[b0732eb] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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| 2 | version="$Id: normalTools.lib,v 1.0 2010/05/19 Exp$"; |
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| 3 | category="Commutative Algebra"; |
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| 4 | info=" |
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| 5 | LIBRARY: locnormal.lib Normalization of affine domains using local methods |
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| 6 | AUTHORS: J. Boehm boehm@mathematik.uni-kl.de |
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| 7 | W. Decker decker@mathematik.uni-kl.de |
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| 8 | S. Laplagne slaplagn@dm.uba.ar |
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| 9 | G. Pfister pfister@mathematik.uni-kl.de |
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| 10 | S. Steidel steidel@mathematik.uni-kl.de |
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| 11 | A. Steenpass steenpass@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 | Suppose A is an affine domain over a perfect field.@* |
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| 16 | This library implements a local-to-global strategy for finding the normalization |
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| 17 | of A. Following [1], the idea is to stratify the singular locus of A, apply the |
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| 18 | normalization algorithm given in [2] locally at each stratum, and put the local |
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| 19 | results together. This approach is inherently parallel.@* |
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| 20 | Furthermore we allow for the optional modular computation of the local results |
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| 21 | as provided by modnormal.lib. See again [1] for details. |
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| 22 | |
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| 23 | REFERENCES: |
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| 24 | |
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| 25 | [1] Janko Boehm, Wolfram Decker, Santiago Laplagne, Gerhard Pfister, Stefan Steidel, |
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| 26 | Andreas Steenpass: Parallel algorithms for normalization, http://arxiv.org/abs/1110.4299, 2011. |
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| 27 | |
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| 28 | [2] Gert-Martin Greuel, Santiago Laplagne, Frank Seelisch: Normalization of Rings, |
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| 29 | Journal of Symbolic Computation 9 (2010), p. 887-901 |
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| 30 | |
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| 31 | KEYWORDS: |
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| 32 | normalization; local methods; modular methods |
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| 33 | |
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| 34 | SEE ALSO: normal_lib, modnormal_lib |
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| 35 | |
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| 36 | PROCEDURES: |
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| 37 | locNormal(I, [...]); normalization of R/I using local methods |
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| 38 | |
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| 39 | "; |
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| 40 | |
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| 41 | LIB "normal.lib"; |
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| 42 | LIB "sing.lib"; |
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| 43 | LIB "modstd.lib"; |
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| 44 | |
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| 45 | static proc changeDenom(ideal U1, poly c1, poly c2, ideal I){ |
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| 46 | // Given a ring in the form 1/c1 * U, it computes a new ideal U2 such that the |
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| 47 | // ring is 1/c2 * U2. |
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| 48 | // The base ring is R, but the computations are to be done in R / I. |
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| 49 | int a; // counter |
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| 50 | def R = basering; |
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| 51 | qring Q = groebner(I); |
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| 52 | ideal U1 = fetch(R, U1); |
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| 53 | poly c1 = fetch(R, c1); |
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| 54 | poly c2 = fetch(R, c2); |
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| 55 | ideal U2 = changeDenomQ(U1, c1, c2); |
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| 56 | setring R; |
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| 57 | ideal U2 = fetch(Q, U2); |
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| 58 | return(U2); |
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| 59 | } |
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| 60 | |
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| 61 | /////////////////////////////////////////////////////////////////////////////// |
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| 62 | |
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| 63 | static proc changeDenomQ(ideal U1, poly c1, poly c2){ |
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| 64 | // Given a ring in the form 1/c1 * U, it computes a new U2 st the ring |
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| 65 | // is 1/c2 * U2. |
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| 66 | // The base ring is already a quotient ring R / I. |
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| 67 | int a; // counter |
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| 68 | ideal U2; |
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| 69 | poly p; |
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| 70 | for(a = 1; a <= ncols(U1); a++){ |
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| 71 | p = lift(c1, c2*U1[a])[1,1]; |
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| 72 | U2[a] = p; |
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| 73 | } |
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| 74 | return(U2); |
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| 75 | } |
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| 76 | |
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| 77 | ///////////////////////////////////////////////////////////////////////////////// |
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| 78 | |
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| 79 | proc locNormal(ideal I, list #) |
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| 80 | "USAGE: locNormal(I [,options]); I = prime ideal, options = list of options. @* |
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| 81 | Optional parameters in list options (can be entered in any order):@* |
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| 82 | modular: use a modular approach for the local computations. The number of primes is |
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| 83 | increased one at a time, starting with 2 primes, until the result stabelizes.@* |
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| 84 | noVerificication: if the modular approach is used, the result will not be verified. |
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| 85 | ASSUME: I is a prime ideal (the algorithm will also work for radical ideals as long as the |
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| 86 | normal command does not detect that the ideal under consideration is not prime). |
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| 87 | RETURN: a list of an ideal U and a universal denominator d such that U/d is the normalization. |
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| 88 | REMARKS: We use the local-to-global algorithm given in [1] to compute the normalization of |
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| 89 | A = R/I, where R is the basering.@* |
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| 90 | The idea is to stratify the singular locus of A, apply the normalization algorithm |
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| 91 | given in [2] locally at each stratum, and put the local results together.@* |
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| 92 | If the option modular is given, the result is returned as a probabilistic result |
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| 93 | or verified, depending on whether the option noVerificication is used or not.@* |
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| 94 | The normalization of A is represented as an R-module by returning a list of U and d, |
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| 95 | where U is an ideal of A and d is an element of A such that U/d is the normalization of A. |
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| 96 | In fact, U and d are returned as an ideal and a polynomial of the base ring R. |
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| 97 | |
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| 98 | References: |
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| 99 | |
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| 100 | [1] Janko Boehm, Wolfram Decker, Santiago Laplagne, Gerhard Pfister, Stefan Steidel, |
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| 101 | Andreas Steenpass: Parallel algorithms for normalization, http://arxiv.org/abs/1110.4299, 2011.@* |
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| 102 | [2] Gert-Martin Greuel, Santiago Laplagne, Frank Seelisch: Normalization of Rings, |
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| 103 | Journal of Symbolic Computation 9 (2010), p. 887-901 |
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| 104 | KEYWORDS: normalization; local methods; modular methods. |
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| 105 | SEE ALSO: normal_lib, modnormal_lib. |
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| 106 | EXAMPLE: example locNormal; shows an example |
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| 107 | " |
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| 108 | { |
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| 109 | // Computes the normalization by localizing at the different components of the singularity. |
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| 110 | int i; |
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| 111 | int totalLocalTime; |
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| 112 | int dbg = printlevel - voice + 2; |
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| 113 | def R = basering; |
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| 114 | |
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| 115 | int totalTime = timer; |
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| 116 | intvec LTimer; |
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| 117 | int t; |
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| 118 | int printTimings=0; |
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| 119 | |
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| 120 | int locmod; |
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| 121 | for ( i=1; i <= size(#); i++ ) |
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| 122 | { |
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| 123 | if ( typeof(#[i]) == "string" ) |
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| 124 | { |
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| 125 | if (#[i]=="modular") { locmod = 1;} |
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| 126 | if (#[i]=="printTimings") { printTimings = 1;} |
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| 127 | } |
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| 128 | } |
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| 129 | |
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| 130 | // Computes the Singular Locus |
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| 131 | list IM = mstd(I); |
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| 132 | I = IM[1]; |
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| 133 | int d = dim(I); |
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| 134 | ideal IMin = IM[2]; |
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| 135 | qring Q = I; // We work in the quotient by the groebner base of the ideal I |
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| 136 | option("redSB"); |
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| 137 | option("returnSB"); |
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| 138 | ideal I = fetch(R, I); |
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| 139 | attrib(I, "isSB", 1); |
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| 140 | ideal IMin = fetch(R, IMin); |
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| 141 | dbprint(dbg, "Computing the jacobian ideal..."); |
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| 142 | ideal J = minor(jacob(IMin), nvars(basering) - d, I); |
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| 143 | t=timer; |
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| 144 | J = modStd(J); |
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| 145 | if (printTimings==1) {"Time for modStd Jacobian "+string(timer-t);} |
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| 146 | |
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| 147 | setring R; |
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| 148 | ideal J = fetch(Q, J); |
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| 149 | // We compute a universal denominator |
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| 150 | poly condu = getSmallest(J); |
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| 151 | J = J, I; |
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| 152 | if(dbg >= 2){ |
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| 153 | "Conductor: ", condu; |
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| 154 | "The original singular locus is"; |
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| 155 | groebner(J); |
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| 156 | if(dbg >= 2){pause();} |
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| 157 | ""; |
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| 158 | } |
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| 159 | t=timer; |
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| 160 | list pd = locIdeals(J); |
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| 161 | dbprint(dbg,pd); |
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| 162 | if (printTimings==1) { |
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| 163 | "Number of maximal components to localize at: ", size(pd); |
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| 164 | ""; |
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| 165 | } |
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| 166 | |
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| 167 | ideal U; |
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| 168 | ideal resT; |
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| 169 | ideal resu; |
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| 170 | poly denomOld; |
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| 171 | poly denom; |
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| 172 | totalLocalTime = timer; |
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| 173 | int maxLocalTime; |
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| 174 | list Lnor; |
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| 175 | list parallelArguments; |
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| 176 | for(i = 1; i <= size(pd); i++){ |
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| 177 | parallelArguments[i] = list(pd[i], I, condu, i, locmod, printTimings, #); |
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| 178 | } |
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| 179 | list parallelResults = parallelWaitAll("locNormal_parallelTask", |
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| 180 | parallelArguments); |
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| 181 | for(i = 1; i <= size(pd); i++){ |
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| 182 | // We sum the result to the previous results. |
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| 183 | resu = resu, parallelResults[i][1]; |
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| 184 | Lnor[i] = parallelResults[i][1]; |
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| 185 | if(parallelResults[i][2] > maxLocalTime) { |
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| 186 | maxLocalTime = parallelResults[i][2]; |
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| 187 | } |
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| 188 | LTimer[i] = parallelResults[i][2]; |
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| 189 | } |
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| 190 | if (printTimings==1) { |
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| 191 | "List of local times: "; LTimer; |
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| 192 | "Maximal local time: "+string(maxLocalTime); |
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| 193 | } |
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| 194 | totalLocalTime = timer - totalLocalTime; |
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| 195 | if (printTimings==1) { |
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| 196 | "Total time local computations: "+string(totalLocalTime); |
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| 197 | } |
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| 198 | |
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| 199 | t=timer; |
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| 200 | resu = modStd(resu); |
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| 201 | if (printTimings==1) { |
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| 202 | "Time for combining the local results, modStd "+string(timer-t); |
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| 203 | } |
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| 204 | |
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| 205 | totalTime = timer - totalTime; |
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| 206 | if (printTimings==1) { |
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| 207 | "Total time locNormal: "+string(totalTime); |
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| 208 | "Simulated parallel time: "+string(totalTime + maxLocalTime - totalLocalTime); |
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| 209 | } |
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| 210 | return(list(resu, condu)); |
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| 211 | } |
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| 212 | |
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| 213 | example |
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| 214 | { "EXAMPLE:"; |
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| 215 | ring R = 0,(x,y,z),dp; |
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| 216 | int k = 4; |
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| 217 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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| 218 | f = subst(f,z,3x-2y+1); |
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| 219 | ring S = 0,(x,y),dp; |
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| 220 | poly f = imap(R,f); |
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| 221 | ideal i = f; |
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| 222 | list L = locNormal(i); |
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| 223 | } |
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| 224 | |
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| 225 | proc locNormal_parallelTask(ideal pdi, ideal I, poly condu, int i, int locmod, |
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| 226 | int printTimings, list #) |
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| 227 | { |
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| 228 | pdi=pdi; |
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| 229 | int t = timer; |
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| 230 | list opt = list(list("inputJ", pdi)) + #; |
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| 231 | if (printTimings==1) { |
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| 232 | "Local component ",i," of degree "+string(deg(pdi))+" and dimension " |
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| 233 | +string(dim(pdi)); |
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| 234 | } |
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| 235 | list n; |
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| 236 | ideal norT; |
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| 237 | poly denomT; |
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| 238 | if (locmod==1) { |
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| 239 | n = modNormal(I,1, opt); |
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| 240 | norT = n[1]; |
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| 241 | denomT = n[2]; |
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| 242 | } else { |
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| 243 | n = normal(I,opt); |
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| 244 | if(size(n[2]) > 1){ |
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| 245 | ERROR("Input was not prime..."); |
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| 246 | } |
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| 247 | norT = n[2][1]; |
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| 248 | denomT = norT[size(norT)]; |
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| 249 | } |
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| 250 | t = timer-t; |
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| 251 | // We compute the normalization of I localized at a component of the Singular |
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| 252 | // Locus |
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| 253 | if (printTimings==1) { |
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| 254 | "Output of normalization of component ", i, ": "; norT; |
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| 255 | ""; |
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| 256 | } |
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| 257 | ideal nor = changeDenom(norT, denomT, condu, I); |
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| 258 | return(list(nor, t)); |
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| 259 | } |
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| 260 | |
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| 261 | static proc locComps(list l) |
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| 262 | { |
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| 263 | int d = size(l); |
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| 264 | int i; |
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| 265 | int j; |
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| 266 | intvec m = 1:d; // 1 = maximal ideal |
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| 267 | ideal IT; |
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| 268 | |
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| 269 | // Check for maximal ideals |
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| 270 | for(i = 1; i<d; i++) |
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| 271 | { |
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| 272 | for(j = i+1; j <= d; j++) |
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| 273 | { |
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| 274 | if(subset(l[i], l[j])) |
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| 275 | { |
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| 276 | m[i] = 0; |
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| 277 | break; |
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| 278 | } |
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| 279 | } |
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| 280 | } |
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| 281 | list outL; |
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| 282 | for(i = 1; i<= d; i++) |
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| 283 | { |
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| 284 | if(m[i] == 1){ |
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| 285 | // Maximal ideal |
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| 286 | IT = l[i]; |
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| 287 | for(j = 1; j <= d; j++){ |
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| 288 | if(j != i) |
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| 289 | { |
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| 290 | if(subset(l[j], l[i])) |
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| 291 | { |
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| 292 | IT = intersect(IT, l[j]); |
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| 293 | } |
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| 294 | } |
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| 295 | } |
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| 296 | outL = insert(outL, IT); |
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| 297 | } |
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| 298 | } |
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| 299 | return(outL); |
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| 300 | } |
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| 301 | |
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| 302 | // I C J ?? |
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| 303 | static proc subset(ideal I, ideal J) |
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| 304 | { |
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| 305 | J = groebner(J); |
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| 306 | return(size(reduce(I, J)) == 0); |
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| 307 | } |
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| 308 | |
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| 309 | |
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| 310 | // Computes the different localizations of the radical of I at all the points of the space. |
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| 311 | static proc locIdeals(ideal I){ |
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| 312 | int i, j; |
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| 313 | I = groebner(I); |
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| 314 | |
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| 315 | // Minimal associated primes of I |
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| 316 | list l = minAssGTZ(I); |
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| 317 | //"Total number of components of the Singular Locus: ", size(l); |
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| 318 | |
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| 319 | int s = size(l); |
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| 320 | int d = dim(I); |
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| 321 | if (d==0) {return(l)}; |
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| 322 | intvec m = (1:d); |
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| 323 | // 1 = maximal ideal |
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| 324 | ideal IT; |
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| 325 | |
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| 326 | // inters will contain all the different intersections of components of I. |
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| 327 | // It is a list of list. The j-th list contains the intersections of j components of I. |
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| 328 | list inters; |
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| 329 | list compIndex; // Indicate the index of the last component in the corresponding intersection |
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| 330 | // This is used to intersect it with the remaining components. |
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| 331 | for(i = 1; i<= d; i++) |
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| 332 | { |
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| 333 | l[i] = groebner(l[i]); |
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| 334 | } |
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| 335 | |
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| 336 | // We add all the components to the list of intersections |
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| 337 | inters[1] = l; |
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| 338 | compIndex[1] = 1..s; |
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| 339 | |
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| 340 | ideal J; |
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| 341 | int e; |
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| 342 | int a; |
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| 343 | |
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| 344 | // Intersections of two or more components |
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| 345 | for(e = 1; e <= d; e++) |
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| 346 | { |
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| 347 | inters[e+1] = list(); |
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| 348 | a = 1; |
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| 349 | for(i = 1; j <= size(inters[e]); i++) |
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| 350 | { |
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| 351 | for(j = compIndex[e][i] + 1; j <= s; j++) |
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| 352 | { |
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| 353 | J = l[j] + inters[e][i]; |
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| 354 | J = groebner(J); |
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| 355 | if(J[1] != 1) |
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| 356 | { |
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| 357 | inters[e+1] = inters[e+1]+list(J); |
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| 358 | if(a == 1) |
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| 359 | { |
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| 360 | compIndex[e+1] = intvec(j); |
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| 361 | } else |
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| 362 | { |
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| 363 | compIndex[e+1][a] = j; |
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| 364 | } |
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| 365 | a++; |
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| 366 | } |
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| 367 | } |
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| 368 | } |
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| 369 | } |
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| 370 | |
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| 371 | list ids; |
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| 372 | for(e = 1; e <= d+1; e++) |
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| 373 | { |
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| 374 | ids = ids + inters[e]; |
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| 375 | } |
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| 376 | return(locComps(ids)); |
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| 377 | } |
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| 378 | |
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| 379 | /////////////////////////////////////////////////////////////////////////// |
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| 380 | // |
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| 381 | // EXAMPLES |
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| 382 | // |
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| 383 | /////////////////////////////////////////////////////////////////////////// |
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| 384 | /* |
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| 385 | // plane curves |
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| 386 | |
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| 387 | ring r24 = 0,(x,y,z),dp; |
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| 388 | int k = 2; |
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| 389 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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| 390 | f = subst(f,z,2x-y+1); |
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| 391 | ring s24 = 0,(x,y),dp; |
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| 392 | poly f = imap(r24,f); |
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| 393 | ideal i = f; |
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| 394 | |
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| 395 | locNormal(i); |
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| 396 | //modNormal(i,1); |
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| 397 | |
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| 398 | |
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| 399 | ring r24 = 0,(x,y,z),dp; |
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| 400 | int k = 3; |
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| 401 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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| 402 | f = subst(f,z,2x-y+1); |
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| 403 | ring s24 = 0,(x,y),dp; |
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| 404 | poly f = imap(r24,f); |
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| 405 | ideal i = f; |
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| 406 | |
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| 407 | locNormal(i); |
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| 408 | //modNormal(i,1,"noVerification"); |
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| 409 | |
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| 410 | |
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| 411 | ring r24 = 0,(x,y,z),dp; |
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| 412 | int k = 4; |
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| 413 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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| 414 | f = subst(f,z,2x-y+1); |
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| 415 | ring s24 = 0,(x,y),dp; |
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| 416 | poly f = imap(r24,f); |
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| 417 | ideal i = f; |
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| 418 | |
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| 419 | locNormal(i); |
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| 420 | //modNormal(i,1,"noVerification"); |
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| 421 | |
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| 422 | |
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| 423 | ring r24 = 0,(x,y,z),dp; |
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| 424 | int k = 5; |
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| 425 | poly f = (x^(k+1)+y^(k+1)+z^(k+1))^2-4*(x^(k+1)*y^(k+1)+y^(k+1)*z^(k+1)+z^(k+1)*x^(k+1)); |
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| 426 | f = subst(f,z,2x-y+1); |
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| 427 | ring s24 = 0,(x,y),dp; |
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| 428 | poly f = imap(r24,f); |
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| 429 | ideal i = f; |
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| 430 | |
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| 431 | locNormal(i); |
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| 432 | |
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| 433 | |
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| 434 | |
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| 435 | ring s24 = 0,(x,y),dp; |
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| 436 | int a=7; |
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| 437 | ideal i = ((x-1)^a-y^3)*((x+1)^a-y^3)*((x)^a-y^3)*((x-2)^a-y^3)*((x+2)^a-y^3)+y^15; |
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| 438 | |
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| 439 | locNormal(i); |
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| 440 | //modNormal(i,1); |
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| 441 | |
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| 442 | |
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| 443 | ring s24 = 0,(x,y),dp; |
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| 444 | int a=7; |
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| 445 | ideal i = ((x-1)^a-y^3)*((x+1)^a-y^3)*((x)^a-y^3)*((x-2)^a-y^3)*((x+2)^a-y^3)+y^15; |
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| 446 | |
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| 447 | locNormal(i); |
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| 448 | //modNormal(i,1); |
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| 449 | |
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| 450 | ring s24 = 0,(x,y),dp; |
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| 451 | int a=7; |
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| 452 | ideal i = ((x-1)^a-y^3)*((x+1)^a-y^3)*((x)^a-y^3)*((x-2)^a-y^3)*((x+2)^a-y^3)+y^15; |
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| 453 | |
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| 454 | locNormal(i); |
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| 455 | //modNormal(i,1,"noVerification"); |
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| 456 | |
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| 457 | |
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| 458 | |
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| 459 | |
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| 460 | ring r=0,(x,y),dp; |
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| 461 | ideal i=9127158539954x10+3212722859346x8y2+228715574724x6y4-34263110700x4y6 |
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| 462 | -5431439286x2y8-201803238y10-134266087241x8-15052058268x6y2+12024807786x4y4 |
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| 463 | +506101284x2y6-202172841y8+761328152x6-128361096x4y2+47970216x2y4-6697080y6 |
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| 464 | -2042158x4+660492x2y2-84366y4+2494x2-474y2-1; |
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| 465 | |
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| 466 | locNormal(i); |
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| 467 | //modNormal(i,1); |
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| 468 | |
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| 469 | |
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| 470 | // surfaces in A3 |
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| 471 | |
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| 472 | |
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| 473 | ring r7 = 0,(x,y,t),dp; |
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| 474 | int a=11; |
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| 475 | ideal i = x*y*(x-y)*(x+y)*(y-1)*t+(x^a-y^2)*(x^10-(y-1)^2); |
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| 476 | locNormal(i); |
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| 477 | //modNormal(i,1,"noVerification"); |
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| 478 | |
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| 479 | ring r7 = 0,(x,y,t),dp; |
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| 480 | int a=12; |
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| 481 | ideal i = x*y*(x-y)*(x+y)*(y-1)*t+(x^a-y^2)*(x^10-(y-1)^2); |
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| 482 | locNormal(i); |
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| 483 | //modNormal(i,1,"noVerification"); |
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| 484 | |
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| 485 | |
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| 486 | ring r7 = 0,(x,y,t),dp; |
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| 487 | int a=13; |
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| 488 | ideal i = x*y*(x-y)*(x+y)*(y-1)*t+(x^a-y^2)*(x^10-(y-1)^2); |
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| 489 | |
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| 490 | locNormal(i); |
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| 491 | modNormal(i,1,"noVerification"); |
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| 492 | |
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| 493 | |
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| 494 | ring r22 = 0,(x,y,z),dp; |
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| 495 | ideal i = z2-(y2-1234x3)^2*(15791x2-y3)*(1231y2-x2*(x+158))*(1357y5-3x11); |
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| 496 | |
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| 497 | locNormal(i); |
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| 498 | //modNormal(i,1,"noVerification"); |
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| 499 | |
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| 500 | |
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| 501 | ring r22 = 0,(x,y,z),dp; |
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| 502 | ideal i = z2-(y2-1234x3)^3*(15791x2-y3)*(1231y2-x2*(x+158))*(1357y5-3x11); |
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| 503 | |
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| 504 | locNormal(i); |
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| 505 | //modNormal(i,1,"noVerification"); |
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| 506 | |
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| 507 | |
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| 508 | ring r23 = 0,(x,y,z),dp; |
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| 509 | ideal i = z5-((13x-17y)*(5x2-7y3)*(3x3-2y2)*(19y2-23x2*(x+29)))^2; |
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| 510 | |
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| 511 | locNormal(i); |
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| 512 | //modNormal(i,1,"noVerification"); |
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| 513 | |
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| 514 | |
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| 515 | // curve in A3 |
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| 516 | |
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| 517 | ring r23 = 0,(x,y,z),dp; |
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| 518 | ideal i = z3-(19y2-23x2*(x+29))^2,x3-(11y2-13z2*(z+1)); |
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| 519 | |
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| 520 | locNormal(i); |
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| 521 | //modNormal(i,1,"noVerification"); |
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| 522 | |
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| 523 | |
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| 524 | ring r23 = 0,(x,y,z),dp; |
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| 525 | ideal i = z3-(19y2-23x2*(x+29))^2,x3-(11y2-13z2*(z+1))^2; |
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| 526 | |
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| 527 | locNormal(i); |
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| 528 | //modNormal(i,1,"noVerification"); |
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| 529 | |
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| 530 | // surface in A4 |
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| 531 | |
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| 532 | ring r23 = 0,(x,y,z,w),dp; |
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| 533 | ideal i = z2-(y3-123456w2)*(15791x2-y3)^2, w*z-(1231y2-x*(111x+158)); |
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| 534 | |
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| 535 | |
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| 536 | locNormal(i); |
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| 537 | //modNormal(i,1,"noVerification"); |
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| 538 | */ |
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| 539 | |
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