[d9adbe] | 1 | //////////////////////////////////////////////////////////////////////////// |
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[4fb2ef5] | 2 | version="version symodstd.lib 4.0.0.0 Dec_2013 "; // $Id$ |
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[d9adbe] | 3 | category = "Commutative Algebra"; |
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| 4 | info=" |
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| 5 | LIBRARY: symodstd.lib Procedures for computing Groebner basis of ideals |
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| 6 | being invariant under certain variable permutations. |
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| 7 | |
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| 8 | AUTHOR: Stefan Steidel, steidel@mathematik.uni-kl.de |
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| 9 | |
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| 10 | OVERVIEW: |
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| 11 | |
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| 12 | A library for computing the Groebner basis of an ideal in the polynomial |
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| 13 | ring over the rational numbers, that is invariant under certain permutations |
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| 14 | of the variables, using the symmetry and modular methods. |
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| 15 | More precisely let I = <f1,...,fr> be an ideal in Q[x(1),...,x(n)] and |
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| 16 | sigma a permutation of order k in Sym(n) such that sigma(I) = I. |
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| 17 | We assume that sigma({f1,...,fr}) = {f1,...,fr}. This can always be obtained |
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| 18 | by adding sigma(fi) to {f1,...,fr}. |
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| 19 | To compute a standard basis of I we apply a modification of the modular |
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| 20 | version of the standard basis algorithm (improving the calculations in |
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| 21 | positive characteristic). Therefore we only allow primes p such that p-1 is |
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| 22 | divisible by k. This guarantees the existance of a k-th primitive root of |
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| 23 | unity in Z/pZ. |
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| 24 | |
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| 25 | PROCEDURES: |
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| 26 | genSymId(I,sigma); compute ideal J such that sigma(J) = J and J includes I |
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| 27 | isSymmetric(I,sigma); check if I is invariant under permutation sigma |
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| 28 | primRoot(p,k); int describing a k-th primitive root of unity in Z/pZ |
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| 29 | eigenvalues(I,sigma); list of eigenvalues of generators of I under sigma |
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| 30 | symmStd(I,sigma); standard basis of I using invariance of I under sigma |
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| 31 | syModStd(I,sigma); SB of I using modular methods and sigma(I) = I |
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| 32 | "; |
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| 33 | |
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| 34 | LIB "brnoeth.lib"; |
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| 35 | LIB "modstd.lib"; |
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| 36 | LIB "parallel.lib"; |
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| 37 | |
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| 38 | //////////////////////////////////////////////////////////////////////////////// |
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| 39 | |
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| 40 | proc genSymId(ideal I, intvec sigma) |
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| 41 | "USAGE: genSymId(I,sigma); I ideal, sigma intvec |
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| 42 | ASSUME: size(sigma) = nvars(basering =: n |
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| 43 | RETURN: ideal J such that sigma(J) = J and J includes I |
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| 44 | NOTE: sigma is a permutation of the variables of the basering, i.e. |
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| 45 | @* sigma: var(i) ----> var(sigma[i]), 1 <= i <= n. |
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| 46 | EXAMPLE: example genSymId; shows an example |
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| 47 | " |
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| 48 | { |
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| 49 | if(nvars(basering) != size(sigma)) |
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| 50 | { |
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| 51 | ERROR("The input is no permutation of the ring-variables!!"); |
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| 52 | } |
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| 53 | |
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| 54 | int i; |
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| 55 | |
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| 56 | ideal perm; |
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| 57 | for(i = 1; i <= size(sigma); i++) |
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| 58 | { |
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| 59 | perm[size(perm)+1] = var(sigma[i]); |
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| 60 | } |
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| 61 | |
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| 62 | map f = basering, perm; |
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| 63 | ideal J = I; |
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| 64 | ideal helpJ = I; |
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| 65 | for(i = 1; i <= order(sigma) - 1; i++) |
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| 66 | { |
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| 67 | helpJ = f(helpJ); |
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| 68 | J = J, helpJ; |
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| 69 | } |
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| 70 | |
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| 71 | return(simplify(simplify(J,4),2)); |
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| 72 | } |
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| 73 | example |
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| 74 | { "EXAMPLE:"; echo = 2; |
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| 75 | ring R = 0,(u,v,w,x,y),dp; |
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| 76 | intvec pi = 2,3,4,5,1; |
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| 77 | ideal I = u2v + x3y - w2; |
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| 78 | genSymId(I,pi); |
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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 | proc isSymmetric(ideal I, intvec sigma) |
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| 84 | "USAGE: isSymmetric(I,sigma); I ideal, sigma intvec |
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| 85 | ASSUME: size(sigma) = nvars(basering) =: n |
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| 86 | RETURN: 1, if the set of generators of I is invariant under sigma; |
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| 87 | @* 0, if the set of generators of I is not invariant under sigma |
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| 88 | NOTE: sigma is a permutation of the variables of the basering, i.e. |
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| 89 | @* sigma: var(i) ----> var(sigma[i]), 1 <= i <= n. |
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| 90 | EXAMPLE: example isSymmetric; shows an example |
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| 91 | " |
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| 92 | { |
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| 93 | if(nvars(basering) != size(sigma)) |
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| 94 | { |
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| 95 | ERROR("The input is no permutation of the ring-variables!!"); |
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| 96 | } |
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| 97 | |
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| 98 | int i, j; |
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| 99 | |
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| 100 | list L; |
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| 101 | for(i = 1; i <= size(I); i++) |
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| 102 | { |
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| 103 | L[size(L)+1] = I[i]; |
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| 104 | } |
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| 105 | |
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| 106 | ideal perm; |
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| 107 | for(i = 1; i <= size(sigma); i++) |
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| 108 | { |
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| 109 | perm[size(perm)+1] = var(sigma[i]); |
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| 110 | } |
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| 111 | |
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| 112 | map f = basering, perm; |
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| 113 | ideal J = f(I); |
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| 114 | |
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| 115 | poly g; |
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| 116 | while(size(L) > 0) |
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| 117 | { |
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| 118 | j = size(L); |
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| 119 | g = L[1]; |
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| 120 | |
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| 121 | for(i = 1; i <= size(J); i++) |
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| 122 | { |
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| 123 | if(g - J[i] == 0) |
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| 124 | { |
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| 125 | L = delete(L, 1); |
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| 126 | break; |
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| 127 | } |
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| 128 | } |
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| 129 | |
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| 130 | if(j == size(L)) |
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| 131 | { |
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| 132 | return(0); |
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| 133 | } |
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| 134 | } |
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| 135 | |
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| 136 | return(1); |
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| 137 | } |
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| 138 | example |
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| 139 | { "EXAMPLE:"; echo = 2; |
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| 140 | ring R = 0,x(1..5),dp; |
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| 141 | ideal I = cyclic(5); |
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| 142 | intvec pi = 2,3,4,5,1; |
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| 143 | isSymmetric(I,pi); |
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| 144 | intvec tau = 2,5,1,4,3; |
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| 145 | isSymmetric(I,tau); |
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| 146 | } |
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| 147 | |
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| 148 | //////////////////////////////////////////////////////////////////////////////// |
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| 149 | |
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| 150 | static proc permute(intvec v, intvec P) |
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| 151 | { |
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| 152 | // permute the intvec v according to the permutation given by P |
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| 153 | |
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| 154 | int s = size(v); |
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| 155 | int n = size(P); |
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| 156 | int i; |
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| 157 | if(s < n) |
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| 158 | { |
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| 159 | for(i = s+1; i <= n; i = i+1) |
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| 160 | { |
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| 161 | v[i] = 0; |
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| 162 | } |
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| 163 | s = size(v); |
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| 164 | } |
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| 165 | intvec auxv = v; |
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| 166 | for(i = 1; i <= n; i = i+1) |
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| 167 | { |
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| 168 | auxv[i] = v[P[i]]; |
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| 169 | } |
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| 170 | |
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| 171 | return(auxv); |
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| 172 | } |
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| 173 | |
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| 174 | //////////////////////////////////////////////////////////////////////////////// |
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| 175 | |
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| 176 | static proc order(intvec sigma) |
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| 177 | { |
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| 178 | // compute the order of sigma in Sym({1,...,n}) with n := size(sigma) |
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| 179 | |
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| 180 | int ORD = 1; |
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| 181 | intvec id = 1..size(sigma); |
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| 182 | intvec tau = sigma; |
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| 183 | |
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| 184 | while(tau != id) |
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| 185 | { |
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| 186 | tau = permute(tau, sigma); |
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| 187 | ORD = ORD + 1; |
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| 188 | } |
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| 189 | |
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| 190 | return(ORD); |
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| 191 | } |
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| 192 | |
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| 193 | //////////////////////////////////////////////////////////////////////////////// |
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| 194 | |
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| 195 | static proc modExpo(bigint x, bigint a, bigint n) |
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| 196 | { |
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| 197 | // compute x^a mod n |
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| 198 | |
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| 199 | bigint z = 1; |
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| 200 | |
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| 201 | while(a != 0) |
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| 202 | { |
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| 203 | while((a mod 2) == 0) |
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| 204 | { |
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| 205 | a = a div 2; |
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| 206 | x = x^2 mod n; |
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| 207 | } |
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| 208 | |
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| 209 | a = a - 1; |
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| 210 | z = (z*x) mod n; |
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| 211 | } |
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| 212 | |
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| 213 | return(z); |
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| 214 | } |
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| 215 | |
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| 216 | //////////////////////////////////////////////////////////////////////////////// |
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| 217 | |
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| 218 | proc primRoot(int p, int k) |
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| 219 | "USAGE: primRoot(p,k); p,k integers |
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| 220 | ASSUME: p is a prime and k divides p-1. |
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| 221 | RETURN: int: a k-th primitive root of unity in Z/pZ |
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| 222 | EXAMPLE: example primRoot; shows an example |
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| 223 | " |
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| 224 | { |
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| 225 | if(k == 2) |
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| 226 | { |
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| 227 | return(-1); |
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| 228 | } |
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| 229 | |
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| 230 | if(p == 0) |
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| 231 | { |
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| 232 | return(0); |
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| 233 | } |
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| 234 | |
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| 235 | int i, j; |
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| 236 | |
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| 237 | if(((p-1) mod k) != 0) |
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| 238 | { |
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| 239 | ERROR("There is no "+string(k)+"-th primitive root of unity " |
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| 240 | +"in Z/"+string(p)+"Z."); |
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| 241 | return(0); |
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| 242 | } |
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| 243 | |
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| 244 | list PF = primefactors(p-1)[1]; |
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| 245 | |
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| 246 | bigint a; |
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| 247 | |
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| 248 | for(i = 2; i <= p-1; i++) |
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| 249 | { |
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| 250 | a = i; |
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| 251 | |
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| 252 | for(j = 1; j <= size(PF); j++) |
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| 253 | { |
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| 254 | if(modExpo(a, (p-1) div PF[j], p) == 1) |
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| 255 | { |
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| 256 | break; |
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| 257 | } |
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| 258 | } |
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| 259 | |
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| 260 | if(j == size(PF)+1) |
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| 261 | { |
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| 262 | a = modExpo(a, (p-1) div k, p); |
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| 263 | |
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| 264 | string str = "int xi = "+string(a); |
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| 265 | execute(str); |
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| 266 | |
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| 267 | return(xi); |
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| 268 | } |
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| 269 | } |
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| 270 | } |
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| 271 | example |
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| 272 | { "EXAMPLE:"; echo = 2; |
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| 273 | primRoot(181,10); |
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| 274 | ring R = 2147482801, x, lp; |
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| 275 | number a = primRoot(2147482801,5); |
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| 276 | a; |
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| 277 | a^2; |
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| 278 | a^3; |
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| 279 | a^4; |
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| 280 | a^5; |
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| 281 | } |
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| 282 | |
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| 283 | //////////////////////////////////////////////////////////////////////////////// |
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| 284 | |
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| 285 | static proc permMat(intvec sigma, list #) |
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| 286 | { |
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| 287 | // compute an intmat such that i-th row describes sigma^i |
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| 288 | |
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| 289 | int i; |
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| 290 | int n = size(sigma); |
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| 291 | |
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| 292 | if(size(#) == 0) |
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| 293 | { |
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| 294 | int ORD = order(sigma); |
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| 295 | } |
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| 296 | else |
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| 297 | { |
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| 298 | int ORD = #[1]; |
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| 299 | } |
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| 300 | |
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| 301 | intmat P[ORD][n]; |
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| 302 | intvec sigmai = sigma; |
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| 303 | for(i = 1; i <= ORD; i++) |
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| 304 | { |
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| 305 | P[i,1..n] = sigmai; |
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| 306 | sigmai = permute(sigmai, sigma); |
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| 307 | } |
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| 308 | |
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| 309 | return(P); |
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| 310 | } |
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| 311 | |
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| 312 | //////////////////////////////////////////////////////////////////////////////// |
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| 313 | |
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| 314 | static proc genTransId(intvec sigma, list #) |
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| 315 | { |
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| 316 | // list L of two ideals such that |
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| 317 | // - L[1] describes the transformation and |
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| 318 | // - L[2] describes the retransformation (inverse mapping). |
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| 319 | // ORD is the order of sigma in Sym({1,...,n}) with n := size(sigma) and |
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| 320 | // sigma: {1,...,n} ---> {1,...,n}: sigma(j) = sigma[j]. Since sigma is a |
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| 321 | // permutation of variables it induces an automorphism phi of the |
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| 322 | // basering, more precisely a linear variable transformation which is |
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| 323 | // generated by this procedure. In terms it holds: |
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| 324 | // |
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| 325 | // phi : basering ---------> basering |
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| 326 | // var(i) |----> L[1][i] |
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| 327 | // L[2][i] <----| var(i) |
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| 328 | |
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| 329 | int n = nvars(basering); |
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| 330 | |
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| 331 | if(n != size(sigma)) |
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| 332 | { |
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| 333 | ERROR("The input is no permutation of the ring-variables!!"); |
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| 334 | } |
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| 335 | |
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| 336 | int i, j, k; |
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| 337 | |
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| 338 | if(size(#) == 0) |
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| 339 | { |
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| 340 | int CHAR = char(basering); |
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| 341 | int ORD = order(sigma); |
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| 342 | |
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| 343 | if((((CHAR - 1) mod ORD) != 0) && (CHAR > 0)) |
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| 344 | { |
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| 345 | ERROR("Basering of characteristic "+string(CHAR)+" has no " |
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| 346 | +string(ORD)+"-th primitive root of unity!!"); |
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| 347 | } |
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| 348 | |
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| 349 | if((CHAR == 0) && (ORD > 2)) |
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| 350 | { |
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| 351 | "======================================== |
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| 352 | ========================================"; |
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| 353 | "If basering really has a "+string(ORD)+"-th " |
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| 354 | +"primitive root of unity then insert it as input!!"; |
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| 355 | "======================================== |
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| 356 | ========================================"; |
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| 357 | return(list()); |
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| 358 | } |
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| 359 | else |
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| 360 | { |
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| 361 | int xi = primRoot(CHAR, ORD); |
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| 362 | number a = xi; |
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| 363 | } |
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| 364 | } |
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| 365 | else |
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| 366 | { |
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| 367 | int ORD = #[1]; |
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| 368 | number a = #[2]; |
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| 369 | } |
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| 370 | |
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| 371 | intmat PERM = permMat(sigma,ORD); |
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| 372 | |
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| 373 | ideal TR, RETR; |
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| 374 | poly s_trans; |
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| 375 | matrix C; |
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| 376 | |
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| 377 | //-------------- retransformation ideal RETR is generated here ----------------- |
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| 378 | for(i = 1; i <= n; i++) |
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| 379 | { |
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| 380 | for(j = 0; j < ORD; j++) |
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| 381 | { |
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| 382 | for(k = 1; k <= ORD; k++) |
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| 383 | { |
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| 384 | // for each variable var(i): |
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| 385 | // s_trans^(j) = sum_{k=1}^{ORD} a^(k*j)*sigma^k(var(i)) |
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| 386 | // for j = 0,...,ORD-1 |
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| 387 | s_trans = s_trans + a^(k*j)*var(PERM[k,i]); |
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| 388 | } |
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| 389 | RETR = RETR + simplify(s_trans, 1); |
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| 390 | s_trans = 0; |
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| 391 | } |
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| 392 | } |
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| 393 | |
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| 394 | //---------------- transformation ideal TR is generated here ------------------- |
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| 395 | for(i = 1; i <= n; i++) |
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| 396 | { |
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| 397 | for(j = 1; j <= size(RETR); j++) |
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| 398 | { |
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| 399 | C = coeffs(RETR[j], var(i)); |
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| 400 | if(nrows(C) > 1) |
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| 401 | { |
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| 402 | // var(j) = RETR[j] = sum_{i in J} c_ij*var(i), J subset {1,...,n}, |
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| 403 | // and therefore var(i) = (sum_{j} s(j)/c_ij)/#(summands) |
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| 404 | s_trans = s_trans + var(j)/(C[nrows(C),1]); |
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| 405 | } |
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| 406 | } |
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| 407 | |
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| 408 | TR = TR + s_trans/number(size(s_trans)); |
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| 409 | s_trans = 0; |
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| 410 | } |
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| 411 | |
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| 412 | return(list(TR,RETR)); |
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| 413 | } |
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| 414 | |
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| 415 | //////////////////////////////////////////////////////////////////////////////// |
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| 416 | |
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| 417 | proc eigenvalues(ideal I, intvec sigma) |
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| 418 | "USAGE: eigenvalues(I,sigma); I ideal, sigma intvec |
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| 419 | ASSUME: size(sigma) = nvars(basering) =: n |
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| 420 | RETURN: list of eigenvalues of generators of I under permutation sigma |
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| 421 | NOTE: sigma is a permutation of the variables of the basering, i.e. |
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| 422 | sigma: var(i) ----> var(sigma[i]), 1 <= i <= n. |
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| 423 | EXAMPLE: example eigenvalues; shows an example |
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| 424 | " |
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| 425 | { |
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| 426 | int i, j; |
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| 427 | |
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| 428 | def A = basering; |
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| 429 | int n = nvars(A); |
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| 430 | |
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| 431 | poly ev; |
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| 432 | list EV; |
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| 433 | poly s, help_var; |
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| 434 | matrix C1, C2; |
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| 435 | |
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| 436 | ideal perm; |
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| 437 | for(i = 1; i <= n; i++) |
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| 438 | { |
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| 439 | perm[size(perm)+1] = var(sigma[i]); |
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| 440 | } |
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| 441 | |
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| 442 | map f = A, perm; |
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| 443 | |
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| 444 | for(i = 1; i <= size(I); i++) |
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| 445 | { |
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| 446 | //-------------- s is the image of I[i] under permutation sigma ---------------- |
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| 447 | s = I[i]; |
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| 448 | s = f(s); |
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| 449 | |
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| 450 | for(j = 1; j <= n; j++) |
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| 451 | { |
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| 452 | C1 = coeffs(I[i], var(j)); |
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| 453 | C2 = coeffs(s, var(j)); |
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| 454 | if(nrows(C1) > 1) |
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| 455 | { |
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| 456 | ev = C2[nrows(C2),1]/C1[nrows(C1),1]; |
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| 457 | |
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| 458 | //------ Furthermore check that I[i] is eigenvector of permutation sigma. ------ |
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| 459 | if(s == ev*I[i]) |
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| 460 | { |
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| 461 | break; |
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| 462 | } |
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| 463 | else |
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| 464 | { |
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| 465 | ERROR("I["+string(i)+"] is no eigenvector " |
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| 466 | +"of permutation sigma!!"); |
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| 467 | } |
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| 468 | } |
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| 469 | } |
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| 470 | |
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| 471 | EV[size(EV)+1] = ev; |
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| 472 | } |
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| 473 | |
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| 474 | return(EV); |
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| 475 | } |
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| 476 | example |
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| 477 | { "EXAMPLE:"; echo = 2; |
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| 478 | ring R = 11, x(1..5), dp; |
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| 479 | poly p1 = x(1)+x(2)+x(3)+x(4)+x(5); |
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| 480 | poly p2 = x(1)+4*x(2)+5*x(3)-2*x(4)+3*x(5); |
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| 481 | poly p3 = x(1)+5*x(2)+3*x(3)+4*x(4)-2*x(5); |
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| 482 | poly p4 = x(1)-2*x(2)+4*x(3)+3*x(4)+5*x(5); |
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| 483 | poly p5 = x(1)+3*x(2)-2*x(3)+5*x(4)+4*x(5); |
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| 484 | ideal I = p1,p2,p3,p4,p5; |
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| 485 | intvec tau = 2,3,4,5,1; |
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| 486 | eigenvalues(I,tau); |
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| 487 | } |
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| 488 | |
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| 489 | //////////////////////////////////////////////////////////////////////////////// |
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| 490 | |
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| 491 | proc symmStd(ideal I, intvec sigma, list #) |
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| 492 | "USAGE: symmStd(I,sigma,#); I ideal, sigma intvec |
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| 493 | ASSUME: size(sigma) = nvars(basering) =: n, basering has an order(sigma)-th |
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| 494 | primitive root of unity a (if char(basering) > 0) and sigma(I) = I |
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| 495 | RETURN: ideal, a standard basis of I |
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| 496 | NOTE: Assuming that the ideal I is invariant under the variable permutation |
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| 497 | sigma and the basering has an order(sigma)-th primitive root of unity |
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| 498 | the procedure uses linear transformation of variables in order to |
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| 499 | improve standard basis computation. |
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| 500 | If char(basering) = 0 all computations are done in the polynomial ring |
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| 501 | over the smallest field extension that has an order(sigma)-th primitive |
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| 502 | root of unity. |
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| 503 | EXAMPLE: example symmStd; shows an example |
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| 504 | " |
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| 505 | { |
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| 506 | if((nvars(basering) != size(sigma)) || (!isSymmetric(I,sigma))) |
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| 507 | { |
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| 508 | ERROR("The input is no permutation of the ring-variables!!"); |
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| 509 | } |
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| 510 | |
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| 511 | option(redSB); |
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| 512 | |
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| 513 | def R = basering; |
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| 514 | int CHAR = char(R); |
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| 515 | int n = nvars(R); |
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| 516 | |
---|
| 517 | int t; |
---|
| 518 | |
---|
| 519 | //-------- (1) Compute the order of variable permutation sigma. ---------------- |
---|
| 520 | int ORD = order(sigma); |
---|
| 521 | if((((CHAR - 1) mod ORD) != 0) && (CHAR > 0)) |
---|
| 522 | { |
---|
| 523 | ERROR("Basering of characteristic "+string(CHAR)+" has no " |
---|
| 524 | +string(ORD)+"-th primitive root of unity!!"); |
---|
| 525 | } |
---|
| 526 | |
---|
| 527 | //-------- (2) Compute the order(sigma)-th primitive root of unity ------------- |
---|
| 528 | //-------- in basering or move to ring extension. ------------- |
---|
| 529 | if((CHAR == 0) && (ORD > 2)) |
---|
| 530 | { |
---|
| 531 | def save_ring=basering; |
---|
| 532 | ring ext_ring=0,p,lp; |
---|
| 533 | def S = changechar(ringlist(ext_ring),save_ring); |
---|
| 534 | setring S; |
---|
| 535 | kill ext_ring; |
---|
| 536 | kill save_ring; |
---|
| 537 | minpoly = rootofUnity(ORD); |
---|
| 538 | ideal I = imap(R, I); |
---|
| 539 | number a = p; |
---|
| 540 | } |
---|
| 541 | else |
---|
| 542 | { |
---|
| 543 | int xi = primRoot(CHAR, ORD); |
---|
| 544 | number a = xi; |
---|
| 545 | } |
---|
| 546 | |
---|
| 547 | //--------- (3) Define the linear transformation of variables with ------------- |
---|
| 548 | //--------- respect to sigma. ------------- |
---|
| 549 | list L = genTransId(sigma,ORD,a); |
---|
| 550 | ideal TR = L[1]; |
---|
| 551 | ideal RETR = L[2]; |
---|
| 552 | |
---|
| 553 | //--------- (4) Compute the eigenvalues of the "new" variables of -------------- |
---|
| 554 | //--------- sigma after transformation. -------------- |
---|
| 555 | list EV = eigenvalues(RETR, sigma); |
---|
| 556 | |
---|
| 557 | //--------- (5) Transformation of the input-ideal is done here. ---------------- |
---|
| 558 | map f = basering, TR; |
---|
| 559 | t = timer; |
---|
| 560 | ideal I_trans = f(I); |
---|
| 561 | if(printlevel >= 11) { "Transformation: "+string(timer - t)+" seconds"; } |
---|
| 562 | |
---|
| 563 | //--------- (6) Compute a standard basis of the transformed ideal. ------------- |
---|
| 564 | t = timer; |
---|
| 565 | if(size(#) > 0) { ideal sI_trans = std(I_trans, #[1]); } |
---|
| 566 | else { ideal sI_trans = std(I_trans); } |
---|
| 567 | if(printlevel >= 11) { "1st Groebner basis: "+string(timer - t)+" seconds"; } |
---|
| 568 | |
---|
| 569 | //--------- (7) Retransformation is done here. --------------------------------- |
---|
| 570 | map g = basering, RETR; |
---|
| 571 | t = timer; |
---|
| 572 | ideal I_retrans = g(sI_trans); |
---|
| 573 | if(printlevel >= 11) { "Reverse Transformation: "+string(timer - t) |
---|
| 574 | +" seconds"; } |
---|
| 575 | |
---|
| 576 | //--------- (8) Compute a standard basis of the retransformaed ideal ----------- |
---|
| 577 | //--------- which is then a standard basis of the input-ideal. ----------- |
---|
| 578 | t = timer; |
---|
| 579 | ideal sI_retrans = std(I_retrans); |
---|
| 580 | if(printlevel >= 11) { "2nd Groebner basis: "+string(timer - t)+" seconds"; } |
---|
| 581 | |
---|
| 582 | if((CHAR == 0) && (ORD > 2)) |
---|
| 583 | { |
---|
| 584 | setring R; |
---|
| 585 | ideal sI_retrans = fetch(S, sI_retrans); |
---|
| 586 | return(sI_retrans); |
---|
| 587 | } |
---|
| 588 | else |
---|
| 589 | { |
---|
| 590 | return(sI_retrans); |
---|
| 591 | } |
---|
| 592 | } |
---|
| 593 | example |
---|
| 594 | { "EXAMPLE:"; echo = 2; |
---|
| 595 | ring R = 0, x(1..4), dp; |
---|
| 596 | ideal I = cyclic(4); |
---|
| 597 | I; |
---|
| 598 | intvec pi = 2,3,4,1; |
---|
| 599 | ideal sI = symmStd(I,pi); |
---|
| 600 | sI; |
---|
| 601 | |
---|
| 602 | ring S = 31, (x,y,z), dp; |
---|
| 603 | ideal J; |
---|
| 604 | J[1] = xy-y2+xz; |
---|
| 605 | J[2] = xy+yz-z2; |
---|
| 606 | J[3] = -x2+xz+yz; |
---|
| 607 | intvec tau = 3,1,2; |
---|
| 608 | ideal sJ = symmStd(J,tau); |
---|
| 609 | sJ; |
---|
| 610 | } |
---|
| 611 | |
---|
| 612 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 613 | |
---|
| 614 | proc divPrimeTest(def II, bigint p, int k) |
---|
| 615 | { |
---|
| 616 | if((p - 1) mod k != 0) { return(0); } |
---|
| 617 | |
---|
| 618 | if(typeof(II) == "string") |
---|
| 619 | { |
---|
| 620 | execute("ideal I = "+II+";"); |
---|
| 621 | } |
---|
| 622 | else |
---|
| 623 | { |
---|
| 624 | ideal I = II; |
---|
| 625 | } |
---|
| 626 | |
---|
| 627 | int i,j; |
---|
| 628 | poly f; |
---|
| 629 | number cnt; |
---|
| 630 | for(i = 1; i <= size(I); i++) |
---|
| 631 | { |
---|
| 632 | f = cleardenom(I[i]); |
---|
| 633 | if(f == 0) { return(0); } |
---|
| 634 | cnt = leadcoef(I[i])/leadcoef(f); |
---|
| 635 | if((numerator(cnt) mod p) == 0) { return(0); } |
---|
| 636 | if((denominator(cnt) mod p) == 0) { return(0); } |
---|
| 637 | for(j = size(f); j > 0; j--) |
---|
| 638 | { |
---|
| 639 | if((leadcoef(f[j]) mod p) == 0) { return(0); } |
---|
| 640 | } |
---|
| 641 | } |
---|
| 642 | return(1); |
---|
| 643 | } |
---|
| 644 | |
---|
| 645 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 646 | |
---|
| 647 | proc divPrimeList(int k, ideal I, int n, list #) |
---|
| 648 | { |
---|
| 649 | // the intvec of n greatest primes p <= 2147483647 (resp. n greatest primes |
---|
| 650 | // < L[size(L)] union with L) such that each (p-1) is divisible by k, and none |
---|
| 651 | // of these primes divides any coefficient occuring in I |
---|
| 652 | // --> similar to procedure primeList in modstd.lib |
---|
| 653 | |
---|
| 654 | intvec L; |
---|
| 655 | int i,p; |
---|
| 656 | int ncores = 1; |
---|
| 657 | |
---|
| 658 | //----------------- Initialize optional parameter ncores --------------------- |
---|
| 659 | if(size(#) > 0) |
---|
| 660 | { |
---|
| 661 | if(size(#) == 1) |
---|
| 662 | { |
---|
| 663 | if(typeof(#[1]) == "int") |
---|
| 664 | { |
---|
| 665 | ncores = #[1]; |
---|
| 666 | # = list(); |
---|
| 667 | } |
---|
| 668 | } |
---|
| 669 | else |
---|
| 670 | { |
---|
| 671 | ncores = #[2]; |
---|
| 672 | } |
---|
| 673 | } |
---|
| 674 | |
---|
| 675 | if(size(#) == 0) |
---|
| 676 | { |
---|
| 677 | p = 2147483647; |
---|
| 678 | while(!divPrimeTest(I,p,k)) |
---|
| 679 | { |
---|
| 680 | p = prime(p-1); |
---|
| 681 | if(p == 2) { ERROR("No more primes."); } |
---|
| 682 | } |
---|
| 683 | L[1] = p; |
---|
| 684 | } |
---|
| 685 | else |
---|
| 686 | { |
---|
| 687 | L = #[1]; |
---|
| 688 | p = prime(L[size(L)]-1); |
---|
| 689 | while(!divPrimeTest(I,p,k)) |
---|
| 690 | { |
---|
| 691 | p = prime(p-1); |
---|
| 692 | if(p == 2) { ERROR("No more primes."); } |
---|
| 693 | } |
---|
| 694 | L[size(L)+1] = p; |
---|
| 695 | } |
---|
| 696 | if(p == 2) { ERROR("No more primes."); } |
---|
| 697 | if(ncores == 1) |
---|
| 698 | { |
---|
| 699 | for(i = 2; i <= n; i++) |
---|
| 700 | { |
---|
| 701 | p = prime(p-1); |
---|
| 702 | while(!divPrimeTest(I,p,k)) |
---|
| 703 | { |
---|
| 704 | p = prime(p-1); |
---|
| 705 | if(p == 2) { ERROR("no more primes"); } |
---|
| 706 | } |
---|
| 707 | L[size(L)+1] = p; |
---|
| 708 | } |
---|
| 709 | } |
---|
| 710 | else |
---|
| 711 | { |
---|
| 712 | int neededSize = size(L)+n-1;; |
---|
| 713 | list parallelResults; |
---|
| 714 | list arguments; |
---|
| 715 | int neededPrimes = neededSize-size(L); |
---|
| 716 | while(neededPrimes > 0) |
---|
| 717 | { |
---|
| 718 | arguments = list(); |
---|
| 719 | for(i = ((neededPrimes div ncores)+1-(neededPrimes%ncores == 0)) |
---|
| 720 | *ncores; i > 0; i--) |
---|
| 721 | { |
---|
| 722 | p = prime(p-1); |
---|
| 723 | if(p == 2) { ERROR("no more primes"); } |
---|
| 724 | arguments[i] = list("I", p, k); |
---|
| 725 | } |
---|
[4fb2ef5] | 726 | parallelResults = parallelWaitAll("divPrimeTest", arguments, 0, |
---|
| 727 | ncores); |
---|
[d9adbe] | 728 | for(i = size(arguments); i > 0; i--) |
---|
| 729 | { |
---|
| 730 | if(parallelResults[i]) |
---|
| 731 | { |
---|
| 732 | L[size(L)+1] = arguments[i][2]; |
---|
| 733 | } |
---|
| 734 | } |
---|
| 735 | neededPrimes = neededSize-size(L); |
---|
| 736 | } |
---|
| 737 | if(size(L) > neededSize) |
---|
| 738 | { |
---|
| 739 | L = L[1..neededSize]; |
---|
| 740 | } |
---|
| 741 | } |
---|
| 742 | return(L); |
---|
| 743 | } |
---|
| 744 | example |
---|
| 745 | { "EXAMPLE:"; echo = 2; |
---|
| 746 | ring r = 0,(x,y,z),dp; |
---|
| 747 | ideal I = 2147483647x+y, z-181; |
---|
| 748 | intvec L = divPrimeList(4,I,10,10); |
---|
| 749 | size(L); |
---|
| 750 | L[1]; |
---|
| 751 | L[size(L)]; |
---|
| 752 | L = divPrimeList(4,I,5,L,5); |
---|
| 753 | size(L); |
---|
| 754 | L[size(L)]; |
---|
| 755 | } |
---|
| 756 | |
---|
| 757 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 758 | |
---|
| 759 | proc spTestSB(ideal I, ideal J, list L, intvec sigma, int variant, list #) |
---|
| 760 | "USAGE: spTestSB(I,J,L,sigma,variant,#); I,J ideals, L intvec of primes, |
---|
| 761 | sigma intvec, variant integer |
---|
| 762 | RETURN: 1 (resp. 0) if for a randomly chosen prime p, that is not in L and |
---|
| 763 | divisible by the order of sigma, J mod p is (resp. is not) a standard |
---|
| 764 | basis of I mod p |
---|
| 765 | EXAMPLE: example spTestSB; shows an example |
---|
| 766 | " |
---|
| 767 | { |
---|
| 768 | int i,j,k,p; |
---|
| 769 | int ORD = order(sigma); |
---|
| 770 | def R = basering; |
---|
| 771 | list r = ringlist(R); |
---|
| 772 | |
---|
| 773 | while(!j) |
---|
| 774 | { |
---|
| 775 | j = 1; |
---|
| 776 | while(((p - 1) mod ORD) != 0) |
---|
| 777 | { |
---|
| 778 | p = prime(random(1000000000,2134567879)); |
---|
| 779 | if(p == 2){ ERROR("no more primes"); } |
---|
| 780 | } |
---|
| 781 | for(i = 1; i <= size(L); i++) |
---|
| 782 | { |
---|
| 783 | if(p == L[i]){ j = 0; break } |
---|
| 784 | } |
---|
| 785 | if(j) |
---|
| 786 | { |
---|
| 787 | for(i = 1; i <= ncols(I); i++) |
---|
| 788 | { |
---|
| 789 | for(k = 2; k <= size(I[i]); k++) |
---|
| 790 | { |
---|
| 791 | if((denominator(leadcoef(I[i][k])) mod p) == 0){ j = 0; break; } |
---|
| 792 | } |
---|
| 793 | if(!j){ break; } |
---|
| 794 | } |
---|
| 795 | } |
---|
| 796 | if(j) |
---|
| 797 | { |
---|
| 798 | if(!primeTest(I,p)) { j = 0; } |
---|
| 799 | } |
---|
| 800 | } |
---|
| 801 | r[1] = p; |
---|
| 802 | def @R = ring(r); |
---|
| 803 | setring @R; |
---|
| 804 | ideal I = imap(R,I); |
---|
| 805 | ideal J = imap(R,J); |
---|
| 806 | attrib(J,"isSB",1); |
---|
| 807 | |
---|
| 808 | int t = timer; |
---|
| 809 | j = 1; |
---|
| 810 | if(isIncluded(I,J) == 0){ j = 0; } |
---|
| 811 | |
---|
| 812 | if(printlevel >= 11) |
---|
| 813 | { |
---|
| 814 | "isIncluded(I,J) takes "+string(timer - t)+" seconds"; |
---|
| 815 | "j = "+string(j); |
---|
| 816 | } |
---|
| 817 | |
---|
| 818 | t = timer; |
---|
| 819 | if(j) |
---|
| 820 | { |
---|
| 821 | if(size(#) > 0) |
---|
| 822 | { |
---|
| 823 | ideal K = smpStd(I,sigma,p,variant,#[1])[1]; |
---|
| 824 | } |
---|
| 825 | else |
---|
| 826 | { |
---|
| 827 | ideal K = smpStd(I,sigma,p,variant)[1]; |
---|
| 828 | } |
---|
| 829 | t = timer; |
---|
| 830 | if(isIncluded(J,K) == 0){ j = 0; } |
---|
| 831 | |
---|
| 832 | if(printlevel >= 11) |
---|
| 833 | { |
---|
| 834 | "isIncluded(J,K) takes "+string(timer - t)+" seconds"; |
---|
| 835 | "j = "+string(j); |
---|
| 836 | } |
---|
| 837 | } |
---|
| 838 | setring R; |
---|
| 839 | return(j); |
---|
| 840 | } |
---|
| 841 | example |
---|
| 842 | { "EXAMPLE:"; echo = 2; |
---|
| 843 | intvec L = 2,3,5; |
---|
| 844 | ring r = 0,(x,y,z),dp; |
---|
| 845 | ideal I = x+1,y+1; |
---|
| 846 | intvec sigma = 2,1,3; |
---|
| 847 | ideal J = x+1,y; |
---|
| 848 | spTestSB(I,J,L,sigma,2); |
---|
| 849 | spTestSB(J,I,L,sigma,2); |
---|
| 850 | } |
---|
| 851 | |
---|
| 852 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 853 | |
---|
| 854 | proc smpStd(ideal I, intvec sigma, int p, int variant, list #) |
---|
| 855 | "USAGE: smpStd(I,sigma,p,#); I ideal, sigma intvec, p integer, variant integer |
---|
| 856 | ASSUME: If size(#) > 0, then #[1] is an intvec describing the Hilbert series. |
---|
| 857 | RETURN: ideal - a standard basis of I mod p, integer - p |
---|
| 858 | NOTE: The procedure computes a standard basis of the ideal I modulo p and |
---|
| 859 | fetches the result to the basering. If size(#) > 0 the Hilbert driven |
---|
| 860 | standard basis computation symmStd(.,.,#[1]) is used in symmStd. |
---|
| 861 | The standard basis computation modulo p does also vary depending on the |
---|
| 862 | integer variant, namely |
---|
| 863 | @* - variant = 1: symmStd(.,.,#[1]) resp. symmStd, |
---|
| 864 | @* - variant = 2: symmStd, |
---|
| 865 | @* - variant = 3: homog. - symmStd(.,.,#[1]) resp. symmStd - dehomog., |
---|
| 866 | @* - variant = 4: fglm. |
---|
| 867 | EXAMPLE: example smpStd; shows an example |
---|
| 868 | " |
---|
| 869 | { |
---|
| 870 | def R0 = basering; |
---|
| 871 | list rl = ringlist(R0); |
---|
| 872 | rl[1] = p; |
---|
| 873 | def @r = ring(rl); |
---|
| 874 | setring @r; |
---|
| 875 | ideal i = fetch(R0,I); |
---|
| 876 | |
---|
| 877 | option(redSB); |
---|
| 878 | |
---|
| 879 | if(variant == 1) |
---|
| 880 | { |
---|
| 881 | if(size(#) > 0) |
---|
| 882 | { |
---|
| 883 | i = symmStd(i, sigma, #[1]); |
---|
| 884 | } |
---|
| 885 | else |
---|
| 886 | { |
---|
| 887 | i = symmStd(i, sigma); |
---|
| 888 | } |
---|
| 889 | } |
---|
| 890 | |
---|
| 891 | if(variant == 2) |
---|
| 892 | { |
---|
| 893 | i = symmStd(i, sigma); |
---|
| 894 | } |
---|
| 895 | |
---|
| 896 | if(variant == 3) |
---|
| 897 | { |
---|
| 898 | list rl = ringlist(@r); |
---|
| 899 | int nvar@r = nvars(@r); |
---|
| 900 | |
---|
| 901 | int k; |
---|
| 902 | intvec w; |
---|
| 903 | for(k = 1; k <= nvar@r; k++) |
---|
| 904 | { |
---|
| 905 | w[k] = deg(var(k)); |
---|
| 906 | } |
---|
| 907 | w[nvar@r + 1] = 1; |
---|
| 908 | |
---|
| 909 | rl[2][nvar@r + 1] = "homvar"; |
---|
| 910 | rl[3][2][2] = w; |
---|
| 911 | |
---|
| 912 | def HomR = ring(rl); |
---|
| 913 | setring HomR; |
---|
| 914 | ideal i = imap(@r, i); |
---|
| 915 | i = homog(i, homvar); |
---|
| 916 | intvec tau = sigma, size(sigma)+1; |
---|
| 917 | |
---|
| 918 | if(size(#) > 0) |
---|
| 919 | { |
---|
| 920 | if(w == 1) |
---|
| 921 | { |
---|
| 922 | i = symmStd(i, tau, #[1]); |
---|
| 923 | } |
---|
| 924 | else |
---|
| 925 | { |
---|
| 926 | i = symmStd(i, tau, #[1], w); |
---|
| 927 | } |
---|
| 928 | } |
---|
| 929 | else |
---|
| 930 | { |
---|
| 931 | i = symmStd(i, tau); |
---|
| 932 | } |
---|
| 933 | |
---|
| 934 | i = subst(i, homvar, 1); |
---|
| 935 | i = simplify(i, 34); |
---|
| 936 | |
---|
| 937 | setring @r; |
---|
| 938 | i = imap(HomR, i); |
---|
| 939 | i = interred(i); |
---|
| 940 | kill HomR; |
---|
| 941 | } |
---|
| 942 | |
---|
| 943 | if(variant == 4) |
---|
| 944 | { |
---|
| 945 | def R1 = changeord(list(list("dp",1:nvars(basering)))); |
---|
| 946 | setring R1; |
---|
| 947 | ideal i = fetch(@r,i); |
---|
| 948 | i = symmStd(i, sigma); |
---|
| 949 | setring @r; |
---|
| 950 | i = fglm(R1,i); |
---|
| 951 | } |
---|
| 952 | |
---|
| 953 | setring R0; |
---|
| 954 | return(list(fetch(@r,i),p)); |
---|
| 955 | } |
---|
| 956 | example |
---|
| 957 | { "EXAMPLE:"; echo = 2; |
---|
| 958 | ring r1 = 0, x(1..4), dp; |
---|
| 959 | ideal I = cyclic(4); |
---|
| 960 | intvec sigma = 2,3,4,1; |
---|
| 961 | int p = 181; |
---|
| 962 | list P = smpStd(I,sigma,p,2); |
---|
| 963 | P; |
---|
| 964 | |
---|
| 965 | ring r2 = 0, x(1..5), lp; |
---|
| 966 | ideal I = cyclic(5); |
---|
| 967 | intvec tau = 2,3,4,5,1; |
---|
| 968 | int q = 31981; |
---|
| 969 | list Q = smpStd(I,tau,q,4); |
---|
| 970 | Q; |
---|
| 971 | } |
---|
| 972 | |
---|
| 973 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 974 | |
---|
| 975 | proc syModStd(ideal I, intvec sigma, list #) |
---|
| 976 | "USAGE: syModStd(I,sigma); I ideal, sigma intvec |
---|
| 977 | ASSUME: size(sigma) = nvars(basering) and sigma(I) = I. If size(#) > 0, then |
---|
| 978 | # contains either 1, 2 or 4 integers such that |
---|
| 979 | @* - #[1] is the number of available processors for the computation, |
---|
| 980 | @* - #[2] is an optional parameter for the exactness of the computation, |
---|
| 981 | if #[2] = 1, the procedure computes a standard basis for sure, |
---|
| 982 | @* - #[3] is the number of primes until the first lifting, |
---|
| 983 | @* - #[4] is the constant number of primes between two liftings until |
---|
| 984 | the computation stops. |
---|
| 985 | RETURN: ideal, a standard basis of I if no warning appears; |
---|
| 986 | NOTE: The procedure computes a standard basis of the ideal I (over the |
---|
| 987 | rational numbers) by using modular methods and the fact that I is |
---|
| 988 | invariant under the variable permutation sigma. |
---|
| 989 | By default the procedure computes a standard basis of I for sure, but |
---|
| 990 | if the optional parameter #[2] = 0, it computes a standard basis of I |
---|
| 991 | with high probability. |
---|
| 992 | The procedure distinguishes between different variants for the standard |
---|
| 993 | basis computation in positive characteristic depending on the ordering |
---|
| 994 | of the basering, the parameter #[2] and if the ideal I is homogeneous. |
---|
| 995 | @* - variant = 1, if I is homogeneous, |
---|
| 996 | @* - variant = 2, if I is not homogeneous, 1-block-ordering, |
---|
| 997 | @* - variant = 3, if I is not homogeneous, complicated ordering (lp or |
---|
| 998 | > 1 block), |
---|
| 999 | @* - variant = 4, if I is not homogeneous, ordering lp, dim(I) = 0. |
---|
| 1000 | EXAMPLE: example syModStd; shows an example |
---|
| 1001 | " |
---|
| 1002 | { |
---|
| 1003 | if((nvars(basering) != size(sigma)) || (!isSymmetric(I,sigma))) |
---|
| 1004 | { |
---|
| 1005 | ERROR("The input is no permutation of the ring-variables!!"); |
---|
| 1006 | } |
---|
| 1007 | |
---|
| 1008 | int TT = timer; |
---|
| 1009 | int RT = rtimer; |
---|
| 1010 | |
---|
| 1011 | def R0 = basering; |
---|
| 1012 | list rl = ringlist(R0); |
---|
| 1013 | if((npars(R0) > 0) || (rl[1] > 0)) |
---|
| 1014 | { |
---|
| 1015 | ERROR("Characteristic of basering should be zero, basering should |
---|
| 1016 | have no parameters."); |
---|
| 1017 | } |
---|
| 1018 | |
---|
| 1019 | int index = 1; |
---|
| 1020 | int i,k,c; |
---|
| 1021 | int j = 1; |
---|
| 1022 | int pTest, sizeTest; |
---|
| 1023 | int en = 2134567879; |
---|
| 1024 | int an = 1000000000; |
---|
| 1025 | bigint N; |
---|
| 1026 | int ORD = order(sigma); |
---|
| 1027 | |
---|
| 1028 | //-------------------- Initialize optional parameters ------------------------ |
---|
| 1029 | if(size(#) > 0) |
---|
| 1030 | { |
---|
| 1031 | if(size(#) == 1) |
---|
| 1032 | { |
---|
| 1033 | int n1 = #[1]; |
---|
| 1034 | int exactness = 1; |
---|
| 1035 | if(n1 >= 10) |
---|
| 1036 | { |
---|
| 1037 | int n2 = n1 + 1; |
---|
| 1038 | int n3 = n1; |
---|
| 1039 | } |
---|
| 1040 | else |
---|
| 1041 | { |
---|
| 1042 | int n2 = 10; |
---|
| 1043 | int n3 = 10; |
---|
| 1044 | } |
---|
| 1045 | } |
---|
| 1046 | if(size(#) == 2) |
---|
| 1047 | { |
---|
| 1048 | int n1 = #[1]; |
---|
| 1049 | int exactness = #[2]; |
---|
| 1050 | if(n1 >= 10) |
---|
| 1051 | { |
---|
| 1052 | int n2 = n1 + 1; |
---|
| 1053 | int n3 = n1; |
---|
| 1054 | } |
---|
| 1055 | else |
---|
| 1056 | { |
---|
| 1057 | int n2 = 10; |
---|
| 1058 | int n3 = 10; |
---|
| 1059 | } |
---|
| 1060 | } |
---|
| 1061 | if(size(#) == 4) |
---|
| 1062 | { |
---|
| 1063 | int n1 = #[1]; |
---|
| 1064 | int exactness = #[2]; |
---|
| 1065 | if(n1 >= #[3]) |
---|
| 1066 | { |
---|
| 1067 | int n2 = n1 + 1; |
---|
| 1068 | } |
---|
| 1069 | else |
---|
| 1070 | { |
---|
| 1071 | int n2 = #[3]; |
---|
| 1072 | } |
---|
| 1073 | if(n1 >= #[4]) |
---|
| 1074 | { |
---|
| 1075 | int n3 = n1; |
---|
| 1076 | } |
---|
| 1077 | else |
---|
| 1078 | { |
---|
| 1079 | int n3 = #[4]; |
---|
| 1080 | } |
---|
| 1081 | } |
---|
| 1082 | } |
---|
| 1083 | else |
---|
| 1084 | { |
---|
| 1085 | int n1 = 1; |
---|
| 1086 | int exactness = 1; |
---|
| 1087 | int n2 = 10; |
---|
| 1088 | int n3 = 10; |
---|
| 1089 | } |
---|
| 1090 | |
---|
| 1091 | if(printlevel >= 10) |
---|
| 1092 | { |
---|
| 1093 | "n1 = "+string(n1)+", n2 = "+string(n2)+", n3 = "+string(n3) |
---|
| 1094 | +", exactness = "+string(exactness); |
---|
| 1095 | } |
---|
| 1096 | |
---|
| 1097 | //-------------------------- Save current options ------------------------------ |
---|
| 1098 | intvec opt = option(get); |
---|
| 1099 | |
---|
| 1100 | option(redSB); |
---|
| 1101 | |
---|
| 1102 | //-------------------- Initialize the list of primes ------------------------- |
---|
| 1103 | int tt = timer; |
---|
| 1104 | int rt = rtimer; |
---|
| 1105 | intvec L = divPrimeList(ORD,I,n2,n1); |
---|
| 1106 | if(printlevel >= 10) |
---|
| 1107 | { |
---|
| 1108 | "CPU-time for divPrimeList: "+string(timer-tt)+" seconds."; |
---|
| 1109 | "Real-time for divPrimeList: "+string(rtimer-rt)+" seconds."; |
---|
| 1110 | } |
---|
| 1111 | |
---|
| 1112 | //--------------------- Decide which variant to take ------------------------- |
---|
| 1113 | int variant; |
---|
| 1114 | int h = homog(I); |
---|
| 1115 | |
---|
| 1116 | tt = timer; |
---|
| 1117 | rt = rtimer; |
---|
| 1118 | |
---|
| 1119 | if(!mixedTest()) |
---|
| 1120 | { |
---|
| 1121 | if(h) |
---|
| 1122 | { |
---|
| 1123 | variant = 1; |
---|
| 1124 | if(printlevel >= 10) { "variant = 1"; } |
---|
| 1125 | |
---|
| 1126 | rl[1] = L[5]; |
---|
| 1127 | def @r = ring(rl); |
---|
| 1128 | setring @r; |
---|
| 1129 | def @s = changeord(list(list("dp",1:nvars(basering)))); |
---|
| 1130 | setring @s; |
---|
| 1131 | ideal I = std(fetch(R0,I)); |
---|
| 1132 | intvec hi = hilb(I,1); |
---|
| 1133 | setring R0; |
---|
| 1134 | kill @r,@s; |
---|
| 1135 | } |
---|
| 1136 | else |
---|
| 1137 | { |
---|
| 1138 | string ordstr_R0 = ordstr(R0); |
---|
| 1139 | int neg = 1 - attrib(R0,"global"); |
---|
| 1140 | |
---|
| 1141 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
| 1142 | { |
---|
| 1143 | variant = 2; |
---|
| 1144 | if(printlevel >= 10) { "variant = 2"; } |
---|
| 1145 | } |
---|
| 1146 | else |
---|
| 1147 | { |
---|
| 1148 | string order; |
---|
| 1149 | if(system("nblocks") <= 2) |
---|
| 1150 | { |
---|
| 1151 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
| 1152 | + find(ordstr_R0, "rp") <= 0) |
---|
| 1153 | { |
---|
| 1154 | order = "simple"; |
---|
| 1155 | } |
---|
| 1156 | } |
---|
| 1157 | |
---|
| 1158 | if((order == "simple") || (size(rl) > 4)) |
---|
| 1159 | { |
---|
| 1160 | variant = 2; |
---|
| 1161 | if(printlevel >= 10) { "variant = 2"; } |
---|
| 1162 | } |
---|
| 1163 | else |
---|
| 1164 | { |
---|
| 1165 | rl[1] = L[5]; |
---|
| 1166 | def @r = ring(rl); |
---|
| 1167 | setring @r; |
---|
| 1168 | |
---|
| 1169 | def @s = changeord(list(list("dp",1:nvars(basering)))); |
---|
| 1170 | setring @s; |
---|
| 1171 | ideal I = std(fetch(R0,I)); |
---|
| 1172 | if(dim(I) == 0) |
---|
| 1173 | { |
---|
| 1174 | variant = 4; |
---|
| 1175 | if(printlevel >= 10) { "variant = 4"; } |
---|
| 1176 | } |
---|
| 1177 | else |
---|
| 1178 | { |
---|
| 1179 | variant = 3; |
---|
| 1180 | if(printlevel >= 10) { "variant = 3"; } |
---|
| 1181 | |
---|
| 1182 | int nvar@r = nvars(@r); |
---|
| 1183 | intvec w; |
---|
| 1184 | for(i = 1; i <= nvar@r; i++) |
---|
| 1185 | { |
---|
| 1186 | w[i] = deg(var(i)); |
---|
| 1187 | } |
---|
| 1188 | w[nvar@r + 1] = 1; |
---|
| 1189 | |
---|
| 1190 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
| 1191 | intvec W = hiRi[2]; |
---|
| 1192 | @s = hiRi[1]; |
---|
| 1193 | setring @s; |
---|
| 1194 | intvec tau = sigma, size(sigma)+1; |
---|
| 1195 | |
---|
| 1196 | Id(1) = symmStd(Id(1),tau); |
---|
| 1197 | intvec hi = hilb(Id(1), 1, W); |
---|
| 1198 | } |
---|
| 1199 | |
---|
| 1200 | setring R0; |
---|
| 1201 | kill @r,@s; |
---|
| 1202 | } |
---|
| 1203 | } |
---|
| 1204 | } |
---|
| 1205 | } |
---|
| 1206 | else |
---|
| 1207 | { |
---|
| 1208 | if(exactness == 1) { return(groebner(I)); } |
---|
| 1209 | if(h) |
---|
| 1210 | { |
---|
| 1211 | variant = 1; |
---|
| 1212 | if(printlevel >= 10) { "variant = 1"; } |
---|
| 1213 | rl[1] = L[5]; |
---|
| 1214 | def @r = ring(rl); |
---|
| 1215 | setring @r; |
---|
| 1216 | def @s = changeord(list(list("dp",1:nvars(basering)))); |
---|
| 1217 | setring @s; |
---|
| 1218 | ideal I = std(fetch(R0,I)); |
---|
| 1219 | intvec hi = hilb(I,1); |
---|
| 1220 | setring R0; |
---|
| 1221 | kill @r,@s; |
---|
| 1222 | } |
---|
| 1223 | else |
---|
| 1224 | { |
---|
| 1225 | string ordstr_R0 = ordstr(R0); |
---|
| 1226 | int neg = 1 - attrib(R0,"global"); |
---|
| 1227 | |
---|
| 1228 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
| 1229 | { |
---|
| 1230 | variant = 2; |
---|
| 1231 | if(printlevel >= 10) { "variant = 2"; } |
---|
| 1232 | } |
---|
| 1233 | else |
---|
| 1234 | { |
---|
| 1235 | string order; |
---|
| 1236 | if(system("nblocks") <= 2) |
---|
| 1237 | { |
---|
| 1238 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
| 1239 | + find(ordstr_R0, "rp") <= 0) |
---|
| 1240 | { |
---|
| 1241 | order = "simple"; |
---|
| 1242 | } |
---|
| 1243 | } |
---|
| 1244 | |
---|
| 1245 | if((order == "simple") || (size(rl) > 4)) |
---|
| 1246 | { |
---|
| 1247 | variant = 2; |
---|
| 1248 | if(printlevel >= 10) { "variant = 2"; } |
---|
| 1249 | } |
---|
| 1250 | else |
---|
| 1251 | { |
---|
| 1252 | variant = 3; |
---|
| 1253 | if(printlevel >= 10) { "variant = 3"; } |
---|
| 1254 | |
---|
| 1255 | rl[1] = L[5]; |
---|
| 1256 | def @r = ring(rl); |
---|
| 1257 | setring @r; |
---|
| 1258 | int nvar@r = nvars(@r); |
---|
| 1259 | intvec w; |
---|
| 1260 | for(i = 1; i <= nvar@r; i++) |
---|
| 1261 | { |
---|
| 1262 | w[i] = deg(var(i)); |
---|
| 1263 | } |
---|
| 1264 | w[nvar@r + 1] = 1; |
---|
| 1265 | |
---|
| 1266 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
| 1267 | intvec W = hiRi[2]; |
---|
| 1268 | def @s = hiRi[1]; |
---|
| 1269 | setring @s; |
---|
| 1270 | intvec tau = sigma, size(sigma)+1; |
---|
| 1271 | |
---|
| 1272 | Id(1) = symmStd(Id(1),tau); |
---|
| 1273 | intvec hi = hilb(Id(1), 1, W); |
---|
| 1274 | |
---|
| 1275 | setring R0; |
---|
| 1276 | kill @r,@s; |
---|
| 1277 | } |
---|
| 1278 | } |
---|
| 1279 | } |
---|
| 1280 | } |
---|
| 1281 | |
---|
| 1282 | list P,T1,T2,T3,LL; |
---|
| 1283 | |
---|
| 1284 | ideal J,K,H; |
---|
| 1285 | |
---|
| 1286 | //----- If there is more than one processor available, we parallelize the ---- |
---|
| 1287 | //----- main standard basis computations in positive characteristic ---- |
---|
| 1288 | |
---|
| 1289 | if(n1 > 1) |
---|
| 1290 | { |
---|
| 1291 | ideal I_for_fork = I; |
---|
| 1292 | export(I_for_fork); // I available for each link |
---|
| 1293 | |
---|
| 1294 | //----- Create n1 links l(1),...,l(n1), open all of them and compute --------- |
---|
| 1295 | //----- standard basis for the primes L[2],...,L[n1 + 1]. --------- |
---|
| 1296 | |
---|
| 1297 | for(i = 1; i <= n1; i++) |
---|
| 1298 | { |
---|
| 1299 | //link l(i) = "MPtcp:fork"; |
---|
| 1300 | link l(i) = "ssi:fork"; |
---|
| 1301 | open(l(i)); |
---|
| 1302 | if((variant == 1) || (variant == 3)) |
---|
| 1303 | { |
---|
| 1304 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), eval(L[i + 1]), |
---|
| 1305 | eval(variant), eval(hi)))); |
---|
| 1306 | } |
---|
| 1307 | if((variant == 2) || (variant == 4)) |
---|
| 1308 | { |
---|
| 1309 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), eval(L[i + 1]), |
---|
| 1310 | eval(variant)))); |
---|
| 1311 | } |
---|
| 1312 | } |
---|
| 1313 | |
---|
| 1314 | int t = timer; |
---|
| 1315 | if((variant == 1) || (variant == 3)) |
---|
| 1316 | { |
---|
| 1317 | P = smpStd(I_for_fork, sigma, L[1], variant, hi); |
---|
| 1318 | } |
---|
| 1319 | if((variant == 2) || (variant == 4)) |
---|
| 1320 | { |
---|
| 1321 | P = smpStd(I_for_fork, sigma, L[1], variant); |
---|
| 1322 | } |
---|
| 1323 | t = timer - t; |
---|
| 1324 | if(t > 60) { t = 60; } |
---|
| 1325 | int i_sleep = system("sh", "sleep "+string(t)); |
---|
| 1326 | T1[1] = P[1]; |
---|
| 1327 | T2[1] = bigint(P[2]); |
---|
| 1328 | index++; |
---|
| 1329 | |
---|
| 1330 | j = j + n1 + 1; |
---|
| 1331 | } |
---|
| 1332 | |
---|
| 1333 | //-------------- Main standard basis computations in positive ---------------- |
---|
| 1334 | //---------------------- characteristic start here --------------------------- |
---|
| 1335 | |
---|
| 1336 | list arguments_farey, results_farey; |
---|
| 1337 | |
---|
| 1338 | while(1) |
---|
| 1339 | { |
---|
| 1340 | tt = timer; rt = rtimer; |
---|
| 1341 | |
---|
| 1342 | if(printlevel >= 10) { "size(L) = "+string(size(L)); } |
---|
| 1343 | |
---|
| 1344 | if(n1 > 1) |
---|
| 1345 | { |
---|
| 1346 | while(j <= size(L) + 1) |
---|
| 1347 | { |
---|
| 1348 | for(i = 1; i <= n1; i++) |
---|
| 1349 | { |
---|
| 1350 | //--- ask if link l(i) is ready otherwise sleep for t seconds --- |
---|
| 1351 | if(status(l(i), "read", "ready")) |
---|
| 1352 | { |
---|
| 1353 | //--- read the result from l(i) --- |
---|
| 1354 | P = read(l(i)); |
---|
| 1355 | T1[index] = P[1]; |
---|
| 1356 | T2[index] = bigint(P[2]); |
---|
| 1357 | index++; |
---|
| 1358 | |
---|
| 1359 | if(j <= size(L)) |
---|
| 1360 | { |
---|
| 1361 | if((variant == 1) || (variant == 3)) |
---|
| 1362 | { |
---|
| 1363 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), |
---|
| 1364 | eval(L[j]), eval(variant), eval(hi)))); |
---|
| 1365 | j++; |
---|
| 1366 | } |
---|
| 1367 | if((variant == 2) || (variant == 4)) |
---|
| 1368 | { |
---|
| 1369 | write(l(i), quote(smpStd(I_for_fork, eval(sigma), |
---|
| 1370 | eval(L[j]), eval(variant)))); |
---|
| 1371 | j++; |
---|
| 1372 | } |
---|
| 1373 | } |
---|
| 1374 | else |
---|
| 1375 | { |
---|
| 1376 | k++; |
---|
| 1377 | close(l(i)); |
---|
| 1378 | } |
---|
| 1379 | } |
---|
| 1380 | } |
---|
| 1381 | //--- k describes the number of closed links --- |
---|
| 1382 | if(k == n1) |
---|
| 1383 | { |
---|
| 1384 | j++; |
---|
| 1385 | } |
---|
| 1386 | i_sleep = system("sh", "sleep "+string(t)); |
---|
| 1387 | } |
---|
| 1388 | } |
---|
| 1389 | else |
---|
| 1390 | { |
---|
| 1391 | while(j <= size(L)) |
---|
| 1392 | { |
---|
| 1393 | if((variant == 1) || (variant == 3)) |
---|
| 1394 | { |
---|
| 1395 | P = smpStd(I, sigma, L[j], variant, hi); |
---|
| 1396 | } |
---|
| 1397 | if((variant == 2) || (variant == 4)) |
---|
| 1398 | { |
---|
| 1399 | P = smpStd(I, sigma, L[j], variant); |
---|
| 1400 | } |
---|
| 1401 | |
---|
| 1402 | T1[index] = P[1]; |
---|
| 1403 | T2[index] = bigint(P[2]); |
---|
| 1404 | index++; |
---|
| 1405 | j++; |
---|
| 1406 | } |
---|
| 1407 | } |
---|
| 1408 | |
---|
| 1409 | if(printlevel >= 10) |
---|
| 1410 | { |
---|
| 1411 | "CPU-time for computing list is "+string(timer - tt)+" seconds."; |
---|
| 1412 | "Real-time for computing list is "+string(rtimer - rt)+" seconds."; |
---|
| 1413 | } |
---|
| 1414 | |
---|
| 1415 | //------------------------ Delete unlucky primes ----------------------------- |
---|
| 1416 | //------------- unlucky if and only if the leading ideal is wrong ------------ |
---|
| 1417 | |
---|
| 1418 | LL = deleteUnluckyPrimes(T1,T2,h); |
---|
| 1419 | T1 = LL[1]; |
---|
| 1420 | T2 = LL[2]; |
---|
| 1421 | |
---|
| 1422 | //------------------- Now all leading ideals are the same -------------------- |
---|
| 1423 | //------------------- Lift results to basering via farey --------------------- |
---|
| 1424 | |
---|
| 1425 | tt = timer; rt = rtimer; |
---|
| 1426 | N = T2[1]; |
---|
| 1427 | for(i = 2; i <= size(T2); i++) { N = N*T2[i]; } |
---|
| 1428 | H = chinrem(T1,T2); |
---|
| 1429 | if(n1 == 1) |
---|
| 1430 | { |
---|
| 1431 | J = farey(H,N); |
---|
| 1432 | } |
---|
| 1433 | else |
---|
| 1434 | { |
---|
| 1435 | for(i = ncols(H); i > 0; i--) |
---|
| 1436 | { |
---|
| 1437 | arguments_farey[i] = list(ideal(H[i]), N); |
---|
| 1438 | } |
---|
[4fb2ef5] | 1439 | results_farey = parallelWaitAll("farey", arguments_farey, 0, n1); |
---|
[d9adbe] | 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 | } |
---|