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