[6fe3a0] | 1 | // Last change 12.02.2001 (Eric Westenberger) |
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[4e461ff] | 2 | /////////////////////////////////////////////////////////////////////////////// |
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[f49f53] | 3 | version="$Id: zeroset.lib,v 1.22 2009-04-08 11:00:08 seelisch Exp $"; |
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[fd3fb7] | 4 | category="Symbolic-numerical solving"; |
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[4e461ff] | 5 | info=" |
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[0bc582c] | 6 | LIBRARY: zeroset.lib Procedures for roots and factorization |
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| 7 | AUTHOR: Thomas Bayer, email: tbayer@mathematik.uni-kl.de,@* |
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| 8 | http://wwwmayr.informatik.tu-muenchen.de/personen/bayert/@* |
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| 9 | Current address: Hochschule Ravensburg-Weingarten |
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[d12655] | 10 | |
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[b9b906] | 11 | OVERVIEW: |
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[9173792] | 12 | Algorithms for finding the zero-set of a zero-dim. ideal in Q(a)[x_1,..,x_n], |
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[731e67e] | 13 | roots and factorization of univariate polynomials over Q(a)[t] |
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| 14 | where a is an algebraic number. Written in the scope of the |
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[b9b906] | 15 | diploma thesis (advisor: Prof. Gert-Martin Greuel) 'Computations of moduli |
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[6fe3a0] | 16 | spaces of semiquasihomogeneous singularities and an implementation in Singular'. |
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[d12655] | 17 | This library is meant as a preliminary extension of the functionality |
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| 18 | of Singular for univariate factorization of polynomials over simple algebraic |
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| 19 | extensions in characteristic 0. |
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[c335c5] | 20 | |
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[0bc582c] | 21 | NOTE: |
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[d12655] | 22 | Subprocedures with postfix 'Main' require that the ring contains a variable |
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| 23 | 'a' and no parameters, and the ideal 'mpoly', where 'minpoly' from the |
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| 24 | basering is stored. |
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[4e461ff] | 25 | |
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| 26 | PROCEDURES: |
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[c335c5] | 27 | Quotient(f, g) quotient q of f w.r.t. g (in f = q*g + remainder) |
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[0bc582c] | 28 | remainder(f,g) remainder of the division of f by g |
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| 29 | roots(f) computes all roots of f in an extension field of Q |
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| 30 | sqfrNorm(f) norm of f (f must be squarefree) |
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| 31 | zeroSet(I) zero-set of the 0-dim. ideal I |
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[4e461ff] | 32 | |
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[d12655] | 33 | AUXILIARY PROCEDURES: |
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[0bc582c] | 34 | egcdMain(f, g) gcd over an algebraic extension field of Q |
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| 35 | factorMain(f) factorization of f over an algebraic extension field |
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| 36 | invertNumberMain(c) inverts an element of an algebraic extension field |
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| 37 | quotientMain(f, g) quotient of f w.r.t. g |
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| 38 | remainderMain(f,g) remainder of the division of f by g |
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| 39 | rootsMain(f) computes all roots of f, might extend the ground field |
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| 40 | sqfrNormMain(f) norm of f (f must be squarefree) |
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| 41 | containedQ(data, f) f in data ? |
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| 42 | sameQ(a, b) a == b (list a,b) |
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[4e461ff] | 43 | "; |
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| 44 | |
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[d12655] | 45 | LIB "primitiv.lib"; |
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| 46 | LIB "primdec.lib"; |
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[4e461ff] | 47 | |
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| 48 | // note : return a ring : ring need not be exported !!! |
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[6fe3a0] | 49 | |
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| 50 | // Artihmetic in Q(a)[x] without built-in procedures |
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[4e461ff] | 51 | // assume basering = Q[x,a] and minpoly is represented by mpoly(a). |
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[6fe3a0] | 52 | // the algorithms are taken from "Polynomial Algorithms in Computer Algebra", |
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[4e461ff] | 53 | // F. Winkler, Springer Verlag Wien, 1996. |
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| 54 | |
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| 55 | |
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| 56 | // To do : |
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[d12655] | 57 | // squarefree factorization |
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[4e461ff] | 58 | // multiplicities |
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| 59 | |
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| 60 | // Improvement : |
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[0bc582c] | 61 | // a main problem is the growth of the coefficients. Try roots(x7 - 1) |
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[9173792] | 62 | // return ideal mpoly ! |
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[4e461ff] | 63 | // mpoly is not monic, comes from primitive_extra |
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| 64 | |
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[d12655] | 65 | // IMPLEMENTATION |
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[4e461ff] | 66 | // |
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| 67 | // In procedures with name 'proc-name'Main a polynomial ring over a simple |
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| 68 | // extension field is represented as Q[x...,a] together with the ideal |
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| 69 | // 'mpoly' (attribute "isSB"). The arithmetic in the extension field is |
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[6fe3a0] | 70 | // implemented in the procedures in the procedures 'MultPolys' (multiplication) |
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[d12655] | 71 | // and 'InvertNumber' (inversion). After addition and substraction one should |
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[4e461ff] | 72 | // apply 'SimplifyPoly' to the result to reduce the result w.r.t. 'mpoly'. |
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[d12655] | 73 | // This is done by reducing each coefficient seperately, which is more |
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[4e461ff] | 74 | // efficient for polynomials with many terms. |
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| 75 | |
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| 76 | |
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| 77 | /////////////////////////////////////////////////////////////////////////////// |
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| 78 | |
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[0bc582c] | 79 | proc roots(poly f) |
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| 80 | "USAGE: roots(f); where f is a polynomial |
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[9173792] | 81 | PURPOSE: compute all roots of f in a finite extension of the ground field |
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[4e461ff] | 82 | without multiplicities. |
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| 83 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
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[f49f53] | 84 | containing a list 'theRoots' and polynomials 'newA' and 'f': |
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[9173792] | 85 | @format |
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[f49f53] | 86 | - 'theRoots' is the list of roots of the polynomial f (no multiplicities) |
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[34b0314] | 87 | - if the ground field is Q(a') and the extension field is Q(a), then |
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[3c4dcc] | 88 | 'newA' is the representation of a' in Q(a). |
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[9173792] | 89 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
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| 90 | then 'newA' = 'a'. |
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| 91 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
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| 92 | - 'f' is the polynomial f in Q(a) (a' being substituted by 'newA') |
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| 93 | @end format |
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[4e461ff] | 94 | ASSUME: ground field to be Q or a simple extension of Q given by a minpoly |
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[f49f53] | 95 | EXAMPLE: example roots; shows an example |
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[4e461ff] | 96 | " |
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| 97 | { |
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| 98 | int dbPrt = printlevel-voice+3; |
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| 99 | |
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| 100 | // create a new ring where par(1) is replaced by the variable |
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| 101 | // with the same name or, if basering does not contain a parameter, |
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| 102 | // with a new variable 'a'. |
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| 103 | |
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| 104 | def ROB = basering; |
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| 105 | def ROR = TransferRing(basering); |
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| 106 | setring ROR; |
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[6fe3a0] | 107 | export(ROR); |
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[4e461ff] | 108 | |
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| 109 | // get the polynomial f and find the roots |
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| 110 | |
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| 111 | poly f = imap(ROB, f); |
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[0bc582c] | 112 | list result = rootsMain(f); // find roots of f |
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[4e461ff] | 113 | |
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[7d56875] | 114 | // store the roots and the new representation of 'a' and transform |
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[4e461ff] | 115 | // the coefficients of f. |
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| 116 | |
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[f49f53] | 117 | list theRoots = result[1]; |
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[4e461ff] | 118 | poly newA = result[2]; |
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| 119 | map F = basering, maxideal(1); |
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| 120 | F[nvars(basering)] = newA; |
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| 121 | poly fn = SimplifyPoly(F(f)); |
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| 122 | |
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| 123 | // create a new ring with minploy = mpoly[1] (from ROR) |
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| 124 | |
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| 125 | def RON = NewBaseRing(); |
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| 126 | setring(RON); |
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[f49f53] | 127 | list theRoots = imap(ROR, theRoots); |
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[4e461ff] | 128 | poly newA = imap(ROR, newA); |
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| 129 | poly f = imap(ROR, fn); |
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[3b77465] | 130 | kill ROR; |
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[f49f53] | 131 | export(theRoots); |
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[4e461ff] | 132 | export(newA); |
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| 133 | export(f); dbprint(dbPrt," |
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[f49f53] | 134 | // 'roots' created a new ring which contains the list 'theRoots' and |
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[4e461ff] | 135 | // the polynomials 'f' and 'newA' |
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| 136 | // To access the roots, newA and the new representation of f, type |
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[f49f53] | 137 | def R = roots(f); setring R; theRoots; newA; f; |
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[4e461ff] | 138 | "); |
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| 139 | return(RON); |
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| 140 | } |
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| 141 | example |
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| 142 | {"EXAMPLE:"; echo = 2; |
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| 143 | ring R = (0,a), x, lp; |
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| 144 | minpoly = a2+1; |
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| 145 | poly f = x3 - a; |
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[0bc582c] | 146 | def R1 = roots(f); |
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[4e461ff] | 147 | setring R1; |
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| 148 | minpoly; |
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| 149 | newA; |
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| 150 | f; |
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[f49f53] | 151 | theRoots; |
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[4e461ff] | 152 | map F; |
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[f49f53] | 153 | F[1] = theRoots[1]; |
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[4e461ff] | 154 | F(f); |
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| 155 | } |
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| 156 | |
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| 157 | /////////////////////////////////////////////////////////////////////////////// |
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| 158 | |
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[0bc582c] | 159 | proc rootsMain(poly f) |
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| 160 | "USAGE: rootsMain(f); where f is a polynomial |
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[9173792] | 161 | PURPOSE: compute all roots of f in a finite extension of the ground field |
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[4e461ff] | 162 | without multiplicities. |
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| 163 | RETURN: list, all entries are polynomials |
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[9173792] | 164 | @format |
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| 165 | _[1] = roots of f, each entry is a polynomial |
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[0bc582c] | 166 | _[2] = 'newA' - if the ground field is Q(b) and the extension field |
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| 167 | is Q(a), then 'newA' is the representation of b in Q(a) |
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[34b0314] | 168 | _[3] = minpoly of the algebraic extension of the ground field |
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[9173792] | 169 | @end format |
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| 170 | ASSUME: basering = Q[x,a] ideal mpoly must be defined, it might be 0! |
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[731e67e] | 171 | NOTE: might change the ideal mpoly!! |
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[f49f53] | 172 | EXAMPLE: example rootsMain; shows an example |
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[4e461ff] | 173 | " |
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| 174 | { |
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| 175 | int i, linFactors, nlinFactors, dbPrt; |
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| 176 | intvec wt = 1,0; // deg(a) = 0 |
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| 177 | list factorList, nlFactors, nlMult, roots, result; |
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| 178 | poly fa, lc; |
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| 179 | |
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| 180 | dbPrt = printlevel-voice+3; |
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| 181 | |
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| 182 | // factor f in Q(a)[t] to obtain the roots lying in Q(a) |
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| 183 | // firstly, find roots of the linear factors, |
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| 184 | // nonlinear factors are processed later |
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| 185 | |
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[0bc582c] | 186 | dbprint(dbPrt, "roots of " + string(f) + ", minimal polynomial = " + string(mpoly[1])); |
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| 187 | factorList = factorMain(f); // Factorize f |
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[4e461ff] | 188 | dbprint(dbPrt, (" prime factors of f are : " + string(factorList[1]))); |
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| 189 | |
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| 190 | linFactors = 0; |
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| 191 | nlinFactors = 0; |
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| 192 | for(i = 2; i <= size(factorList[1]); i = i + 1) { // find linear and nonlinear factors |
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| 193 | fa = factorList[1][i]; |
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| 194 | if(deg(fa, wt) == 1) { |
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| 195 | linFactors++; // get the root from the linear factor |
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| 196 | lc = LeadTerm(fa, 1)[3]; |
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[0bc582c] | 197 | fa = MultPolys(invertNumberMain(lc), fa); // make factor monic |
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[4e461ff] | 198 | roots[linFactors] = var(1) - fa; // fa is monic !! |
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| 199 | } |
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| 200 | else { // ignore nonlinear factors |
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| 201 | nlinFactors++; |
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| 202 | nlFactors[nlinFactors] = factorList[1][i]; |
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| 203 | nlMult[nlinFactors] = factorList[2][i]; |
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| 204 | } |
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| 205 | } |
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[34b0314] | 206 | if(linFactors == size(factorList[1]) - 1) { // all roots of f are contained in the ground field |
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[4e461ff] | 207 | result[1] = roots; |
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| 208 | result[2] = var(2); |
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| 209 | result[3] = mpoly[1]; |
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| 210 | return(result); |
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| 211 | } |
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| 212 | |
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[34b0314] | 213 | // process the nonlinear factors, i.e., extend the ground field |
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[4e461ff] | 214 | // where a nonlinear factor (irreducible) is a minimal polynomial |
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| 215 | // compute the primitive element of this extension |
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| 216 | |
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| 217 | ideal primElem, minPolys, Fid; |
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| 218 | list partSol; |
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| 219 | map F, Xchange; |
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| 220 | poly f1, newA, mp, oldMinPoly; |
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| 221 | |
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| 222 | Fid = mpoly; |
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| 223 | F[1] = var(1); |
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| 224 | Xchange[1] = var(2); // the variables have to be exchanged |
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| 225 | Xchange[2] = var(1); // for the use of 'primitive' |
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| 226 | |
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| 227 | if(nlinFactors == 1) { // one nl factor |
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| 228 | |
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| 229 | // compute the roots of the nonlinear (irreducible, monic) factor f1 of f |
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| 230 | // by extending the basefield by a' with minimal polynomial f1 |
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[0bc582c] | 231 | // Then call roots(f1) to find the roots of f1 over the new base field |
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[4e461ff] | 232 | |
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| 233 | f1 = nlFactors[1]; |
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| 234 | if(mpoly[1] != 0) { |
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| 235 | mp = mpoly[1]; |
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| 236 | minPolys = Xchange(mp), Xchange(f1); |
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| 237 | primElem = primitive_extra(minPolys); // no random coord. change |
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| 238 | mpoly = std(primElem[1]); |
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| 239 | F = basering, maxideal(1); |
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| 240 | F[2] = primElem[2]; // transfer all to the new representation |
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| 241 | newA = primElem[2]; // new representation of a |
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| 242 | f1 = SimplifyPoly(F(f1)); //reduce(F(f1), mpoly); |
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| 243 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
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| 244 | } |
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| 245 | else { |
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| 246 | mpoly = std(Xchange(f1)); |
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| 247 | newA = var(2); |
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| 248 | } |
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| 249 | result[3] = mpoly[1]; |
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| 250 | oldMinPoly = mpoly[1]; |
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[0bc582c] | 251 | partSol = rootsMain(f1); // find roots of f1 over extended field |
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[4e461ff] | 252 | |
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| 253 | if(oldMinPoly != partSol[3]) { // minpoly has changed ? |
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| 254 | // all previously computed roots must be transformed |
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| 255 | // because the minpoly has changed |
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| 256 | result[3] = partSol[3]; // new minpoly |
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| 257 | F[2] = partSol[2]; // new representation of algebraic number |
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| 258 | if(size(roots) > 0) {roots = SimplifyData(F(roots)); } |
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| 259 | newA = SimplifyPoly(F(newA)); // F(newA); |
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| 260 | } |
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| 261 | roots = roots + partSol[1]; // add roots |
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| 262 | result[2] = newA; |
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| 263 | result[1] = roots; |
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| 264 | } |
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| 265 | else { // more than one nonlinear (irreducible) factor (f_1,...,f_r) |
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[0bc582c] | 266 | // solve each of them by rootsMain(f_i), append their roots |
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[4e461ff] | 267 | // change the minpoly and transform all previously computed |
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| 268 | // roots if necessary. |
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| 269 | // Note that the for-loop is more or less book-keeping |
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| 270 | |
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| 271 | newA = var(2); |
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| 272 | result[2] = newA; |
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| 273 | for(i = 1; i <= size(nlFactors); i = i + 1) { |
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| 274 | oldMinPoly = mpoly[1]; |
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[0bc582c] | 275 | partSol = rootsMain(nlFactors[i]); // main work |
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[4e461ff] | 276 | nlFactors[i] = 0; // delete factor |
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| 277 | result[3] = partSol[3]; // store minpoly |
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| 278 | |
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| 279 | // book-keeping starts here as in the case 1 nonlinear factor |
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| 280 | |
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| 281 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
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| 282 | F = basering, maxideal(1); |
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| 283 | F[2] = partSol[2]; // transfer all to the new representation |
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| 284 | newA = SimplifyPoly(F(newA)); // F(newA); new representation of a |
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| 285 | result[2] = newA; |
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| 286 | if(i < size(nlFactors)) { |
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| 287 | nlFactors = SimplifyData(F(nlFactors)); |
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| 288 | } // transform remaining factors |
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| 289 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
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| 290 | } |
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| 291 | roots = roots + partSol[1]; // transform roots |
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| 292 | result[1] = roots; |
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| 293 | } // end more than one nl factor |
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| 294 | |
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| 295 | } |
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| 296 | return(result); |
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| 297 | } |
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| 298 | |
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| 299 | /////////////////////////////////////////////////////////////////////////////// |
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| 300 | |
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[0bc582c] | 301 | proc zeroSet(ideal I, list #) |
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| 302 | "USAGE: zeroSet(I [,opt] ); I=ideal, opt=integer |
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[9173792] | 303 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
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[34b0314] | 304 | of the ground field. |
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[4e461ff] | 305 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
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[9173792] | 306 | containing a list 'zeroset', a polynomial 'newA', and an |
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| 307 | ideal 'id': |
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| 308 | @format |
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| 309 | - 'zeroset' is the list of the zeros of the ideal I, each zero is an ideal. |
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[0bc582c] | 310 | - if the ground field is Q(b) and the extension field is Q(a), then |
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| 311 | 'newA' is the representation of b in Q(a). |
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[9173792] | 312 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
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| 313 | then 'newA' = 'a'. |
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[3c4dcc] | 314 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
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[9173792] | 315 | - 'id' is the ideal I in Q(a)[x_1,...] (a' substituted by 'newA') |
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| 316 | @end format |
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[d12655] | 317 | ASSUME: dim(I) = 0, and ground field to be Q or a simple extension of Q given |
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| 318 | by a minpoly. |
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[0bc582c] | 319 | OPTIONS: opt = 0: no primary decomposition (default) |
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| 320 | opt > 0: primary decomposition |
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| 321 | NOTE: If I contains an algebraic number (parameter) then I must be |
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[35f23d] | 322 | transformed w.r.t. 'newA' in the new ring. |
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[0bc582c] | 323 | EXAMPLE: example zeroSet; shows an example |
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[4e461ff] | 324 | " |
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| 325 | { |
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| 326 | int primaryDecQ, dbPrt; |
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| 327 | list rp; |
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| 328 | |
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| 329 | dbPrt = printlevel-voice+2; |
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| 330 | |
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| 331 | if(size(#) > 0) { primaryDecQ = #[1]; } |
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| 332 | else { primaryDecQ = 0; } |
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| 333 | |
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[3c4dcc] | 334 | // create a new ring 'ZSR' with one additional variable instead of the |
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[6fe3a0] | 335 | // parameter |
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[3c4dcc] | 336 | // if the basering does not contain a parameter then 'a' is used as the |
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[6fe3a0] | 337 | // additional variable. |
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[4e461ff] | 338 | |
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| 339 | def RZSB = basering; |
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[6fe3a0] | 340 | def ZSR = TransferRing(RZSB); |
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[4e461ff] | 341 | setring ZSR; |
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| 342 | |
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| 343 | // get ideal I and find the zero-set |
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| 344 | |
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| 345 | ideal id = std(imap(RZSB, I)); |
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[6fe3a0] | 346 | // print(dim(id)); |
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| 347 | if(dim(id) > 1) { // new variable adjoined to ZSR |
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| 348 | ERROR(" ideal not zerodimensional "); |
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| 349 | } |
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[4e461ff] | 350 | |
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[0bc582c] | 351 | list result = zeroSetMain(id, primaryDecQ); |
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[4e461ff] | 352 | |
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| 353 | // store the zero-set, minimal polynomial and the new representative of 'a' |
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| 354 | |
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[f49f53] | 355 | list theZeroset = result[1]; |
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[4e461ff] | 356 | poly newA = result[2]; |
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| 357 | poly minPoly = result[3][1]; |
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| 358 | |
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[3c4dcc] | 359 | // transform the generators of the ideal I w.r.t. the new representation |
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[6fe3a0] | 360 | // of 'a' |
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[4e461ff] | 361 | |
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| 362 | map F = basering, maxideal(1); |
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| 363 | F[nvars(basering)] = newA; |
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| 364 | id = SimplifyData(F(id)); |
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| 365 | |
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| 366 | // create a new ring with minpoly = minPoly |
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| 367 | |
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| 368 | def RZBN = NewBaseRing(); |
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| 369 | setring RZBN; |
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| 370 | |
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[f49f53] | 371 | list theZeroset = imap(ZSR, theZeroset); |
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[4e461ff] | 372 | poly newA = imap(ZSR, newA); |
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| 373 | ideal id = imap(ZSR, id); |
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[3b77465] | 374 | kill ZSR; |
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[4e461ff] | 375 | |
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| 376 | export(id); |
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[f49f53] | 377 | export(theZeroset); |
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[4e461ff] | 378 | export(newA); |
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| 379 | dbprint(dbPrt," |
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[f49f53] | 380 | // 'zeroSet' created a new ring which contains the list 'theZeroset', the ideal |
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[4e461ff] | 381 | // 'id' and the polynomial 'newA'. 'id' is the ideal of the input transformed |
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| 382 | // w.r.t. 'newA'. |
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| 383 | // To access the zero-set, 'newA' and the new representation of the ideal, type |
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[f49f53] | 384 | def R = zeroSet(I); setring R; theZeroset; newA; id; |
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[4e461ff] | 385 | "); |
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[6fe3a0] | 386 | setring RZSB; |
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[4e461ff] | 387 | return(RZBN); |
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| 388 | } |
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| 389 | example |
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| 390 | {"EXAMPLE:"; echo = 2; |
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| 391 | ring R = (0,a), (x,y,z), lp; |
---|
| 392 | minpoly = a2 + 1; |
---|
| 393 | ideal I = x2 - 1/2, a*z - 1, y - 2; |
---|
[0bc582c] | 394 | def T = zeroSet(I); |
---|
[4e461ff] | 395 | setring T; |
---|
| 396 | minpoly; |
---|
| 397 | newA; |
---|
| 398 | id; |
---|
[f49f53] | 399 | theZeroset; |
---|
| 400 | map F1 = basering, theZeroset[1]; |
---|
| 401 | map F2 = basering, theZeroset[2]; |
---|
[4e461ff] | 402 | F1(id); |
---|
| 403 | F2(id); |
---|
| 404 | } |
---|
| 405 | |
---|
| 406 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 407 | |
---|
[0bc582c] | 408 | proc invertNumberMain(poly f) |
---|
| 409 | "USAGE: invertNumberMain(f); where f is a polynomial |
---|
[731e67e] | 410 | PURPOSE: compute 1/f if f is a number in Q(a), i.e., f is represented by a |
---|
[4e461ff] | 411 | polynomial in Q[a]. |
---|
| 412 | RETURN: poly 1/f |
---|
[9173792] | 413 | ASSUME: basering = Q[x_1,...,x_n,a], ideal mpoly must be defined and != 0 ! |
---|
[980120f] | 414 | NOTE: outdated, use / instead |
---|
[4e461ff] | 415 | " |
---|
| 416 | { |
---|
| 417 | if(diff(f, var(1)) != 0) { ERROR("number must not contain variable !");} |
---|
| 418 | |
---|
| 419 | int n = nvars(basering); |
---|
| 420 | def RINB = basering; |
---|
| 421 | string ringSTR = "ring RINR = 0, " + string(var(n)) + ", dp;"; |
---|
| 422 | execute(ringSTR); // new ring = Q[a] |
---|
| 423 | |
---|
| 424 | list gcdList; |
---|
| 425 | poly f, g, inv; |
---|
| 426 | |
---|
| 427 | f = imap(RINB, f); |
---|
| 428 | g = imap(RINB, mpoly)[1]; |
---|
| 429 | |
---|
| 430 | if(diff(f, var(1)) != 0) { inv = extgcd(f, g)[2]; } // f contains var(1) |
---|
| 431 | else { inv = 1/f;} // f element in Q |
---|
| 432 | |
---|
| 433 | setring(RINB); |
---|
| 434 | return(imap(RINR, inv)); |
---|
| 435 | } |
---|
| 436 | |
---|
| 437 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 438 | |
---|
| 439 | proc MultPolys(poly f, poly g) |
---|
| 440 | "USAGE: MultPolys(f, g); poly f,g |
---|
| 441 | PURPOSE: multiply the polynomials f and g and reduce them w.r.t. mpoly |
---|
| 442 | RETURN: poly f*g |
---|
| 443 | ASSUME: basering = Q[x,a], ideal mpoly must be defined, it might be 0 ! |
---|
| 444 | " |
---|
| 445 | { |
---|
| 446 | return(SimplifyPoly(f * g)); |
---|
| 447 | } |
---|
| 448 | |
---|
| 449 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 450 | |
---|
| 451 | proc LeadTerm(poly f, int i) |
---|
| 452 | "USAGE: LeadTerm(f); poly f, int i |
---|
[9173792] | 453 | PURPOSE: compute the leading coef and term of f w.r.t var(i), where the last |
---|
[4e461ff] | 454 | ring variable is treated as a parameter. |
---|
| 455 | RETURN: list of polynomials |
---|
| 456 | _[1] = leading term |
---|
| 457 | _[2] = leading monomial |
---|
| 458 | _[3] = leading coefficient |
---|
| 459 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
| 460 | " |
---|
| 461 | { |
---|
| 462 | list result; |
---|
| 463 | matrix co = coef(f, var(i)); |
---|
| 464 | result[1] = co[1, 1]*co[2, 1]; |
---|
| 465 | result[2] = co[1, 1]; |
---|
| 466 | result[3] = co[2, 1]; |
---|
| 467 | return(result); |
---|
| 468 | } |
---|
| 469 | |
---|
| 470 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 471 | |
---|
[c335c5] | 472 | proc Quotient(poly f, poly g) |
---|
| 473 | "USAGE: Quotient(f, g); where f,g are polynomials; |
---|
[6fe3a0] | 474 | PURPOSE: compute the quotient q and remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
[4e461ff] | 475 | RETURN: list of polynomials |
---|
[6fe3a0] | 476 | @format |
---|
| 477 | _[1] = quotient q |
---|
| 478 | _[2] = remainder r |
---|
| 479 | @end format |
---|
[4e461ff] | 480 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
[f3c31a] | 481 | NOTE: This procedure is outdated, and should no longer be used. Use div and mod |
---|
| 482 | instead. |
---|
[f49f53] | 483 | EXAMPLE: example Quotient; shows an example |
---|
[4e461ff] | 484 | " |
---|
| 485 | { |
---|
[6fe3a0] | 486 | def QUOB = basering; |
---|
| 487 | def QUOR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
| 488 | setring QUOR; |
---|
| 489 | export(QUOR); |
---|
| 490 | poly f = imap(QUOB, f); |
---|
| 491 | poly g = imap(QUOB, g); |
---|
[0bc582c] | 492 | list result = quotientMain(f, g); |
---|
[6fe3a0] | 493 | |
---|
| 494 | setring(QUOB); |
---|
| 495 | list result = imap(QUOR, result); |
---|
[3b77465] | 496 | kill QUOR; |
---|
[6fe3a0] | 497 | return(result); |
---|
[4e461ff] | 498 | } |
---|
| 499 | example |
---|
| 500 | {"EXAMPLE:"; echo = 2; |
---|
| 501 | ring R = (0,a), x, lp; |
---|
| 502 | minpoly = a2+1; |
---|
| 503 | poly f = x4 - 2; |
---|
| 504 | poly g = x - a; |
---|
[c335c5] | 505 | list qr = Quotient(f, g); |
---|
[4e461ff] | 506 | qr; |
---|
| 507 | qr[1]*g + qr[2] - f; |
---|
| 508 | } |
---|
| 509 | |
---|
[0bc582c] | 510 | proc quotientMain(poly f, poly g) |
---|
| 511 | "USAGE: quotientMain(f, g); where f,g are polynomials |
---|
[731e67e] | 512 | PURPOSE: compute the quotient q and remainder r s.th. f = g*q + r, deg(r) < deg(g) |
---|
[4e461ff] | 513 | RETURN: list of polynomials |
---|
[6fe3a0] | 514 | @format |
---|
| 515 | _[1] = quotient q |
---|
| 516 | _[2] = remainder r |
---|
| 517 | @end format |
---|
| 518 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
| 519 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
[980120f] | 520 | NOTE: outdated, use div/mod instead |
---|
[4e461ff] | 521 | " |
---|
| 522 | { |
---|
[6fe3a0] | 523 | if(g == 0) { ERROR("Division by zero !");} |
---|
| 524 | |
---|
| 525 | def QMB = basering; |
---|
| 526 | def QMR = NewBaseRing(); |
---|
| 527 | setring QMR; |
---|
| 528 | poly f, g, h; |
---|
| 529 | h = imap(QMB, f) / imap(QMB, g); |
---|
| 530 | setring QMB; |
---|
| 531 | return(list(imap(QMR, h), 0)); |
---|
[4e461ff] | 532 | } |
---|
| 533 | |
---|
| 534 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 535 | |
---|
[0bc582c] | 536 | proc remainder(poly f, poly g) |
---|
| 537 | "USAGE: remainder(f, g); where f,g are polynomials |
---|
[9173792] | 538 | PURPOSE: compute the remainder of the division of f by g, i.e. a polynomial r |
---|
| 539 | s.t. f = g*q + r, deg(r) < deg(g). |
---|
[4e461ff] | 540 | RETURN: poly |
---|
| 541 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
[980120f] | 542 | NOTE: outdated, use mod/reduce instead |
---|
[4e461ff] | 543 | " |
---|
| 544 | { |
---|
| 545 | def REMB = basering; |
---|
| 546 | def REMR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
| 547 | setring(REMR); |
---|
[6fe3a0] | 548 | export(REMR); |
---|
[4e461ff] | 549 | poly f = imap(REMB, f); |
---|
| 550 | poly g = imap(REMB, g); |
---|
[0bc582c] | 551 | poly h = remainderMain(f, g); |
---|
[4e461ff] | 552 | |
---|
| 553 | setring(REMB); |
---|
| 554 | poly r = imap(REMR, h); |
---|
[3b77465] | 555 | kill REMR; |
---|
[4e461ff] | 556 | return(r); |
---|
| 557 | } |
---|
| 558 | example |
---|
| 559 | {"EXAMPLE:"; echo = 2; |
---|
| 560 | ring R = (0,a), x, lp; |
---|
| 561 | minpoly = a2+1; |
---|
| 562 | poly f = x4 - 1; |
---|
| 563 | poly g = x3 - 1; |
---|
[0bc582c] | 564 | remainder(f, g); |
---|
[4e461ff] | 565 | } |
---|
| 566 | |
---|
[0bc582c] | 567 | proc remainderMain(poly f, poly g) |
---|
| 568 | "USAGE: remainderMain(f, g); where f,g are polynomials |
---|
[9173792] | 569 | PURPOSE: compute the remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
[4e461ff] | 570 | RETURN: poly |
---|
[9173792] | 571 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
| 572 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
[980120f] | 573 | NOTE: outdated, use mod/reduce instead |
---|
[4e461ff] | 574 | " |
---|
| 575 | { |
---|
| 576 | int dg; |
---|
| 577 | intvec wt = 1,0;; |
---|
| 578 | poly lc, g1, r; |
---|
| 579 | |
---|
| 580 | if(deg(g, wt) == 0) { return(0); } |
---|
| 581 | |
---|
| 582 | lc = LeadTerm(g, 1)[3]; |
---|
[0bc582c] | 583 | g1 = MultPolys(invertNumberMain(lc), g); // make g monic |
---|
[4e461ff] | 584 | |
---|
| 585 | return(SimplifyPoly(reduce(f, std(g1)))); |
---|
| 586 | } |
---|
| 587 | |
---|
| 588 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 589 | |
---|
[0bc582c] | 590 | proc egcdMain(poly f, poly g) |
---|
| 591 | "USAGE: egcdMain(f, g); where f,g are polynomials |
---|
[9173792] | 592 | PURPOSE: compute the polynomial gcd of f and g over Q(a)[x] |
---|
[4e461ff] | 593 | RETURN: poly |
---|
[9173792] | 594 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
| 595 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
[980120f] | 596 | NOTE: outdated, use gcd instead |
---|
[4e461ff] | 597 | EXAMPLE: example EGCD; shows an example |
---|
| 598 | " |
---|
| 599 | { |
---|
[c1986a] | 600 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gcd |
---|
[4e461ff] | 601 | int i = 1; |
---|
| 602 | poly r1, r2, r; |
---|
| 603 | |
---|
| 604 | r1 = f; |
---|
| 605 | r2 = g; |
---|
| 606 | |
---|
| 607 | while(r2 != 0) { |
---|
[0bc582c] | 608 | r = remainderMain(r1, r2); |
---|
[4e461ff] | 609 | r1 = r2; |
---|
| 610 | r2 = r; |
---|
| 611 | } |
---|
| 612 | return(r1); |
---|
| 613 | } |
---|
| 614 | |
---|
| 615 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 616 | |
---|
| 617 | proc MEGCD(poly f, poly g, int varIndex) |
---|
| 618 | "USAGE: MEGCD(f, g, i); poly f, g; int i |
---|
[9173792] | 619 | PURPOSE: compute the polynomial gcd of f and g in the i'th variable |
---|
[4e461ff] | 620 | RETURN: poly |
---|
[d12655] | 621 | ASSUME: f, g are polynomials in var(i), last variable is the algebraic number |
---|
[f49f53] | 622 | EXAMPLE: example MEGCD; shows an example |
---|
[4e461ff] | 623 | " |
---|
| 624 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gc |
---|
| 625 | // not used ! |
---|
| 626 | { |
---|
| 627 | string @str, @sf, @sg, @mp, @parName; |
---|
| 628 | |
---|
| 629 | def @RGCDB = basering; |
---|
| 630 | |
---|
| 631 | @sf = string(f); |
---|
| 632 | @sg = string(g); |
---|
| 633 | @mp = string(minpoly); |
---|
| 634 | |
---|
| 635 | if(npars(basering) == 0) { @parName = "0";} |
---|
| 636 | else { @parName = "(0, " + parstr(basering) + ")"; } |
---|
| 637 | @str = "ring @RGCD = " + @parName + ", " + string(var(varIndex)) + ", dp;"; |
---|
| 638 | execute(@str); |
---|
| 639 | if(@mp != "0") { execute ("minpoly = " + @mp + ";"); } |
---|
| 640 | execute("poly @f = " + @sf + ";"); |
---|
| 641 | execute("poly @g = " + @sg + ";"); |
---|
[6fe3a0] | 642 | export(@RGCD); |
---|
[4f5b07] | 643 | poly @h = gcd(@f, @g); |
---|
[4e461ff] | 644 | setring(@RGCDB); |
---|
| 645 | poly h = imap(@RGCD, @h); |
---|
[3b77465] | 646 | kill @RGCD; |
---|
[4e461ff] | 647 | return(h); |
---|
| 648 | } |
---|
| 649 | |
---|
| 650 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 651 | |
---|
[0bc582c] | 652 | proc sqfrNorm(poly f) |
---|
| 653 | "USAGE: sqfrNorm(f); where f is a polynomial |
---|
[9173792] | 654 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
| 655 | RETURN: list with 3 entries |
---|
| 656 | @format |
---|
| 657 | _[1] = squarefree norm of g (poly) |
---|
| 658 | _[2] = g (= f(x - s*a)) (poly) |
---|
| 659 | _[3] = s (int) |
---|
| 660 | @end format |
---|
[4e461ff] | 661 | ASSUME: f must be squarefree, basering = Q(a)[x] and minpoly != 0. |
---|
| 662 | NOTE: the norm is an element of Q[x] |
---|
[f49f53] | 663 | EXAMPLE: example sqfrNorm; shows an example |
---|
[4e461ff] | 664 | " |
---|
| 665 | { |
---|
| 666 | def SNB = basering; |
---|
[3c4dcc] | 667 | def SNR = TransferRing(SNB); // new ring with parameter 'a' |
---|
[6fe3a0] | 668 | // replaced by a variable |
---|
[4e461ff] | 669 | setring SNR; |
---|
| 670 | poly f = imap(SNB, f); |
---|
[0bc582c] | 671 | list result = sqfrNormMain(f); // squarefree norm of f |
---|
[4e461ff] | 672 | |
---|
[6fe3a0] | 673 | setring SNB; |
---|
[4e461ff] | 674 | list result = imap(SNR, result); |
---|
[6fe3a0] | 675 | kill SNR; |
---|
[4e461ff] | 676 | return(result); |
---|
| 677 | } |
---|
| 678 | example |
---|
| 679 | {"EXAMPLE:"; echo = 2; |
---|
| 680 | ring R = (0,a), x, lp; |
---|
| 681 | minpoly = a2+1; |
---|
| 682 | poly f = x4 - 2*x + 1; |
---|
[0bc582c] | 683 | sqfrNorm(f); |
---|
[4e461ff] | 684 | } |
---|
| 685 | |
---|
[0bc582c] | 686 | proc sqfrNormMain(poly f) |
---|
| 687 | "USAGE: sqfrNorm(f); where f is a polynomial |
---|
[9173792] | 688 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
| 689 | RETURN: list with 3 entries |
---|
| 690 | @format |
---|
| 691 | _[1] = squarefree norm of g (poly) |
---|
| 692 | _[2] = g (= f(x - s*a)) (poly) |
---|
| 693 | _[3] = s (int) |
---|
| 694 | @end format |
---|
| 695 | ASSUME: f must be squarefree, basering = Q[x,a] and ideal mpoly is equal to |
---|
[731e67e] | 696 | 'minpoly', this represents the ring Q(a)[x] together with 'minpoly'. |
---|
[4e461ff] | 697 | NOTE: the norm is an element of Q[x] |
---|
[f49f53] | 698 | EXAMPLE: example SqfrNorm; shows an example |
---|
[4e461ff] | 699 | " |
---|
| 700 | { |
---|
[6fe3a0] | 701 | def SNRMB = basering; |
---|
[4e461ff] | 702 | int s = 0; |
---|
| 703 | intvec wt = 1,0; |
---|
| 704 | ideal mapId; |
---|
| 705 | // list result; |
---|
| 706 | poly g, N, N1, h; |
---|
| 707 | string ringSTR; |
---|
| 708 | |
---|
| 709 | mapId[1] = var(1) - var(2); // linear transformation |
---|
| 710 | mapId[2] = var(2); |
---|
[6fe3a0] | 711 | map Fs = SNRMB, mapId; |
---|
[4e461ff] | 712 | |
---|
| 713 | N = resultant(f, mpoly[1], var(2)); // norm of f |
---|
| 714 | N1 = diff(N, var(1)); |
---|
| 715 | g = f; |
---|
| 716 | |
---|
| 717 | ringSTR = "ring SNRM1 = 0, " + string(var(1)) + ", dp;"; // univariate ring |
---|
| 718 | execute(ringSTR); |
---|
| 719 | poly N, N1, h; // N, N1 do not contain 'a', use built-in gcd |
---|
| 720 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
| 721 | setring(SNRMB); |
---|
| 722 | h = imap(SNRM1, h); |
---|
| 723 | while(deg(h, wt) != 0) { // while norm is not squarefree |
---|
| 724 | s = s + 1; |
---|
| 725 | g = reduce(Fs(g), mpoly); |
---|
| 726 | N = reduce(resultant(g, mpoly[1], var(2)), mpoly); // norm of g |
---|
| 727 | N1 = reduce(diff(N, var(1)), mpoly); |
---|
| 728 | setring(SNRM1); |
---|
| 729 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
| 730 | setring(SNRMB); |
---|
| 731 | h = imap(SNRM1, h); |
---|
| 732 | } |
---|
| 733 | return(list(N, g, s)); |
---|
| 734 | } |
---|
| 735 | |
---|
| 736 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 737 | |
---|
[0bc582c] | 738 | proc factorMain(poly f) |
---|
| 739 | "USAGE: factorMain(f); where f is a polynomial |
---|
[9173792] | 740 | PURPOSE: compute the factorization of the squarefree poly f over Q(a)[t], |
---|
[d12655] | 741 | minpoly = p(a). |
---|
[9173792] | 742 | RETURN: list with 2 entries |
---|
| 743 | @format |
---|
| 744 | _[1] = factors, first is a constant |
---|
| 745 | _[2] = multiplicities (not yet implemented) |
---|
| 746 | @end format |
---|
| 747 | ASSUME: basering = Q[x,a], representing Q(a)[x]. An ideal mpoly must |
---|
| 748 | be defined, representing the minimal polynomial (it might be 0!). |
---|
[980120f] | 749 | NOTE: outdated, use factorize instead |
---|
[f49f53] | 750 | EXAMPLE: example Factor; shows an example |
---|
[4e461ff] | 751 | " |
---|
[c1986a] | 752 | { |
---|
[4e461ff] | 753 | // extend this by a squarefree factorization !! |
---|
| 754 | // multiplicities are not valid !! |
---|
| 755 | int i, s; |
---|
| 756 | list normList, factorList, quo_rem; |
---|
| 757 | poly f1, h, h1, H, g, leadCoef, invCoeff; |
---|
| 758 | ideal fac1, fac2; |
---|
| 759 | map F; |
---|
| 760 | |
---|
| 761 | // if no minimal polynomial is defined then use 'factorize' |
---|
| 762 | // FactorOverQ is wrapped around 'factorize' |
---|
| 763 | |
---|
| 764 | if(mpoly[1] == 0) { |
---|
| 765 | // print(" factorize : deg = " + string(deg(f, intvec(1,0)))); |
---|
| 766 | factorList = factorize(f); // FactorOverQ(f); |
---|
| 767 | return(factorList); |
---|
| 768 | } |
---|
| 769 | |
---|
| 770 | // if mpoly != 0 and f does not contain the algebraic number, a root of |
---|
| 771 | // f might be contained in Q(a). Hence one must not use 'factorize'. |
---|
| 772 | |
---|
| 773 | fac1[1] = 1; |
---|
| 774 | fac2[1] = 1; |
---|
[0bc582c] | 775 | normList = sqfrNormMain(f); |
---|
[4e461ff] | 776 | // print(" factorize : deg = " + string(deg(normList[1], intvec(1,0)))); |
---|
| 777 | factorList = factorize(normList[1]); // factor squarefree norm of f over Q[x] |
---|
| 778 | g = normList[2]; |
---|
| 779 | s = normList[3]; |
---|
| 780 | F[1] = var(1) + s*var(2); // inverse transformation |
---|
| 781 | F[2] = var(2); |
---|
| 782 | fac1[1] = factorList[1][1]; |
---|
| 783 | fac2[1] = factorList[2][1]; |
---|
| 784 | for(i = 2; i <= size(factorList[1]); i = i + 1) { |
---|
| 785 | H = factorList[1][i]; |
---|
[0bc582c] | 786 | h = egcdMain(H, g); |
---|
| 787 | quo_rem = quotientMain(g, h); |
---|
[4e461ff] | 788 | g = quo_rem[1]; |
---|
| 789 | fac1[i] = SimplifyPoly(F(h)); |
---|
| 790 | fac2[i] = 1; // to be changed later |
---|
| 791 | } |
---|
| 792 | return(list(fac1, fac2)); |
---|
| 793 | } |
---|
| 794 | |
---|
| 795 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 796 | |
---|
[0bc582c] | 797 | proc zeroSetMain(ideal I, int primDecQ) |
---|
| 798 | "USAGE: zeroSetMain(ideal I, int opt); ideal I, int opt |
---|
[9173792] | 799 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a simple extension |
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[34b0314] | 800 | of the ground field. |
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[4e461ff] | 801 | RETURN: list |
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[d12655] | 802 | - 'f' is the polynomial f in Q(a) (a' being substituted by newA) |
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[4e461ff] | 803 | _[1] = zero-set (list), is the list of the zero-set of the ideal I, |
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| 804 | each entry is an ideal. |
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[34b0314] | 805 | _[2] = 'newA'; if the ground field is Q(a') and the extension field |
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[4e461ff] | 806 | is Q(a), then 'newA' is the representation of a' in Q(a). |
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| 807 | If the basering contains a parameter 'a' and the minpoly |
---|
| 808 | remains unchanged then 'newA' = 'a'. If the basering does not |
---|
| 809 | contain a parameter then 'newA' = 'a' (default). |
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[d12655] | 810 | _[3] = 'mpoly' (ideal), the minimal polynomial of the simple extension |
---|
| 811 | of the ground field. |
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| 812 | ASSUME: basering = K[x_1,x_2,...,x_n] where K = Q or a simple extension of Q |
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| 813 | given by a minpoly; dim(I) = 0. |
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[4e461ff] | 814 | NOTE: opt = 0 no primary decomposition |
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[d12655] | 815 | opt > 0 use a primary decomposition |
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[f49f53] | 816 | EXAMPLE: example zeroSetMain; shows an example |
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[4e461ff] | 817 | " |
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| 818 | { |
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[0bc582c] | 819 | // main work is done in zeroSetMainWork, here the zero-set of each ideal from the |
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| 820 | // primary decompostion is coputed by menas of zeroSetMainWork, and then the |
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[4e461ff] | 821 | // minpoly and the parameter representing the algebraic extension are |
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| 822 | // transformed according to 'newA', i.e., only bookeeping is done. |
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| 823 | |
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[6fe3a0] | 824 | def altring=basering; |
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[4e461ff] | 825 | int i, j, n, noMP, dbPrt; |
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| 826 | intvec w; |
---|
| 827 | list currentSol, result, idealList, primDecList, zeroSet; |
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| 828 | ideal J; |
---|
| 829 | map Fa; |
---|
| 830 | poly newA, oldMinPoly; |
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| 831 | |
---|
| 832 | dbPrt = printlevel-voice+2; |
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[0bc582c] | 833 | dbprint(dbPrt, "zeroSet of " + string(I) + ", minpoly = " + string(minpoly)); |
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[4e461ff] | 834 | |
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| 835 | n = nvars(basering) - 1; |
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| 836 | for(i = 1; i <= n; i++) { w[i] = 1;} |
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| 837 | w[n + 1] = 0; |
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| 838 | |
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[0bc582c] | 839 | if(primDecQ == 0) { return(zeroSetMainWork(I, w, 0)); } |
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[4e461ff] | 840 | |
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| 841 | newA = var(n + 1); |
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| 842 | if(mpoly[1] == 0) { noMP = 1;} |
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| 843 | else {noMP = 0;} |
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| 844 | |
---|
| 845 | primDecList = primdecGTZ(I); // primary decomposition |
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| 846 | dbprint(dbPrt, "primary decomposition consists of " + string(size(primDecList)) + " primary ideals "); |
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| 847 | // idealList = PDSort(idealList); // high degrees first |
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| 848 | |
---|
| 849 | for(i = 1; i <= size(primDecList); i = i + 1) { |
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| 850 | idealList[i] = primDecList[i][2]; // use prime component |
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| 851 | dbprint(dbPrt, string(i) + " " + string(idealList[i])); |
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| 852 | } |
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| 853 | |
---|
| 854 | // compute the zero-set of each primary ideal and join them. |
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[34b0314] | 855 | // If necessary, change the ground field and transform the zero-set |
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[4e461ff] | 856 | |
---|
| 857 | dbprint(dbPrt, " |
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| 858 | find the zero-set of each primary ideal, form the union |
---|
| 859 | and keep track of the minimal polynomials "); |
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| 860 | |
---|
| 861 | for(i = 1; i <= size(idealList); i = i + 1) { |
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| 862 | J = idealList[i]; |
---|
| 863 | idealList[i] = 0; |
---|
| 864 | oldMinPoly = mpoly[1]; |
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| 865 | dbprint(dbPrt, " ideal#" + string(i) + " of " + string(size(idealList)) + " = " + string(J)); |
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[0bc582c] | 866 | currentSol = zeroSetMainWork(J, w, 0); |
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[4e461ff] | 867 | |
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| 868 | if(oldMinPoly != currentSol[3]) { // change minpoly and transform solutions |
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| 869 | dbprint(dbPrt, " change minpoly to " + string(currentSol[3][1])); |
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| 870 | dbprint(dbPrt, " new representation of algebraic number = " + string(currentSol[2])); |
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| 871 | if(!noMP) { // transform the algebraic number a |
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| 872 | Fa = basering, maxideal(1); |
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| 873 | Fa[n + 1] = currentSol[2]; |
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| 874 | newA = SimplifyPoly(Fa(newA)); // new representation of a |
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| 875 | if(size(zeroSet) > 0) {zeroSet = SimplifyZeroset(Fa(zeroSet)); } |
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| 876 | if(i < size(idealList)) { idealList = SimplifyZeroset(Fa(idealList)); } |
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| 877 | } |
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| 878 | else { noMP = 0;} |
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| 879 | } |
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| 880 | zeroSet = zeroSet + currentSol[1]; // add new elements |
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| 881 | } |
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| 882 | return(list(zeroSet, newA, mpoly)); |
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| 883 | } |
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| 884 | |
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| 885 | /////////////////////////////////////////////////////////////////////////////// |
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| 886 | |
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[0bc582c] | 887 | proc zeroSetMainWork(ideal id, intvec wt, int sVars) |
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| 888 | "USAGE: zeroSetMainWork(I, wt, sVars); |
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[9173792] | 889 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
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[34b0314] | 890 | of the ground field (without multiplicities). |
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[4e461ff] | 891 | RETURN: list, all entries are polynomials |
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| 892 | _[1] = zeros, each entry is an ideal |
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[34b0314] | 893 | _[2] = newA; if the ground field is Q(a') this is the rep. of a' w.r.t. a |
---|
| 894 | _[3] = minpoly of the algebraic extension of the ground field (ideal) |
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[4e461ff] | 895 | _[4] = name of algebraic number (default = 'a') |
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| 896 | ASSUME: basering = Q[x_1,x_2,...,x_n,a] |
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[9173792] | 897 | ideal mpoly must be defined, it might be 0! |
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[4e461ff] | 898 | NOTE: might change 'mpoly' !! |
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[f49f53] | 899 | EXAMPLE: example IdealSolve; shows an example |
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[4e461ff] | 900 | " |
---|
| 901 | { |
---|
[6fe3a0] | 902 | def altring=basering; |
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[4e461ff] | 903 | int i, j, k, nrSols, n, noMP; |
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| 904 | ideal I, generators, gens, solid, partsolid; |
---|
| 905 | list linSol, linearSolution, nLinSol, nonlinSolutions, partSol, sol, solutions, result; |
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| 906 | list linIndex, nlinIndex, index; |
---|
| 907 | map Fa, Fsubs; |
---|
| 908 | poly oldMinPoly, newA; |
---|
| 909 | |
---|
| 910 | if(mpoly[1] == 0) { noMP = 1;} |
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| 911 | else { noMP = 0;} |
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| 912 | n = nvars(basering) - 1; |
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| 913 | newA = var(n + 1); |
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| 914 | |
---|
| 915 | I = std(id); |
---|
| 916 | |
---|
| 917 | // find linear solutions of univariate generators |
---|
| 918 | |
---|
| 919 | linSol = LinearZeroSetMain(I, wt); |
---|
| 920 | generators = linSol[3]; // they are a standardbasis |
---|
| 921 | linIndex = linSol[2]; |
---|
| 922 | linearSolution = linSol[1]; |
---|
| 923 | if(size(linIndex) + sVars == n) { // all variables solved |
---|
| 924 | solid = SubsMapIdeal(linearSolution, linIndex, 0); |
---|
| 925 | result[1] = list(solid); |
---|
| 926 | result[2] = var(n + 1); |
---|
| 927 | result[3] = mpoly; |
---|
| 928 | return(result); |
---|
| 929 | } |
---|
| 930 | |
---|
| 931 | // find roots of the nonlinear univariate polynomials of generators |
---|
| 932 | // if necessary, transform linear solutions w.r.t. newA |
---|
| 933 | |
---|
| 934 | oldMinPoly = mpoly[1]; |
---|
| 935 | nLinSol = NonLinearZeroSetMain(generators, wt); // find solutions of univariate generators |
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| 936 | nonlinSolutions = nLinSol[1]; // store solutions |
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| 937 | nlinIndex = nLinSol[4]; // and index of solved variables |
---|
| 938 | generators = nLinSol[5]; // new generators |
---|
| 939 | |
---|
| 940 | // change minpoly if necessary and transform the ideal and the partial solutions |
---|
| 941 | |
---|
| 942 | if(oldMinPoly != nLinSol[3]) { |
---|
| 943 | newA = nLinSol[2]; |
---|
| 944 | if(!noMP && size(linearSolution) > 0) { // transform the algebraic number a |
---|
| 945 | Fa = basering, maxideal(1); |
---|
| 946 | Fa[n + 1] = newA; |
---|
| 947 | linearSolution = SimplifyData(Fa(linearSolution)); // ... |
---|
| 948 | } |
---|
| 949 | } |
---|
| 950 | |
---|
| 951 | // check if all variables are solved. |
---|
| 952 | |
---|
| 953 | if(size(linIndex) + size(nlinIndex) == n - sVars) { |
---|
| 954 | solutions = MergeSolutions(linearSolution, linIndex, nonlinSolutions, nlinIndex, list(), n); |
---|
| 955 | } |
---|
| 956 | |
---|
| 957 | else { |
---|
| 958 | |
---|
| 959 | // some variables are not solved. |
---|
| 960 | // substitute each partial solution in generators and find the |
---|
| 961 | // zero set of the resulting ideal by recursive application |
---|
[0bc582c] | 962 | // of zeroSetMainWork ! |
---|
[4e461ff] | 963 | |
---|
| 964 | index = linIndex + nlinIndex; |
---|
| 965 | nrSols = 0; |
---|
| 966 | for(i = 1; i <= size(nonlinSolutions); i = i + 1) { |
---|
| 967 | sol = linearSolution + nonlinSolutions[i]; |
---|
| 968 | solid = SubsMapIdeal(sol, index, 1); |
---|
| 969 | Fsubs = basering, solid; |
---|
| 970 | gens = std(SimplifyData(Fsubs(generators))); // substitute partial solution |
---|
| 971 | oldMinPoly = mpoly[1]; |
---|
[0bc582c] | 972 | partSol = zeroSetMainWork(gens, wt, size(index) + sVars); |
---|
[4e461ff] | 973 | |
---|
| 974 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
---|
| 975 | Fa = basering, maxideal(1); |
---|
| 976 | Fa[n + 1] = partSol[2]; // a -> p(a), representation of a w.r.t. new minpoly |
---|
| 977 | newA = reduce(Fa(newA), mpoly); |
---|
| 978 | generators = std(SimplifyData(Fa(generators))); |
---|
| 979 | if(size(linearSolution) > 0) { linearSolution = SimplifyData(Fa(linearSolution));} |
---|
| 980 | if(size(nonlinSolutions) > 0) { |
---|
| 981 | nonlinSolutions = SimplifyZeroset(Fa(nonlinSolutions)); |
---|
| 982 | } |
---|
| 983 | sol = linearSolution + nonlinSolutions[i]; |
---|
| 984 | } |
---|
| 985 | |
---|
| 986 | for(j = 1; j <= size(partSol[1]); j++) { // for all partial solutions |
---|
| 987 | partsolid = partSol[1][j]; |
---|
| 988 | for(k = 1; k <= size(index); k++) { |
---|
| 989 | partsolid[index[k]] = sol[k]; |
---|
| 990 | } |
---|
| 991 | nrSols++; |
---|
| 992 | solutions[nrSols] = partsolid; |
---|
| 993 | } |
---|
| 994 | } |
---|
| 995 | |
---|
| 996 | } // end else |
---|
| 997 | return(list(solutions, newA, mpoly)); |
---|
| 998 | } |
---|
| 999 | |
---|
| 1000 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1001 | |
---|
| 1002 | proc LinearZeroSetMain(ideal I, intvec wt) |
---|
| 1003 | "USAGE: LinearZeroSetMain(I, wt) |
---|
| 1004 | PURPOSE: solve the univariate linear polys in I |
---|
| 1005 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
| 1006 | RETURN: list |
---|
| 1007 | _[1] = partial solution of I |
---|
| 1008 | _[2] = index of solved vars |
---|
| 1009 | _[3] = new generators (standardbasis) |
---|
| 1010 | " |
---|
| 1011 | { |
---|
[6fe3a0] | 1012 | def altring=basering; |
---|
[4e461ff] | 1013 | int i, ok, n, found, nrSols; |
---|
| 1014 | ideal generators, newGens; |
---|
| 1015 | list result, index, totalIndex, vars, sol, temp; |
---|
| 1016 | map F; |
---|
| 1017 | poly f; |
---|
| 1018 | |
---|
| 1019 | result[1] = index; // sol[1] should be the empty list |
---|
| 1020 | n = nvars(basering) - 1; |
---|
| 1021 | generators = I; // might be wrong, use index ! |
---|
| 1022 | ok = 1; |
---|
| 1023 | nrSols = 0; |
---|
| 1024 | while(ok) { |
---|
| 1025 | found = 0; |
---|
| 1026 | for(i = 1; i <= size(generators); i = i + 1) { |
---|
| 1027 | f = generators[i]; |
---|
| 1028 | vars = Variables(f, n); |
---|
| 1029 | if(size(vars) == 1 && deg(f, wt) == 1) { // univariate,linear |
---|
| 1030 | nrSols++; found++; |
---|
| 1031 | index[nrSols] = vars[1]; |
---|
[0bc582c] | 1032 | sol[nrSols] = var(vars[1]) - MultPolys(invertNumberMain(LeadTerm(f, vars[1])[3]), f); |
---|
[4e461ff] | 1033 | } |
---|
| 1034 | } |
---|
| 1035 | if(found > 0) { |
---|
| 1036 | F = basering, SubsMapIdeal(sol, index, 1); |
---|
| 1037 | newGens = std(SimplifyData(F(generators))); // substitute, simplify alg. number |
---|
| 1038 | if(size(newGens) == 0) {ok = 0;} |
---|
| 1039 | generators = newGens; |
---|
| 1040 | } |
---|
| 1041 | else { |
---|
| 1042 | ok = 0; |
---|
| 1043 | } |
---|
| 1044 | } |
---|
| 1045 | if(nrSols > 0) { result[1] = sol;} |
---|
| 1046 | result[2] = index; |
---|
| 1047 | result[3] = generators; |
---|
| 1048 | return(result); |
---|
| 1049 | } |
---|
| 1050 | |
---|
| 1051 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1052 | |
---|
| 1053 | proc NonLinearZeroSetMain(ideal I, intvec wt) |
---|
[0bc582c] | 1054 | "USAGE: NonLinearZeroSetMain(I, wt); |
---|
[9173792] | 1055 | PURPOSE: solves the (nonlinear) univariate polynomials in I |
---|
[34b0314] | 1056 | of the ground field (without multiplicities). |
---|
[4e461ff] | 1057 | RETURN: list, all entries are polynomials |
---|
| 1058 | _[1] = list of solutions |
---|
| 1059 | _[2] = newA |
---|
| 1060 | _[3] = minpoly |
---|
| 1061 | _[4] - index of solved variables |
---|
| 1062 | _[5] = new representation of I |
---|
| 1063 | ASSUME: basering = Q[x_1,x_2,...,x_n,a], ideal 'mpoly' must be defined, |
---|
| 1064 | it might be 0 ! |
---|
| 1065 | NOTE: might change 'mpoly' !! |
---|
| 1066 | " |
---|
| 1067 | { |
---|
| 1068 | int i, nrSols, ok, n; |
---|
| 1069 | ideal generators; |
---|
| 1070 | list result, sols, index, vars, partSol; |
---|
| 1071 | map F; |
---|
| 1072 | poly f, newA; |
---|
| 1073 | string ringSTR; |
---|
| 1074 | |
---|
| 1075 | def NLZR = basering; |
---|
[6fe3a0] | 1076 | export(NLZR); |
---|
[4e461ff] | 1077 | |
---|
| 1078 | n = nvars(basering) - 1; |
---|
| 1079 | |
---|
| 1080 | generators = I; |
---|
| 1081 | newA = var(n + 1); |
---|
| 1082 | result[2] = newA; // default |
---|
| 1083 | nrSols = 0; |
---|
| 1084 | ok = 1; |
---|
| 1085 | i = 1; |
---|
| 1086 | while(ok) { |
---|
| 1087 | |
---|
| 1088 | // test if the i-th generator of I is univariate |
---|
| 1089 | |
---|
| 1090 | f = generators[i]; |
---|
| 1091 | vars = Variables(f, n); |
---|
| 1092 | if(size(vars) == 1) { |
---|
| 1093 | generators[i] = 0; |
---|
| 1094 | generators = simplify(generators, 2); // remove 0 |
---|
| 1095 | nrSols++; |
---|
| 1096 | index[nrSols] = vars[1]; // store index of solved variable |
---|
| 1097 | |
---|
| 1098 | // create univariate ring |
---|
| 1099 | |
---|
| 1100 | ringSTR = "ring RIS1 = 0, (" + string(var(vars[1])) + ", " + string(var(n+1)) + "), lp;"; |
---|
| 1101 | execute(ringSTR); |
---|
| 1102 | ideal mpoly = std(imap(NLZR, mpoly)); |
---|
| 1103 | list roots; |
---|
| 1104 | poly f = imap(NLZR, f); |
---|
[6fe3a0] | 1105 | export(RIS1); |
---|
[4e461ff] | 1106 | export(mpoly); |
---|
[0bc582c] | 1107 | roots = rootsMain(f); |
---|
[6fe3a0] | 1108 | |
---|
[4e461ff] | 1109 | // get "old" basering with new minpoly |
---|
| 1110 | |
---|
| 1111 | setring(NLZR); |
---|
| 1112 | partSol = imap(RIS1, roots); |
---|
[3b77465] | 1113 | kill RIS1; |
---|
[4e461ff] | 1114 | if(mpoly[1] != partSol[3]) { // change minpoly |
---|
| 1115 | mpoly = std(partSol[3]); |
---|
| 1116 | F = NLZR, maxideal(1); |
---|
| 1117 | F[n + 1] = partSol[2]; |
---|
| 1118 | if(size(sols) > 0) {sols = SimplifyZeroset(F(sols)); } |
---|
| 1119 | newA = reduce(F(newA), mpoly); // normal form |
---|
| 1120 | result[2] = newA; |
---|
| 1121 | generators = SimplifyData(F(generators)); // does not remove 0's |
---|
| 1122 | } |
---|
| 1123 | sols = ExtendSolutions(sols, partSol[1]); |
---|
| 1124 | } // end univariate |
---|
| 1125 | else { |
---|
| 1126 | i = i + 1; |
---|
| 1127 | } |
---|
| 1128 | if(i > size(generators)) { ok = 0;} |
---|
| 1129 | } |
---|
| 1130 | result[1] = sols; |
---|
| 1131 | result[3] = mpoly; |
---|
| 1132 | result[4] = index; |
---|
| 1133 | result[5] = std(generators); |
---|
| 1134 | |
---|
[3b77465] | 1135 | kill NLZR; |
---|
[4e461ff] | 1136 | return(result); |
---|
| 1137 | } |
---|
| 1138 | |
---|
| 1139 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1140 | |
---|
| 1141 | static proc ExtendSolutions(list solutions, list newSolutions) |
---|
| 1142 | "USAGE: ExtendSolutions(sols, newSols); list sols, newSols; |
---|
[9173792] | 1143 | PURPOSE: extend the entries of 'sols' by the entries of 'newSols', |
---|
[4e461ff] | 1144 | each entry of 'newSols' is a number. |
---|
| 1145 | RETURN: list |
---|
| 1146 | ASSUME: basering = Q[x_1,...,x_n,a], ideal 'mpoly' must be defined, |
---|
| 1147 | it might be 0 ! |
---|
| 1148 | NOTE: used by 'NonLinearZeroSetMain' |
---|
| 1149 | " |
---|
| 1150 | { |
---|
| 1151 | int i, j, k, n, nrSols; |
---|
| 1152 | list newSols, temp; |
---|
| 1153 | |
---|
| 1154 | nrSols = size(solutions); |
---|
| 1155 | if(nrSols > 0) {n = size(solutions[1]);} |
---|
| 1156 | else { |
---|
| 1157 | n = 0; |
---|
| 1158 | nrSols = 1; |
---|
| 1159 | } |
---|
| 1160 | k = 0; |
---|
| 1161 | for(i = 1; i <= nrSols; i++) { |
---|
| 1162 | for(j = 1; j <= size(newSolutions); j++) { |
---|
| 1163 | k++; |
---|
| 1164 | if(n == 0) { temp[1] = newSolutions[j];} |
---|
| 1165 | else { |
---|
| 1166 | temp = solutions[i]; |
---|
| 1167 | temp[n + 1] = newSolutions[j]; |
---|
| 1168 | } |
---|
| 1169 | newSols[k] = temp; |
---|
| 1170 | } |
---|
| 1171 | } |
---|
| 1172 | return(newSols); |
---|
| 1173 | } |
---|
| 1174 | |
---|
| 1175 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1176 | |
---|
| 1177 | static proc MergeSolutions(list sol1, list index1, list sol2, list index2) |
---|
| 1178 | "USAGE: MergeSolutions(sol1, index1, sol2, index2); all parameters are lists |
---|
| 1179 | RETURN: list |
---|
| 1180 | PURPOSE: create a list of solutions of size n, each entry of 'sol2' must |
---|
[d12655] | 1181 | have size n. 'sol1' is one partial solution (from 'LinearZeroSetMain') |
---|
| 1182 | 'sol2' is a list of partial solutions (from 'NonLinearZeroSetMain') |
---|
[4e461ff] | 1183 | ASSUME: 'sol2' is not empty |
---|
[0bc582c] | 1184 | NOTE: used by 'zeroSetMainWork' |
---|
[4e461ff] | 1185 | { |
---|
| 1186 | int i, j, k, m; |
---|
| 1187 | ideal sol; |
---|
| 1188 | list newSols; |
---|
| 1189 | |
---|
| 1190 | m = 0; |
---|
| 1191 | for(i = 1; i <= size(sol2); i++) { |
---|
| 1192 | m++; |
---|
| 1193 | newSols[m] = SubsMapIdeal(sol1 + sol2[i], index1 + index2, 0); |
---|
| 1194 | } |
---|
| 1195 | return(newSols); |
---|
| 1196 | } |
---|
| 1197 | |
---|
| 1198 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1199 | |
---|
| 1200 | static proc SubsMapIdeal(list sol, list index, int opt) |
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| 1201 | "USAGE: SubsMapIdeal(sol,index,opt); list sol, index; int opt; |
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[9173792] | 1202 | PURPOSE: built an ideal I as follows. |
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[4e461ff] | 1203 | if i is contained in 'index' then set I[i] = sol[i] |
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| 1204 | if i is not contained in 'index' then |
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| 1205 | - opt = 0: set I[i] = 0 |
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| 1206 | - opt = 1: set I[i] = var(i) |
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| 1207 | if opt = 1 and n = nvars(basering) then set I[n] = var(n). |
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| 1208 | RETURN: ideal |
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| 1209 | ASSUME: size(sol) = size(index) <= nvars(basering) |
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| 1210 | " |
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| 1211 | { |
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| 1212 | int k = 0; |
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| 1213 | ideal I; |
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| 1214 | for(int i = 1; i <= nvars(basering) - 1; i = i + 1) { // built subs. map |
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[0bc582c] | 1215 | if(containedQ(index, i)) { |
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[4e461ff] | 1216 | k++; |
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| 1217 | I[index[k]] = sol[k]; |
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| 1218 | } |
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| 1219 | else { |
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| 1220 | if(opt) { I[i] = var(i); } |
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| 1221 | else { I[i] = 0; } |
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| 1222 | } |
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| 1223 | } |
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| 1224 | if(opt) {I[nvars(basering)] = var(nvars(basering));} |
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| 1225 | return(I); |
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| 1226 | } |
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| 1227 | |
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| 1228 | /////////////////////////////////////////////////////////////////////////////// |
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| 1229 | |
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| 1230 | proc SimplifyZeroset(data) |
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| 1231 | "USAGE: SimplifyZeroset(data); list data |
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[9173792] | 1232 | PURPOSE: reduce the entries of the elements of 'data' w.r.t. the ideal 'mpoly' |
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[4e461ff] | 1233 | 'data' is a list of ideals/lists. |
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| 1234 | RETURN: list |
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| 1235 | ASSUME: basering = Q[x_1,...,x_n,a], order = lp |
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| 1236 | 'data' is a list of ideals |
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| 1237 | ideal 'mpoly' must be defined, it might be 0 ! |
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| 1238 | " |
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| 1239 | { |
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| 1240 | int i; |
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| 1241 | list result; |
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| 1242 | |
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| 1243 | for(i = 1; i <= size(data); i++) { |
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| 1244 | result[i] = SimplifyData(data[i]); |
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| 1245 | } |
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| 1246 | return(result); |
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| 1247 | } |
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| 1248 | |
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| 1249 | /////////////////////////////////////////////////////////////////////////////// |
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| 1250 | |
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| 1251 | proc Variables(poly f, int n) |
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| 1252 | "USAGE: Variables(f,n); poly f; int n; |
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[9173792] | 1253 | PURPOSE: list of variables among var(1),...,var(n) which occur in f. |
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[4e461ff] | 1254 | RETURN: list |
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| 1255 | ASSUME: n <= nvars(basering) |
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| 1256 | " |
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| 1257 | { |
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| 1258 | int i, nrV; |
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| 1259 | list index; |
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| 1260 | |
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| 1261 | nrV = 0; |
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| 1262 | for(i = 1; i <= n; i = i + 1) { |
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| 1263 | if(diff(f, var(i)) != 0) { nrV++; index[nrV] = i; } |
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| 1264 | } |
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| 1265 | return(index); |
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| 1266 | } |
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| 1267 | |
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| 1268 | /////////////////////////////////////////////////////////////////////////////// |
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| 1269 | |
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[0bc582c] | 1270 | proc containedQ(data, f, list #) |
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| 1271 | "USAGE: containedQ(data, f [, opt]); data=list; f=any type; opt=integer |
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[9173792] | 1272 | PURPOSE: test if f is an element of data. |
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[4e461ff] | 1273 | RETURN: int |
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[9173792] | 1274 | 0 if f not contained in data |
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| 1275 | 1 if f contained in data |
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| 1276 | OPTIONS: opt = 0 : use '==' for comparing f with elements from data@* |
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[0bc582c] | 1277 | opt = 1 : use @code{sameQ} for comparing f with elements from data |
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[4e461ff] | 1278 | " |
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| 1279 | { |
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| 1280 | int opt, i, found; |
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| 1281 | if(size(#) > 0) { opt = #[1];} |
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| 1282 | else { opt = 0; } |
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| 1283 | i = 1; |
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| 1284 | found = 0; |
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| 1285 | |
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| 1286 | while((!found) && (i <= size(data))) { |
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| 1287 | if(opt == 0) { |
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| 1288 | if(f == data[i]) { found = 1;} |
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| 1289 | else {i = i + 1;} |
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| 1290 | } |
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| 1291 | else { |
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[0bc582c] | 1292 | if(sameQ(f, data[i])) { found = 1;} |
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[4e461ff] | 1293 | else {i = i + 1;} |
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| 1294 | } |
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| 1295 | } |
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| 1296 | return(found); |
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| 1297 | } |
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| 1298 | |
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| 1299 | ////////////////////////////////////////////////////////////////////////////// |
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| 1300 | |
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[0bc582c] | 1301 | proc sameQ(a, b) |
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| 1302 | "USAGE: sameQ(a, b); a,b=list/intvec |
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[9173792] | 1303 | PURPOSE: test a == b elementwise, i.e., a[i] = b[i]. |
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[4e461ff] | 1304 | RETURN: int |
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| 1305 | 0 if a != b |
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| 1306 | 1 if a == b |
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| 1307 | " |
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| 1308 | { |
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| 1309 | if(typeof(a) == typeof(b)) { |
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| 1310 | if(typeof(a) == "list" || typeof(a) == "intvec") { |
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| 1311 | if(size(a) == size(b)) { |
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| 1312 | int i = 1; |
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| 1313 | int ok = 1; |
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| 1314 | while(ok && (i <= size(a))) { |
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| 1315 | if(a[i] == b[i]) { i = i + 1;} |
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| 1316 | else {ok = 0;} |
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| 1317 | } |
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| 1318 | return(ok); |
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| 1319 | } |
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| 1320 | else { return(0); } |
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| 1321 | } |
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| 1322 | else { return(a == b);} |
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| 1323 | } |
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| 1324 | else { return(0);} |
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| 1325 | } |
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| 1326 | |
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| 1327 | /////////////////////////////////////////////////////////////////////////////// |
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| 1328 | |
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| 1329 | static proc SimplifyPoly(poly f) |
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| 1330 | "USAGE: SimplifyPoly(f); poly f |
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[9173792] | 1331 | PURPOSE: reduces the coefficients of f w.r.t. the ideal 'moly' if they contain |
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[4e461ff] | 1332 | the algebraic number 'a'. |
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| 1333 | RETURN: poly |
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| 1334 | ASSUME: basering = Q[x_1,...,x_n,a] |
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| 1335 | ideal mpoly must be defined, it might be 0 ! |
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[980120f] | 1336 | NOTE: outdated, use reduce instead |
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[4e461ff] | 1337 | " |
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| 1338 | { |
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| 1339 | matrix coMx; |
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| 1340 | poly f1, vp; |
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| 1341 | |
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| 1342 | vp = 1; |
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| 1343 | for(int i = 1; i < nvars(basering); i++) { vp = vp * var(i);} |
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| 1344 | |
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| 1345 | coMx = coef(f, vp); |
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| 1346 | f1 = 0; |
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| 1347 | for(i = 1; i <= ncols(coMx); i++) { |
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| 1348 | f1 = f1 + coMx[1, i] * reduce(coMx[2, i], mpoly); |
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| 1349 | } |
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| 1350 | return(f1); |
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| 1351 | } |
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| 1352 | |
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| 1353 | /////////////////////////////////////////////////////////////////////////////// |
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| 1354 | |
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| 1355 | static proc SimplifyData(data) |
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| 1356 | "USAGE: SimplifyData(data); ideal/list data; |
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[9173792] | 1357 | PURPOSE: reduces the entries of 'data' w.r.t. the ideal 'mpoly' if they contain |
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[d12655] | 1358 | the algebraic number 'a' |
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[4e461ff] | 1359 | RETURN: ideal/list |
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| 1360 | ASSUME: basering = Q[x_1,...,x_n,a] |
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| 1361 | ideal 'mpoly' must be defined, it might be 0 ! |
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| 1362 | " |
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| 1363 | { |
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[6fe3a0] | 1364 | def altring=basering; |
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[4e461ff] | 1365 | int n; |
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| 1366 | poly f; |
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| 1367 | |
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| 1368 | if(typeof(data) == "ideal") { n = ncols(data); } |
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| 1369 | else { n = size(data);} |
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| 1370 | |
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| 1371 | for(int i = 1; i <= n; i++) { |
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| 1372 | f = data[i]; |
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| 1373 | data[i] = SimplifyPoly(f); |
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| 1374 | } |
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| 1375 | return(data); |
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| 1376 | } |
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| 1377 | |
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| 1378 | /////////////////////////////////////////////////////////////////////////////// |
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| 1379 | |
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| 1380 | static proc TransferRing(R) |
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| 1381 | "USAGE: TransferRing(R); |
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[9173792] | 1382 | PURPOSE: creates a new ring containing the same variables as R, but without |
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[4e461ff] | 1383 | parameters. If R contains a parameter then this parameter is added |
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[d12655] | 1384 | as the last variable and 'minpoly' is represented by the ideal 'mpoly' |
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| 1385 | If the basering does not contain a parameter then 'a' is added and |
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| 1386 | 'mpoly' = 0. |
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[4e461ff] | 1387 | RETURN: ring |
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| 1388 | ASSUME: R = K[x_1,...,x_n] where K = Q or K = Q(a). |
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| 1389 | NOTE: Creates the ring needed for all prodecures with name 'proc-name'Main |
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| 1390 | " |
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| 1391 | { |
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[6fe3a0] | 1392 | def altring=basering; |
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[4e461ff] | 1393 | string ringSTR, parName, minPoly; |
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| 1394 | |
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| 1395 | setring(R); |
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| 1396 | |
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| 1397 | if(npars(basering) == 0) { |
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| 1398 | parName = "a"; |
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| 1399 | minPoly = "0"; |
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| 1400 | } |
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| 1401 | else { |
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| 1402 | parName = parstr(basering); |
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| 1403 | minPoly = string(minpoly); |
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| 1404 | } |
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| 1405 | ringSTR = "ring TR = 0, (" + varstr(basering) + "," + parName + "), lp;"; |
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| 1406 | |
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| 1407 | execute(ringSTR); |
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| 1408 | execute("ideal mpoly = std(" + minPoly + ");"); |
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| 1409 | export(mpoly); |
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[6fe3a0] | 1410 | setring altring; |
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[4e461ff] | 1411 | return(TR); |
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| 1412 | } |
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| 1413 | |
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| 1414 | /////////////////////////////////////////////////////////////////////////////// |
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| 1415 | |
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| 1416 | static proc NewBaseRing() |
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| 1417 | "USAGE: NewBaseRing(); |
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[9173792] | 1418 | PURPOSE: creates a new ring, the last variable is added as a parameter. |
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[4e461ff] | 1419 | minpoly is set to mpoly[1]. |
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| 1420 | RETURN: ring |
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| 1421 | ASSUME: basering = Q[x_1,...,x_n, a], 'mpoly' must be defined |
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| 1422 | " |
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| 1423 | { |
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| 1424 | int n = nvars(basering); |
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| 1425 | int MP; |
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| 1426 | string ringSTR, parName, varString; |
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| 1427 | |
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| 1428 | def BR = basering; |
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| 1429 | if(mpoly[1] != 0) { |
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| 1430 | parName = "(0, " + string(var(n)) + ")"; |
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| 1431 | MP = 1; |
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| 1432 | } |
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| 1433 | else { |
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| 1434 | parName = "0"; |
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| 1435 | MP = 0; |
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| 1436 | } |
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| 1437 | |
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| 1438 | |
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| 1439 | for(int i = 1; i < n - 1; i++) { |
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| 1440 | varString = varString + string(var(i)) + ","; |
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| 1441 | } |
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| 1442 | varString = varString + string(var(n-1)); |
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| 1443 | |
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| 1444 | ringSTR = "ring TR = " + parName + ", (" + varString + "), lp;"; |
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| 1445 | execute(ringSTR); |
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| 1446 | if(MP) { minpoly = number(imap(BR, mpoly)[1]); } |
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[6fe3a0] | 1447 | setring BR; |
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[4e461ff] | 1448 | return(TR); |
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| 1449 | } |
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| 1450 | |
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| 1451 | /////////////////////////////////////////////////////////////////////////////// |
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| 1452 | |
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| 1453 | /* |
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| 1454 | Examples: |
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| 1455 | |
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| 1456 | |
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| 1457 | // order = 20; |
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| 1458 | ring S1 = 0, (s(1..3)), lp; |
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| 1459 | ideal I = s(2)*s(3), s(1)^2*s(2)+s(1)^2*s(3)-1, s(1)^2*s(3)^2-s(3), s(2)^4-s(3)^4+s(1)^2, s(1)^4+s(2)^3-s(3)^3, s(3)^5-s(1)^2*s(3); |
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[d12655] | 1460 | ideal mpoly = std(0); |
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[4e461ff] | 1461 | |
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| 1462 | // order = 10 |
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| 1463 | ring S2 = 0, (s(1..5)), lp; |
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| 1464 | ideal I = s(2)+s(3)-s(5), s(4)^2-s(5), s(1)*s(5)+s(3)*s(4)-s(4)*s(5), s(1)*s(4)+s(3)-s(5), s(3)^2-2*s(3)*s(5), s(1)*s(3)-s(1)*s(5)+s(4)*s(5), s(1)^2+s(4)^2-2*s(5), -s(1)+s(5)^3, s(3)*s(5)^2+s(4)-s(5)^3, s(1)*s(5)^2-1; |
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[d12655] | 1465 | ideal mpoly = std(0); |
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[4e461ff] | 1466 | |
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| 1467 | //order = 126 |
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| 1468 | ring S3 = 0, (s(1..5)), lp; |
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| 1469 | ideal I = s(3)*s(4), s(2)*s(4), s(1)*s(3), s(1)*s(2), s(3)^3+s(4)^3-1, s(2)^3+s(4)^3-1, s(1)^3-s(4)^3, s(4)^4-s(4), s(1)*s(4)^3-s(1), s(5)^7-1; |
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[d12655] | 1470 | ideal mpoly = std(0); |
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[4e461ff] | 1471 | |
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| 1472 | // order = 192 |
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| 1473 | ring S4 = 0, (s(1..4)), lp; |
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| 1474 | ideal I = s(2)*s(3)^2*s(4)+s(1)*s(3)*s(4)^2, s(2)^2*s(3)*s(4)+s(1)*s(2)*s(4)^2, s(1)*s(3)^3+s(2)*s(4)^3, s(1)*s(2)*s(3)^2+s(1)^2*s(3)*s(4), s(1)^2*s(3)^2-s(2)^2*s(4)^2, s(1)*s(2)^2*s(3)+s(1)^2*s(2)*s(4), s(1)^3*s(3)+s(2)^3*s(4), s(2)^4-s(3)^4, s(1)*s(2)^3+s(3)*s(4)^3, s(1)^2*s(2)^2-s(3)^2*s(4)^2, s(1)^3*s(2)+s(3)^3*s(4), s(1)^4-s(4)^4, s(3)^5*s(4)-s(3)*s(4)^5, s(3)^8+14*s(3)^4*s(4)^4+s(4)^8-1, 15*s(2)*s(3)*s(4)^7-s(1)*s(4)^8+s(1), 15*s(3)^4*s(4)^5+s(4)^9-s(4), 16*s(3)*s(4)^9-s(3)*s(4), 16*s(2)*s(4)^9-s(2)*s(4), 16*s(1)*s(3)*s(4)^8-s(1)*s(3), 16*s(1)*s(2)*s(4)^8-s(1)*s(2), 16*s(1)*s(4)^10-15*s(2)*s(3)*s(4)-16*s(1)*s(4)^2, 16*s(1)^2*s(4)^9-15*s(1)*s(2)*s(3)-16*s(1)^2*s(4), 16*s(4)^13+15*s(3)^4*s(4)-16*s(4)^5; |
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[d12655] | 1475 | ideal mpoly = std(0); |
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[4e461ff] | 1476 | |
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| 1477 | ring R = (0,a), (x,y,z), lp; |
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| 1478 | minpoly = a2 + 1; |
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| 1479 | ideal I1 = x2 - 1/2, a*z - 1, y - 2; |
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| 1480 | ideal I2 = x3 - 1/2, a*z2 - 3, y - 2*a; |
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[35f23d] | 1481 | |
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[d12655] | 1482 | */ |
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