of the normalization |
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493 | { |
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494 | if (printTimings==1) {"normalCheck: computes the new ring structure and checks if the ring is normal";} |
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495 | |
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496 | def R = basering; |
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497 | poly D = U[1]; // universal denominator |
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498 | |
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499 | if (printTimings==1) {"Computing the new ring structure";} |
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500 | list ele = Normal::computeRing(U, I, "noRed"); |
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501 | |
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502 | def origEre = ele[1]; |
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503 | setring origEre; |
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504 | if (printTimings==1) {"Number of variables: ", nvars(basering);} |
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505 | |
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506 | if (printTimings==1) {"Computing the groebner base of the relations...";} |
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507 | norid = modStd(norid); |
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508 | |
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509 | if (printTimings==1) {"Computing the jacobian ideal...";} |
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510 | ideal J = jacobIdOne(norid,printTimings); |
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511 | ideal JI = J + norid; |
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512 | if (printTimings==1) {"Computing the radical...";} |
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513 | ideal JR = radical(JI); |
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514 | poly testP = getSmallest(JR); // Choses the polynomial of smallest degree |
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515 | |
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516 | qring Q = norid; |
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517 | ideal J = fetch(origEre, JR); |
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518 | poly D = fetch(origEre, testP); |
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519 | if (printTimings==1) {"Computing the quotient (DJ : J)...";} |
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520 | ideal oldU = 1; |
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521 | ideal U = quotient(D*J, J); |
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522 | U = groebner(U); |
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523 | |
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524 | // ----------------- Grauer-Remmert criterion check ----------------------- |
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525 | // We check if the equality in Grauert - Remmert criterion is satisfied. |
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526 | int isNormal = Normal::checkInclusions(D*oldU, U); |
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527 | setring R; |
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528 | return(isNormal); |
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529 | } |
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530 | |
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531 | /////////////////////////////////////////////////////////////////////////// |
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532 | // |
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533 | // EXAMPLES |
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534 | // |
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535 | /////////////////////////////////////////////////////////////////////////// |
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536 | /* |
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537 | // plane curves |
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538 | |
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539 | ring r24 = 0,(x,y,z),dp; |
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540 | int k = 2; |
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541 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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542 | f = subst(f,z,2x-y+1); |
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543 | ring s24 = 0,(x,y),dp; |
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544 | poly f = imap(r24,f); |
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545 | ideal i = f; |
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546 | |
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547 | //locNormal(i); |
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548 | modNormal(i,1); |
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549 | |
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550 | |
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551 | ring r24 = 0,(x,y,z),dp; |
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552 | int k = 3; |
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553 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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554 | f = subst(f,z,2x-y+1); |
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555 | ring s24 = 0,(x,y),dp; |
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556 | poly f = imap(r24,f); |
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557 | ideal i = f; |
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558 | |
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559 | //locNormal(i); |
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560 | modNormal(i,1,"noVerification"); |
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561 | |
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562 | |
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563 | ring r24 = 0,(x,y,z),dp; |
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564 | int k = 4; |
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565 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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566 | f = subst(f,z,2x-y+1); |
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567 | ring s24 = 0,(x,y),dp; |
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568 | poly f = imap(r24,f); |
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569 | ideal i = f; |
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570 | |
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571 | //locNormal(i); |
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572 | modNormal(i,1,"noVerification"); |
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573 | |
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574 | |
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575 | ring r24 = 0,(x,y,z),dp; |
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576 | int k = 5; |
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577 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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578 | f = subst(f,z,2x-y+1); |
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579 | ring s24 = 0,(x,y),dp; |
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580 | poly f = imap(r24,f); |
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581 | ideal i = f; |
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582 | |
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583 | //locNormal(i); |
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584 | modNormal(i,1); |
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585 | |
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586 | |
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587 | ring s24 = 0,(x,y),dp; |
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588 | int a=7; |
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589 | ideal i = ((x-1)^a-y^3)*((x+1)^a-y^3)*((x)^a-y^3)*((x-2)^a-y^3)*((x+2)^a-y^3)+y^15; |
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590 | |
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591 | //locNormal(i); |
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592 | modNormal(i,1); |
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593 | |
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594 | |
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595 | ring s24 = 0,(x,y),dp; |
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596 | int a=8; |
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597 | ideal i = ((x-1)^a-y^3)*((x+1)^a-y^3)*((x)^a-y^3)*((x-2)^a-y^3)*((x+2)^a-y^3)+y^15; |
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598 | |
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599 | //locNormal(i); |
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600 | modNormal(i,1); |
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601 | |
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602 | ring s24 = 0,(x,y),dp; |
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603 | int a=9; |
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604 | ideal i = ((x-1)^a-y^3)*((x+1)^a-y^3)*((x)^a-y^3)*((x-2)^a-y^3)*((x+2)^a-y^3)+y^15; |
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605 | |
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606 | //locNormal(i); |
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607 | modNormal(i,1,"noVerification"); |
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608 | |
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609 | |
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610 | |
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611 | |
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612 | ring r=0,(x,y),dp; |
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613 | ideal i=9127158539954x10+3212722859346x8y2+228715574724x6y4-34263110700x4y6 |
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614 | -5431439286x2y8-201803238y10-134266087241x8-15052058268x6y2+12024807786x4y4 |
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615 | +506101284x2y6-202172841y8+761328152x6-128361096x4y2+47970216x2y4-6697080y6 |
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616 | -2042158x4+660492x2y2-84366y4+2494x2-474y2-1; |
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617 | |
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618 | //locNormal(i); |
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619 | modNormal(i,1); |
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620 | |
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621 | |
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622 | // surfaces in A3 |
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623 | |
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624 | |
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625 | ring r7 = 0,(x,y,t),dp; |
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626 | int a=11; |
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627 | ideal i = x*y*(x-y)*(x+y)*(y-1)*t+(x^a-y^2)*(x^10-(y-1)^2); |
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628 | //locNormal(i); |
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629 | modNormal(i,1,"noVerification"); |
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630 | |
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631 | ring r7 = 0,(x,y,t),dp; |
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632 | int a=12; |
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633 | ideal i = x*y*(x-y)*(x+y)*(y-1)*t+(x^a-y^2)*(x^10-(y-1)^2); |
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634 | //locNormal(i); |
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635 | modNormal(i,1,"noVerification"); |
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636 | |
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637 | |
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638 | ring r7 = 0,(x,y,t),dp; |
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639 | int a=13; |
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640 | ideal i = x*y*(x-y)*(x+y)*(y-1)*t+(x^a-y^2)*(x^10-(y-1)^2); |
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641 | |
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642 | //locNormal(i); |
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643 | modNormal(i,1,"noVerification"); |
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644 | |
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645 | |
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646 | ring r22 = 0,(x,y,z),dp; |
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647 | ideal i = z2-(y2-1234x3)^2*(15791x2-y3)*(1231y2-x2*(x+158))*(1357y5-3x11); |
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648 | |
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649 | //locNormal(i); |
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650 | modNormal(i,1,"noVerification"); |
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651 | |
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652 | |
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653 | ring r22 = 0,(x,y,z),dp; |
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654 | ideal i = z2-(y2-1234x3)^3*(15791x2-y3)*(1231y2-x2*(x+158))*(1357y5-3x11); |
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655 | |
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656 | //locNormal(i); |
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657 | modNormal(i,1,"noVerification"); |
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658 | |
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659 | |
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660 | ring r23 = 0,(x,y,z),dp; |
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661 | ideal i = z5-((13x-17y)*(5x2-7y3)*(3x3-2y2)*(19y2-23x2*(x+29)))^2; |
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662 | |
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663 | //locNormal(i); |
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664 | modNormal(i,1,"noVerification"); |
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665 | |
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666 | |
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667 | // curve in A3 |
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668 | |
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669 | ring r23 = 0,(x,y,z),dp; |
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670 | ideal i = z3-(19y2-23x2*(x+29))^2,x3-(11y2-13z2*(z+1)); |
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671 | |
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672 | //locNormal(i); |
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673 | modNormal(i,1,"noVerification"); |
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674 | |
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675 | |
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676 | ring r23 = 0,(x,y,z),dp; |
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677 | ideal i = z3-(19y2-23x2*(x+29))^2,x3-(11y2-13z2*(z+1))^2; |
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678 | |
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679 | //locNormal(i); |
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680 | modNormal(i,1,"noVerification"); |
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681 | |
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682 | // surface in A4 |
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683 | |
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684 | ring r23 = 0,(x,y,z,w),dp; |
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685 | ideal i = z2-(y3-123456w2)*(15791x2-y3)^2, w*z-(1231y2-x*(111x+158)); |
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686 | |
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687 | |
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688 | //locNormal(i); |
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689 | modNormal(i,1,"noVerification"); |
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690 | |
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691 | */ |
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692 | |
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693 | |
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