1 | //////////////////////////////////////////////////////////////////////////// |
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2 | category = "Commutative Algebra"; |
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3 | info=" |
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4 | LIBRARY: symodstd.lib Procedures for computing Groebner basis of ideals |
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5 | being invariant under certain variable permutations. |
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6 | |
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7 | AUTHOR: Stefan Steidel, steidel@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 computing the Groebner basis of an ideal in the polynomial |
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12 | ring over the rational numbers, that is invariant under certain permutations |
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13 | of the variables, using the symmetry and modular methods. |
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14 | More precisely let I = <f1,...,fr> be an ideal in Q[x(1),...,x(n)] and |
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15 | sigma a permutation of order k in Sym(n) such that sigma(I) = I. |
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16 | We assume that sigma({f1,...,fr}) = {f1,...,fr}. This can always be obtained |
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17 | by adding sigma(fi) to {f1,...,fr}. |
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18 | To compute a standard basis of I we apply a modification of the modular |
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19 | version of the standard basis algorithm (improving the calculations in |
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20 | positive characteristic). Therefore we only allow primes p such that p-1 is |
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21 | divisible by k. This guarantees the existance of a k-th primitive root of |
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22 | unity in Z/pZ. |
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23 | |
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24 | PROCEDURES: |
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25 | genSymId(I,sigma); compute ideal J such that sigma(J) = J and J includes I |
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26 | isSymmetric(I,sigma); check if I is invariant under permutation sigma |
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27 | primRoot(p,k); int describing a k-th primitive root of unity in Z/pZ |
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28 | eigenvalues(I,sigma); list of eigenvalues of generators of I under sigma |
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29 | symmStd(I,sigma); standard basis of I using invariance of I under sigma |
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30 | syModStd(I,sigma); SB of I using modular methods and sigma(I) = I |
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31 | "; |
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32 | |
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33 | LIB "brnoeth.lib"; |
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34 | LIB "modstd.lib"; |
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35 | LIB "parallel.lib"; |
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36 | |
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37 | //////////////////////////////////////////////////////////////////////////////// |
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38 | |
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39 | proc genSymId(ideal I, intvec sigma) |
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40 | "USAGE: genSymId(I,sigma); I ideal, sigma intvec |
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41 | ASSUME: size(sigma) = nvars(basering =: n |
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42 | RETURN: ideal J such that sigma(J) = J and J includes I |
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43 | NOTE: sigma is a permutation of the variables of the basering, i.e. |
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44 | @* sigma: var(i) ----> var(sigma[i]), 1 <= i <= n. |
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45 | EXAMPLE: example genSymId; shows an example |
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46 | " |
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47 | { |
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48 | if(nvars(basering) != size(sigma)) |
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49 | { |
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50 | ERROR("The input is no permutation of the ring-variables!!"); |
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51 | } |
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52 | |
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53 | int i; |
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54 | |
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55 | ideal perm; |
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56 | for(i = 1; i <= size(sigma); i++) |
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57 | { |
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58 | perm[size(perm)+1] = var(sigma[i]); |
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59 | } |
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60 | |
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61 | map f = basering, perm; |
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62 | ideal J = I; |
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63 | ideal helpJ = I; |
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64 | for(i = 1; i <= order(sigma) - 1; i++) |
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65 | { |
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66 | helpJ = f(helpJ); |
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67 | J = J, helpJ; |
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68 | } |
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69 | |
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70 | return(simplify(simplify(J,4),2)); |
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71 | } |
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72 | example |
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73 | { "EXAMPLE:"; echo = 2; |
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74 | ring R = 0,(u,v,w,x,y),dp; |
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75 | intvec pi = 2,3,4,5,1; |
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76 | ideal I = u2v + x3y - w2; |
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77 | genSymId(I,pi); |
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78 | } |
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79 | |
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80 | //////////////////////////////////////////////////////////////////////////////// |
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81 | |
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82 | proc isSymmetric(ideal I, intvec sigma) |
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83 | "USAGE: isSymmetric(I,sigma); I ideal, sigma intvec |
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84 | ASSUME: size(sigma) = nvars(basering) =: n |
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85 | RETURN: 1, if the set of generators of I is invariant under sigma; |
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86 | @* 0, if the set of generators of I is not invariant under sigma |
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87 | NOTE: sigma is a permutation of the variables of the basering, i.e. |
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88 | @* sigma: var(i) ----> var(sigma[i]), 1 <= i <= n. |
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89 | EXAMPLE: example isSymmetric; shows an example |
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90 | " |
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91 | { |
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92 | if(nvars(basering) != size(sigma)) |
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93 | { |
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94 | ERROR("The input is no permutation of the ring-variables!!"); |
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95 | } |
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96 | |
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97 | int i, j; |
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98 | |
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99 | list L; |
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100 | for(i = 1; i <= size(I); i++) |
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101 | { |
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102 | L[size(L)+1] = I[i]; |
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103 | } |
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104 | |
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105 | ideal perm; |
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106 | for(i = 1; i <= size(sigma); i++) |
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107 | { |
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108 | perm[size(perm)+1] = var(sigma[i]); |
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109 | } |
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110 | |
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111 | map f = basering, perm; |
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112 | ideal J = f(I); |
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113 | |
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114 | poly g; |
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115 | while(size(L) > 0) |
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116 | { |
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117 | j = size(L); |
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118 | g = L[1]; |
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119 | |
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120 | for(i = 1; i <= size(J); i++) |
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121 | { |
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122 | if(g - J[i] == 0) |
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123 | { |
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124 | L = delete(L, 1); |
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125 | break; |
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126 | } |
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127 | } |
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128 | |
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129 | if(j == size(L)) |
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130 | { |
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131 | return(0); |
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132 | } |
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133 | } |
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134 | |
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135 | return(1); |
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136 | } |
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137 | example |
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138 | { "EXAMPLE:"; echo = 2; |
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139 | ring R = 0,x(1..5),dp; |
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140 | ideal I = cyclic(5); |
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141 | intvec pi = 2,3,4,5,1; |
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142 | isSymmetric(I,pi); |
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143 | intvec tau = 2,5,1,4,3; |
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144 | isSymmetric(I,tau); |
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145 | } |
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146 | |
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147 | //////////////////////////////////////////////////////////////////////////////// |
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148 | |
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149 | static proc permute(intvec v, intvec P) |
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150 | { |
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151 | // permute the intvec v according to the permutation given by P |
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152 | |
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153 | int s = size(v); |
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154 | int n = size(P); |
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155 | int i; |
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156 | if(s < n) |
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157 | { |
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158 | for(i = s+1; i <= n; i = i+1) |
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159 | { |
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160 | v[i] = 0; |
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161 | } |
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162 | s = size(v); |
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163 | } |
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164 | intvec auxv = v; |
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165 | for(i = 1; i <= n; i = i+1) |
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166 | { |
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167 | auxv[i] = v[P[i]]; |
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168 | } |
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169 | |
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170 | return(auxv); |
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171 | } |
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172 | |
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173 | //////////////////////////////////////////////////////////////////////////////// |
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174 | |
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175 | static proc order(intvec sigma) |
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176 | { |
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177 | // compute the order of sigma in Sym({1,...,n}) with n := size(sigma) |
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178 | |
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179 | int ORD = 1; |
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180 | intvec id = 1..size(sigma); |
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181 | intvec tau = sigma; |
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182 | |
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183 | while(tau != id) |
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184 | { |
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185 | tau = permute(tau, sigma); |
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186 | ORD = ORD + 1; |
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187 | } |
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188 | |
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189 | return(ORD); |
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190 | } |
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191 | |
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192 | //////////////////////////////////////////////////////////////////////////////// |
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193 | |
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194 | static proc modExpo(bigint x, bigint a, bigint n) |
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195 | { |
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196 | // compute x^a mod n |
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197 | |
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198 | bigint z = 1; |
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199 | |
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200 | while(a != 0) |
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201 | { |
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202 | while((a mod 2) == 0) |
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203 | { |
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204 | a = a div 2; |
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205 | x = x^2 mod n; |
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206 | } |
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207 | |
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208 | a = a - 1; |
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209 | z = (z*x) mod n; |
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210 | } |
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211 | |
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212 | return(z); |
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213 | } |
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214 | |
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215 | //////////////////////////////////////////////////////////////////////////////// |
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216 | |
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217 | proc primRoot(int p, int k) |
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218 | "USAGE: primRoot(p,k); p,k integers |
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219 | ASSUME: p is a prime and k divides p-1. |
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220 | RETURN: int: a k-th primitive root of unity in Z/pZ |
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221 | EXAMPLE: example primRoot; shows an example |
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222 | " |
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223 | { |
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224 | if(k == 2) |
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225 | { |
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226 | return(-1); |
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227 | } |
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228 | |
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229 | if(p == 0) |
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230 | { |
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231 | return(0); |
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232 | } |
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233 | |
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234 | int i, j; |
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235 | |
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236 | if(((p-1) mod k) != 0) |
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237 | { |
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238 | ERROR("There is no "+string(k)+"-th primitive root of unity " |
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239 | +"in Z/"+string(p)+"Z."); |
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240 | return(0); |
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241 | } |
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242 | |
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243 | list PF = primefactors(p-1)[1]; |
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244 | |
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245 | bigint a; |
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246 | |
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247 | for(i = 2; i <= p-1; i++) |
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248 | { |
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249 | a = i; |
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250 | |
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251 | for(j = 1; j <= size(PF); j++) |
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252 | { |
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253 | if(modExpo(a, (p-1) div PF[j], p) == 1) |
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254 | { |
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255 | break; |
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256 | } |
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257 | } |
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258 | |
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259 | if(j == size(PF)+1) |
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260 | { |
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261 | a = modExpo(a, (p-1) div k, p); |
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262 | |
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263 | string str = "int xi = "+string(a); |
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264 | execute(str); |
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265 | |
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266 | return(xi); |
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267 | } |
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268 | } |
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269 | } |
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270 | example |
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271 | { "EXAMPLE:"; echo = 2; |
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272 | primRoot(181,10); |
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273 | ring R = 2147482801, x, lp; |
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274 | number a = primRoot(2147482801,5); |
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275 | a; |
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276 | a^2; |
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277 | a^3; |
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278 | a^4; |
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279 | a^5; |
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280 | } |
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281 | |
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282 | //////////////////////////////////////////////////////////////////////////////// |
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283 | |
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284 | static proc permMat(intvec sigma, list #) |
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285 | { |
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286 | // compute an intmat such that i-th row describes sigma^i |
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287 | |
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288 | int i; |
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289 | int n = size(sigma); |
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290 | |
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291 | if(size(#) == 0) |
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292 | { |
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293 | int ORD = order(sigma); |
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294 | } |
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295 | else |
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296 | { |
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297 | int ORD = #[1]; |
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298 | } |
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299 | |
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300 | intmat P[ORD][n]; |
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301 | intvec sigmai = sigma; |
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302 | for(i = 1; i <= ORD; i++) |
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303 | { |
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304 | P[i,1..n] = sigmai; |
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305 | sigmai = permute(sigmai, sigma); |
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306 | } |
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307 | |
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308 | return(P); |
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309 | } |
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310 | |
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311 | //////////////////////////////////////////////////////////////////////////////// |
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312 | |
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313 | static proc genTransId(intvec sigma, list #) |
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314 | { |
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315 | // list L of two ideals such that |
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316 | // - L[1] describes the transformation and |
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317 | // - L[2] describes the retransformation (inverse mapping). |
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318 | // ORD is the order of sigma in Sym({1,...,n}) with n := size(sigma) and |
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319 | // sigma: {1,...,n} ---> {1,...,n}: sigma(j) = sigma[j]. Since sigma is a |
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320 | // permutation of variables it induces an automorphism phi of the |
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321 | // basering, more precisely a linear variable transformation which is |
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322 | // generated by this procedure. In terms it holds: |
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323 | // |
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324 | // phi : basering ---------> basering |
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325 | // var(i) |----> L[1][i] |
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326 | // L[2][i] <----| var(i) |
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327 | |
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328 | int n = nvars(basering); |
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329 | |
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330 | if(n != size(sigma)) |
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331 | { |
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332 | ERROR("The input is no permutation of the ring-variables!!"); |
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333 | } |
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334 | |
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335 | int i, j, k; |
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336 | |
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337 | if(size(#) == 0) |
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338 | { |
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339 | int CHAR = char(basering); |
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340 | int ORD = order(sigma); |
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341 | |
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342 | if((((CHAR - 1) mod ORD) != 0) && (CHAR > 0)) |
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343 | { |
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344 | ERROR("Basering of characteristic "+string(CHAR)+" has no " |
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345 | +string(ORD)+"-th primitive root of unity!!"); |
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346 | } |
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347 | |
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348 | if((CHAR == 0) && (ORD > 2)) |
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349 | { |
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350 | "======================================== |
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351 | ========================================"; |
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352 | "If basering really has a "+string(ORD)+"-th " |
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353 | +"primitive root of unity then insert it as input!!"; |
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354 | "======================================== |
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355 | ========================================"; |
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356 | return(list()); |
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357 | } |
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358 | else |
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359 | { |
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360 | int xi = primRoot(CHAR, ORD); |
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361 | number a = xi; |
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362 | } |
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363 | } |
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364 | else |
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365 | { |
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366 | int ORD = #[1]; |
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367 | number a = #[2]; |
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368 | } |
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369 | |
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370 | intmat PERM = permMat(sigma,ORD); |
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371 | |
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372 | ideal TR, RETR; |
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373 | poly s_trans; |
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374 | matrix C; |
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375 | |
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376 | //-------------- retransformation ideal RETR is generated here ----------------- |
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377 | for(i = 1; i <= n; i++) |
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378 | { |
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379 | for(j = 0; j < ORD; j++) |
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380 | { |
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381 | for(k = 1; k <= ORD; k++) |
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382 | { |
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383 | // for each variable var(i): |
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384 | // s_trans^(j) = sum_{k=1}^{ORD} a^(k*j)*sigma^k(var(i)) |
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385 | // for j = 0,...,ORD-1 |
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386 | s_trans = s_trans + a^(k*j)*var(PERM[k,i]); |
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387 | } |
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388 | RETR = RETR + simplify(s_trans, 1); |
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389 | s_trans = 0; |
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390 | } |
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391 | } |
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392 | |
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393 | //---------------- transformation ideal TR is generated here ------------------- |
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394 | for(i = 1; i <= n; i++) |
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395 | { |
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396 | for(j = 1; j <= size(RETR); j++) |
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397 | { |
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398 | C = coeffs(RETR[j], var(i)); |
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399 | if(nrows(C) > 1) |
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400 | { |
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401 | // var(j) = RETR[j] = sum_{i in J} c_ij*var(i), J subset {1,...,n}, |
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402 | // and therefore var(i) = (sum_{j} s(j)/c_ij)/#(summands) |
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403 | s_trans = s_trans + var(j)/(C[nrows(C),1]); |
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404 | } |
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405 | } |
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406 | |
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407 | TR = TR + s_trans/number(size(s_trans)); |
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408 | s_trans = 0; |
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409 | } |
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410 | |
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411 | return(list(TR,RETR)); |
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412 | } |
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413 | |
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414 | //////////////////////////////////////////////////////////////////////////////// |
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415 | |
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416 | proc eigenvalues(ideal I, intvec sigma) |
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417 | "USAGE: eigenvalues(I,sigma); I ideal, sigma intvec |
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418 | ASSUME: size(sigma) = nvars(basering) =: n |
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419 | RETURN: list of eigenvalues of generators of I under permutation sigma |
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420 | NOTE: sigma is a permutation of the variables of the basering, i.e. |
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421 | sigma: var(i) ----> var(sigma[i]), 1 <= i <= n. |
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422 | EXAMPLE: example eigenvalues; shows an example |
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423 | " |
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424 | { |
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425 | int i, j; |
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426 | |
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427 | def A = basering; |
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428 | int n = nvars(A); |
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429 | |
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430 | poly ev; |
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431 | list EV; |
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432 | poly s, help_var; |
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433 | matrix C1, C2; |
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434 | |
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435 | ideal perm; |
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436 | for(i = 1; i <= n; i++) |
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437 | { |
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438 | perm[size(perm)+1] = var(sigma[i]); |
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439 | } |
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440 | |
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441 | map f = A, perm; |
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442 | |
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443 | for(i = 1; i <= size(I); i++) |
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444 | { |
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445 | //-------------- s is the image of I[i] under permutation sigma ---------------- |
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446 | s = I[i]; |
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447 | s = f(s); |
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448 | |
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449 | for(j = 1; j <= n; j++) |
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450 | { |
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451 | C1 = coeffs(I[i], var(j)); |
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452 | C2 = coeffs(s, var(j)); |
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453 | if(nrows(C1) > 1) |
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454 | { |
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455 | ev = C2[nrows(C2),1]/C1[nrows(C1),1]; |
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456 | |
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457 | //------ Furthermore check that I[i] is eigenvector of permutation sigma. ------ |
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458 | if(s == ev*I[i]) |
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459 | { |
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460 | break; |
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461 | } |
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462 | else |
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463 | { |
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464 | ERROR("I["+string(i)+"] is no eigenvector " |
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465 | +"of permutation sigma!!"); |
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466 | } |
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467 | } |
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468 | } |
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469 | |
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470 | EV[size(EV)+1] = ev; |
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471 | } |
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472 | |
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473 | return(EV); |
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474 | } |
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475 | example |
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476 | { "EXAMPLE:"; echo = 2; |
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477 | ring R = 11, x(1..5), dp; |
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478 | poly p1 = x(1)+x(2)+x(3)+x(4)+x(5); |
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479 | poly p2 = x(1)+4*x(2)+5*x(3)-2*x(4)+3*x(5); |
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480 | poly p3 = x(1)+5*x(2)+3*x(3)+4*x(4)-2*x(5); |
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481 | poly p4 = x(1)-2*x(2)+4*x(3)+3*x(4)+5*x(5); |
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482 | poly p5 = x(1)+3*x(2)-2*x(3)+5*x(4)+4*x(5); |
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483 | ideal I = p1,p2,p3,p4,p5; |
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484 | intvec tau = 2,3,4,5,1; |
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485 | eigenvalues(I,tau); |
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486 | } |
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487 | |
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488 | //////////////////////////////////////////////////////////////////////////////// |
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489 | |
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490 | proc symmStd(ideal I, intvec sigma, list #) |
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491 | "USAGE: symmStd(I,sigma,#); I ideal, sigma intvec |
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492 | ASSUME: size(sigma) = nvars(basering) =: n, basering has an order(sigma)-th |
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493 | primitive root of unity a (if char(basering) > 0) and sigma(I) = I |
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494 | RETURN: ideal, a standard basis of I |
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495 | NOTE: Assuming that the ideal I is invariant under the variable permutation |
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496 | sigma and the basering has an order(sigma)-th primitive root of unity |
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497 | the procedure uses linear transformation of variables in order to |
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498 | improve standard basis computation. |
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499 | If char(basering) = 0 all computations are done in the polynomial ring |
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500 | over the smallest field extension that has an order(sigma)-th primitive |
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501 | root of unity. |
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502 | EXAMPLE: example symmStd; shows an example |
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503 | " |
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504 | { |
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505 | if((nvars(basering) != size(sigma)) || (!isSymmetric(I,sigma))) |
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506 | { |
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507 | ERROR("The input is no permutation of the ring-variables!!"); |
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508 | } |
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509 | |
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510 | option(redSB); |
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511 | |
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512 | def R = basering; |
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513 | int CHAR = char(R); |
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514 | int n = nvars(R); |
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515 | |
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516 | int t; |
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517 | |
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518 | //-------- (1) Compute the order of variable permutation sigma. ---------------- |
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519 | int ORD = order(sigma); |
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520 | if((((CHAR - 1) mod ORD) != 0) && (CHAR > 0)) |
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521 | { |
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522 | ERROR("Basering of characteristic "+string(CHAR)+" has no " |
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523 | +string(ORD)+"-th primitive root of unity!!"); |
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524 | } |
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525 | |
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526 | //-------- (2) Compute the order(sigma)-th primitive root of unity ------------- |
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527 | //-------- in basering or move to ring extension. ------------- |
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528 | if((CHAR == 0) && (ORD > 2)) |
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529 | { |
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530 | def save_ring=basering; |
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531 | ring ext_ring=0,p,lp; |
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532 | def S = changechar(ringlist(ext_ring),save_ring); |
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533 | setring S; |
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534 | kill ext_ring; |
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535 | kill save_ring; |
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536 | minpoly = rootofUnity(ORD); |
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537 | ideal I = imap(R, I); |
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538 | number a = p; |
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539 | } |
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540 | else |
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541 | { |
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542 | int xi = primRoot(CHAR, ORD); |
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543 | number a = xi; |
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544 | } |
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545 | |
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546 | //--------- (3) Define the linear transformation of variables with ------------- |
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547 | //--------- respect to sigma. ------------- |
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548 | list L = genTransId(sigma,ORD,a); |
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549 | ideal TR = L[1]; |
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550 | ideal RETR = L[2]; |
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551 | |
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552 | //--------- (4) Compute the eigenvalues of the "new" variables of -------------- |
---|
553 | //--------- sigma after transformation. -------------- |
---|
554 | list EV = eigenvalues(RETR, sigma); |
---|
555 | |
---|
556 | //--------- (5) Transformation of the input-ideal is done here. ---------------- |
---|
557 | map f = basering, TR; |
---|
558 | t = timer; |
---|
559 | ideal I_trans = f(I); |
---|
560 | if(printlevel >= 11) { "Transformation: "+string(timer - t)+" seconds"; } |
---|
561 | |
---|
562 | //--------- (6) Compute a standard basis of the transformed ideal. ------------- |
---|
563 | t = timer; |
---|
564 | if(size(#) > 0) { ideal sI_trans = std(I_trans, #[1]); } |
---|
565 | else { ideal sI_trans = std(I_trans); } |
---|
566 | if(printlevel >= 11) { "1st Groebner basis: "+string(timer - t)+" seconds"; } |
---|
567 | |
---|
568 | //--------- (7) Retransformation is done here. --------------------------------- |
---|
569 | map g = basering, RETR; |
---|
570 | t = timer; |
---|
571 | ideal I_retrans = g(sI_trans); |
---|
572 | if(printlevel >= 11) { "Reverse Transformation: "+string(timer - t) |
---|
573 | +" seconds"; } |
---|
574 | |
---|
575 | //--------- (8) Compute a standard basis of the retransformaed ideal ----------- |
---|
576 | //--------- which is then a standard basis of the input-ideal. ----------- |
---|
577 | t = timer; |
---|
578 | ideal sI_retrans = std(I_retrans); |
---|
579 | if(printlevel >= 11) { "2nd Groebner basis: "+string(timer - t)+" seconds"; } |
---|
580 | |
---|
581 | if((CHAR == 0) && (ORD > 2)) |
---|
582 | { |
---|
583 | setring R; |
---|
584 | ideal sI_retrans = fetch(S, sI_retrans); |
---|
585 | return(sI_retrans); |
---|
586 | } |
---|
587 | else |
---|
588 | { |
---|
589 | return(sI_retrans); |
---|
590 | } |
---|
591 | } |
---|
592 | example |
---|
593 | { "EXAMPLE:"; echo = 2; |
---|
594 | ring R = 0, x(1..4), dp; |
---|
595 | ideal I = cyclic(4); |
---|
596 | I; |
---|
597 | intvec pi = 2,3,4,1; |
---|
598 | ideal sI = symmStd(I,pi); |
---|
599 | sI; |
---|
600 | |
---|
601 | ring S = 31, (x,y,z), dp; |
---|
602 | ideal J; |
---|
603 | J[1] = xy-y2+xz; |
---|
604 | J[2] = xy+yz-z2; |
---|
605 | J[3] = -x2+xz+yz; |
---|
606 | intvec tau = 3,1,2; |
---|
607 | ideal sJ = symmStd(J,tau); |
---|
608 | sJ; |
---|
609 | } |
---|
610 | |
---|
611 | //////////////////////////////////////////////////////////////////////////////// |
---|
612 | |
---|
613 | proc divPrimeTest(def II, bigint p, int k) |
---|
614 | { |
---|
615 | if((p - 1) mod k != 0) { return(0); } |
---|
616 | |
---|
617 | if(typeof(II) == "string") |
---|
618 | { |
---|
619 | execute("ideal I = "+II+";"); |
---|
620 | } |
---|
621 | else |
---|
622 | { |
---|
623 | ideal I = II; |
---|
624 | } |
---|
625 | |
---|
626 | int i,j; |
---|
627 | poly f; |
---|
628 | number cnt; |
---|
629 | for(i = 1; i <= size(I); i++) |
---|
630 | { |
---|
631 | f = cleardenom(I[i]); |
---|
632 | if(f == 0) { return(0); } |
---|
633 | cnt = leadcoef(I[i])/leadcoef(f); |
---|
634 | if((numerator(cnt) mod p) == 0) { return(0); } |
---|
635 | if((denominator(cnt) mod p) == 0) { return(0); } |
---|
636 | for(j = size(f); j > 0; j--) |
---|
637 | { |
---|
638 | if((leadcoef(f[j]) mod p) == 0) { return(0); } |
---|
639 | } |
---|
640 | } |
---|
641 | return(1); |
---|
642 | } |
---|
643 | |
---|
644 | //////////////////////////////////////////////////////////////////////////////// |
---|
645 | |
---|
646 | proc divPrimeList(int k, ideal I, int n, list #) |
---|
647 | { |
---|
648 | // the intvec of n greatest primes p <= 2147483647 (resp. n greatest primes |
---|
649 | // < L[size(L)] union with L) such that each (p-1) is divisible by k, and none |
---|
650 | // of these primes divides any coefficient occuring in I |
---|
651 | // --> similar to procedure primeList in modstd.lib |
---|
652 | |
---|
653 | intvec L; |
---|
654 | int i,p; |
---|
655 | int ncores = 1; |
---|
656 | |
---|
657 | //----------------- Initialize optional parameter ncores --------------------- |
---|
658 | if(size(#) > 0) |
---|
659 | { |
---|
660 | if(size(#) == 1) |
---|
661 | { |
---|
662 | if(typeof(#[1]) == "int") |
---|
663 | { |
---|
664 | ncores = #[1]; |
---|
665 | # = list(); |
---|
666 | } |
---|
667 | } |
---|
668 | else |
---|
669 | { |
---|
670 | ncores = #[2]; |
---|
671 | } |
---|
672 | } |
---|
673 | |
---|
674 | if(size(#) == 0) |
---|
675 | { |
---|
676 | p = 2147483647; |
---|
677 | while(!divPrimeTest(I,p,k)) |
---|
678 | { |
---|
679 | p = prime(p-1); |
---|
680 | if(p == 2) { ERROR("No more primes."); } |
---|
681 | } |
---|
682 | L[1] = p; |
---|
683 | } |
---|
684 | else |
---|
685 | { |
---|
686 | L = #[1]; |
---|
687 | p = prime(L[size(L)]-1); |
---|
688 | while(!divPrimeTest(I,p,k)) |
---|
689 | { |
---|
690 | p = prime(p-1); |
---|
691 | if(p == 2) { ERROR("No more primes."); } |
---|
692 | } |
---|
693 | L[size(L)+1] = p; |
---|
694 | } |
---|
695 | if(p == 2) { ERROR("No more primes."); } |
---|
696 | if(ncores == 1) |
---|
697 | { |
---|
698 | for(i = 2; i <= n; i++) |
---|
699 | { |
---|
700 | p = prime(p-1); |
---|
701 | while(!divPrimeTest(I,p,k)) |
---|
702 | { |
---|
703 | p = prime(p-1); |
---|
704 | if(p == 2) { ERROR("no more primes"); } |
---|
705 | } |
---|
706 | L[size(L)+1] = p; |
---|
707 | } |
---|
708 | } |
---|
709 | else |
---|
710 | { |
---|
711 | int neededSize = size(L)+n-1;; |
---|
712 | list parallelResults; |
---|
713 | list arguments; |
---|
714 | int neededPrimes = neededSize-size(L); |
---|
715 | while(neededPrimes > 0) |
---|
716 | { |
---|
717 | arguments = list(); |
---|
718 | for(i = ((neededPrimes div ncores)+1-(neededPrimes%ncores == 0)) |
---|
719 | *ncores; i > 0; i--) |
---|
720 | { |
---|
721 | p = prime(p-1); |
---|
722 | if(p == 2) { ERROR("no more primes"); } |
---|
723 | arguments[i] = list("I", p, k); |
---|
724 | } |
---|
725 | parallelResults = parallelWaitAll("divPrimeTest", arguments, |
---|
726 | list(list(list(ncores)))); |
---|
727 | for(i = size(arguments); i > 0; i--) |
---|
728 | { |
---|
729 | if(parallelResults[i]) |
---|
730 | { |
---|
731 | L[size(L)+1] = arguments[i][2]; |
---|
732 | } |
---|
733 | } |
---|
734 | neededPrimes = neededSize-size(L); |
---|
735 | } |
---|
736 | if(size(L) > neededSize) |
---|
737 | { |
---|
738 | L = L[1..neededSize]; |
---|
739 | } |
---|
740 | } |
---|
741 | return(L); |
---|
742 | } |
---|
743 | example |
---|
744 | { "EXAMPLE:"; echo = 2; |
---|
745 | ring r = 0,(x,y,z),dp; |
---|
746 | ideal I = 2147483647x+y, z-181; |
---|
747 | intvec L = divPrimeList(4,I,10,10); |
---|
748 | size(L); |
---|
749 | L[1]; |
---|
750 | L[size(L)]; |
---|
751 | L = divPrimeList(4,I,5,L,5); |
---|
752 | size(L); |
---|
753 | L[size(L)]; |
---|
754 | } |
---|
755 | |
---|
756 | //////////////////////////////////////////////////////////////////////////////// |
---|
757 | |
---|
758 | proc spTestSB(ideal I, ideal J, list L, intvec sigma, int variant, list #) |
---|
759 | "USAGE: spTestSB(I,J,L,sigma,variant,#); I,J ideals, L intvec of primes, |
---|
760 | sigma intvec, variant integer |
---|
761 | RETURN: 1 (resp. 0) if for a randomly chosen prime p, that is not in L and |
---|
762 | divisible by the order of sigma, J mod p is (resp. is not) a standard |
---|
763 | basis of I mod p |
---|
764 | EXAMPLE: example spTestSB; shows an example |
---|
765 | " |
---|
766 | { |
---|
767 | int i,j,k,p; |
---|
768 | int ORD = order(sigma); |
---|
769 | def R = basering; |
---|
770 | list r = ringlist(R); |
---|
771 | |
---|
772 | while(!j) |
---|
773 | { |
---|
774 | j = 1; |
---|
775 | while(((p - 1) mod ORD) != 0) |
---|
776 | { |
---|
777 | p = prime(random(1000000000,2134567879)); |
---|
778 | if(p == 2){ ERROR("no more primes"); } |
---|
779 | } |
---|
780 | for(i = 1; i <= size(L); i++) |
---|
781 | { |
---|
782 | if(p == L[i]){ j = 0; break } |
---|
783 | } |
---|
784 | if(j) |
---|
785 | { |
---|
786 | for(i = 1; i <= ncols(I); i++) |
---|
787 | { |
---|
788 | for(k = 2; k <= size(I[i]); k++) |
---|
789 | { |
---|
790 | if((denominator(leadcoef(I[i][k])) mod p) == 0){ j = 0; break; } |
---|
791 | } |
---|
792 | if(!j){ break; } |
---|
793 | } |
---|
794 | } |
---|
795 | if(j) |
---|
796 | { |
---|
797 | if(!primeTest(I,p)) { j = 0; } |
---|
798 | } |
---|
799 | } |
---|
800 | r[1] = p; |
---|
801 | def @R = ring(r); |
---|
802 | setring @R; |
---|
803 | ideal I = imap(R,I); |
---|
804 | ideal J = imap(R,J); |
---|
805 | attrib(J,"isSB",1); |
---|
806 | |
---|
807 | int t = timer; |
---|
808 | j = 1; |
---|
809 | if(isIncluded(I,J) == 0){ j = 0; } |
---|
810 | |
---|
811 | if(printlevel >= 11) |
---|
812 | { |
---|
813 | "isIncluded(I,J) takes "+string(timer - t)+" seconds"; |
---|
814 | "j = "+string(j); |
---|
815 | } |
---|
816 | |
---|
817 | t = timer; |
---|
818 | if(j) |
---|
819 | { |
---|
820 | if(size(#) > 0) |
---|
821 | { |
---|
822 | ideal K = smpStd(I,sigma,p,variant,#[1])[1]; |
---|
823 | } |
---|
824 | else |
---|
825 | { |
---|
826 | ideal K = smpStd(I,sigma,p,variant)[1]; |
---|
827 | } |
---|
828 | t = timer; |
---|
829 | if(isIncluded(J,K) == 0){ j = 0; } |
---|
830 | |
---|
831 | if(printlevel >= 11) |
---|
832 | { |
---|
833 | "isIncluded(J,K) takes "+string(timer - t)+" seconds"; |
---|
834 | "j = "+string(j); |
---|
835 | } |
---|
836 | } |
---|
837 | setring R; |
---|
838 | return(j); |
---|
839 | } |
---|
840 | example |
---|
841 | { "EXAMPLE:"; echo = 2; |
---|
842 | intvec L = 2,3,5; |
---|
843 | ring r = 0,(x,y,z),dp; |
---|
844 | ideal I = x+1,y+1; |
---|
845 | intvec sigma = 2,1,3; |
---|
846 | ideal J = x+1,y; |
---|
847 | spTestSB(I,J,L,sigma,2); |
---|
848 | spTestSB(J,I,L,sigma,2); |
---|
849 | } |
---|
850 | |
---|
851 | //////////////////////////////////////////////////////////////////////////////// |
---|
852 | |
---|
853 | proc smpStd(ideal I, intvec sigma, int p, int variant, list #) |
---|
854 | "USAGE: smpStd(I,sigma,p,#); I ideal, sigma intvec, p integer, variant integer |
---|
855 | ASSUME: If size(#) > 0, then #[1] is an intvec describing the Hilbert series. |
---|
856 | RETURN: ideal - a standard basis of I mod p, integer - p |
---|
857 | NOTE: The procedure computes a standard basis of the ideal I modulo p and |
---|
858 | fetches the result to the basering. If size(#) > 0 the Hilbert driven |
---|
859 | standard basis computation symmStd(.,.,#[1]) is used in symmStd. |
---|
860 | The standard basis computation modulo p does also vary depending on the |
---|
861 | integer variant, namely |
---|
862 | @* - variant = 1: symmStd(.,.,#[1]) resp. symmStd, |
---|
863 | @* - variant = 2: symmStd, |
---|
864 | @* - variant = 3: homog. - symmStd(.,.,#[1]) resp. symmStd - dehomog., |
---|
865 | @* - variant = 4: fglm. |
---|
866 | EXAMPLE: example smpStd; shows an example |
---|
867 | " |
---|
868 | { |
---|
869 | def R0 = basering; |
---|
870 | list rl = ringlist(R0); |
---|
871 | rl[1] = p; |
---|
872 | def @r = ring(rl); |
---|
873 | setring @r; |
---|
874 | ideal i = fetch(R0,I); |
---|
875 | |
---|
876 | option(redSB); |
---|
877 | |
---|
878 | if(variant == 1) |
---|
879 | { |
---|
880 | if(size(#) > 0) |
---|
881 | { |
---|
882 | i = symmStd(i, sigma, #[1]); |
---|
883 | } |
---|
884 | else |
---|
885 | { |
---|
886 | i = symmStd(i, sigma); |
---|
887 | } |
---|
888 | } |
---|
889 | |
---|
890 | if(variant == 2) |
---|
891 | { |
---|
892 | i = symmStd(i, sigma); |
---|
893 | } |
---|
894 | |
---|
895 | if(variant == 3) |
---|
896 | { |
---|
897 | list rl = ringlist(@r); |
---|
898 | int nvar@r = nvars(@r); |
---|
899 | |
---|
900 | int k; |
---|
901 | intvec w; |
---|
902 | for(k = 1; k <= nvar@r; k++) |
---|
903 | { |
---|
904 | w[k] = deg(var(k)); |
---|
905 | } |
---|
906 | w[nvar@r + 1] = 1; |
---|
907 | |
---|
908 | rl[2][nvar@r + 1] = "homvar"; |
---|
909 | rl[3][2][2] = w; |
---|
910 | |
---|
911 | def HomR = ring(rl); |
---|
912 | setring HomR; |
---|
913 | ideal i = imap(@r, i); |
---|
914 | i = homog(i, homvar); |
---|
915 | intvec tau = sigma, size(sigma)+1; |
---|
916 | |
---|
917 | if(size(#) > 0) |
---|
918 | { |
---|
919 | if(w == 1) |
---|
920 | { |
---|
921 | i = symmStd(i, tau, #[1]); |
---|
922 | } |
---|
923 | else |
---|
924 | { |
---|
925 | i = symmStd(i, tau, #[1], w); |
---|
926 | } |
---|
927 | } |
---|
928 | else |
---|
929 | { |
---|
930 | i = symmStd(i, tau); |
---|
931 | } |
---|
932 | |
---|
933 | i = subst(i, homvar, 1); |
---|
934 | i = simplify(i, 34); |
---|
935 | |
---|
936 | setring @r; |
---|
937 | i = imap(HomR, i); |
---|
938 | i = interred(i); |
---|
939 | kill HomR; |
---|
940 | } |
---|
941 | |
---|
942 | if(variant == 4) |
---|
943 | { |
---|
944 | def R1 = changeord(list(list("dp",1:nvars(basering)))); |
---|
945 | setring R1; |
---|
946 | ideal i = fetch(@r,i); |
---|
947 | i = symmStd(i, sigma); |
---|
948 | setring @r; |
---|
949 | i = fglm(R1,i); |
---|
950 | } |
---|
951 | |
---|
952 | setring R0; |
---|
953 | return(list(fetch(@r,i),p)); |
---|
954 | } |
---|
955 | example |
---|
956 | { "EXAMPLE:"; echo = 2; |
---|
957 | ring r1 = 0, x(1..4), dp; |
---|
958 | ideal I = cyclic(4); |
---|
959 | intvec sigma = 2,3,4,1; |
---|
960 | int p = 181; |
---|
961 | list P = smpStd(I,sigma,p,2); |
---|
962 | P; |
---|
963 | |
---|
964 | ring r2 = 0, x(1..5), lp; |
---|
965 | ideal I = cyclic(5); |
---|
966 | intvec tau = 2,3,4,5,1; |
---|
967 | int q = 31981; |
---|
968 | list Q = smpStd(I,tau,q,4); |
---|
969 | Q; |
---|
970 | } |
---|
971 | |
---|
972 | //////////////////////////////////////////////////////////////////////////////// |
---|
973 | |
---|
974 | proc syModStd(ideal I, intvec sigma, list #) |
---|
975 | "USAGE: syModStd(I,sigma); I ideal, sigma intvec |
---|
976 | ASSUME: size(sigma) = nvars(basering) and sigma(I) = I. If size(#) > 0, then |
---|
977 | # contains either 1, 2 or 4 integers such that |
---|
978 | @* - #[1] is the number of available processors for the computation, |
---|
979 | @* - #[2] is an optional parameter for the exactness of the computation, |
---|
980 | if #[2] = 1, the procedure computes a standard basis for sure, |
---|
981 | @* - #[3] is the number of primes until the first lifting, |
---|
982 | @* - #[4] is the constant number of primes between two liftings until |
---|
983 | the computation stops. |
---|
984 | RETURN: ideal, a standard basis of I if no warning appears; |
---|
985 | NOTE: The procedure computes a standard basis of the ideal I (over the |
---|
986 | rational numbers) by using modular methods and the fact that I is |
---|
987 | invariant under the variable permutation sigma. |
---|
988 | By default the procedure computes a standard basis of I for sure, but |
---|
989 | if the optional parameter #[2] = 0, it computes a standard basis of I |
---|
990 | with high probability. |
---|
991 | The procedure distinguishes between different variants for the standard |
---|
992 | basis computation in positive characteristic depending on the ordering |
---|
993 | of the basering, the parameter #[2] and if the ideal I is homogeneous. |
---|
994 | @* - variant = 1, if I is homogeneous, |
---|
995 | @* - variant = 2, if I is not homogeneous, 1-block-ordering, |
---|
996 | @* - variant = 3, if I is not homogeneous, complicated ordering (lp or |
---|
997 | > 1 block), |
---|
998 | @* - variant = 4, if I is not homogeneous, ordering lp, dim(I) = 0. |
---|
999 | EXAMPLE: example syModStd; shows an example |
---|
1000 | " |
---|
1001 | { |
---|
1002 | if((nvars(basering) != size(sigma)) || (!isSymmetric(I,sigma))) |
---|
1003 | { |
---|
1004 | ERROR("The input is no permutation of the ring-variables!!"); |
---|
1005 | } |
---|
1006 | |
---|
1007 | int TT = timer; |
---|
1008 | int RT = rtimer; |
---|
1009 | |
---|
1010 | def R0 = basering; |
---|
1011 | list rl = ringlist(R0); |
---|
1012 | if((npars(R0) > 0) || (rl[1] > 0)) |
---|
1013 | { |
---|
1014 | ERROR("Characteristic of basering should be zero, basering should |
---|
1015 | have no parameters."); |
---|
1016 | } |
---|
1017 | |
---|
1018 | int index = 1; |
---|
1019 | int i,k,c; |
---|
1020 | int j = 1; |
---|
1021 | int pTest, sizeTest; |
---|
1022 | int en = 2134567879; |
---|
1023 | int an = 1000000000; |
---|
1024 | bigint N; |
---|
1025 | int ORD = order(sigma); |
---|
1026 | |
---|
1027 | //-------------------- Initialize optional parameters ------------------------ |
---|
1028 | if(size(#) > 0) |
---|
1029 | { |
---|
1030 | if(size(#) == 1) |
---|
1031 | { |
---|
1032 | int n1 = #[1]; |
---|
1033 | int exactness = 1; |
---|
1034 | if(n1 >= 10) |
---|
1035 | { |
---|
1036 | int n2 = n1 + 1; |
---|
1037 | int n3 = n1; |
---|
1038 | } |
---|
1039 | else |
---|
1040 | { |
---|
1041 | int n2 = 10; |
---|
1042 | int n3 = 10; |
---|
1043 | } |
---|
1044 | } |
---|
1045 | if(size(#) == 2) |
---|
1046 | { |
---|
1047 | int n1 = #[1]; |
---|
1048 | int exactness = #[2]; |
---|
1049 | if(n1 >= 10) |
---|
1050 | { |
---|
1051 | int n2 = n1 + 1; |
---|
1052 | int n3 = n1; |
---|
1053 | } |
---|
1054 | else |
---|
1055 | { |
---|
1056 | int n2 = 10; |
---|
1057 | int n3 = 10; |
---|
1058 | } |
---|
1059 | } |
---|
1060 | if(size(#) == 4) |
---|
1061 | { |
---|
1062 | int n1 = #[1]; |
---|
1063 | int exactness = #[2]; |
---|
1064 | if(n1 >= #[3]) |
---|
1065 | { |
---|
1066 | int n2 = n1 + 1; |
---|
1067 | } |
---|
1068 | else |
---|
1069 | { |
---|
1070 | int n2 = #[3]; |
---|
1071 | } |
---|
1072 | if(n1 >= #[4]) |
---|
1073 | { |
---|
1074 | int n3 = n1; |
---|
1075 | } |
---|
1076 | else |
---|
1077 | { |
---|
1078 | int n3 = #[4]; |
---|
1079 | } |
---|
1080 | } |
---|
1081 | } |
---|
1082 | else |
---|
1083 | { |
---|
1084 | int n1 = 1; |
---|
1085 | int exactness = 1; |
---|
1086 | int n2 = 10; |
---|
1087 | int n3 = 10; |
---|
1088 | } |
---|
1089 | |
---|
1090 | if(printlevel >= 10) |
---|
1091 | { |
---|
1092 | "n1 = "+string(n1)+", n2 = "+string(n2)+", n3 = "+string(n3) |
---|
1093 | +", exactness = "+string(exactness); |
---|
1094 | } |
---|
1095 | |
---|
1096 | //-------------------------- Save current options ------------------------------ |
---|
1097 | intvec opt = option(get); |
---|
1098 | |
---|
1099 | option(redSB); |
---|
1100 | |
---|
1101 | //-------------------- Initialize the list of primes ------------------------- |
---|
1102 | int tt = timer; |
---|
1103 | int rt = rtimer; |
---|
1104 | intvec L = divPrimeList(ORD,I,n2,n1); |
---|
1105 | if(printlevel >= 10) |
---|
1106 | { |
---|
1107 | "CPU-time for divPrimeList: "+string(timer-tt)+" seconds."; |
---|
1108 | "Real-time for divPrimeList: "+string(rtimer-rt)+" seconds."; |
---|
1109 | } |
---|
1110 | |
---|
1111 | //--------------------- Decide which variant to take ------------------------- |
---|
1112 | int variant; |
---|
1113 | int h = homog(I); |
---|
1114 | |
---|
1115 | tt = timer; |
---|
1116 | rt = rtimer; |
---|
1117 | |
---|
1118 | if(!mixedTest()) |
---|
1119 | { |
---|
1120 | if(h) |
---|
1121 | { |
---|
1122 | variant = 1; |
---|
1123 | if(printlevel >= 10) { "variant = 1"; } |
---|
1124 | |
---|
1125 | rl[1] = L[5]; |
---|
1126 | def @r = ring(rl); |
---|
1127 | setring @r; |
---|
1128 | def @s = changeord(list(list("dp",1:nvars(basering)))); |
---|
1129 | setring @s; |
---|
1130 | ideal I = std(fetch(R0,I)); |
---|
1131 | intvec hi = hilb(I,1); |
---|
1132 | setring R0; |
---|
1133 | kill @r,@s; |
---|
1134 | } |
---|
1135 | else |
---|
1136 | { |
---|
1137 | string ordstr_R0 = ordstr(R0); |
---|
1138 | int neg = 1 - attrib(R0,"global"); |
---|
1139 | |
---|
1140 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
1141 | { |
---|
1142 | variant = 2; |
---|
1143 | if(printlevel >= 10) { "variant = 2"; } |
---|
1144 | } |
---|
1145 | else |
---|
1146 | { |
---|
1147 | string order; |
---|
1148 | if(system("nblocks") <= 2) |
---|
1149 | { |
---|
1150 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
1151 | + find(ordstr_R0, "rp") <= 0) |
---|
1152 | { |
---|
1153 | order = "simple"; |
---|
1154 | } |
---|
1155 | } |
---|
1156 | |
---|
1157 | if((order == "simple") || (size(rl) > 4)) |
---|
1158 | { |
---|
1159 | variant = 2; |
---|
1160 | if(printlevel >= 10) { "variant = 2"; } |
---|
1161 | } |
---|
1162 | else |
---|
1163 | { |
---|
1164 | rl[1] = L[5]; |
---|
1165 | def @r = ring(rl); |
---|
1166 | setring @r; |
---|
1167 | |
---|
1168 | def @s = changeord(list(list("dp",1:nvars(basering)))); |
---|
1169 | setring @s; |
---|
1170 | ideal I = std(fetch(R0,I)); |
---|
1171 | if(dim(I) == 0) |
---|
1172 | { |
---|
1173 | variant = 4; |
---|
1174 | if(printlevel >= 10) { "variant = 4"; } |
---|
1175 | } |
---|
1176 | else |
---|
1177 | { |
---|
1178 | variant = 3; |
---|
1179 | if(printlevel >= 10) { "variant = 3"; } |
---|
1180 | |
---|
1181 | int nvar@r = nvars(@r); |
---|
1182 | intvec w; |
---|
1183 | for(i = 1; i <= nvar@r; i++) |
---|
1184 | { |
---|
1185 | w[i] = deg(var(i)); |
---|
1186 | } |
---|
1187 | w[nvar@r + 1] = 1; |
---|
1188 | |
---|
1189 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
1190 | intvec W = hiRi[2]; |
---|
1191 | @s = hiRi[1]; |
---|
1192 | setring @s; |
---|
1193 | intvec tau = sigma, size(sigma)+1; |
---|
1194 | |
---|
1195 | Id(1) = symmStd(Id(1),tau); |
---|
1196 | intvec hi = hilb(Id(1), 1, W); |
---|
1197 | } |
---|
1198 | |
---|
1199 | setring R0; |
---|
1200 | kill @r,@s; |
---|
1201 | } |
---|
1202 | } |
---|
1203 | } |
---|
1204 | } |
---|
1205 | else |
---|
1206 | { |
---|
1207 | if(exactness == 1) { return(groebner(I)); } |
---|
1208 | if(h) |
---|
1209 | { |
---|
1210 | variant = 1; |
---|
1211 | if(printlevel >= 10) { "variant = 1"; } |
---|
1212 | rl[1] = L[5]; |
---|
1213 | def @r = ring(rl); |
---|
1214 | setring @r; |
---|
1215 | def @s = changeord(list(list("dp",1:nvars(basering)))); |
---|
1216 | setring @s; |
---|
1217 | ideal I = std(fetch(R0,I)); |
---|
1218 | intvec hi = hilb(I,1); |
---|
1219 | setring R0; |
---|
1220 | kill @r,@s; |
---|
1221 | } |
---|
1222 | else |
---|
1223 | { |
---|
1224 | string ordstr_R0 = ordstr(R0); |
---|
1225 | int neg = 1 - attrib(R0,"global"); |
---|
1226 | |
---|
1227 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
1228 | { |
---|
1229 | variant = 2; |
---|
1230 | if(printlevel >= 10) { "variant = 2"; } |
---|
1231 | } |
---|
1232 | else |
---|
1233 | { |
---|
1234 | string order; |
---|
1235 | if(system("nblocks") <= 2) |
---|
1236 | { |
---|
1237 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
1238 | + find(ordstr_R0, "rp") <= 0) |
---|
1239 | { |
---|
1240 | order = "simple"; |
---|
1241 | } |
---|
1242 | } |
---|
1243 | |
---|
1244 | if((order == "simple") || (size(rl) > 4)) |
---|
1245 | { |
---|
1246 | variant = 2; |
---|
1247 | if(printlevel >= 10) { "variant = 2"; } |
---|
1248 | } |
---|
1249 | else |
---|
1250 | { |
---|
1251 | variant = 3; |
---|
1252 | if(printlevel >= 10) { "variant = 3"; } |
---|
1253 | |
---|
1254 | rl[1] = L[5]; |
---|
1255 | def @r = ring(rl); |
---|
1256 | setring @r; |
---|
1257 | int nvar@r = nvars(@r); |
---|
1258 | intvec w; |
---|
1259 | for(i = 1; i <= nvar@r; i++) |
---|
1260 | { |
---|
1261 | w[i] = deg(var(i)); |
---|
1262 | } |
---|
1263 | w[nvar@r + 1] = 1; |
---|
1264 | |
---|
1265 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
1266 | intvec W = hiRi[2]; |
---|
1267 | def @s = hiRi[1]; |
---|
1268 | setring @s; |
---|
1269 | intvec tau = sigma, size(sigma)+1; |
---|
1270 | |
---|
1271 | Id(1) = symmStd(Id(1),tau); |
---|
1272 | intvec hi = hilb(Id(1), 1, W); |
---|
1273 | |
---|
1274 | setring R0; |
---|
1275 | kill @r,@s; |
---|
1276 | } |
---|
1277 | } |
---|
1278 | } |
---|
1279 | } |
---|
1280 | |
---|
1281 | list P,T1,T2,T3,LL; |
---|
1282 | |
---|
1283 | ideal J,K,H; |
---|
1284 | |
---|
1285 | //----- If there is more than one processor available, we parallelize the ---- |
---|
1286 | //----- main standard basis computations in positive characteristic ---- |
---|
1287 | |
---|
1288 | if(n1 > 1) |
---|
1289 | { |
---|
1290 | ideal I_for_fork = I; |
---|
1291 | export(I_for_fork); // I available for each link |
---|
1292 | |
---|
1293 | //----- Create n1 links l(1),...,l(n1), open all of them and compute --------- |
---|
1294 | //----- standard basis for the primes L[2],...,L[n1 + 1]. --------- |
---|
1295 | |
---|
1296 | for(i = 1; i <= n1; i++) |
---|
1297 | { |
---|
1298 | //link l(i) = "MPtcp:fork"; |
---|
1299 | link l(i) = "ssi:fork"; |
---|
1300 | open(l(i)); |
---|
1301 | if((variant == 1) || (variant == 3)) |
---|
1302 | { |
---|
1303 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), eval(L[i + 1]), |
---|
1304 | eval(variant), eval(hi)))); |
---|
1305 | } |
---|
1306 | if((variant == 2) || (variant == 4)) |
---|
1307 | { |
---|
1308 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), eval(L[i + 1]), |
---|
1309 | eval(variant)))); |
---|
1310 | } |
---|
1311 | } |
---|
1312 | |
---|
1313 | int t = timer; |
---|
1314 | if((variant == 1) || (variant == 3)) |
---|
1315 | { |
---|
1316 | P = smpStd(I_for_fork, sigma, L[1], variant, hi); |
---|
1317 | } |
---|
1318 | if((variant == 2) || (variant == 4)) |
---|
1319 | { |
---|
1320 | P = smpStd(I_for_fork, sigma, L[1], variant); |
---|
1321 | } |
---|
1322 | t = timer - t; |
---|
1323 | if(t > 60) { t = 60; } |
---|
1324 | int i_sleep = system("sh", "sleep "+string(t)); |
---|
1325 | T1[1] = P[1]; |
---|
1326 | T2[1] = bigint(P[2]); |
---|
1327 | index++; |
---|
1328 | |
---|
1329 | j = j + n1 + 1; |
---|
1330 | } |
---|
1331 | |
---|
1332 | //-------------- Main standard basis computations in positive ---------------- |
---|
1333 | //---------------------- characteristic start here --------------------------- |
---|
1334 | |
---|
1335 | list arguments_farey, results_farey; |
---|
1336 | |
---|
1337 | while(1) |
---|
1338 | { |
---|
1339 | tt = timer; rt = rtimer; |
---|
1340 | |
---|
1341 | if(printlevel >= 10) { "size(L) = "+string(size(L)); } |
---|
1342 | |
---|
1343 | if(n1 > 1) |
---|
1344 | { |
---|
1345 | while(j <= size(L) + 1) |
---|
1346 | { |
---|
1347 | for(i = 1; i <= n1; i++) |
---|
1348 | { |
---|
1349 | //--- ask if link l(i) is ready otherwise sleep for t seconds --- |
---|
1350 | if(status(l(i), "read", "ready")) |
---|
1351 | { |
---|
1352 | //--- read the result from l(i) --- |
---|
1353 | P = read(l(i)); |
---|
1354 | T1[index] = P[1]; |
---|
1355 | T2[index] = bigint(P[2]); |
---|
1356 | index++; |
---|
1357 | |
---|
1358 | if(j <= size(L)) |
---|
1359 | { |
---|
1360 | if((variant == 1) || (variant == 3)) |
---|
1361 | { |
---|
1362 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), |
---|
1363 | eval(L[j]), eval(variant), eval(hi)))); |
---|
1364 | j++; |
---|
1365 | } |
---|
1366 | if((variant == 2) || (variant == 4)) |
---|
1367 | { |
---|
1368 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), |
---|
1369 | eval(L[j]), eval(variant)))); |
---|
1370 | j++; |
---|
1371 | } |
---|
1372 | } |
---|
1373 | else |
---|
1374 | { |
---|
1375 | k++; |
---|
1376 | close(l(i)); |
---|
1377 | } |
---|
1378 | } |
---|
1379 | } |
---|
1380 | //--- k describes the number of closed links --- |
---|
1381 | if(k == n1) |
---|
1382 | { |
---|
1383 | j++; |
---|
1384 | } |
---|
1385 | i_sleep = system("sh", "sleep "+string(t)); |
---|
1386 | } |
---|
1387 | } |
---|
1388 | else |
---|
1389 | { |
---|
1390 | while(j <= size(L)) |
---|
1391 | { |
---|
1392 | if((variant == 1) || (variant == 3)) |
---|
1393 | { |
---|
1394 | P = smpStd(I, sigma, L[j], variant, hi); |
---|
1395 | } |
---|
1396 | if((variant == 2) || (variant == 4)) |
---|
1397 | { |
---|
1398 | P = smpStd(I, sigma, L[j], variant); |
---|
1399 | } |
---|
1400 | |
---|
1401 | T1[index] = P[1]; |
---|
1402 | T2[index] = bigint(P[2]); |
---|
1403 | index++; |
---|
1404 | j++; |
---|
1405 | } |
---|
1406 | } |
---|
1407 | |
---|
1408 | if(printlevel >= 10) |
---|
1409 | { |
---|
1410 | "CPU-time for computing list is "+string(timer - tt)+" seconds."; |
---|
1411 | "Real-time for computing list is "+string(rtimer - rt)+" seconds."; |
---|
1412 | } |
---|
1413 | |
---|
1414 | //------------------------ Delete unlucky primes ----------------------------- |
---|
1415 | //------------- unlucky if and only if the leading ideal is wrong ------------ |
---|
1416 | |
---|
1417 | LL = deleteUnluckyPrimes(T1,T2,h); |
---|
1418 | T1 = LL[1]; |
---|
1419 | T2 = LL[2]; |
---|
1420 | |
---|
1421 | //------------------- Now all leading ideals are the same -------------------- |
---|
1422 | //------------------- Lift results to basering via farey --------------------- |
---|
1423 | |
---|
1424 | tt = timer; rt = rtimer; |
---|
1425 | N = T2[1]; |
---|
1426 | for(i = 2; i <= size(T2); i++) { N = N*T2[i]; } |
---|
1427 | H = chinrem(T1,T2); |
---|
1428 | if(n1 == 1) |
---|
1429 | { |
---|
1430 | J = farey(H,N); |
---|
1431 | } |
---|
1432 | else |
---|
1433 | { |
---|
1434 | for(i = ncols(H); i > 0; i--) |
---|
1435 | { |
---|
1436 | arguments_farey[i] = list(ideal(H[i]), N); |
---|
1437 | } |
---|
1438 | results_farey = parallelWaitAll("farey", arguments_farey, |
---|
1439 | list(list(list(n1)))); |
---|
1440 | for(i = ncols(H); i > 0; i--) |
---|
1441 | { |
---|
1442 | J[i] = results_farey[i][1]; |
---|
1443 | } |
---|
1444 | } |
---|
1445 | if(printlevel >= 10) |
---|
1446 | { |
---|
1447 | "CPU-time for lifting-process is "+string(timer - tt)+" seconds."; |
---|
1448 | "Real-time for lifting-process is "+string(rtimer - rt)+" seconds."; |
---|
1449 | } |
---|
1450 | |
---|
1451 | //---------------- Test if we already have a standard basis of I -------------- |
---|
1452 | |
---|
1453 | tt = timer; rt = rtimer; |
---|
1454 | if((variant == 1) || (variant == 3)) |
---|
1455 | { |
---|
1456 | pTest = spTestSB(I,J,L,sigma,variant,hi); |
---|
1457 | } |
---|
1458 | if((variant == 2) || (variant == 4)) |
---|
1459 | { |
---|
1460 | pTest = spTestSB(I,J,L,sigma,variant); |
---|
1461 | } |
---|
1462 | |
---|
1463 | if(printlevel >= 10) |
---|
1464 | { |
---|
1465 | "CPU-time for pTest is "+string(timer - tt)+" seconds."; |
---|
1466 | "Real-time for pTest is "+string(rtimer - rt)+" seconds."; |
---|
1467 | } |
---|
1468 | |
---|
1469 | if(pTest) |
---|
1470 | { |
---|
1471 | if(printlevel >= 10) |
---|
1472 | { |
---|
1473 | "CPU-time for computation without final tests is " |
---|
1474 | +string(timer - TT)+" seconds."; |
---|
1475 | "Real-time for computation without final tests is " |
---|
1476 | +string(rtimer - RT)+" seconds."; |
---|
1477 | } |
---|
1478 | |
---|
1479 | attrib(J,"isSB",1); |
---|
1480 | |
---|
1481 | if(exactness == 0) |
---|
1482 | { |
---|
1483 | option(set, opt); |
---|
1484 | if(n1 > 1) { kill I_for_fork; } |
---|
1485 | return(J); |
---|
1486 | } |
---|
1487 | |
---|
1488 | if(exactness == 1) |
---|
1489 | { |
---|
1490 | tt = timer; rt = rtimer; |
---|
1491 | sizeTest = 1 - isIncluded(I,J,n1); |
---|
1492 | |
---|
1493 | if(printlevel >= 10) |
---|
1494 | { |
---|
1495 | "CPU-time for checking if I subset <G> is " |
---|
1496 | +string(timer - tt)+" seconds."; |
---|
1497 | "Real-time for checking if I subset <G> is " |
---|
1498 | +string(rtimer - rt)+" seconds."; |
---|
1499 | } |
---|
1500 | |
---|
1501 | if(sizeTest == 0) |
---|
1502 | { |
---|
1503 | tt = timer; rt = rtimer; |
---|
1504 | K = std(J); |
---|
1505 | |
---|
1506 | if(printlevel >= 10) |
---|
1507 | { |
---|
1508 | "CPU-time for last std-computation is " |
---|
1509 | +string(timer - tt)+" seconds."; |
---|
1510 | "Real-time for last std-computation is " |
---|
1511 | +string(rtimer - rt)+" seconds."; |
---|
1512 | } |
---|
1513 | |
---|
1514 | if(size(reduce(K,J)) == 0) |
---|
1515 | { |
---|
1516 | option(set, opt); |
---|
1517 | if(n1 > 1) { kill I_for_fork; } |
---|
1518 | return(J); |
---|
1519 | } |
---|
1520 | } |
---|
1521 | } |
---|
1522 | } |
---|
1523 | |
---|
1524 | //-------------- We do not already have a standard basis of I ---------------- |
---|
1525 | //----------- Therefore do the main computation for more primes -------------- |
---|
1526 | |
---|
1527 | T1 = H; |
---|
1528 | T2 = N; |
---|
1529 | index = 2; |
---|
1530 | |
---|
1531 | j = size(L) + 1; |
---|
1532 | tt = timer; rt = rtimer; |
---|
1533 | L = divPrimeList(ORD,I,n3,L,n1); |
---|
1534 | if(printlevel >= 10) |
---|
1535 | { |
---|
1536 | "CPU-time for divPrimeList: "+string(timer-tt)+" seconds."; |
---|
1537 | "Real-time for divPrimeList: "+string(rtimer-rt)+" seconds."; |
---|
1538 | } |
---|
1539 | |
---|
1540 | if(n1 > 1) |
---|
1541 | { |
---|
1542 | for(i = 1; i <= n1; i++) |
---|
1543 | { |
---|
1544 | open(l(i)); |
---|
1545 | if((variant == 1) || (variant == 3)) |
---|
1546 | { |
---|
1547 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), eval(L[j+i-1]), |
---|
1548 | eval(variant), eval(hi)))); |
---|
1549 | } |
---|
1550 | if((variant == 2) || (variant == 4)) |
---|
1551 | { |
---|
1552 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), eval(L[j+i-1]), |
---|
1553 | eval(variant)))); |
---|
1554 | } |
---|
1555 | } |
---|
1556 | j = j + n1; |
---|
1557 | k = 0; |
---|
1558 | } |
---|
1559 | } |
---|
1560 | } |
---|
1561 | example |
---|
1562 | { "EXAMPLE:"; echo = 2; |
---|
1563 | ring R1 = 0, (x,y,z), dp; |
---|
1564 | ideal I; |
---|
1565 | I[1] = -2xyz4+xz5+xz; |
---|
1566 | I[2] = -2xyz4+yz5+yz; |
---|
1567 | intvec sigma = 2,1,3; |
---|
1568 | ideal sI = syModStd(I,sigma); |
---|
1569 | sI; |
---|
1570 | |
---|
1571 | ring R2 = 0, x(1..4), dp; |
---|
1572 | ideal I = cyclic(4); |
---|
1573 | I; |
---|
1574 | intvec pi = 2,3,4,1; |
---|
1575 | ideal sJ1 = syModStd(I,pi,1); |
---|
1576 | ideal sJ2 = syModStd(I,pi,1,0); |
---|
1577 | size(reduce(sJ1,sJ2)); |
---|
1578 | size(reduce(sJ2,sJ1)); |
---|
1579 | } |
---|