[d68d30] | 1 | //////////////////////////////////////////////////////////////////////////////// |
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[7f30e2] | 2 | version="$Id: modstd.lib 14375 2011-08-23 09:29:47Z steidel $"; |
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[d68d30] | 3 | category = "Commutative Algebra"; |
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| 4 | info=" |
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[4cc3cac] | 5 | LIBRARY: modstd.lib Groebner basis of ideals |
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[d68d30] | 6 | |
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| 7 | AUTHORS: A. Hashemi Amir.Hashemi@lip6.fr |
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| 8 | @* G. Pfister pfister@mathematik.uni-kl.de |
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| 9 | @* H. Schoenemann hannes@mathematik.uni-kl.de |
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[7f30e2] | 10 | @* A. Steenpass steenpass@mathematik.uni-kl.de |
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[d68d30] | 11 | @* S. Steidel steidel@mathematik.uni-kl.de |
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| 12 | |
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| 13 | OVERVIEW: |
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| 14 | |
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| 15 | A library for computing the Groebner basis of an ideal in the polynomial |
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| 16 | ring over the rational numbers using modular methods. The procedures are |
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| 17 | inspired by the following paper: |
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| 18 | Elizabeth A. Arnold: Modular algorithms for computing Groebner bases. |
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| 19 | Journal of Symbolic Computation 35, 403-419 (2003). |
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| 20 | |
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| 21 | PROCEDURES: |
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| 22 | modStd(I); standard basis of I using modular methods (chinese remainder) |
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| 23 | modS(I,L); liftings to Q of standard bases of I mod p for p in L |
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[4cc3cac] | 24 | modHenselStd(I); standard basis of I using modular methods (hensel lifting) |
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[d68d30] | 25 | "; |
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| 26 | |
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| 27 | LIB "poly.lib"; |
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| 28 | LIB "ring.lib"; |
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[7f30e2] | 29 | LIB "parallel.lib"; |
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[d68d30] | 30 | |
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| 31 | //////////////////////////////////////////////////////////////////////////////// |
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| 32 | |
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[0c40ae] | 33 | proc mixedTest() |
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| 34 | "USAGE: mixedTest(); |
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| 35 | RETURN: 1 if ordering of basering is mixed, 0 else |
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| 36 | EXAMPLE: example mixedTest(); shows an example |
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| 37 | " |
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[4cc3cac] | 38 | { |
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| 39 | int i,p,m; |
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| 40 | for(i = 1; i <= nvars(basering); i++) |
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| 41 | { |
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| 42 | if(var(i) > 1) |
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| 43 | { |
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| 44 | p++; |
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| 45 | } |
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| 46 | else |
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| 47 | { |
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| 48 | m++; |
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| 49 | } |
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| 50 | } |
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| 51 | if((p > 0) && (m > 0)) { return(1); } |
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| 52 | return(0); |
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| 53 | } |
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[0c40ae] | 54 | example |
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| 55 | { "EXAMPLE:"; echo = 2; |
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| 56 | ring R1 = 0,(x,y,z),dp; |
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| 57 | mixedTest(); |
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| 58 | ring R2 = 31,(x(1..4),y(1..3)),(ds(4),lp(3)); |
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| 59 | mixedTest(); |
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| 60 | ring R3 = 181,x(1..9),(dp(5),lp(4)); |
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| 61 | mixedTest(); |
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| 62 | } |
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[4cc3cac] | 63 | |
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| 64 | //////////////////////////////////////////////////////////////////////////////// |
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| 65 | |
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[d68d30] | 66 | static proc redFork(ideal I, ideal J, int n) |
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| 67 | { |
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| 68 | attrib(J,"isSB",1); |
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| 69 | return(reduce(I,J,1)); |
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| 70 | } |
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| 71 | |
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| 72 | //////////////////////////////////////////////////////////////////////////////// |
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| 73 | |
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| 74 | proc isIncluded(ideal I, ideal J, list #) |
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| 75 | "USAGE: isIncluded(I,J); I,J ideals |
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| 76 | RETURN: 1 if J includes I, |
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[0c40ae] | 77 | @* 0 if there is an element f in I which does not reduce to 0 w.r.t. J. |
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[d68d30] | 78 | EXAMPLE: example isIncluded; shows an example |
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| 79 | " |
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| 80 | { |
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[4cc3cac] | 81 | def R = basering; |
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| 82 | setring R; |
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[27d4bb] | 83 | |
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[d68d30] | 84 | attrib(J,"isSB",1); |
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| 85 | int i,j,k; |
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| 86 | |
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| 87 | if(size(#) > 0) |
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| 88 | { |
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| 89 | int n = #[1]; |
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[4cc3cac] | 90 | if(n >= ncols(I)) { n = ncols(I); } |
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[c512d9] | 91 | if(n > 1) |
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[d68d30] | 92 | { |
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| 93 | for(i = 1; i <= n - 1; i++) |
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| 94 | { |
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[c512d9] | 95 | //link l(i) = "MPtcp:fork"; |
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| 96 | link l(i) = "ssi:fork"; |
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[d68d30] | 97 | open(l(i)); |
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[27d4bb] | 98 | |
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[d68d30] | 99 | write(l(i), quote(redFork(eval(I[ncols(I)-i]), eval(J), 1))); |
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| 100 | } |
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| 101 | |
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| 102 | int t = timer; |
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| 103 | if(reduce(I[ncols(I)], J, 1) != 0) |
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| 104 | { |
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| 105 | for(i = 1; i <= n - 1; i++) |
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| 106 | { |
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| 107 | close(l(i)); |
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| 108 | } |
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| 109 | return(0); |
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| 110 | } |
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| 111 | t = timer - t; |
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| 112 | if(t > 60) { t = 60; } |
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| 113 | int i_sleep = system("sh", "sleep "+string(t)); |
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| 114 | |
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| 115 | j = ncols(I) - n; |
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| 116 | |
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| 117 | while(j >= 0) |
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| 118 | { |
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| 119 | for(i = 1; i <= n - 1; i++) |
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| 120 | { |
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| 121 | if(status(l(i), "read", "ready")) |
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[27d4bb] | 122 | { |
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[d68d30] | 123 | if(read(l(i)) != 0) |
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| 124 | { |
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| 125 | for(i = 1; i <= n - 1; i++) |
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| 126 | { |
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| 127 | close(l(i)); |
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| 128 | } |
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| 129 | return(0); |
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| 130 | } |
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| 131 | else |
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| 132 | { |
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| 133 | if(j >= 1) |
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| 134 | { |
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| 135 | write(l(i), quote(redFork(eval(I[j]), eval(J), 1))); |
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| 136 | j--; |
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| 137 | } |
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| 138 | else |
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| 139 | { |
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| 140 | k++; |
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| 141 | close(l(i)); |
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| 142 | } |
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| 143 | } |
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| 144 | } |
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| 145 | } |
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| 146 | if(k == n - 1) |
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| 147 | { |
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| 148 | j--; |
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| 149 | } |
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| 150 | i_sleep = system("sh", "sleep "+string(t)); |
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| 151 | } |
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| 152 | return(1); |
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| 153 | } |
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| 154 | } |
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| 155 | |
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| 156 | for(i = ncols(I); i >= 1; i--) |
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| 157 | { |
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| 158 | if(reduce(I[i],J,1) != 0){ return(0); } |
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| 159 | } |
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| 160 | return(1); |
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| 161 | } |
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| 162 | example |
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| 163 | { "EXAMPLE:"; echo = 2; |
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| 164 | ring r=0,(x,y,z),dp; |
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| 165 | ideal I = x+1,x+y+1; |
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| 166 | ideal J = x+1,y; |
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| 167 | isIncluded(I,J); |
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| 168 | isIncluded(J,I); |
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| 169 | isIncluded(I,J,4); |
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| 170 | |
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| 171 | ring R = 0, x(1..5), dp; |
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| 172 | ideal I1 = cyclic(4); |
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| 173 | ideal I2 = I1,x(5)^2; |
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| 174 | isIncluded(I1,I2,4); |
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| 175 | } |
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| 176 | |
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| 177 | //////////////////////////////////////////////////////////////////////////////// |
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| 178 | |
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| 179 | proc pTestSB(ideal I, ideal J, list L, int variant, list #) |
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| 180 | "USAGE: pTestSB(I,J,L,variant,#); I,J ideals, L intvec of primes, variant int |
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| 181 | RETURN: 1 (resp. 0) if for a randomly chosen prime p that is not in L |
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| 182 | J mod p is (resp. is not) a standard basis of I mod p |
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| 183 | EXAMPLE: example pTestSB; shows an example |
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| 184 | " |
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| 185 | { |
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| 186 | int i,j,k,p; |
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| 187 | def R = basering; |
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| 188 | list r = ringlist(R); |
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| 189 | |
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| 190 | while(!j) |
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| 191 | { |
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| 192 | j = 1; |
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| 193 | p = prime(random(1000000000,2134567879)); |
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| 194 | for(i = 1; i <= size(L); i++) |
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| 195 | { |
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| 196 | if(p == L[i]) { j = 0; break; } |
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| 197 | } |
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| 198 | if(j) |
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| 199 | { |
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| 200 | for(i = 1; i <= ncols(I); i++) |
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| 201 | { |
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| 202 | for(k = 2; k <= size(I[i]); k++) |
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| 203 | { |
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| 204 | if((denominator(leadcoef(I[i][k])) mod p) == 0) { j = 0; break; } |
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| 205 | } |
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| 206 | if(!j){ break; } |
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| 207 | } |
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| 208 | } |
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| 209 | if(j) |
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| 210 | { |
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| 211 | if(!primeTest(I,p)) { j = 0; } |
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| 212 | } |
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| 213 | } |
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| 214 | r[1] = p; |
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| 215 | def @R = ring(r); |
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| 216 | setring @R; |
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| 217 | ideal I = imap(R,I); |
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| 218 | ideal J = imap(R,J); |
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| 219 | attrib(J,"isSB",1); |
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| 220 | |
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| 221 | int t = timer; |
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| 222 | j = 1; |
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| 223 | if(isIncluded(I,J) == 0) { j = 0; } |
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| 224 | |
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[b4f7eb] | 225 | if(printlevel >= 11) |
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[d68d30] | 226 | { |
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| 227 | "isIncluded(I,J) takes "+string(timer - t)+" seconds"; |
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| 228 | "j = "+string(j); |
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| 229 | } |
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| 230 | |
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| 231 | t = timer; |
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| 232 | if(j) |
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| 233 | { |
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| 234 | if(size(#) > 0) |
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| 235 | { |
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| 236 | ideal K = modpStd(I,p,variant,#[1])[1]; |
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| 237 | } |
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| 238 | else |
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| 239 | { |
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| 240 | ideal K = groebner(I); |
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| 241 | } |
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| 242 | t = timer; |
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| 243 | if(isIncluded(J,K) == 0) { j = 0; } |
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| 244 | |
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[b4f7eb] | 245 | if(printlevel >= 11) |
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[d68d30] | 246 | { |
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[7f30e2] | 247 | "isIncluded(J,K) takes "+string(timer - t)+" seconds"; |
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[d68d30] | 248 | "j = "+string(j); |
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| 249 | } |
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| 250 | } |
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| 251 | setring R; |
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| 252 | return(j); |
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| 253 | } |
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| 254 | example |
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| 255 | { "EXAMPLE:"; echo = 2; |
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| 256 | intvec L = 2,3,5; |
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| 257 | ring r = 0,(x,y,z),dp; |
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| 258 | ideal I = x+1,x+y+1; |
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| 259 | ideal J = x+1,y; |
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| 260 | pTestSB(I,I,L,2); |
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| 261 | pTestSB(I,J,L,2); |
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| 262 | } |
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| 263 | |
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| 264 | //////////////////////////////////////////////////////////////////////////////// |
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| 265 | |
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| 266 | proc deleteUnluckyPrimes(list T, list L, int ho, list #) |
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| 267 | "USAGE: deleteUnluckyPrimes(T,L,ho,#); T/L list of polys/primes, ho integer |
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[27d4bb] | 268 | RETURN: lists T,L(,M),lT with T/L(/M) list of polys/primes(/type of #), |
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[4cc3cac] | 269 | lT ideal |
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[d68d30] | 270 | NOTE: - if ho = 1, the polynomials in T are homogeneous, else ho = 0, |
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[0c40ae] | 271 | @* - lT is prevalent, i.e. the most appearing leading ideal in T |
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[d68d30] | 272 | EXAMPLE: example deleteUnluckyPrimes; shows an example |
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| 273 | " |
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| 274 | { |
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| 275 | ho = ((ho)||(ord_test(basering) == -1)); |
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| 276 | int j,k,c; |
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| 277 | intvec hl,hc; |
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| 278 | ideal cT,lT,cK; |
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| 279 | lT = lead(T[size(T)]); |
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| 280 | attrib(lT,"isSB",1); |
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| 281 | if(!ho) |
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| 282 | { |
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| 283 | for(j = 1; j < size(T); j++) |
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| 284 | { |
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| 285 | cT = lead(T[j]); |
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| 286 | attrib(cT,"isSB",1); |
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| 287 | if((size(reduce(cT,lT))!=0)||(size(reduce(lT,cT))!=0)) |
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| 288 | { |
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| 289 | cK = cT; |
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| 290 | c++; |
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| 291 | } |
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| 292 | } |
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[ed7a55c] | 293 | if(c > size(T) div 2){ lT = cK; } |
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[d68d30] | 294 | } |
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| 295 | else |
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| 296 | { |
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| 297 | hl = hilb(lT,1); |
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| 298 | for(j = 1; j < size(T); j++) |
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| 299 | { |
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| 300 | cT = lead(T[j]); |
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| 301 | attrib(cT,"isSB",1); |
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| 302 | hc = hilb(cT,1); |
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| 303 | if(hl == hc) |
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| 304 | { |
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| 305 | for(k = 1; k <= size(lT); k++) |
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| 306 | { |
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| 307 | if(lT[k] < cT[k]) { lT = cT; c++; break; } |
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| 308 | if(lT[k] > cT[k]) { c++; break; } |
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| 309 | } |
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| 310 | } |
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| 311 | else |
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| 312 | { |
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[d19fa2] | 313 | if(hc < hl){ lT = cT; hl = hilb(lT,1); c++; } |
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[d68d30] | 314 | } |
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| 315 | } |
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| 316 | } |
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| 317 | |
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| 318 | int addList; |
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| 319 | if(size(#) > 0) { list M = #; addList = 1; } |
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| 320 | j = 1; |
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| 321 | attrib(lT,"isSB",1); |
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| 322 | while((j <= size(T))&&(c > 0)) |
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| 323 | { |
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| 324 | cT = lead(T[j]); |
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| 325 | attrib(cT,"isSB",1); |
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| 326 | if((size(reduce(cT,lT)) != 0)||(size(reduce(lT,cT)) != 0)) |
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| 327 | { |
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| 328 | T = delete(T,j); |
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| 329 | if(j == 1) |
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| 330 | { |
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| 331 | L = L[2..size(L)]; |
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| 332 | if(addList == 1) { M = M[2..size(M)]; } |
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| 333 | } |
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| 334 | else |
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| 335 | { |
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| 336 | if(j == size(L)) |
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| 337 | { |
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| 338 | L = L[1..size(L)-1]; |
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| 339 | if(addList == 1) { M = M[1..size(M)-1]; } |
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| 340 | } |
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| 341 | else |
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| 342 | { |
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| 343 | L = L[1..j-1],L[j+1..size(L)]; |
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| 344 | if(addList == 1) { M = M[1..j-1],M[j+1..size(M)]; } |
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| 345 | } |
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| 346 | } |
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| 347 | j--; |
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| 348 | } |
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| 349 | j++; |
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| 350 | } |
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| 351 | |
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| 352 | for(j = 1; j <= size(L); j++) |
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| 353 | { |
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| 354 | L[j] = bigint(L[j]); |
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| 355 | } |
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| 356 | |
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| 357 | if(addList == 0) { return(list(T,L,lT)); } |
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| 358 | if(addList == 1) { return(list(T,L,M,lT)); } |
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| 359 | } |
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| 360 | example |
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| 361 | { "EXAMPLE:"; echo = 2; |
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| 362 | list L = 2,3,5,7,11; |
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| 363 | ring r = 0,(y,x),Dp; |
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| 364 | ideal I1 = 2y2x,y6; |
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| 365 | ideal I2 = yx2,y3x,x5,y6; |
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| 366 | ideal I3 = y2x,x3y,x5,y6; |
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| 367 | ideal I4 = y2x,11x3y,x5; |
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| 368 | ideal I5 = y2x,yx3,x5,7y6; |
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| 369 | list T = I1,I2,I3,I4,I5; |
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| 370 | deleteUnluckyPrimes(T,L,1); |
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| 371 | list P = poly(x),poly(x2),poly(x3),poly(x4),poly(x5); |
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| 372 | deleteUnluckyPrimes(T,L,1,P); |
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| 373 | } |
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| 374 | |
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| 375 | //////////////////////////////////////////////////////////////////////////////// |
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| 376 | |
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[7f30e2] | 377 | proc primeTest(def II, bigint p) |
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[d68d30] | 378 | { |
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[7f30e2] | 379 | if(typeof(II) == "string") |
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| 380 | { |
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| 381 | execute("ideal I = "+II+";"); |
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| 382 | } |
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| 383 | else |
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| 384 | { |
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| 385 | ideal I = II; |
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| 386 | } |
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| 387 | |
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[d68d30] | 388 | int i,j; |
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[7f30e2] | 389 | poly f; |
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| 390 | number cnt; |
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[d68d30] | 391 | for(i = 1; i <= size(I); i++) |
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| 392 | { |
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[7f30e2] | 393 | f = cleardenom(I[i]); |
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| 394 | if(f == 0) { return(0); } |
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| 395 | cnt = leadcoef(I[i])/leadcoef(f); |
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| 396 | if((numerator(cnt) mod p) == 0) { return(0); } |
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| 397 | if((denominator(cnt) mod p) == 0) { return(0); } |
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| 398 | for(j = size(f); j > 0; j--) |
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[d68d30] | 399 | { |
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[7f30e2] | 400 | if((leadcoef(f[j]) mod p) == 0) { return(0); } |
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[d68d30] | 401 | } |
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| 402 | } |
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| 403 | return(1); |
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| 404 | } |
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| 405 | |
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| 406 | //////////////////////////////////////////////////////////////////////////////// |
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| 407 | |
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| 408 | proc primeList(ideal I, int n, list #) |
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[7f30e2] | 409 | "USAGE: primeList(I,n[,ncores]); ( resp. primeList(I,n[,L,ncores]); ) I ideal, |
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| 410 | n integer |
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| 411 | RETURN: the intvec of n greatest primes <= 2147483647 (resp. n greatest primes |
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[4cc3cac] | 412 | < L[size(L)] union with L) such that none of these primes divides any |
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[d68d30] | 413 | coefficient occuring in I |
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[7f30e2] | 414 | NOTE: The number of cores to use can be defined by ncores, default is 1. |
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[d68d30] | 415 | EXAMPLE: example primList; shows an example |
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| 416 | " |
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| 417 | { |
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| 418 | intvec L; |
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| 419 | int i,p; |
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[7f30e2] | 420 | int ncores = 1; |
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| 421 | |
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| 422 | //----------------- Initialize optional parameter ncores --------------------- |
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| 423 | if(size(#) > 0) |
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| 424 | { |
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| 425 | if(size(#) == 1) |
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| 426 | { |
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| 427 | if(typeof(#[1]) == "int") |
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| 428 | { |
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| 429 | ncores = #[1]; |
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| 430 | # = list(); |
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| 431 | } |
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| 432 | } |
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| 433 | else |
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| 434 | { |
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| 435 | ncores = #[2]; |
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| 436 | } |
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| 437 | } |
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| 438 | |
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[d68d30] | 439 | if(size(#) == 0) |
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| 440 | { |
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| 441 | p = 2147483647; |
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| 442 | while(!primeTest(I,p)) |
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| 443 | { |
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| 444 | p = prime(p-1); |
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| 445 | if(p == 2) { ERROR("no more primes"); } |
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| 446 | } |
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| 447 | L[1] = p; |
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| 448 | } |
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| 449 | else |
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| 450 | { |
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| 451 | L = #[1]; |
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| 452 | p = prime(L[size(L)]-1); |
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| 453 | while(!primeTest(I,p)) |
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| 454 | { |
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| 455 | p = prime(p-1); |
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| 456 | if(p == 2) { ERROR("no more primes"); } |
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| 457 | } |
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| 458 | L[size(L)+1] = p; |
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| 459 | } |
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| 460 | if(p == 2) { ERROR("no more primes"); } |
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[7f30e2] | 461 | if(ncores == 1) |
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[d68d30] | 462 | { |
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[7f30e2] | 463 | for(i = 2; i <= n; i++) |
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[d68d30] | 464 | { |
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| 465 | p = prime(p-1); |
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[7f30e2] | 466 | while(!primeTest(I,p)) |
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| 467 | { |
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| 468 | p = prime(p-1); |
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| 469 | if(p == 2) { ERROR("no more primes"); } |
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| 470 | } |
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| 471 | L[size(L)+1] = p; |
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| 472 | } |
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| 473 | } |
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| 474 | else |
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| 475 | { |
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| 476 | int neededSize = size(L)+n-1;; |
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| 477 | list parallelResults; |
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| 478 | list arguments; |
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| 479 | int neededPrimes = neededSize-size(L); |
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| 480 | while(neededPrimes > 0) |
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| 481 | { |
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| 482 | arguments = list(); |
---|
| 483 | for(i = ((neededPrimes div ncores)+1-(neededPrimes%ncores == 0)) |
---|
| 484 | *ncores; i > 0; i--) |
---|
| 485 | { |
---|
| 486 | p = prime(p-1); |
---|
| 487 | if(p == 2) { ERROR("no more primes"); } |
---|
| 488 | arguments[i] = list("I", p); |
---|
| 489 | } |
---|
| 490 | parallelResults = parallelWaitAll("primeTest", arguments, |
---|
| 491 | list(list(list(ncores)))); |
---|
| 492 | for(i = size(arguments); i > 0; i--) |
---|
| 493 | { |
---|
| 494 | if(parallelResults[i]) |
---|
| 495 | { |
---|
| 496 | L[size(L)+1] = arguments[i][2]; |
---|
| 497 | } |
---|
| 498 | } |
---|
| 499 | neededPrimes = neededSize-size(L); |
---|
| 500 | } |
---|
| 501 | if(size(L) > neededSize) |
---|
| 502 | { |
---|
| 503 | L = L[1..neededSize]; |
---|
[d68d30] | 504 | } |
---|
| 505 | } |
---|
| 506 | return(L); |
---|
| 507 | } |
---|
| 508 | example |
---|
| 509 | { "EXAMPLE:"; echo = 2; |
---|
| 510 | ring r = 0,(x,y,z),dp; |
---|
| 511 | ideal I = 2147483647x+y, z-181; |
---|
| 512 | intvec L = primeList(I,10); |
---|
| 513 | size(L); |
---|
| 514 | L[1]; |
---|
| 515 | L[size(L)]; |
---|
| 516 | L = primeList(I,5,L); |
---|
| 517 | size(L); |
---|
| 518 | L[size(L)]; |
---|
| 519 | } |
---|
| 520 | |
---|
| 521 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 522 | |
---|
[78a42b1] | 523 | static proc liftstd1(ideal I) |
---|
[d68d30] | 524 | { |
---|
| 525 | def R = basering; |
---|
| 526 | list rl = ringlist(R); |
---|
| 527 | list ordl = rl[3]; |
---|
| 528 | |
---|
| 529 | int i; |
---|
| 530 | for(i = 1; i <= size(ordl); i++) |
---|
| 531 | { |
---|
| 532 | if((ordl[i][1] == "C") || (ordl[i][1] == "c")) |
---|
| 533 | { |
---|
| 534 | ordl = delete(ordl, i); |
---|
| 535 | break; |
---|
| 536 | } |
---|
| 537 | } |
---|
| 538 | |
---|
| 539 | ordl = insert(ordl, list("c", 0)); |
---|
| 540 | rl[3] = ordl; |
---|
| 541 | def newR = ring(rl); |
---|
| 542 | setring newR; |
---|
| 543 | ideal I = imap(R,I); |
---|
| 544 | |
---|
[7f30e2] | 545 | intvec opt = option(get); |
---|
[d68d30] | 546 | option(none); |
---|
| 547 | option(prompt); |
---|
| 548 | |
---|
| 549 | module M; |
---|
| 550 | for(i = 1; i <= size(I); i++) |
---|
| 551 | { |
---|
| 552 | M = M + module(I[i]*gen(1) + gen(i+1)); |
---|
| 553 | M = M + module(gen(i+1)); |
---|
| 554 | } |
---|
| 555 | |
---|
| 556 | module sM = std(M); |
---|
| 557 | |
---|
| 558 | ideal sI; |
---|
| 559 | if(attrib(R,"global")) |
---|
| 560 | { |
---|
| 561 | for(i = size(I)+1; i <= size(sM); i++) |
---|
| 562 | { |
---|
| 563 | sI[size(sI)+1] = sM[i][1]; |
---|
| 564 | } |
---|
| 565 | matrix T = submat(sM,2..nrows(sM),size(I)+1..ncols(sM)); |
---|
| 566 | } |
---|
| 567 | else |
---|
| 568 | { |
---|
[78a42b1] | 569 | //"=========================================================="; |
---|
| 570 | //"WARNING: Algorithm is not applicable if ordering is mixed."; |
---|
| 571 | //"=========================================================="; |
---|
[d68d30] | 572 | for(i = 1; i <= size(sM)-size(I); i++) |
---|
| 573 | { |
---|
| 574 | sI[size(sI)+1] = sM[i][1]; |
---|
| 575 | } |
---|
| 576 | matrix T = submat(sM,2..nrows(sM),1..ncols(sM)-size(I)); |
---|
| 577 | } |
---|
| 578 | |
---|
| 579 | setring R; |
---|
[7f30e2] | 580 | option(set, opt); |
---|
[d68d30] | 581 | return(imap(newR,sI),imap(newR,T)); |
---|
| 582 | } |
---|
| 583 | example |
---|
| 584 | { "EXAMPLE:"; echo = 2; |
---|
| 585 | ring R = 0,(x,y,z),dp; |
---|
| 586 | poly f = x3+y7+z2+xyz; |
---|
| 587 | ideal i = jacob(f); |
---|
| 588 | matrix T; |
---|
| 589 | ideal sm = liftstd(i,T); |
---|
| 590 | sm; |
---|
| 591 | print(T); |
---|
| 592 | matrix(sm) - matrix(i)*T; |
---|
| 593 | |
---|
| 594 | |
---|
[78a42b1] | 595 | ring S = 32003, x(1..5), lp; |
---|
| 596 | ideal I = cyclic(5); |
---|
[d68d30] | 597 | ideal sI; |
---|
| 598 | matrix T; |
---|
| 599 | sI,T = liftstd1(I); |
---|
| 600 | matrix(sI) - matrix(I)*T; |
---|
| 601 | } |
---|
| 602 | |
---|
| 603 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 604 | |
---|
| 605 | proc modpStd(ideal I, int p, int variant, list #) |
---|
| 606 | "USAGE: modpStd(I,p,variant,#); I ideal, p integer, variant integer |
---|
| 607 | ASSUME: If size(#) > 0, then #[1] is an intvec describing the Hilbert series. |
---|
[27d4bb] | 608 | RETURN: ideal - a standard basis of I mod p, integer - p |
---|
[d68d30] | 609 | NOTE: The procedure computes a standard basis of the ideal I modulo p and |
---|
| 610 | fetches the result to the basering. If size(#) > 0 the Hilbert driven |
---|
| 611 | standard basis computation std(.,#[1]) is used instead of groebner. |
---|
| 612 | The standard basis computation modulo p does also vary depending on the |
---|
| 613 | integer variant, namely |
---|
[7f30e2] | 614 | @* - variant = 1: std(.,#[1]) resp. groebner, |
---|
| 615 | @* - variant = 2: groebner, |
---|
| 616 | @* - variant = 3: homog. - std(.,#[1]) resp. groebner - dehomog., |
---|
| 617 | @* - variant = 4: fglm. |
---|
[d68d30] | 618 | EXAMPLE: example modpStd; shows an example |
---|
| 619 | " |
---|
| 620 | { |
---|
| 621 | def R0 = basering; |
---|
| 622 | list rl = ringlist(R0); |
---|
| 623 | rl[1] = p; |
---|
| 624 | def @r = ring(rl); |
---|
| 625 | setring @r; |
---|
| 626 | ideal i = fetch(R0,I); |
---|
| 627 | |
---|
| 628 | option(redSB); |
---|
| 629 | |
---|
[7f30e2] | 630 | if(variant == 1) |
---|
[d68d30] | 631 | { |
---|
| 632 | if(size(#) > 0) |
---|
| 633 | { |
---|
| 634 | i = std(i, #[1]); |
---|
| 635 | } |
---|
| 636 | else |
---|
| 637 | { |
---|
| 638 | i = groebner(i); |
---|
| 639 | } |
---|
| 640 | } |
---|
[27d4bb] | 641 | |
---|
[7f30e2] | 642 | if(variant == 2) |
---|
[d68d30] | 643 | { |
---|
| 644 | i = groebner(i); |
---|
| 645 | } |
---|
| 646 | |
---|
[7f30e2] | 647 | if(variant == 3) |
---|
[d68d30] | 648 | { |
---|
| 649 | list rl = ringlist(@r); |
---|
| 650 | int nvar@r = nvars(@r); |
---|
| 651 | |
---|
| 652 | int k; |
---|
| 653 | intvec w; |
---|
| 654 | for(k = 1; k <= nvar@r; k++) |
---|
| 655 | { |
---|
| 656 | w[k] = deg(var(k)); |
---|
| 657 | } |
---|
| 658 | w[nvar@r + 1] = 1; |
---|
| 659 | |
---|
| 660 | rl[2][nvar@r + 1] = "homvar"; |
---|
| 661 | rl[3][2][2] = w; |
---|
| 662 | |
---|
| 663 | def HomR = ring(rl); |
---|
| 664 | setring HomR; |
---|
| 665 | ideal i = imap(@r, i); |
---|
| 666 | i = homog(i, homvar); |
---|
| 667 | |
---|
| 668 | if(size(#) > 0) |
---|
| 669 | { |
---|
| 670 | if(w == 1) |
---|
| 671 | { |
---|
| 672 | i = std(i, #[1]); |
---|
| 673 | } |
---|
| 674 | else |
---|
| 675 | { |
---|
| 676 | i = std(i, #[1], w); |
---|
| 677 | } |
---|
| 678 | } |
---|
| 679 | else |
---|
| 680 | { |
---|
| 681 | i = groebner(i); |
---|
| 682 | } |
---|
| 683 | |
---|
| 684 | i = subst(i, homvar, 1); |
---|
| 685 | i = simplify(i, 34); |
---|
| 686 | |
---|
| 687 | setring @r; |
---|
| 688 | i = imap(HomR, i); |
---|
| 689 | i = interred(i); |
---|
| 690 | kill HomR; |
---|
| 691 | } |
---|
[7f30e2] | 692 | |
---|
| 693 | if(variant == 4) |
---|
| 694 | { |
---|
| 695 | def R1 = changeord("dp"); |
---|
| 696 | setring R1; |
---|
| 697 | ideal i = fetch(@r,i); |
---|
| 698 | i = std(i); |
---|
| 699 | setring @r; |
---|
| 700 | i = fglm(R1,i); |
---|
| 701 | } |
---|
[27d4bb] | 702 | |
---|
[d68d30] | 703 | setring R0; |
---|
| 704 | return(list(fetch(@r,i),p)); |
---|
| 705 | } |
---|
| 706 | example |
---|
| 707 | { "EXAMPLE:"; echo = 2; |
---|
| 708 | ring r = 0, x(1..4), dp; |
---|
| 709 | ideal I = cyclic(4); |
---|
| 710 | int p = 181; |
---|
| 711 | list P = modpStd(I,p,5); |
---|
| 712 | P; |
---|
| 713 | |
---|
| 714 | int q = 32003; |
---|
| 715 | list Q = modpStd(I,q,2); |
---|
| 716 | Q; |
---|
| 717 | } |
---|
| 718 | |
---|
| 719 | ////////////////////////////// main procedures ///////////////////////////////// |
---|
| 720 | |
---|
| 721 | proc modStd(ideal I, list #) |
---|
| 722 | "USAGE: modStd(I); I ideal |
---|
| 723 | ASSUME: If size(#) > 0, then # contains either 1, 2 or 4 integers such that |
---|
[c512d9] | 724 | @* - #[1] is the number of available processors for the computation, |
---|
[d68d30] | 725 | @* - #[2] is an optional parameter for the exactness of the computation, |
---|
| 726 | if #[2] = 1, the procedure computes a standard basis for sure, |
---|
| 727 | @* - #[3] is the number of primes until the first lifting, |
---|
| 728 | @* - #[4] is the constant number of primes between two liftings until |
---|
| 729 | the computation stops. |
---|
| 730 | RETURN: a standard basis of I if no warning appears; |
---|
| 731 | NOTE: The procedure computes a standard basis of I (over the rational |
---|
[7f30e2] | 732 | numbers) by using modular methods. |
---|
[27d4bb] | 733 | By default the procedure computes a standard basis of I for sure, but |
---|
| 734 | if the optional parameter #[2] = 0, it computes a standard basis of I |
---|
[7f30e2] | 735 | with high probability. |
---|
[d68d30] | 736 | The procedure distinguishes between different variants for the standard |
---|
| 737 | basis computation in positive characteristic depending on the ordering |
---|
| 738 | of the basering, the parameter #[2] and if the ideal I is homogeneous. |
---|
[7f30e2] | 739 | @* - variant = 1, if I is homogeneous, |
---|
| 740 | @* - variant = 2, if I is not homogeneous, 1-block-ordering, |
---|
[d68d30] | 741 | @* - variant = 3, if I is not homogeneous, complicated ordering (lp or |
---|
[7f30e2] | 742 | > 1 block), |
---|
| 743 | @* - variant = 4, if I is not homogeneous, ordering lp, dim(I) = 0. |
---|
[d68d30] | 744 | EXAMPLE: example modStd; shows an example |
---|
| 745 | " |
---|
| 746 | { |
---|
| 747 | int TT = timer; |
---|
| 748 | int RT = rtimer; |
---|
| 749 | |
---|
| 750 | def R0 = basering; |
---|
| 751 | list rl = ringlist(R0); |
---|
| 752 | if((npars(R0) > 0) || (rl[1] > 0)) |
---|
| 753 | { |
---|
[27d4bb] | 754 | ERROR("Characteristic of basering should be zero, basering should |
---|
[4cc3cac] | 755 | have no parameters."); |
---|
[d68d30] | 756 | } |
---|
| 757 | |
---|
| 758 | int index = 1; |
---|
| 759 | int i,k,c; |
---|
| 760 | int j = 1; |
---|
[4cc3cac] | 761 | int pTest, sizeTest; |
---|
[d68d30] | 762 | int en = 2134567879; |
---|
| 763 | int an = 1000000000; |
---|
| 764 | bigint N; |
---|
| 765 | |
---|
| 766 | //-------------------- Initialize optional parameters ------------------------ |
---|
| 767 | if(size(#) > 0) |
---|
| 768 | { |
---|
| 769 | if(size(#) == 1) |
---|
| 770 | { |
---|
| 771 | int n1 = #[1]; |
---|
[4cc3cac] | 772 | int exactness = 1; |
---|
[fd0df1] | 773 | if(n1 >= 10) |
---|
| 774 | { |
---|
| 775 | int n2 = n1 + 1; |
---|
| 776 | int n3 = n1; |
---|
| 777 | } |
---|
| 778 | else |
---|
| 779 | { |
---|
| 780 | int n2 = 10; |
---|
| 781 | int n3 = 10; |
---|
| 782 | } |
---|
[d68d30] | 783 | } |
---|
| 784 | if(size(#) == 2) |
---|
| 785 | { |
---|
| 786 | int n1 = #[1]; |
---|
| 787 | int exactness = #[2]; |
---|
[fd0df1] | 788 | if(n1 >= 10) |
---|
| 789 | { |
---|
| 790 | int n2 = n1 + 1; |
---|
| 791 | int n3 = n1; |
---|
| 792 | } |
---|
| 793 | else |
---|
| 794 | { |
---|
| 795 | int n2 = 10; |
---|
| 796 | int n3 = 10; |
---|
| 797 | } |
---|
[d68d30] | 798 | } |
---|
| 799 | if(size(#) == 4) |
---|
| 800 | { |
---|
| 801 | int n1 = #[1]; |
---|
| 802 | int exactness = #[2]; |
---|
[fd0df1] | 803 | if(n1 >= #[3]) |
---|
| 804 | { |
---|
| 805 | int n2 = n1 + 1; |
---|
| 806 | } |
---|
| 807 | else |
---|
| 808 | { |
---|
| 809 | int n2 = #[3]; |
---|
| 810 | } |
---|
| 811 | if(n1 >= #[4]) |
---|
| 812 | { |
---|
| 813 | int n3 = n1; |
---|
| 814 | } |
---|
| 815 | else |
---|
| 816 | { |
---|
| 817 | int n3 = #[4]; |
---|
| 818 | } |
---|
[d68d30] | 819 | } |
---|
| 820 | } |
---|
| 821 | else |
---|
| 822 | { |
---|
| 823 | int n1 = 1; |
---|
[4cc3cac] | 824 | int exactness = 1; |
---|
[d68d30] | 825 | int n2 = 10; |
---|
| 826 | int n3 = 10; |
---|
| 827 | } |
---|
| 828 | |
---|
[fd0df1] | 829 | if(printlevel >= 10) |
---|
| 830 | { |
---|
[7f30e2] | 831 | "n1 = "+string(n1)+", n2 = "+string(n2)+", n3 = "+string(n3) |
---|
| 832 | +", exactness = "+string(exactness); |
---|
[fd0df1] | 833 | } |
---|
| 834 | |
---|
[d68d30] | 835 | //------------------------- Save current options ----------------------------- |
---|
| 836 | intvec opt = option(get); |
---|
[27d4bb] | 837 | |
---|
[d68d30] | 838 | option(redSB); |
---|
| 839 | |
---|
| 840 | //-------------------- Initialize the list of primes ------------------------- |
---|
[7f30e2] | 841 | int tt = timer; |
---|
| 842 | int rt = rtimer; |
---|
| 843 | intvec L = primeList(I,n2,n1); |
---|
| 844 | if(printlevel >= 10) |
---|
| 845 | { |
---|
| 846 | "CPU-time for primeList: "+string(timer-tt)+" seconds."; |
---|
| 847 | "Real-time for primeList: "+string(rtimer-rt)+" seconds."; |
---|
| 848 | } |
---|
[d68d30] | 849 | L[5] = prime(random(an,en)); |
---|
| 850 | |
---|
| 851 | //--------------------- Decide which variant to take ------------------------- |
---|
| 852 | int variant; |
---|
| 853 | int h = homog(I); |
---|
| 854 | |
---|
[7f30e2] | 855 | tt = timer; |
---|
| 856 | rt = rtimer; |
---|
[d68d30] | 857 | |
---|
[4cc3cac] | 858 | if(!mixedTest()) |
---|
[d68d30] | 859 | { |
---|
[4cc3cac] | 860 | if(h) |
---|
| 861 | { |
---|
[7f30e2] | 862 | variant = 1; |
---|
| 863 | if(printlevel >= 10) { "variant = 1"; } |
---|
| 864 | |
---|
[4cc3cac] | 865 | rl[1] = L[5]; |
---|
| 866 | def @r = ring(rl); |
---|
| 867 | setring @r; |
---|
| 868 | def @s = changeord("dp"); |
---|
| 869 | setring @s; |
---|
| 870 | ideal I = std(fetch(R0,I)); |
---|
| 871 | intvec hi = hilb(I,1); |
---|
| 872 | setring R0; |
---|
| 873 | kill @r,@s; |
---|
| 874 | } |
---|
| 875 | else |
---|
| 876 | { |
---|
| 877 | string ordstr_R0 = ordstr(R0); |
---|
| 878 | int neg = 1 - attrib(R0,"global"); |
---|
| 879 | |
---|
| 880 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
| 881 | { |
---|
[7f30e2] | 882 | variant = 2; |
---|
| 883 | if(printlevel >= 10) { "variant = 2"; } |
---|
[4cc3cac] | 884 | } |
---|
| 885 | else |
---|
| 886 | { |
---|
| 887 | string order; |
---|
| 888 | if(system("nblocks") <= 2) |
---|
| 889 | { |
---|
[27d4bb] | 890 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
[4cc3cac] | 891 | + find(ordstr_R0, "rp") <= 0) |
---|
| 892 | { |
---|
| 893 | order = "simple"; |
---|
| 894 | } |
---|
| 895 | } |
---|
| 896 | |
---|
| 897 | if((order == "simple") || (size(rl) > 4)) |
---|
| 898 | { |
---|
[7f30e2] | 899 | variant = 2; |
---|
| 900 | if(printlevel >= 10) { "variant = 2"; } |
---|
[4cc3cac] | 901 | } |
---|
| 902 | else |
---|
| 903 | { |
---|
| 904 | rl[1] = L[5]; |
---|
| 905 | def @r = ring(rl); |
---|
| 906 | setring @r; |
---|
[7f30e2] | 907 | |
---|
| 908 | def @s = changeord("dp"); |
---|
| 909 | setring @s; |
---|
| 910 | ideal I = std(fetch(R0,I)); |
---|
| 911 | if(dim(I) == 0) |
---|
[4cc3cac] | 912 | { |
---|
[7f30e2] | 913 | variant = 4; |
---|
| 914 | if(printlevel >= 10) { "variant = 4"; } |
---|
[4cc3cac] | 915 | } |
---|
[7f30e2] | 916 | else |
---|
| 917 | { |
---|
| 918 | variant = 3; |
---|
| 919 | if(printlevel >= 10) { "variant = 3"; } |
---|
| 920 | |
---|
| 921 | int nvar@r = nvars(@r); |
---|
| 922 | intvec w; |
---|
| 923 | for(i = 1; i <= nvar@r; i++) |
---|
| 924 | { |
---|
| 925 | w[i] = deg(var(i)); |
---|
| 926 | } |
---|
| 927 | w[nvar@r + 1] = 1; |
---|
[4cc3cac] | 928 | |
---|
[7f30e2] | 929 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
| 930 | intvec W = hiRi[2]; |
---|
| 931 | @s = hiRi[1]; |
---|
| 932 | setring @s; |
---|
[4cc3cac] | 933 | |
---|
[7f30e2] | 934 | Id(1) = std(Id(1)); |
---|
| 935 | intvec hi = hilb(Id(1), 1, W); |
---|
| 936 | } |
---|
| 937 | |
---|
[4cc3cac] | 938 | setring R0; |
---|
| 939 | kill @r,@s; |
---|
| 940 | } |
---|
| 941 | } |
---|
| 942 | } |
---|
[d68d30] | 943 | } |
---|
| 944 | else |
---|
| 945 | { |
---|
[4cc3cac] | 946 | if(exactness == 1) { return(groebner(I)); } |
---|
| 947 | if(h) |
---|
| 948 | { |
---|
[fd0df1] | 949 | variant = 1; |
---|
| 950 | if(printlevel >= 10) { "variant = 1"; } |
---|
[4cc3cac] | 951 | rl[1] = L[5]; |
---|
| 952 | def @r = ring(rl); |
---|
| 953 | setring @r; |
---|
| 954 | def @s = changeord("dp"); |
---|
| 955 | setring @s; |
---|
| 956 | ideal I = std(fetch(R0,I)); |
---|
| 957 | intvec hi = hilb(I,1); |
---|
| 958 | setring R0; |
---|
| 959 | kill @r,@s; |
---|
| 960 | } |
---|
| 961 | else |
---|
[d68d30] | 962 | { |
---|
| 963 | string ordstr_R0 = ordstr(R0); |
---|
| 964 | int neg = 1 - attrib(R0,"global"); |
---|
| 965 | |
---|
| 966 | if((find(ordstr_R0, "M") > 0) || (find(ordstr_R0, "a") > 0) || neg) |
---|
| 967 | { |
---|
| 968 | variant = 2; |
---|
| 969 | if(printlevel >= 10) { "variant = 2"; } |
---|
| 970 | } |
---|
| 971 | else |
---|
| 972 | { |
---|
| 973 | string order; |
---|
| 974 | if(system("nblocks") <= 2) |
---|
| 975 | { |
---|
[27d4bb] | 976 | if(find(ordstr_R0, "M") + find(ordstr_R0, "lp") |
---|
[4cc3cac] | 977 | + find(ordstr_R0, "rp") <= 0) |
---|
[d68d30] | 978 | { |
---|
| 979 | order = "simple"; |
---|
| 980 | } |
---|
| 981 | } |
---|
| 982 | |
---|
| 983 | if((order == "simple") || (size(rl) > 4)) |
---|
| 984 | { |
---|
| 985 | variant = 2; |
---|
| 986 | if(printlevel >= 10) { "variant = 2"; } |
---|
| 987 | } |
---|
| 988 | else |
---|
| 989 | { |
---|
| 990 | variant = 3; |
---|
| 991 | if(printlevel >= 10) { "variant = 3"; } |
---|
[27d4bb] | 992 | |
---|
[d68d30] | 993 | rl[1] = L[5]; |
---|
| 994 | def @r = ring(rl); |
---|
| 995 | setring @r; |
---|
| 996 | int nvar@r = nvars(@r); |
---|
| 997 | intvec w; |
---|
| 998 | for(i = 1; i <= nvar@r; i++) |
---|
| 999 | { |
---|
| 1000 | w[i] = deg(var(i)); |
---|
| 1001 | } |
---|
| 1002 | w[nvar@r + 1] = 1; |
---|
| 1003 | |
---|
| 1004 | list hiRi = hilbRing(fetch(R0,I),w); |
---|
| 1005 | intvec W = hiRi[2]; |
---|
| 1006 | def @s = hiRi[1]; |
---|
| 1007 | setring @s; |
---|
| 1008 | |
---|
| 1009 | Id(1) = std(Id(1)); |
---|
| 1010 | intvec hi = hilb(Id(1), 1, W); |
---|
| 1011 | |
---|
| 1012 | setring R0; |
---|
| 1013 | kill @r,@s; |
---|
| 1014 | } |
---|
| 1015 | } |
---|
| 1016 | } |
---|
| 1017 | } |
---|
[27d4bb] | 1018 | |
---|
[d68d30] | 1019 | list P,T1,T2,T3,LL; |
---|
| 1020 | |
---|
| 1021 | ideal J,K,H; |
---|
| 1022 | |
---|
| 1023 | //----- If there is more than one processor available, we parallelize the ---- |
---|
| 1024 | //----- main standard basis computations in positive characteristic ---- |
---|
| 1025 | |
---|
| 1026 | if(n1 > 1) |
---|
| 1027 | { |
---|
| 1028 | ideal I_for_fork = I; |
---|
| 1029 | export(I_for_fork); // I available for each link |
---|
| 1030 | |
---|
| 1031 | //----- Create n1 links l(1),...,l(n1), open all of them and compute --------- |
---|
| 1032 | //----- standard basis for the primes L[2],...,L[n1 + 1]. --------- |
---|
| 1033 | |
---|
| 1034 | for(i = 1; i <= n1; i++) |
---|
| 1035 | { |
---|
[c512d9] | 1036 | //link l(i) = "MPtcp:fork"; |
---|
| 1037 | link l(i) = "ssi:fork"; |
---|
[d68d30] | 1038 | open(l(i)); |
---|
[7f30e2] | 1039 | if((variant == 1) || (variant == 3)) |
---|
[d68d30] | 1040 | { |
---|
[27d4bb] | 1041 | write(l(i), quote(modpStd(I_for_fork, eval(L[i + 1]), |
---|
[4cc3cac] | 1042 | eval(variant), eval(hi)))); |
---|
[d68d30] | 1043 | } |
---|
[7f30e2] | 1044 | if((variant == 2) || (variant == 4)) |
---|
[d68d30] | 1045 | { |
---|
[27d4bb] | 1046 | write(l(i), quote(modpStd(I_for_fork, eval(L[i + 1]), |
---|
[4cc3cac] | 1047 | eval(variant)))); |
---|
[d68d30] | 1048 | } |
---|
| 1049 | } |
---|
| 1050 | |
---|
| 1051 | int t = timer; |
---|
[7f30e2] | 1052 | if((variant == 1) || (variant == 3)) |
---|
[d68d30] | 1053 | { |
---|
| 1054 | P = modpStd(I_for_fork, L[1], variant, hi); |
---|
| 1055 | } |
---|
[7f30e2] | 1056 | if((variant == 2) || (variant == 4)) |
---|
[d68d30] | 1057 | { |
---|
| 1058 | P = modpStd(I_for_fork, L[1], variant); |
---|
| 1059 | } |
---|
| 1060 | t = timer - t; |
---|
| 1061 | if(t > 60) { t = 60; } |
---|
| 1062 | int i_sleep = system("sh", "sleep "+string(t)); |
---|
| 1063 | T1[1] = P[1]; |
---|
| 1064 | T2[1] = bigint(P[2]); |
---|
| 1065 | index++; |
---|
| 1066 | |
---|
| 1067 | j = j + n1 + 1; |
---|
| 1068 | } |
---|
| 1069 | |
---|
| 1070 | //-------------- Main standard basis computations in positive ---------------- |
---|
| 1071 | //---------------------- characteristic start here --------------------------- |
---|
| 1072 | |
---|
[7f30e2] | 1073 | list arguments_farey, results_farey; |
---|
| 1074 | |
---|
[d68d30] | 1075 | while(1) |
---|
| 1076 | { |
---|
| 1077 | tt = timer; rt = rtimer; |
---|
[fd0df1] | 1078 | |
---|
| 1079 | if(printlevel >= 10) { "size(L) = "+string(size(L)); } |
---|
| 1080 | |
---|
[d68d30] | 1081 | if(n1 > 1) |
---|
| 1082 | { |
---|
| 1083 | while(j <= size(L) + 1) |
---|
| 1084 | { |
---|
| 1085 | for(i = 1; i <= n1; i++) |
---|
| 1086 | { |
---|
[4cc3cac] | 1087 | //--- ask if link l(i) is ready otherwise sleep for t seconds --- |
---|
[27d4bb] | 1088 | if(status(l(i), "read", "ready")) |
---|
[d68d30] | 1089 | { |
---|
[4cc3cac] | 1090 | //--- read the result from l(i) --- |
---|
[27d4bb] | 1091 | P = read(l(i)); |
---|
[d68d30] | 1092 | T1[index] = P[1]; |
---|
| 1093 | T2[index] = bigint(P[2]); |
---|
| 1094 | index++; |
---|
| 1095 | |
---|
| 1096 | if(j <= size(L)) |
---|
| 1097 | { |
---|
[7f30e2] | 1098 | if((variant == 1) || (variant == 3)) |
---|
[d68d30] | 1099 | { |
---|
[27d4bb] | 1100 | write(l(i), quote(modpStd(I_for_fork, eval(L[j]), |
---|
[4cc3cac] | 1101 | eval(variant), eval(hi)))); |
---|
[d68d30] | 1102 | j++; |
---|
| 1103 | } |
---|
[7f30e2] | 1104 | if((variant == 2) || (variant == 4)) |
---|
[d68d30] | 1105 | { |
---|
[27d4bb] | 1106 | write(l(i), quote(modpStd(I_for_fork, |
---|
[4cc3cac] | 1107 | eval(L[j]), eval(variant)))); |
---|
[d68d30] | 1108 | j++; |
---|
| 1109 | } |
---|
| 1110 | } |
---|
| 1111 | else |
---|
| 1112 | { |
---|
| 1113 | k++; |
---|
| 1114 | close(l(i)); |
---|
| 1115 | } |
---|
| 1116 | } |
---|
| 1117 | } |
---|
[4cc3cac] | 1118 | //--- k describes the number of closed links --- |
---|
[27d4bb] | 1119 | if(k == n1) |
---|
[d68d30] | 1120 | { |
---|
| 1121 | j++; |
---|
| 1122 | } |
---|
| 1123 | i_sleep = system("sh", "sleep "+string(t)); |
---|
| 1124 | } |
---|
| 1125 | } |
---|
| 1126 | else |
---|
| 1127 | { |
---|
| 1128 | while(j <= size(L)) |
---|
| 1129 | { |
---|
[7f30e2] | 1130 | if((variant == 1) || (variant == 3)) |
---|
[d68d30] | 1131 | { |
---|
| 1132 | P = modpStd(I, L[j], variant, hi); |
---|
| 1133 | } |
---|
[7f30e2] | 1134 | if((variant == 2) || (variant == 4)) |
---|
[d68d30] | 1135 | { |
---|
| 1136 | P = modpStd(I, L[j], variant); |
---|
| 1137 | } |
---|
| 1138 | |
---|
| 1139 | T1[index] = P[1]; |
---|
| 1140 | T2[index] = bigint(P[2]); |
---|
| 1141 | index++; |
---|
| 1142 | j++; |
---|
| 1143 | } |
---|
| 1144 | } |
---|
| 1145 | |
---|
| 1146 | if(printlevel >= 10) |
---|
| 1147 | { |
---|
| 1148 | "CPU-time for computing list is "+string(timer - tt)+" seconds."; |
---|
| 1149 | "Real-time for computing list is "+string(rtimer - rt)+" seconds."; |
---|
| 1150 | } |
---|
| 1151 | |
---|
| 1152 | //------------------------ Delete unlucky primes ----------------------------- |
---|
| 1153 | //------------- unlucky if and only if the leading ideal is wrong ------------ |
---|
| 1154 | |
---|
[4cc3cac] | 1155 | LL = deleteUnluckyPrimes(T1,T2,h); |
---|
[d68d30] | 1156 | T1 = LL[1]; |
---|
| 1157 | T2 = LL[2]; |
---|
| 1158 | |
---|
| 1159 | //------------------- Now all leading ideals are the same -------------------- |
---|
| 1160 | //------------------- Lift results to basering via farey --------------------- |
---|
| 1161 | |
---|
[7f30e2] | 1162 | tt = timer; rt = rtimer; |
---|
[d68d30] | 1163 | N = T2[1]; |
---|
[b4f7eb] | 1164 | for(i = 2; i <= size(T2); i++) { N = N*T2[i]; } |
---|
[d68d30] | 1165 | H = chinrem(T1,T2); |
---|
[7f30e2] | 1166 | if(n1 == 1) |
---|
| 1167 | { |
---|
| 1168 | J = farey(H,N); |
---|
| 1169 | } |
---|
| 1170 | else |
---|
| 1171 | { |
---|
| 1172 | for(i = ncols(H); i > 0; i--) |
---|
| 1173 | { |
---|
| 1174 | arguments_farey[i] = list(ideal(H[i]), N); |
---|
| 1175 | } |
---|
| 1176 | results_farey = parallelWaitAll("farey", arguments_farey, |
---|
| 1177 | list(list(list(n1)))); |
---|
| 1178 | for(i = ncols(H); i > 0; i--) |
---|
| 1179 | { |
---|
| 1180 | J[i] = results_farey[i][1]; |
---|
| 1181 | } |
---|
| 1182 | } |
---|
| 1183 | if(printlevel >= 10) |
---|
| 1184 | { |
---|
| 1185 | "CPU-time for lifting-process is "+string(timer - tt)+" seconds."; |
---|
| 1186 | "Real-time for lifting-process is "+string(rtimer - rt)+" seconds."; |
---|
| 1187 | } |
---|
[d68d30] | 1188 | |
---|
| 1189 | //---------------- Test if we already have a standard basis of I -------------- |
---|
| 1190 | |
---|
| 1191 | tt = timer; rt = rtimer; |
---|
[7f30e2] | 1192 | if((variant == 1) || (variant == 3)) |
---|
[27d4bb] | 1193 | { |
---|
| 1194 | pTest = pTestSB(I,J,L,variant,hi); |
---|
[4cc3cac] | 1195 | } |
---|
[7f30e2] | 1196 | if((variant == 2) || (variant == 4)) |
---|
[27d4bb] | 1197 | { |
---|
| 1198 | pTest = pTestSB(I,J,L,variant); |
---|
[4cc3cac] | 1199 | } |
---|
[d68d30] | 1200 | |
---|
| 1201 | if(printlevel >= 10) |
---|
| 1202 | { |
---|
| 1203 | "CPU-time for pTest is "+string(timer - tt)+" seconds."; |
---|
| 1204 | "Real-time for pTest is "+string(rtimer - rt)+" seconds."; |
---|
| 1205 | } |
---|
[27d4bb] | 1206 | |
---|
[d68d30] | 1207 | if(pTest) |
---|
| 1208 | { |
---|
| 1209 | if(printlevel >= 10) |
---|
| 1210 | { |
---|
[b4f7eb] | 1211 | "CPU-time for computation without final tests is " |
---|
| 1212 | +string(timer - TT)+" seconds."; |
---|
| 1213 | "Real-time for computation without final tests is " |
---|
| 1214 | +string(rtimer - RT)+" seconds."; |
---|
[d68d30] | 1215 | } |
---|
| 1216 | |
---|
| 1217 | attrib(J,"isSB",1); |
---|
[7f30e2] | 1218 | |
---|
| 1219 | if(exactness == 0) |
---|
[d68d30] | 1220 | { |
---|
[7f30e2] | 1221 | option(set, opt); |
---|
| 1222 | if(n1 > 1) { kill I_for_fork; } |
---|
| 1223 | return(J); |
---|
[d68d30] | 1224 | } |
---|
[7f30e2] | 1225 | |
---|
| 1226 | if(exactness == 1) |
---|
[d68d30] | 1227 | { |
---|
[7f30e2] | 1228 | tt = timer; rt = rtimer; |
---|
| 1229 | sizeTest = 1 - isIncluded(I,J,n1); |
---|
| 1230 | |
---|
| 1231 | if(printlevel >= 10) |
---|
[d68d30] | 1232 | { |
---|
[7f30e2] | 1233 | "CPU-time for checking if I subset <G> is " |
---|
| 1234 | +string(timer - tt)+" seconds."; |
---|
| 1235 | "Real-time for checking if I subset <G> is " |
---|
| 1236 | +string(rtimer - rt)+" seconds."; |
---|
[d68d30] | 1237 | } |
---|
[7f30e2] | 1238 | |
---|
| 1239 | if(sizeTest == 0) |
---|
[d68d30] | 1240 | { |
---|
| 1241 | tt = timer; rt = rtimer; |
---|
| 1242 | K = std(J); |
---|
| 1243 | |
---|
| 1244 | if(printlevel >= 10) |
---|
| 1245 | { |
---|
[b4f7eb] | 1246 | "CPU-time for last std-computation is " |
---|
| 1247 | +string(timer - tt)+" seconds."; |
---|
| 1248 | "Real-time for last std-computation is " |
---|
| 1249 | +string(rtimer - rt)+" seconds."; |
---|
[d68d30] | 1250 | } |
---|
| 1251 | |
---|
| 1252 | if(size(reduce(K,J)) == 0) |
---|
| 1253 | { |
---|
| 1254 | option(set, opt); |
---|
| 1255 | if(n1 > 1) { kill I_for_fork; } |
---|
| 1256 | return(J); |
---|
| 1257 | } |
---|
| 1258 | } |
---|
| 1259 | } |
---|
| 1260 | } |
---|
| 1261 | |
---|
| 1262 | //-------------- We do not already have a standard basis of I ---------------- |
---|
| 1263 | //----------- Therefore do the main computation for more primes -------------- |
---|
| 1264 | |
---|
| 1265 | T1 = H; |
---|
| 1266 | T2 = N; |
---|
| 1267 | index = 2; |
---|
| 1268 | |
---|
| 1269 | j = size(L) + 1; |
---|
[7f30e2] | 1270 | tt = timer; rt = rtimer; |
---|
| 1271 | L = primeList(I,n3,L,n1); |
---|
| 1272 | if(printlevel >= 10) |
---|
| 1273 | { |
---|
| 1274 | "CPU-time for primeList: "+string(timer-tt)+" seconds."; |
---|
| 1275 | "Real-time for primeList: "+string(rtimer-rt)+" seconds."; |
---|
| 1276 | } |
---|
[d68d30] | 1277 | |
---|
| 1278 | if(n1 > 1) |
---|
| 1279 | { |
---|
| 1280 | for(i = 1; i <= n1; i++) |
---|
| 1281 | { |
---|
| 1282 | open(l(i)); |
---|
[7f30e2] | 1283 | if((variant == 1) || (variant == 3)) |
---|
[d68d30] | 1284 | { |
---|
[27d4bb] | 1285 | write(l(i), quote(modpStd(I_for_fork, eval(L[j+i-1]), |
---|
[4cc3cac] | 1286 | eval(variant), eval(hi)))); |
---|
[d68d30] | 1287 | } |
---|
[7f30e2] | 1288 | if((variant == 2) || (variant == 4)) |
---|
[d68d30] | 1289 | { |
---|
[27d4bb] | 1290 | write(l(i), quote(modpStd(I_for_fork, eval(L[j+i-1]), |
---|
[4cc3cac] | 1291 | eval(variant)))); |
---|
[d68d30] | 1292 | } |
---|
| 1293 | } |
---|
| 1294 | j = j + n1; |
---|
| 1295 | k = 0; |
---|
| 1296 | } |
---|
| 1297 | } |
---|
| 1298 | } |
---|
| 1299 | example |
---|
| 1300 | { "EXAMPLE:"; echo = 2; |
---|
[4cc3cac] | 1301 | ring R1 = 0,(x,y,z,t),dp; |
---|
[d68d30] | 1302 | ideal I = 3x3+x2+1, 11y5+y3+2, 5z4+z2+4; |
---|
| 1303 | ideal J = modStd(I); |
---|
| 1304 | J; |
---|
| 1305 | I = homog(I,t); |
---|
| 1306 | J = modStd(I); |
---|
| 1307 | J; |
---|
| 1308 | |
---|
[4cc3cac] | 1309 | ring R2 = 0,(x,y,z),ds; |
---|
[d68d30] | 1310 | ideal I = jacob(x5+y6+z7+xyz); |
---|
[4cc3cac] | 1311 | ideal J1 = modStd(I,1,0); |
---|
[d68d30] | 1312 | J1; |
---|
| 1313 | |
---|
[4cc3cac] | 1314 | ring R3 = 0,x(1..4),lp; |
---|
[d68d30] | 1315 | ideal I = cyclic(4); |
---|
[14fb6ea] | 1316 | ideal J1 = modStd(I,1); |
---|
[4cc3cac] | 1317 | ideal J2 = modStd(I,1,0); |
---|
[ba2be4] | 1318 | size(reduce(J1,J2)); |
---|
| 1319 | size(reduce(J2,J1)); |
---|
[d68d30] | 1320 | } |
---|
| 1321 | |
---|
| 1322 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1323 | |
---|
| 1324 | proc modS(ideal I, list L, list #) |
---|
| 1325 | "USAGE: modS(I,L); I ideal, L intvec of primes |
---|
| 1326 | if size(#)>0 std is used instead of groebner |
---|
| 1327 | RETURN: an ideal which is with high probability a standard basis |
---|
| 1328 | NOTE: This procedure is designed for fast experiments. |
---|
| 1329 | It is not tested whether the result is a standard basis. |
---|
| 1330 | It is not tested whether the result generates I. |
---|
| 1331 | EXAMPLE: example modS; shows an example |
---|
| 1332 | " |
---|
| 1333 | { |
---|
| 1334 | int j; |
---|
| 1335 | bigint N = 1; |
---|
| 1336 | def R0 = basering; |
---|
| 1337 | ideal J; |
---|
| 1338 | list T; |
---|
| 1339 | list rl = ringlist(R0); |
---|
[27d4bb] | 1340 | if((npars(R0)>0) || (rl[1]>0)) |
---|
| 1341 | { |
---|
| 1342 | ERROR("Characteristic of basering should be zero."); |
---|
[4cc3cac] | 1343 | } |
---|
[d68d30] | 1344 | for(j = 1; j <= size(L); j++) |
---|
| 1345 | { |
---|
| 1346 | N = N*L[j]; |
---|
| 1347 | rl[1] = L[j]; |
---|
| 1348 | def @r = ring(rl); |
---|
| 1349 | setring @r; |
---|
| 1350 | ideal I = fetch(R0,I); |
---|
| 1351 | if(size(#) > 0) |
---|
| 1352 | { |
---|
| 1353 | I = std(I); |
---|
| 1354 | } |
---|
| 1355 | else |
---|
| 1356 | { |
---|
| 1357 | I = groebner(I); |
---|
| 1358 | } |
---|
| 1359 | setring R0; |
---|
| 1360 | T[j] = fetch(@r,I); |
---|
| 1361 | kill @r; |
---|
| 1362 | } |
---|
[27d4bb] | 1363 | L = deleteUnluckyPrimes(T,L,homog(I)); |
---|
[4cc3cac] | 1364 | // unlucky if and only if the leading ideal is wrong |
---|
[d68d30] | 1365 | J = farey(chinrem(L[1],L[2]),N); |
---|
| 1366 | attrib(J,"isSB",1); |
---|
| 1367 | return(J); |
---|
| 1368 | } |
---|
| 1369 | example |
---|
| 1370 | { "EXAMPLE:"; echo = 2; |
---|
| 1371 | list L = 3,5,11,13,181,32003; |
---|
| 1372 | ring r = 0,(x,y,z,t),dp; |
---|
| 1373 | ideal I = 3x3+x2+1,11y5+y3+2,5z4+z2+4; |
---|
| 1374 | I = homog(I,t); |
---|
| 1375 | ideal J = modS(I,L); |
---|
| 1376 | J; |
---|
| 1377 | } |
---|
| 1378 | |
---|
| 1379 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1380 | |
---|
| 1381 | proc modHenselStd(ideal I, list #) |
---|
| 1382 | "USAGE: modHenselStd(I); |
---|
| 1383 | RETURN: a standard basis of I; |
---|
| 1384 | NOTE: The procedure computes a standard basis of I (over the rational |
---|
| 1385 | numbers) by using modular computations and Hensellifting. |
---|
| 1386 | For further experiments see procedure modS. |
---|
| 1387 | EXAMPLE: example modHenselStd; shows an example |
---|
| 1388 | " |
---|
| 1389 | { |
---|
| 1390 | int i,j; |
---|
| 1391 | |
---|
| 1392 | bigint p = 2134567879; |
---|
| 1393 | if(size(#)!=0) { p=#[1]; } |
---|
| 1394 | while(!primeTest(I,p)) |
---|
| 1395 | { |
---|
| 1396 | p = prime(random(2000000000,2134567879)); |
---|
| 1397 | } |
---|
| 1398 | |
---|
| 1399 | def R = basering; |
---|
| 1400 | module F,PrevG,PrevZ,Z2; |
---|
| 1401 | ideal testG,testG1,G1,G2,G3,Gp; |
---|
| 1402 | list L = p; |
---|
| 1403 | list rl = ringlist(R); |
---|
| 1404 | rl[1] = int(p); |
---|
| 1405 | |
---|
| 1406 | def S = ring(rl); |
---|
| 1407 | setring S; |
---|
[7f30e2] | 1408 | intvec opt = option(get); |
---|
[d68d30] | 1409 | option(redSB); |
---|
| 1410 | module Z,M,Z2; |
---|
| 1411 | ideal I = imap(R,I); |
---|
| 1412 | ideal Gp,G1,G2,G3; |
---|
| 1413 | Gp,Z = liftstd1(I); |
---|
| 1414 | attrib(Gp,"isSB",1); |
---|
| 1415 | module ZZ = syz(I); |
---|
| 1416 | attrib(ZZ,"isSB",1); |
---|
| 1417 | Z = reduce(Z,ZZ); |
---|
| 1418 | |
---|
| 1419 | setring R; |
---|
| 1420 | Gp = imap(S,Gp); |
---|
| 1421 | PrevZ = imap(S,Z); |
---|
| 1422 | PrevG = module(Gp); |
---|
| 1423 | F = module(I); |
---|
| 1424 | testG = farey(Gp,p); |
---|
| 1425 | attrib(testG,"isSB",1); |
---|
| 1426 | while(1) |
---|
| 1427 | { |
---|
| 1428 | i++; |
---|
| 1429 | G1 = ideal(1/(p^i) * sum(F*PrevZ,(-1)*PrevG)); |
---|
| 1430 | setring S; |
---|
| 1431 | G1 = imap(R,G1); |
---|
| 1432 | G2 = reduce(G1,Gp); |
---|
| 1433 | G3 = sum(G1,(-1)*G2); |
---|
| 1434 | M = lift(Gp,G3); |
---|
| 1435 | Z2 = (-1)*Z*M; |
---|
| 1436 | |
---|
| 1437 | setring R; |
---|
| 1438 | G2 = imap(S,G2); |
---|
| 1439 | Z2 = imap(S,Z2); |
---|
| 1440 | PrevG = sum(PrevG, module(p^i*G2)); |
---|
| 1441 | PrevZ = sum(PrevZ, multiply(poly(p^i),Z2)); |
---|
| 1442 | testG1 = farey(ideal(PrevG),p^(i+1)); |
---|
| 1443 | attrib(testG1,"isSB",1); |
---|
| 1444 | if(size(reduce(testG1,testG)) == 0) |
---|
| 1445 | { |
---|
| 1446 | if(size(reduce(I,testG1)) == 0) // I is in testG1 |
---|
| 1447 | { |
---|
| 1448 | if(pTestSB(I,testG1,L,2)) |
---|
| 1449 | { |
---|
| 1450 | G3 = std(testG1); // testG1 is SB |
---|
| 1451 | if(size(reduce(G3,testG1)) == 0) |
---|
| 1452 | { |
---|
[7f30e2] | 1453 | option(set, opt); |
---|
[d68d30] | 1454 | return(G3); |
---|
| 1455 | } |
---|
| 1456 | } |
---|
| 1457 | } |
---|
| 1458 | } |
---|
| 1459 | testG = testG1; |
---|
| 1460 | attrib(testG,"isSB",1); |
---|
| 1461 | } |
---|
| 1462 | } |
---|
| 1463 | example |
---|
| 1464 | { "EXAMPLE:"; echo = 2; |
---|
| 1465 | ring r = 0,(x,y,z),dp; |
---|
| 1466 | ideal I = 3x3+x2+1,11y5+y3+2,5z4+z2+4; |
---|
| 1467 | ideal J = modHenselStd(I); |
---|
| 1468 | J; |
---|
| 1469 | } |
---|
| 1470 | |
---|
| 1471 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1472 | |
---|
| 1473 | static proc sum(list #) |
---|
| 1474 | { |
---|
| 1475 | if(typeof(#[1])=="ideal") |
---|
| 1476 | { |
---|
| 1477 | ideal M; |
---|
| 1478 | } |
---|
| 1479 | else |
---|
| 1480 | { |
---|
| 1481 | module M; |
---|
| 1482 | } |
---|
| 1483 | |
---|
| 1484 | int i; |
---|
| 1485 | for(i = 1; i <= ncols(#[1]); i++) { M[i] = #[1][i] + #[2][i]; } |
---|
| 1486 | return(M); |
---|
| 1487 | } |
---|
| 1488 | |
---|
| 1489 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1490 | |
---|
| 1491 | static proc multiply(poly p, list #) |
---|
| 1492 | { |
---|
| 1493 | if(typeof(#[1])=="ideal") |
---|
| 1494 | { |
---|
| 1495 | ideal M; |
---|
| 1496 | } |
---|
| 1497 | else |
---|
| 1498 | { |
---|
| 1499 | module M; |
---|
| 1500 | } |
---|
| 1501 | |
---|
| 1502 | int i; |
---|
| 1503 | for(i = 1; i <= ncols(#[1]); i++) { M[i] = p * #[1][i]; } |
---|
| 1504 | return(M); |
---|
| 1505 | } |
---|
| 1506 | |
---|
| 1507 | |
---|
| 1508 | ////////////////////////////// further examples //////////////////////////////// |
---|
| 1509 | |
---|
| 1510 | /* |
---|
| 1511 | ring r = 0, (x,y,z), lp; |
---|
| 1512 | poly s1 = 5x3y2z+3y3x2z+7xy2z2; |
---|
| 1513 | poly s2 = 3xy2z2+x5+11y2z2; |
---|
| 1514 | poly s3 = 4xyz+7x3+12y3+1; |
---|
| 1515 | poly s4 = 3x3-4y3+yz2; |
---|
| 1516 | ideal i = s1, s2, s3, s4; |
---|
| 1517 | |
---|
| 1518 | ring r = 0, (x,y,z), lp; |
---|
| 1519 | poly s1 = 2xy4z2+x3y2z-x2y3z+2xyz2+7y3+7; |
---|
| 1520 | poly s2 = 2x2y4z+x2yz2-xy2z2+2x2yz-12x+12y; |
---|
| 1521 | poly s3 = 2y5z+x2y2z-xy3z-xy3+y4+2y2z; |
---|
| 1522 | poly s4 = 3xy4z3+x2y2z-xy3z+4y3z2+3xyz3+4z2-x+y; |
---|
| 1523 | ideal i = s1, s2, s3, s4; |
---|
| 1524 | |
---|
| 1525 | ring r = 0, (x,y,z), lp; |
---|
| 1526 | poly s1 = 8x2y2 + 5xy3 + 3x3z + x2yz; |
---|
| 1527 | poly s2 = x5 + 2y3z2 + 13y2z3 + 5yz4; |
---|
| 1528 | poly s3 = 8x3 + 12y3 + xz2 + 3; |
---|
| 1529 | poly s4 = 7x2y4 + 18xy3z2 + y3z3; |
---|
| 1530 | ideal i = s1, s2, s3, s4; |
---|
| 1531 | |
---|
| 1532 | int n = 6; |
---|
| 1533 | ring r = 0,(x(1..n)),lp; |
---|
| 1534 | ideal i = cyclic(n); |
---|
| 1535 | ring s = 0, (x(1..n),t), lp; |
---|
| 1536 | ideal i = imap(r,i); |
---|
| 1537 | i = homog(i,t); |
---|
| 1538 | |
---|
| 1539 | ring r = 0, (x(1..4),s), (dp(4),dp); |
---|
| 1540 | poly s1 = 1 + s^2*x(1)*x(3) + s^8*x(2)*x(3) + s^19*x(1)*x(2)*x(4); |
---|
| 1541 | poly s2 = x(1) + s^8 *x(1)* x(2)* x(3) + s^19* x(2)* x(4); |
---|
| 1542 | poly s3 = x(2) + s^10*x(3)*x(4) + s^11*x(1)*x(4); |
---|
| 1543 | poly s4 = x(3) + s^4*x(1)*x(2) + s^19*x(1)*x(3)*x(4) +s^24*x(2)*x(3)*x(4); |
---|
| 1544 | poly s5 = x(4) + s^31* x(1)* x(2)* x(3)* x(4); |
---|
| 1545 | ideal i = s1, s2, s3, s4, s5; |
---|
| 1546 | |
---|
| 1547 | ring r = 0, (x,y,z), ds; |
---|
| 1548 | int a = 16; |
---|
| 1549 | int b = 15; |
---|
| 1550 | int c = 4; |
---|
| 1551 | int t = 1; |
---|
[4cc3cac] | 1552 | poly f = x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3 |
---|
| 1553 | +x^(c-2)*y^c*(y2+t*x)^2; |
---|
[d68d30] | 1554 | ideal i = jacob(f); |
---|
| 1555 | |
---|
| 1556 | ring r = 0, (x,y,z), ds; |
---|
| 1557 | int a = 25; |
---|
| 1558 | int b = 25; |
---|
| 1559 | int c = 5; |
---|
| 1560 | int t = 1; |
---|
[4cc3cac] | 1561 | poly f = x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3 |
---|
| 1562 | +x^(c-2)*y^c*(y2+t*x)^2; |
---|
[d68d30] | 1563 | ideal i = jacob(f),f; |
---|
| 1564 | |
---|
| 1565 | ring r = 0, (x,y,z), ds; |
---|
| 1566 | int a = 10; |
---|
| 1567 | poly f = xyz*(x+y+z)^2 +(x+y+z)^3 +x^a+y^a+z^a; |
---|
| 1568 | ideal i = jacob(f); |
---|
| 1569 | |
---|
| 1570 | ring r = 0, (x,y,z), ds; |
---|
| 1571 | int a = 6; |
---|
| 1572 | int b = 8; |
---|
| 1573 | int c = 10; |
---|
| 1574 | int alpha = 5; |
---|
| 1575 | int beta = 5; |
---|
| 1576 | int t = 1; |
---|
[4cc3cac] | 1577 | poly f = x^a+y^b+z^c+x^alpha*y^(beta-5)+x^(alpha-2)*y^(beta-3) |
---|
| 1578 | +x^(alpha-3)*y^(beta-4)*z^2+x^(alpha-4)*y^(beta-4)*(y^2+t*x)^2; |
---|
[d68d30] | 1579 | ideal i = jacob(f); |
---|
| 1580 | */ |
---|
| 1581 | |
---|