[fba6f18] | 1 | /**************************************** |
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| 2 | * Computer Algebra System SINGULAR * |
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| 3 | ****************************************/ |
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[1f414c8] | 4 | /** |
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| 5 | * ABSTRACT: numbers in an algebraic extension field K[a] / < f(a) > |
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| 6 | * Assuming that we have a coeffs object cf, then these numbers |
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| 7 | * are polynomials in the polynomial ring K[a] represented by |
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| 8 | * cf->extRing. |
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| 9 | * IMPORTANT ASSUMPTIONS: |
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| 10 | * 1.) So far we assume that cf->extRing is a valid polynomial |
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| 11 | * ring in exactly one variable, i.e., K[a], where K is allowed |
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| 12 | * to be any field (representable in SINGULAR and which may |
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| 13 | * itself be some extension field, thus allowing for extension |
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| 14 | * towers). |
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| 15 | * 2.) Moreover, this implementation assumes that |
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[dd668f] | 16 | * cf->extRing->qideal is not NULL but an ideal with at |
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[1f414c8] | 17 | * least one non-zero generator which may be accessed by |
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[dd668f] | 18 | * cf->extRing->qideal->m[0] and which represents the minimal |
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[1f414c8] | 19 | * polynomial f(a) of the extension variable 'a' in K[a]. |
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| 20 | * 3.) As soon as an std method for polynomial rings becomes |
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| 21 | * availabe, all reduction steps modulo f(a) should be replaced |
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| 22 | * by a call to std. Moreover, in this situation one can finally |
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| 23 | * move from K[a] / < f(a) > to |
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| 24 | * K[a_1, ..., a_s] / I, with I some zero-dimensional ideal |
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| 25 | * in K[a_1, ..., a_s] given by a lex |
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| 26 | * Gröbner basis. |
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| 27 | * The code in algext.h and algext.cc is then capable of |
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| 28 | * computing in K[a_1, ..., a_s] / I. |
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| 29 | **/ |
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[fba6f18] | 30 | |
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| 31 | #include "config.h" |
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| 32 | #include <misc/auxiliary.h> |
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| 33 | |
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| 34 | #include <omalloc/omalloc.h> |
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| 35 | |
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| 36 | #include <reporter/reporter.h> |
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| 37 | |
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| 38 | #include <coeffs/coeffs.h> |
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| 39 | #include <coeffs/numbers.h> |
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[0fb5991] | 40 | #include <coeffs/longrat.h> |
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[fba6f18] | 41 | |
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| 42 | #include <polys/monomials/ring.h> |
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| 43 | #include <polys/monomials/p_polys.h> |
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| 44 | #include <polys/simpleideals.h> |
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| 45 | |
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[dc79bd] | 46 | #include <polys/PolyEnumerator.h> |
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| 47 | |
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[146c603] | 48 | #ifdef HAVE_FACTORY |
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| 49 | #include <factory/factory.h> |
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[dc79bd] | 50 | #include <polys/clapconv.h> |
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| 51 | #include <polys/clapsing.h> |
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[146c603] | 52 | #endif |
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| 53 | |
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[dc79bd] | 54 | |
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| 55 | #include <polys/ext_fields/algext.h> |
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[331fd0] | 56 | #define TRANSEXT_PRIVATES 1 |
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[dc79bd] | 57 | #include <polys/ext_fields/transext.h> |
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[1f414c8] | 58 | |
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| 59 | #ifdef LDEBUG |
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| 60 | #define naTest(a) naDBTest(a,__FILE__,__LINE__,cf) |
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| 61 | BOOLEAN naDBTest(number a, const char *f, const int l, const coeffs r); |
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| 62 | #else |
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[644f81] | 63 | #define naTest(a) ((void)(TRUE)) |
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[1f414c8] | 64 | #endif |
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| 65 | |
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| 66 | /// Our own type! |
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| 67 | static const n_coeffType ID = n_algExt; |
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| 68 | |
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| 69 | /* polynomial ring in which our numbers live */ |
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| 70 | #define naRing cf->extRing |
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| 71 | |
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| 72 | /* coeffs object in which the coefficients of our numbers live; |
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| 73 | * methods attached to naCoeffs may be used to compute with the |
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| 74 | * coefficients of our numbers, e.g., use naCoeffs->nAdd to add |
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| 75 | * coefficients of our numbers */ |
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| 76 | #define naCoeffs cf->extRing->cf |
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| 77 | |
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| 78 | /* minimal polynomial */ |
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[dd668f] | 79 | #define naMinpoly naRing->qideal->m[0] |
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[fba6f18] | 80 | |
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| 81 | /// forward declarations |
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[3c0498] | 82 | BOOLEAN naGreaterZero(number a, const coeffs cf); |
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[fba6f18] | 83 | BOOLEAN naGreater(number a, number b, const coeffs cf); |
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| 84 | BOOLEAN naEqual(number a, number b, const coeffs cf); |
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| 85 | BOOLEAN naIsOne(number a, const coeffs cf); |
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| 86 | BOOLEAN naIsMOne(number a, const coeffs cf); |
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| 87 | BOOLEAN naIsZero(number a, const coeffs cf); |
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[2f3764] | 88 | number naInit(long i, const coeffs cf); |
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[fba6f18] | 89 | int naInt(number &a, const coeffs cf); |
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| 90 | number naNeg(number a, const coeffs cf); |
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| 91 | number naInvers(number a, const coeffs cf); |
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| 92 | number naAdd(number a, number b, const coeffs cf); |
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| 93 | number naSub(number a, number b, const coeffs cf); |
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| 94 | number naMult(number a, number b, const coeffs cf); |
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| 95 | number naDiv(number a, number b, const coeffs cf); |
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| 96 | void naPower(number a, int exp, number *b, const coeffs cf); |
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| 97 | number naCopy(number a, const coeffs cf); |
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[ce1f78] | 98 | void naWriteLong(number &a, const coeffs cf); |
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| 99 | void naWriteShort(number &a, const coeffs cf); |
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[fba6f18] | 100 | number naRePart(number a, const coeffs cf); |
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| 101 | number naImPart(number a, const coeffs cf); |
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[0fb5991] | 102 | number naGetDenom(number &a, const coeffs cf); |
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| 103 | number naGetNumerator(number &a, const coeffs cf); |
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[ba2359] | 104 | number naGcd(number a, number b, const coeffs cf); |
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[36ef6e0] | 105 | //number naLcm(number a, number b, const coeffs cf); |
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[ba2359] | 106 | int naSize(number a, const coeffs cf); |
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[fba6f18] | 107 | void naDelete(number *a, const coeffs cf); |
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[03f7b5] | 108 | void naCoeffWrite(const coeffs cf, BOOLEAN details); |
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[dc79bd] | 109 | //number naIntDiv(number a, number b, const coeffs cf); |
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[fba6f18] | 110 | const char * naRead(const char *s, number *a, const coeffs cf); |
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[03f7b5] | 111 | |
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[fba6f18] | 112 | static BOOLEAN naCoeffIsEqual(const coeffs cf, n_coeffType n, void * param); |
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| 113 | |
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[20c540] | 114 | |
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| 115 | /// returns NULL if p == NULL, otherwise makes p monic by dividing |
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| 116 | /// by its leading coefficient (only done if this is not already 1); |
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| 117 | /// this assumes that we are over a ground field so that division |
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| 118 | /// is well-defined; |
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| 119 | /// modifies p |
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| 120 | // void p_Monic(poly p, const ring r); |
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| 121 | |
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| 122 | /// assumes that p and q are univariate polynomials in r, |
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| 123 | /// mentioning the same variable; |
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| 124 | /// assumes a global monomial ordering in r; |
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| 125 | /// assumes that not both p and q are NULL; |
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| 126 | /// returns the gcd of p and q; |
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| 127 | /// leaves p and q unmodified |
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| 128 | // poly p_Gcd(const poly p, const poly q, const ring r); |
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| 129 | |
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| 130 | /* returns NULL if p == NULL, otherwise makes p monic by dividing |
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| 131 | by its leading coefficient (only done if this is not already 1); |
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| 132 | this assumes that we are over a ground field so that division |
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| 133 | is well-defined; |
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| 134 | modifies p */ |
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| 135 | static inline void p_Monic(poly p, const ring r) |
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| 136 | { |
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| 137 | if (p == NULL) return; |
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| 138 | number n = n_Init(1, r->cf); |
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| 139 | if (p->next==NULL) { p_SetCoeff(p,n,r); return; } |
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| 140 | poly pp = p; |
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| 141 | number lc = p_GetCoeff(p, r); |
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| 142 | if (n_IsOne(lc, r->cf)) return; |
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| 143 | number lcInverse = n_Invers(lc, r->cf); |
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| 144 | p_SetCoeff(p, n, r); // destroys old leading coefficient! |
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| 145 | pIter(p); |
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| 146 | while (p != NULL) |
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| 147 | { |
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| 148 | number n = n_Mult(p_GetCoeff(p, r), lcInverse, r->cf); |
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| 149 | n_Normalize(n,r->cf); |
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| 150 | p_SetCoeff(p, n, r); // destroys old leading coefficient! |
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| 151 | pIter(p); |
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| 152 | } |
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| 153 | n_Delete(&lcInverse, r->cf); |
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| 154 | p = pp; |
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| 155 | } |
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| 156 | |
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| 157 | /// see p_Gcd; |
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| 158 | /// additional assumption: deg(p) >= deg(q); |
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| 159 | /// must destroy p and q (unless one of them is returned) |
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| 160 | static inline poly p_GcdHelper(poly &p, poly &q, const ring r) |
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| 161 | { |
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| 162 | while (q != NULL) |
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| 163 | { |
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| 164 | p_PolyDiv(p, q, FALSE, r); |
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| 165 | // swap p and q: |
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| 166 | poly& t = q; |
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| 167 | q = p; |
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| 168 | p = t; |
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| 169 | |
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| 170 | } |
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| 171 | return p; |
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| 172 | } |
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| 173 | |
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| 174 | /* assumes that p and q are univariate polynomials in r, |
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| 175 | mentioning the same variable; |
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| 176 | assumes a global monomial ordering in r; |
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| 177 | assumes that not both p and q are NULL; |
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| 178 | returns the gcd of p and q; |
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| 179 | leaves p and q unmodified */ |
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| 180 | static inline poly p_Gcd(const poly p, const poly q, const ring r) |
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| 181 | { |
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| 182 | assume((p != NULL) || (q != NULL)); |
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| 183 | |
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| 184 | poly a = p; poly b = q; |
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| 185 | if (p_Deg(a, r) < p_Deg(b, r)) { a = q; b = p; } |
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| 186 | a = p_Copy(a, r); b = p_Copy(b, r); |
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| 187 | |
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| 188 | /* We have to make p monic before we return it, so that if the |
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| 189 | gcd is a unit in the ground field, we will actually return 1. */ |
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| 190 | a = p_GcdHelper(a, b, r); |
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| 191 | p_Monic(a, r); |
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| 192 | return a; |
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| 193 | } |
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| 194 | |
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| 195 | /* see p_ExtGcd; |
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| 196 | additional assumption: deg(p) >= deg(q); |
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| 197 | must destroy p and q (unless one of them is returned) */ |
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| 198 | static inline poly p_ExtGcdHelper(poly &p, poly &pFactor, poly &q, poly &qFactor, |
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| 199 | ring r) |
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| 200 | { |
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| 201 | if (q == NULL) |
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| 202 | { |
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| 203 | qFactor = NULL; |
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| 204 | pFactor = p_ISet(1, r); |
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| 205 | p_SetCoeff(pFactor, n_Invers(p_GetCoeff(p, r), r->cf), r); |
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| 206 | p_Monic(p, r); |
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| 207 | return p; |
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| 208 | } |
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| 209 | else |
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| 210 | { |
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| 211 | poly pDivQ = p_PolyDiv(p, q, TRUE, r); |
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| 212 | poly ppFactor = NULL; poly qqFactor = NULL; |
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| 213 | poly theGcd = p_ExtGcdHelper(q, qqFactor, p, ppFactor, r); |
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| 214 | pFactor = ppFactor; |
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| 215 | qFactor = p_Add_q(qqFactor, |
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| 216 | p_Neg(p_Mult_q(pDivQ, p_Copy(ppFactor, r), r), r), |
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| 217 | r); |
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| 218 | return theGcd; |
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| 219 | } |
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| 220 | } |
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| 221 | |
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| 222 | /* assumes that p and q are univariate polynomials in r, |
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| 223 | mentioning the same variable; |
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| 224 | assumes a global monomial ordering in r; |
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| 225 | assumes that not both p and q are NULL; |
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| 226 | returns the gcd of p and q; |
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| 227 | moreover, afterwards pFactor and qFactor contain appropriate |
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| 228 | factors such that gcd(p, q) = p * pFactor + q * qFactor; |
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| 229 | leaves p and q unmodified */ |
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| 230 | poly p_ExtGcd(poly p, poly &pFactor, poly q, poly &qFactor, ring r) |
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| 231 | { |
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| 232 | assume((p != NULL) || (q != NULL)); |
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| 233 | poly a = p; poly b = q; BOOLEAN aCorrespondsToP = TRUE; |
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| 234 | if (p_Deg(a, r) < p_Deg(b, r)) |
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| 235 | { a = q; b = p; aCorrespondsToP = FALSE; } |
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| 236 | a = p_Copy(a, r); b = p_Copy(b, r); |
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| 237 | poly aFactor = NULL; poly bFactor = NULL; |
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| 238 | poly theGcd = p_ExtGcdHelper(a, aFactor, b, bFactor, r); |
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| 239 | if (aCorrespondsToP) { pFactor = aFactor; qFactor = bFactor; } |
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| 240 | else { pFactor = bFactor; qFactor = aFactor; } |
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| 241 | return theGcd; |
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| 242 | } |
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| 243 | |
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| 244 | |
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| 245 | |
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[fba6f18] | 246 | #ifdef LDEBUG |
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[ba2359] | 247 | BOOLEAN naDBTest(number a, const char *f, const int l, const coeffs cf) |
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[fba6f18] | 248 | { |
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[1f414c8] | 249 | assume(getCoeffType(cf) == ID); |
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[f0b01f] | 250 | if (a == NULL) return TRUE; |
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[c28ecf] | 251 | p_Test((poly)a, naRing); |
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[3c0498] | 252 | if((((poly)a)!=naMinpoly) |
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| 253 | && p_Totaldegree((poly)a, naRing) >= p_Totaldegree(naMinpoly, naRing)) |
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[58f11d0] | 254 | { |
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| 255 | Print("deg >= deg(minpoly) in %s:%d\n",f,l); |
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| 256 | return FALSE; |
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| 257 | } |
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[fba6f18] | 258 | return TRUE; |
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| 259 | } |
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| 260 | #endif |
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| 261 | |
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[4a2260e] | 262 | void heuristicReduce(poly &p, poly reducer, const coeffs cf); |
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| 263 | void definiteReduce(poly &p, poly reducer, const coeffs cf); |
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[ba2359] | 264 | |
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[488808e] | 265 | /* returns the bottom field in this field extension tower; if the tower |
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| 266 | is flat, i.e., if there is no extension, then r itself is returned; |
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| 267 | as a side-effect, the counter 'height' is filled with the height of |
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| 268 | the extension tower (in case the tower is flat, 'height' is zero) */ |
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| 269 | static coeffs nCoeff_bottom(const coeffs r, int &height) |
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| 270 | { |
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| 271 | assume(r != NULL); |
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| 272 | coeffs cf = r; |
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| 273 | height = 0; |
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| 274 | while (nCoeff_is_Extension(cf)) |
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| 275 | { |
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[6ccdd3a] | 276 | assume(cf->extRing != NULL); assume(cf->extRing->cf != NULL); |
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| 277 | cf = cf->extRing->cf; |
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[488808e] | 278 | height++; |
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| 279 | } |
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| 280 | return cf; |
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| 281 | } |
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| 282 | |
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[fba6f18] | 283 | BOOLEAN naIsZero(number a, const coeffs cf) |
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| 284 | { |
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| 285 | naTest(a); |
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| 286 | return (a == NULL); |
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| 287 | } |
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| 288 | |
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[c28ecf] | 289 | void naDelete(number * a, const coeffs cf) |
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[fba6f18] | 290 | { |
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[c28ecf] | 291 | if (*a == NULL) return; |
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[3c0498] | 292 | if (((poly)*a)==naMinpoly) { *a=NULL;return;} |
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[c28ecf] | 293 | poly aAsPoly = (poly)(*a); |
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| 294 | p_Delete(&aAsPoly, naRing); |
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[fba6f18] | 295 | *a = NULL; |
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| 296 | } |
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| 297 | |
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[2c7f28] | 298 | BOOLEAN naEqual(number a, number b, const coeffs cf) |
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[fba6f18] | 299 | { |
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| 300 | naTest(a); naTest(b); |
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[3c0498] | 301 | |
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[fba6f18] | 302 | /// simple tests |
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| 303 | if (a == b) return TRUE; |
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| 304 | if ((a == NULL) && (b != NULL)) return FALSE; |
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| 305 | if ((b == NULL) && (a != NULL)) return FALSE; |
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| 306 | |
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| 307 | /// deg test |
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| 308 | int aDeg = 0; |
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[2c7f28] | 309 | if (a != NULL) aDeg = p_Totaldegree((poly)a, naRing); |
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[fba6f18] | 310 | int bDeg = 0; |
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[2c7f28] | 311 | if (b != NULL) bDeg = p_Totaldegree((poly)b, naRing); |
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[fba6f18] | 312 | if (aDeg != bDeg) return FALSE; |
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[3c0498] | 313 | |
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[fba6f18] | 314 | /// subtraction test |
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| 315 | number c = naSub(a, b, cf); |
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| 316 | BOOLEAN result = naIsZero(c, cf); |
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[e852125] | 317 | naDelete(&c, cf); |
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[fba6f18] | 318 | return result; |
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| 319 | } |
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| 320 | |
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[0fb5991] | 321 | number naCopy(number a, const coeffs cf) |
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[fba6f18] | 322 | { |
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| 323 | naTest(a); |
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[ba2359] | 324 | if (a == NULL) return NULL; |
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[3c0498] | 325 | if (((poly)a)==naMinpoly) return a; |
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[fba6f18] | 326 | return (number)p_Copy((poly)a, naRing); |
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| 327 | } |
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| 328 | |
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[0fb5991] | 329 | number naGetNumerator(number &a, const coeffs cf) |
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| 330 | { |
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| 331 | return naCopy(a, cf); |
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| 332 | } |
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| 333 | |
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[fba6f18] | 334 | number naGetDenom(number &a, const coeffs cf) |
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| 335 | { |
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| 336 | naTest(a); |
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| 337 | return naInit(1, cf); |
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| 338 | } |
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| 339 | |
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| 340 | BOOLEAN naIsOne(number a, const coeffs cf) |
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| 341 | { |
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| 342 | naTest(a); |
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[2c7f28] | 343 | poly aAsPoly = (poly)a; |
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[bca341] | 344 | if ((a==NULL) || (!p_IsConstant(aAsPoly, naRing))) return FALSE; |
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[2c7f28] | 345 | return n_IsOne(p_GetCoeff(aAsPoly, naRing), naCoeffs); |
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[fba6f18] | 346 | } |
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| 347 | |
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| 348 | BOOLEAN naIsMOne(number a, const coeffs cf) |
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| 349 | { |
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| 350 | naTest(a); |
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[2c7f28] | 351 | poly aAsPoly = (poly)a; |
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[bca341] | 352 | if ((a==NULL) || (!p_IsConstant(aAsPoly, naRing))) return FALSE; |
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[2c7f28] | 353 | return n_IsMOne(p_GetCoeff(aAsPoly, naRing), naCoeffs); |
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[fba6f18] | 354 | } |
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| 355 | |
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| 356 | /// this is in-place, modifies a |
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| 357 | number naNeg(number a, const coeffs cf) |
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| 358 | { |
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| 359 | naTest(a); |
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| 360 | if (a != NULL) a = (number)p_Neg((poly)a, naRing); |
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| 361 | return a; |
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| 362 | } |
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| 363 | |
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| 364 | number naImPart(number a, const coeffs cf) |
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| 365 | { |
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| 366 | naTest(a); |
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| 367 | return NULL; |
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| 368 | } |
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| 369 | |
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[61b2e16] | 370 | number naInit_bigint(number longratBigIntNumber, const coeffs src, const coeffs cf) |
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| 371 | { |
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| 372 | assume( cf != NULL ); |
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| 373 | |
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| 374 | const ring A = cf->extRing; |
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| 375 | |
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| 376 | assume( A != NULL ); |
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| 377 | |
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| 378 | const coeffs C = A->cf; |
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| 379 | |
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| 380 | assume( C != NULL ); |
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| 381 | |
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| 382 | number n = n_Init_bigint(longratBigIntNumber, src, C); |
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| 383 | |
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| 384 | if ( n_IsZero(n, C) ) |
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| 385 | { |
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| 386 | n_Delete(&n, C); |
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| 387 | return NULL; |
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| 388 | } |
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[3c0498] | 389 | |
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[61b2e16] | 390 | return (number)p_NSet(n, A); |
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| 391 | } |
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| 392 | |
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| 393 | |
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| 394 | |
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[2f3764] | 395 | number naInit(long i, const coeffs cf) |
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[fba6f18] | 396 | { |
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| 397 | if (i == 0) return NULL; |
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| 398 | else return (number)p_ISet(i, naRing); |
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| 399 | } |
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| 400 | |
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| 401 | int naInt(number &a, const coeffs cf) |
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| 402 | { |
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| 403 | naTest(a); |
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[2c7f28] | 404 | poly aAsPoly = (poly)a; |
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[1090a98] | 405 | if(aAsPoly == NULL) |
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| 406 | return 0; |
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| 407 | if (!p_IsConstant(aAsPoly, naRing)) |
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| 408 | return 0; |
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| 409 | assume( aAsPoly != NULL ); |
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[2c7f28] | 410 | return n_Int(p_GetCoeff(aAsPoly, naRing), naCoeffs); |
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[fba6f18] | 411 | } |
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| 412 | |
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[afda22] | 413 | /* TRUE iff (a != 0 and (b == 0 or deg(a) > deg(b) or (deg(a)==deg(b) && lc(a)>lc(b))) */ |
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[fba6f18] | 414 | BOOLEAN naGreater(number a, number b, const coeffs cf) |
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| 415 | { |
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[2c7f28] | 416 | naTest(a); naTest(b); |
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[afda22] | 417 | if (naIsZero(a, cf)) |
---|
| 418 | { |
---|
| 419 | if (naIsZero(b, cf)) return FALSE; |
---|
| 420 | return !n_GreaterZero(pGetCoeff((poly)b),cf); |
---|
| 421 | } |
---|
| 422 | if (naIsZero(b, cf)) |
---|
| 423 | { |
---|
| 424 | return n_GreaterZero(pGetCoeff((poly)a),cf); |
---|
| 425 | } |
---|
| 426 | int aDeg = p_Totaldegree((poly)a, naRing); |
---|
| 427 | int bDeg = p_Totaldegree((poly)b, naRing); |
---|
| 428 | if (aDeg>bDeg) return TRUE; |
---|
| 429 | if (aDeg<bDeg) return FALSE; |
---|
| 430 | return n_Greater(pGetCoeff((poly)a),pGetCoeff((poly)b),cf); |
---|
[fba6f18] | 431 | } |
---|
| 432 | |
---|
| 433 | /* TRUE iff a != 0 and (LC(a) > 0 or deg(a) > 0) */ |
---|
| 434 | BOOLEAN naGreaterZero(number a, const coeffs cf) |
---|
| 435 | { |
---|
| 436 | naTest(a); |
---|
| 437 | if (a == NULL) return FALSE; |
---|
| 438 | if (n_GreaterZero(p_GetCoeff((poly)a, naRing), naCoeffs)) return TRUE; |
---|
[2c7f28] | 439 | if (p_Totaldegree((poly)a, naRing) > 0) return TRUE; |
---|
[fba6f18] | 440 | return FALSE; |
---|
| 441 | } |
---|
| 442 | |
---|
[03f7b5] | 443 | void naCoeffWrite(const coeffs cf, BOOLEAN details) |
---|
[fba6f18] | 444 | { |
---|
[a55ef0] | 445 | assume( cf != NULL ); |
---|
[3c0498] | 446 | |
---|
[a55ef0] | 447 | const ring A = cf->extRing; |
---|
[3c0498] | 448 | |
---|
[a55ef0] | 449 | assume( A != NULL ); |
---|
| 450 | assume( A->cf != NULL ); |
---|
[3c0498] | 451 | |
---|
[03f7b5] | 452 | n_CoeffWrite(A->cf, details); |
---|
[3c0498] | 453 | |
---|
[a55ef0] | 454 | // rWrite(A); |
---|
[3c0498] | 455 | |
---|
[a55ef0] | 456 | const int P = rVar(A); |
---|
| 457 | assume( P > 0 ); |
---|
[3c0498] | 458 | |
---|
[a55ef0] | 459 | Print("// %d parameter : ", P); |
---|
[3c0498] | 460 | |
---|
[a55ef0] | 461 | for (int nop=0; nop < P; nop ++) |
---|
| 462 | Print("%s ", rRingVar(nop, A)); |
---|
[3c0498] | 463 | |
---|
[03f7b5] | 464 | PrintLn(); |
---|
[3c0498] | 465 | |
---|
[dd668f] | 466 | const ideal I = A->qideal; |
---|
[a55ef0] | 467 | |
---|
| 468 | assume( I != NULL ); |
---|
| 469 | assume( IDELEMS(I) == 1 ); |
---|
[3c0498] | 470 | |
---|
[03f7b5] | 471 | |
---|
| 472 | if ( details ) |
---|
| 473 | { |
---|
| 474 | PrintS("// minpoly : ("); |
---|
| 475 | p_Write0( I->m[0], A); |
---|
| 476 | PrintS(")"); |
---|
| 477 | } |
---|
| 478 | else |
---|
| 479 | PrintS("// minpoly : ..."); |
---|
[3c0498] | 480 | |
---|
[03f7b5] | 481 | PrintLn(); |
---|
[3c0498] | 482 | |
---|
[a55ef0] | 483 | /* |
---|
| 484 | char *x = rRingVar(0, A); |
---|
| 485 | |
---|
[fba6f18] | 486 | Print("// Coefficients live in the extension field K[%s]/<f(%s)>\n", x, x); |
---|
| 487 | Print("// with the minimal polynomial f(%s) = %s\n", x, |
---|
[dd668f] | 488 | p_String(A->qideal->m[0], A)); |
---|
[a55ef0] | 489 | PrintS("// and K: "); |
---|
| 490 | */ |
---|
[fba6f18] | 491 | } |
---|
| 492 | |
---|
[ba2359] | 493 | number naAdd(number a, number b, const coeffs cf) |
---|
[fba6f18] | 494 | { |
---|
[ba2359] | 495 | naTest(a); naTest(b); |
---|
| 496 | if (a == NULL) return naCopy(b, cf); |
---|
| 497 | if (b == NULL) return naCopy(a, cf); |
---|
| 498 | poly aPlusB = p_Add_q(p_Copy((poly)a, naRing), |
---|
| 499 | p_Copy((poly)b, naRing), naRing); |
---|
[0fb5991] | 500 | definiteReduce(aPlusB, naMinpoly, cf); |
---|
| 501 | return (number)aPlusB; |
---|
[fba6f18] | 502 | } |
---|
| 503 | |
---|
| 504 | number naSub(number a, number b, const coeffs cf) |
---|
| 505 | { |
---|
[ba2359] | 506 | naTest(a); naTest(b); |
---|
| 507 | if (b == NULL) return naCopy(a, cf); |
---|
| 508 | poly minusB = p_Neg(p_Copy((poly)b, naRing), naRing); |
---|
[0fb5991] | 509 | if (a == NULL) return (number)minusB; |
---|
[ba2359] | 510 | poly aMinusB = p_Add_q(p_Copy((poly)a, naRing), minusB, naRing); |
---|
[0fb5991] | 511 | definiteReduce(aMinusB, naMinpoly, cf); |
---|
[ba2359] | 512 | return (number)aMinusB; |
---|
| 513 | } |
---|
| 514 | |
---|
| 515 | number naMult(number a, number b, const coeffs cf) |
---|
| 516 | { |
---|
| 517 | naTest(a); naTest(b); |
---|
| 518 | if (a == NULL) return NULL; |
---|
| 519 | if (b == NULL) return NULL; |
---|
| 520 | poly aTimesB = p_Mult_q(p_Copy((poly)a, naRing), |
---|
| 521 | p_Copy((poly)b, naRing), naRing); |
---|
[0fb5991] | 522 | definiteReduce(aTimesB, naMinpoly, cf); |
---|
| 523 | return (number)aTimesB; |
---|
[ba2359] | 524 | } |
---|
| 525 | |
---|
| 526 | number naDiv(number a, number b, const coeffs cf) |
---|
| 527 | { |
---|
| 528 | naTest(a); naTest(b); |
---|
| 529 | if (b == NULL) WerrorS(nDivBy0); |
---|
| 530 | if (a == NULL) return NULL; |
---|
| 531 | poly bInverse = (poly)naInvers(b, cf); |
---|
[865b20] | 532 | if(bInverse != NULL) // b is non-zero divisor! |
---|
| 533 | { |
---|
| 534 | poly aDivB = p_Mult_q(p_Copy((poly)a, naRing), bInverse, naRing); |
---|
| 535 | definiteReduce(aDivB, naMinpoly, cf); |
---|
| 536 | return (number)aDivB; |
---|
| 537 | } |
---|
| 538 | return NULL; |
---|
[ba2359] | 539 | } |
---|
| 540 | |
---|
| 541 | /* 0^0 = 0; |
---|
| 542 | for |exp| <= 7 compute power by a simple multiplication loop; |
---|
| 543 | for |exp| >= 8 compute power along binary presentation of |exp|, e.g. |
---|
| 544 | p^13 = p^1 * p^4 * p^8, where we utilise that |
---|
[2c7f28] | 545 | p^(2^(k+1)) = p^(2^k) * p^(2^k); |
---|
[ba2359] | 546 | intermediate reduction modulo the minimal polynomial is controlled by |
---|
| 547 | the in-place method heuristicReduce(poly, poly, coeffs); see there. |
---|
| 548 | */ |
---|
| 549 | void naPower(number a, int exp, number *b, const coeffs cf) |
---|
| 550 | { |
---|
| 551 | naTest(a); |
---|
[3c0498] | 552 | |
---|
[ba2359] | 553 | /* special cases first */ |
---|
| 554 | if (a == NULL) |
---|
| 555 | { |
---|
[0fb5991] | 556 | if (exp >= 0) *b = NULL; |
---|
[ba2359] | 557 | else WerrorS(nDivBy0); |
---|
| 558 | } |
---|
[35e86e] | 559 | else if (exp == 0) { *b = naInit(1, cf); return; } |
---|
| 560 | else if (exp == 1) { *b = naCopy(a, cf); return; } |
---|
| 561 | else if (exp == -1) { *b = naInvers(a, cf); return; } |
---|
[3c0498] | 562 | |
---|
[ba2359] | 563 | int expAbs = exp; if (expAbs < 0) expAbs = -expAbs; |
---|
[3c0498] | 564 | |
---|
[2c7f28] | 565 | /* now compute a^expAbs */ |
---|
[ba2359] | 566 | poly pow; poly aAsPoly = (poly)a; |
---|
| 567 | if (expAbs <= 7) |
---|
| 568 | { |
---|
| 569 | pow = p_Copy(aAsPoly, naRing); |
---|
| 570 | for (int i = 2; i <= expAbs; i++) |
---|
| 571 | { |
---|
| 572 | pow = p_Mult_q(pow, p_Copy(aAsPoly, naRing), naRing); |
---|
| 573 | heuristicReduce(pow, naMinpoly, cf); |
---|
| 574 | } |
---|
| 575 | definiteReduce(pow, naMinpoly, cf); |
---|
| 576 | } |
---|
| 577 | else |
---|
| 578 | { |
---|
| 579 | pow = p_ISet(1, naRing); |
---|
| 580 | poly factor = p_Copy(aAsPoly, naRing); |
---|
| 581 | while (expAbs != 0) |
---|
| 582 | { |
---|
| 583 | if (expAbs & 1) |
---|
| 584 | { |
---|
| 585 | pow = p_Mult_q(pow, p_Copy(factor, naRing), naRing); |
---|
| 586 | heuristicReduce(pow, naMinpoly, cf); |
---|
| 587 | } |
---|
| 588 | expAbs = expAbs / 2; |
---|
| 589 | if (expAbs != 0) |
---|
| 590 | { |
---|
[f681a60] | 591 | factor = p_Mult_q(factor, p_Copy(factor, naRing), naRing); |
---|
[ba2359] | 592 | heuristicReduce(factor, naMinpoly, cf); |
---|
| 593 | } |
---|
| 594 | } |
---|
[0fb5991] | 595 | p_Delete(&factor, naRing); |
---|
[ba2359] | 596 | definiteReduce(pow, naMinpoly, cf); |
---|
| 597 | } |
---|
[3c0498] | 598 | |
---|
[ba2359] | 599 | /* invert if original exponent was negative */ |
---|
| 600 | number n = (number)pow; |
---|
| 601 | if (exp < 0) |
---|
| 602 | { |
---|
| 603 | number m = naInvers(n, cf); |
---|
| 604 | naDelete(&n, cf); |
---|
| 605 | n = m; |
---|
| 606 | } |
---|
| 607 | *b = n; |
---|
| 608 | } |
---|
| 609 | |
---|
[2c7f28] | 610 | /* may reduce p modulo the reducer by calling definiteReduce; |
---|
[ba2359] | 611 | the decision is made based on the following heuristic |
---|
| 612 | (which should also only be changed here in this method): |
---|
[4a2260e] | 613 | if (deg(p) > 10*deg(reducer) then perform reduction; |
---|
| 614 | modifies p */ |
---|
| 615 | void heuristicReduce(poly &p, poly reducer, const coeffs cf) |
---|
[ba2359] | 616 | { |
---|
| 617 | #ifdef LDEBUG |
---|
[c28ecf] | 618 | p_Test((poly)p, naRing); |
---|
| 619 | p_Test((poly)reducer, naRing); |
---|
[ba2359] | 620 | #endif |
---|
[2c7f28] | 621 | if (p_Totaldegree(p, naRing) > 10 * p_Totaldegree(reducer, naRing)) |
---|
[ba2359] | 622 | definiteReduce(p, reducer, cf); |
---|
| 623 | } |
---|
| 624 | |
---|
[ce1f78] | 625 | void naWriteLong(number &a, const coeffs cf) |
---|
[ba2359] | 626 | { |
---|
| 627 | naTest(a); |
---|
| 628 | if (a == NULL) |
---|
| 629 | StringAppendS("0"); |
---|
| 630 | else |
---|
| 631 | { |
---|
| 632 | poly aAsPoly = (poly)a; |
---|
| 633 | /* basically, just write aAsPoly using p_Write, |
---|
| 634 | but use brackets around the output, if a is not |
---|
[6ccdd3a] | 635 | a constant living in naCoeffs = cf->extRing->cf */ |
---|
[fd01a8] | 636 | BOOLEAN useBrackets = !(p_IsConstant(aAsPoly, naRing)); |
---|
[ba2359] | 637 | if (useBrackets) StringAppendS("("); |
---|
[ce1f78] | 638 | p_String0Long(aAsPoly, naRing, naRing); |
---|
| 639 | if (useBrackets) StringAppendS(")"); |
---|
| 640 | } |
---|
| 641 | } |
---|
| 642 | |
---|
| 643 | void naWriteShort(number &a, const coeffs cf) |
---|
| 644 | { |
---|
| 645 | naTest(a); |
---|
| 646 | if (a == NULL) |
---|
| 647 | StringAppendS("0"); |
---|
| 648 | else |
---|
| 649 | { |
---|
| 650 | poly aAsPoly = (poly)a; |
---|
| 651 | /* basically, just write aAsPoly using p_Write, |
---|
| 652 | but use brackets around the output, if a is not |
---|
| 653 | a constant living in naCoeffs = cf->extRing->cf */ |
---|
| 654 | BOOLEAN useBrackets = !(p_IsConstant(aAsPoly, naRing)); |
---|
| 655 | if (useBrackets) StringAppendS("("); |
---|
| 656 | p_String0Short(aAsPoly, naRing, naRing); |
---|
[ba2359] | 657 | if (useBrackets) StringAppendS(")"); |
---|
| 658 | } |
---|
| 659 | } |
---|
| 660 | |
---|
| 661 | const char * naRead(const char *s, number *a, const coeffs cf) |
---|
| 662 | { |
---|
| 663 | poly aAsPoly; |
---|
| 664 | const char * result = p_Read(s, aAsPoly, naRing); |
---|
[58f11d0] | 665 | definiteReduce(aAsPoly, naMinpoly, cf); |
---|
[ba2359] | 666 | *a = (number)aAsPoly; |
---|
| 667 | return result; |
---|
[fba6f18] | 668 | } |
---|
| 669 | |
---|
[36ef6e0] | 670 | #if 0 |
---|
[ba2359] | 671 | /* implemented by the rule lcm(a, b) = a * b / gcd(a, b) */ |
---|
| 672 | number naLcm(number a, number b, const coeffs cf) |
---|
| 673 | { |
---|
| 674 | naTest(a); naTest(b); |
---|
| 675 | if (a == NULL) return NULL; |
---|
| 676 | if (b == NULL) return NULL; |
---|
| 677 | number theProduct = (number)p_Mult_q(p_Copy((poly)a, naRing), |
---|
| 678 | p_Copy((poly)b, naRing), naRing); |
---|
| 679 | /* note that theProduct needs not be reduced w.r.t. naMinpoly; |
---|
| 680 | but the final division will take care of the necessary reduction */ |
---|
| 681 | number theGcd = naGcd(a, b, cf); |
---|
[0fb5991] | 682 | return naDiv(theProduct, theGcd, cf); |
---|
[ba2359] | 683 | } |
---|
[36ef6e0] | 684 | #endif |
---|
[ba2359] | 685 | |
---|
[6ccdd3a] | 686 | /* expects *param to be castable to AlgExtInfo */ |
---|
[ba2359] | 687 | static BOOLEAN naCoeffIsEqual(const coeffs cf, n_coeffType n, void * param) |
---|
| 688 | { |
---|
[1f414c8] | 689 | if (ID != n) return FALSE; |
---|
[6ccdd3a] | 690 | AlgExtInfo *e = (AlgExtInfo *)param; |
---|
[ba2359] | 691 | /* for extension coefficient fields we expect the underlying |
---|
[2c7f28] | 692 | polynomial rings to be IDENTICAL, i.e. the SAME OBJECT; |
---|
[ba2359] | 693 | this expectation is based on the assumption that we have properly |
---|
| 694 | registered cf and perform reference counting rather than creating |
---|
| 695 | multiple copies of the same coefficient field/domain/ring */ |
---|
[4d94c97] | 696 | if (naRing == e->r) |
---|
| 697 | return TRUE; |
---|
[ba2359] | 698 | /* (Note that then also the minimal ideals will necessarily be |
---|
| 699 | the same, as they are attached to the ring.) */ |
---|
[4d94c97] | 700 | |
---|
| 701 | // NOTE: Q(a)[x] && Q(a)[y] should better share the _same_ Q(a)... |
---|
[7e9f12] | 702 | if( rEqual(naRing, e->r, TRUE) ) // also checks the equality of qideals |
---|
[4d94c97] | 703 | { |
---|
[dd668f] | 704 | const ideal mi = naRing->qideal; |
---|
[4d94c97] | 705 | assume( IDELEMS(mi) == 1 ); |
---|
[7e9f12] | 706 | const ideal ii = e->r->qideal; |
---|
[4d94c97] | 707 | assume( IDELEMS(ii) == 1 ); |
---|
| 708 | |
---|
| 709 | // TODO: the following should be extended for 2 *equal* rings... |
---|
[7e9f12] | 710 | assume( p_EqualPolys(mi->m[0], ii->m[0], naRing, e->r) ); |
---|
| 711 | |
---|
| 712 | rDelete(e->r); |
---|
| 713 | |
---|
| 714 | return TRUE; |
---|
[4d94c97] | 715 | } |
---|
| 716 | |
---|
[3c0498] | 717 | return FALSE; |
---|
| 718 | |
---|
[ba2359] | 719 | } |
---|
| 720 | |
---|
| 721 | int naSize(number a, const coeffs cf) |
---|
| 722 | { |
---|
| 723 | if (a == NULL) return -1; |
---|
| 724 | /* this has been taken from the old implementation of field extensions, |
---|
| 725 | where we computed the sum of the degree and the number of terms in |
---|
| 726 | (poly)a; so we leave it at that, for the time being; |
---|
| 727 | maybe, the number of terms alone is a better measure? */ |
---|
| 728 | poly aAsPoly = (poly)a; |
---|
| 729 | int theDegree = 0; int noOfTerms = 0; |
---|
| 730 | while (aAsPoly != NULL) |
---|
| 731 | { |
---|
| 732 | noOfTerms++; |
---|
[2c7f28] | 733 | int d = p_GetExp(aAsPoly, 1, naRing); |
---|
[ba2359] | 734 | if (d > theDegree) theDegree = d; |
---|
| 735 | pIter(aAsPoly); |
---|
| 736 | } |
---|
| 737 | return theDegree + noOfTerms; |
---|
| 738 | } |
---|
| 739 | |
---|
| 740 | /* performs polynomial division and overrides p by the remainder |
---|
[4a2260e] | 741 | of division of p by the reducer; |
---|
| 742 | modifies p */ |
---|
| 743 | void definiteReduce(poly &p, poly reducer, const coeffs cf) |
---|
[ba2359] | 744 | { |
---|
| 745 | #ifdef LDEBUG |
---|
[c28ecf] | 746 | p_Test((poly)p, naRing); |
---|
| 747 | p_Test((poly)reducer, naRing); |
---|
[ba2359] | 748 | #endif |
---|
[bca341] | 749 | if ((p!=NULL) && (p_GetExp(p,1,naRing)>=p_GetExp(reducer,1,naRing))) |
---|
| 750 | { |
---|
| 751 | p_PolyDiv(p, reducer, FALSE, naRing); |
---|
| 752 | } |
---|
[ba2359] | 753 | } |
---|
| 754 | |
---|
[146c603] | 755 | void naNormalize(number &a, const coeffs cf) |
---|
| 756 | { |
---|
| 757 | poly aa=(poly)a; |
---|
[3c0498] | 758 | if (aa!=naMinpoly) |
---|
| 759 | definiteReduce(aa,naMinpoly,cf); |
---|
[146c603] | 760 | a=(number)aa; |
---|
| 761 | } |
---|
| 762 | |
---|
| 763 | #ifdef HAVE_FACTORY |
---|
| 764 | number naConvFactoryNSingN( const CanonicalForm n, const coeffs cf) |
---|
| 765 | { |
---|
| 766 | if (n.isZero()) return NULL; |
---|
| 767 | poly p=convFactoryPSingP(n,naRing); |
---|
| 768 | return (number)p; |
---|
| 769 | } |
---|
[2d2e40] | 770 | CanonicalForm naConvSingNFactoryN( number n, BOOLEAN /*setChar*/, const coeffs cf ) |
---|
[146c603] | 771 | { |
---|
| 772 | naTest(n); |
---|
| 773 | if (n==NULL) return CanonicalForm(0); |
---|
| 774 | |
---|
| 775 | return convSingPFactoryP((poly)n,naRing); |
---|
| 776 | } |
---|
| 777 | #endif |
---|
| 778 | |
---|
| 779 | |
---|
[cfb500] | 780 | /* IMPORTANT NOTE: Since an algebraic field extension is again a field, |
---|
| 781 | the gcd of two elements is not very interesting. (It |
---|
| 782 | is actually any unit in the field, i.e., any non- |
---|
| 783 | zero element.) Note that the below method does not operate |
---|
| 784 | in this strong sense but rather computes the gcd of |
---|
| 785 | two given elements in the underlying polynomial ring. */ |
---|
[ba2359] | 786 | number naGcd(number a, number b, const coeffs cf) |
---|
| 787 | { |
---|
| 788 | naTest(a); naTest(b); |
---|
| 789 | if ((a == NULL) && (b == NULL)) WerrorS(nDivBy0); |
---|
[dc79bd] | 790 | const ring R = naRing; |
---|
| 791 | return (number) singclap_gcd(p_Copy((poly)a, R), p_Copy((poly)b, R), R); |
---|
| 792 | // return (number)p_Gcd((poly)a, (poly)b, naRing); |
---|
[ba2359] | 793 | } |
---|
| 794 | |
---|
| 795 | number naInvers(number a, const coeffs cf) |
---|
| 796 | { |
---|
| 797 | naTest(a); |
---|
| 798 | if (a == NULL) WerrorS(nDivBy0); |
---|
[865b20] | 799 | |
---|
[dc79bd] | 800 | poly aFactor = NULL; poly mFactor = NULL; poly theGcd = NULL; |
---|
| 801 | // singclap_extgcd! |
---|
| 802 | const BOOLEAN ret = singclap_extgcd ((poly)a, naMinpoly, theGcd, aFactor, mFactor, naRing); |
---|
| 803 | |
---|
| 804 | assume( !ret ); |
---|
| 805 | |
---|
| 806 | // if( ret ) theGcd = p_ExtGcd((poly)a, aFactor, naMinpoly, mFactor, naRing); |
---|
[865b20] | 807 | |
---|
[4a2260e] | 808 | naTest((number)theGcd); naTest((number)aFactor); naTest((number)mFactor); |
---|
[f0b01f] | 809 | p_Delete(&mFactor, naRing); |
---|
[865b20] | 810 | |
---|
| 811 | // /* the gcd must be 1 since naMinpoly is irreducible and a != NULL: */ |
---|
| 812 | // assume(naIsOne((number)theGcd, cf)); |
---|
| 813 | |
---|
| 814 | if( !naIsOne((number)theGcd, cf) ) |
---|
| 815 | { |
---|
| 816 | WerrorS("zero divisor found - your minpoly is not irreducible"); |
---|
| 817 | p_Delete(&aFactor, naRing); aFactor = NULL; |
---|
| 818 | } |
---|
| 819 | p_Delete(&theGcd, naRing); |
---|
| 820 | |
---|
[f0b01f] | 821 | return (number)(aFactor); |
---|
[ba2359] | 822 | } |
---|
| 823 | |
---|
| 824 | /* assumes that src = Q, dst = Q(a) */ |
---|
| 825 | number naMap00(number a, const coeffs src, const coeffs dst) |
---|
| 826 | { |
---|
[2c7f28] | 827 | if (n_IsZero(a, src)) return NULL; |
---|
[6ccdd3a] | 828 | assume(src == dst->extRing->cf); |
---|
| 829 | poly result = p_One(dst->extRing); |
---|
[3c4a33] | 830 | p_SetCoeff(result, n_Copy(a, src), dst->extRing); |
---|
[ba2359] | 831 | return (number)result; |
---|
| 832 | } |
---|
| 833 | |
---|
| 834 | /* assumes that src = Z/p, dst = Q(a) */ |
---|
| 835 | number naMapP0(number a, const coeffs src, const coeffs dst) |
---|
| 836 | { |
---|
[2c7f28] | 837 | if (n_IsZero(a, src)) return NULL; |
---|
[ba2359] | 838 | /* mapping via intermediate int: */ |
---|
| 839 | int n = n_Int(a, src); |
---|
[6ccdd3a] | 840 | number q = n_Init(n, dst->extRing->cf); |
---|
| 841 | poly result = p_One(dst->extRing); |
---|
| 842 | p_SetCoeff(result, q, dst->extRing); |
---|
[ba2359] | 843 | return (number)result; |
---|
| 844 | } |
---|
| 845 | |
---|
[c14846c] | 846 | #if 0 |
---|
[ba2359] | 847 | /* assumes that either src = Q(a), dst = Q(a), or |
---|
[2c7f28] | 848 | src = Z/p(a), dst = Z/p(a) */ |
---|
[ba2359] | 849 | number naCopyMap(number a, const coeffs src, const coeffs dst) |
---|
| 850 | { |
---|
| 851 | return naCopy(a, dst); |
---|
| 852 | } |
---|
[c14846c] | 853 | #endif |
---|
[ba2359] | 854 | |
---|
[2d2e40] | 855 | number naCopyExt(number a, const coeffs src, const coeffs) |
---|
[331fd0] | 856 | { |
---|
| 857 | fraction fa=(fraction)a; |
---|
| 858 | return (number)p_Copy(NUM(fa),src->extRing); |
---|
| 859 | } |
---|
| 860 | |
---|
[ba2359] | 861 | /* assumes that src = Q, dst = Z/p(a) */ |
---|
| 862 | number naMap0P(number a, const coeffs src, const coeffs dst) |
---|
| 863 | { |
---|
[2c7f28] | 864 | if (n_IsZero(a, src)) return NULL; |
---|
[6ccdd3a] | 865 | int p = rChar(dst->extRing); |
---|
[79020f] | 866 | |
---|
| 867 | number q = nlModP(a, src, dst->extRing->cf); |
---|
| 868 | |
---|
| 869 | poly result = p_NSet(q, dst->extRing); |
---|
[3c0498] | 870 | |
---|
[ba2359] | 871 | return (number)result; |
---|
| 872 | } |
---|
| 873 | |
---|
| 874 | /* assumes that src = Z/p, dst = Z/p(a) */ |
---|
| 875 | number naMapPP(number a, const coeffs src, const coeffs dst) |
---|
| 876 | { |
---|
[2c7f28] | 877 | if (n_IsZero(a, src)) return NULL; |
---|
[6ccdd3a] | 878 | assume(src == dst->extRing->cf); |
---|
| 879 | poly result = p_One(dst->extRing); |
---|
[3c4a33] | 880 | p_SetCoeff(result, n_Copy(a, src), dst->extRing); |
---|
[ba2359] | 881 | return (number)result; |
---|
| 882 | } |
---|
| 883 | |
---|
| 884 | /* assumes that src = Z/u, dst = Z/p(a), where u != p */ |
---|
| 885 | number naMapUP(number a, const coeffs src, const coeffs dst) |
---|
| 886 | { |
---|
[2c7f28] | 887 | if (n_IsZero(a, src)) return NULL; |
---|
[ba2359] | 888 | /* mapping via intermediate int: */ |
---|
| 889 | int n = n_Int(a, src); |
---|
[6ccdd3a] | 890 | number q = n_Init(n, dst->extRing->cf); |
---|
| 891 | poly result = p_One(dst->extRing); |
---|
| 892 | p_SetCoeff(result, q, dst->extRing); |
---|
[ba2359] | 893 | return (number)result; |
---|
| 894 | } |
---|
| 895 | |
---|
[0654122] | 896 | nMapFunc naSetMap(const coeffs src, const coeffs dst) |
---|
[ba2359] | 897 | { |
---|
[488808e] | 898 | /* dst is expected to be an algebraic field extension */ |
---|
[1f414c8] | 899 | assume(getCoeffType(dst) == ID); |
---|
[da0565b] | 900 | |
---|
| 901 | if( src == dst ) return ndCopyMap; |
---|
[3c0498] | 902 | |
---|
[488808e] | 903 | int h = 0; /* the height of the extension tower given by dst */ |
---|
| 904 | coeffs bDst = nCoeff_bottom(dst, h); /* the bottom field in the tower dst */ |
---|
[331fd0] | 905 | coeffs bSrc = nCoeff_bottom(src, h); /* the bottom field in the tower src */ |
---|
[3c0498] | 906 | |
---|
[488808e] | 907 | /* for the time being, we only provide maps if h = 1 and if b is Q or |
---|
| 908 | some field Z/pZ: */ |
---|
[331fd0] | 909 | if (h==0) |
---|
[488808e] | 910 | { |
---|
[331fd0] | 911 | if (nCoeff_is_Q(src) && nCoeff_is_Q(bDst)) |
---|
| 912 | return naMap00; /// Q --> Q(a) |
---|
| 913 | if (nCoeff_is_Zp(src) && nCoeff_is_Q(bDst)) |
---|
| 914 | return naMapP0; /// Z/p --> Q(a) |
---|
| 915 | if (nCoeff_is_Q(src) && nCoeff_is_Zp(bDst)) |
---|
| 916 | return naMap0P; /// Q --> Z/p(a) |
---|
| 917 | if (nCoeff_is_Zp(src) && nCoeff_is_Zp(bDst)) |
---|
| 918 | { |
---|
| 919 | if (src->ch == dst->ch) return naMapPP; /// Z/p --> Z/p(a) |
---|
| 920 | else return naMapUP; /// Z/u --> Z/p(a) |
---|
| 921 | } |
---|
[488808e] | 922 | } |
---|
| 923 | if (h != 1) return NULL; |
---|
[331fd0] | 924 | if ((!nCoeff_is_Zp(bDst)) && (!nCoeff_is_Q(bDst))) return NULL; |
---|
[488808e] | 925 | if ((!nCoeff_is_Zp(bSrc)) && (!nCoeff_is_Q(bSrc))) return NULL; |
---|
[3c0498] | 926 | |
---|
[488808e] | 927 | if (nCoeff_is_Q(bSrc) && nCoeff_is_Q(bDst)) |
---|
[ba2359] | 928 | { |
---|
[2c7f28] | 929 | if (strcmp(rRingVar(0, src->extRing), |
---|
| 930 | rRingVar(0, dst->extRing)) == 0) |
---|
[331fd0] | 931 | { |
---|
| 932 | if (src->type==n_algExt) |
---|
[c14846c] | 933 | return ndCopyMap; // naCopyMap; /// Q(a) --> Q(a) |
---|
[331fd0] | 934 | else |
---|
| 935 | return naCopyExt; |
---|
| 936 | } |
---|
[ba2359] | 937 | else |
---|
[331fd0] | 938 | return NULL; /// Q(b) --> Q(a) |
---|
[ba2359] | 939 | } |
---|
[3c0498] | 940 | |
---|
[488808e] | 941 | if (nCoeff_is_Zp(bSrc) && nCoeff_is_Zp(bDst)) |
---|
[ba2359] | 942 | { |
---|
[6f6b9d] | 943 | if (strcmp(rRingVar(0,src->extRing),rRingVar(0,dst->extRing))==0) |
---|
[331fd0] | 944 | { |
---|
| 945 | if (src->type==n_algExt) |
---|
[c14846c] | 946 | return ndCopyMap; // naCopyMap; /// Z/p(a) --> Z/p(a) |
---|
[331fd0] | 947 | else |
---|
| 948 | return naCopyExt; |
---|
| 949 | } |
---|
[ba2359] | 950 | else |
---|
[331fd0] | 951 | return NULL; /// Z/p(b) --> Z/p(a) |
---|
[ba2359] | 952 | } |
---|
[3c0498] | 953 | |
---|
[ba2359] | 954 | return NULL; /// default |
---|
| 955 | } |
---|
[fba6f18] | 956 | |
---|
[da5d77] | 957 | static int naParDeg(number a, const coeffs cf) |
---|
[48a41a] | 958 | { |
---|
| 959 | if (a == NULL) return -1; |
---|
| 960 | poly aa=(poly)a; |
---|
| 961 | return cf->extRing->pFDeg(aa,cf->extRing); |
---|
| 962 | } |
---|
| 963 | |
---|
[7fee876] | 964 | /// return the specified parameter as a number in the given alg. field |
---|
| 965 | static number naParameter(const int iParameter, const coeffs cf) |
---|
| 966 | { |
---|
| 967 | assume(getCoeffType(cf) == ID); |
---|
| 968 | |
---|
| 969 | const ring R = cf->extRing; |
---|
[3c0498] | 970 | assume( R != NULL ); |
---|
[7fee876] | 971 | assume( 0 < iParameter && iParameter <= rVar(R) ); |
---|
| 972 | |
---|
| 973 | poly p = p_One(R); p_SetExp(p, iParameter, 1, R); p_Setm(p, R); |
---|
| 974 | |
---|
[3c0498] | 975 | return (number) p; |
---|
[7fee876] | 976 | } |
---|
| 977 | |
---|
| 978 | |
---|
[3c0498] | 979 | /// if m == var(i)/1 => return i, |
---|
[7fee876] | 980 | int naIsParam(number m, const coeffs cf) |
---|
| 981 | { |
---|
| 982 | assume(getCoeffType(cf) == ID); |
---|
| 983 | |
---|
| 984 | const ring R = cf->extRing; |
---|
[3c0498] | 985 | assume( R != NULL ); |
---|
[7fee876] | 986 | |
---|
[3c0498] | 987 | return p_Var( (poly)m, R ); |
---|
[7fee876] | 988 | } |
---|
| 989 | |
---|
[dc79bd] | 990 | |
---|
| 991 | static void naClearContent(ICoeffsEnumerator& numberCollectionEnumerator, number& c, const coeffs cf) |
---|
[de88371] | 992 | { |
---|
| 993 | assume(cf != NULL); |
---|
| 994 | assume(getCoeffType(cf) == ID); |
---|
[dc79bd] | 995 | assume(nCoeff_is_Q_algext(cf)); // only over (Q[a]/m(a)), while the default impl. is used over Zp[a]/m(a) ! |
---|
| 996 | |
---|
| 997 | const ring R = cf->extRing; |
---|
| 998 | assume(R != NULL); |
---|
| 999 | const coeffs Q = R->cf; |
---|
| 1000 | assume(Q != NULL); |
---|
| 1001 | assume(nCoeff_is_Q(Q)); |
---|
| 1002 | |
---|
| 1003 | numberCollectionEnumerator.Reset(); |
---|
| 1004 | |
---|
| 1005 | if( !numberCollectionEnumerator.MoveNext() ) // empty zero polynomial? |
---|
| 1006 | { |
---|
| 1007 | c = n_Init(1, cf); |
---|
| 1008 | return; |
---|
| 1009 | } |
---|
| 1010 | |
---|
| 1011 | naTest(numberCollectionEnumerator.Current()); |
---|
| 1012 | |
---|
| 1013 | // part 1, find a small candidate for gcd |
---|
| 1014 | int s1; int s=2147483647; // max. int |
---|
| 1015 | |
---|
| 1016 | const BOOLEAN lc_is_pos=naGreaterZero(numberCollectionEnumerator.Current(),cf); |
---|
| 1017 | |
---|
| 1018 | int normalcount = 0; |
---|
| 1019 | |
---|
| 1020 | poly cand1, cand; |
---|
| 1021 | |
---|
| 1022 | do |
---|
| 1023 | { |
---|
| 1024 | number& n = numberCollectionEnumerator.Current(); |
---|
| 1025 | naNormalize(n, cf); ++normalcount; |
---|
| 1026 | |
---|
| 1027 | naTest(n); |
---|
| 1028 | |
---|
| 1029 | cand1 = (poly)n; |
---|
| 1030 | |
---|
| 1031 | s1 = p_Deg(cand1, R); // naSize? |
---|
| 1032 | if (s>s1) |
---|
| 1033 | { |
---|
| 1034 | cand = cand1; |
---|
| 1035 | s = s1; |
---|
| 1036 | } |
---|
| 1037 | } while (numberCollectionEnumerator.MoveNext() ); |
---|
| 1038 | |
---|
| 1039 | // assume( nlGreaterZero(cand,cf) ); // cand may be a negative integer! |
---|
| 1040 | |
---|
| 1041 | cand = p_Copy(cand, R); |
---|
| 1042 | // part 2: compute gcd(cand,all coeffs) |
---|
| 1043 | |
---|
| 1044 | numberCollectionEnumerator.Reset(); |
---|
| 1045 | |
---|
| 1046 | int length = 0; |
---|
| 1047 | while (numberCollectionEnumerator.MoveNext() ) |
---|
| 1048 | { |
---|
| 1049 | number& n = numberCollectionEnumerator.Current(); |
---|
| 1050 | ++length; |
---|
| 1051 | |
---|
| 1052 | if( (--normalcount) <= 0) |
---|
| 1053 | naNormalize(n, cf); |
---|
| 1054 | |
---|
| 1055 | naTest(n); |
---|
| 1056 | |
---|
| 1057 | // p_InpGcd(cand, (poly)n, R); |
---|
| 1058 | |
---|
| 1059 | cand = singclap_gcd(cand, p_Copy((poly)n, R), R); |
---|
| 1060 | |
---|
| 1061 | // cand1 = p_Gcd(cand,(poly)n, R); p_Delete(&cand, R); cand = cand1; |
---|
| 1062 | |
---|
| 1063 | assume( naGreaterZero((number)cand, cf) ); // ??? |
---|
| 1064 | /* |
---|
| 1065 | if(p_IsConstant(cand,R)) |
---|
| 1066 | { |
---|
| 1067 | c = cand; |
---|
| 1068 | |
---|
| 1069 | if(!lc_is_pos) |
---|
| 1070 | { |
---|
| 1071 | // make the leading coeff positive |
---|
| 1072 | c = nlNeg(c, cf); |
---|
| 1073 | numberCollectionEnumerator.Reset(); |
---|
| 1074 | |
---|
| 1075 | while (numberCollectionEnumerator.MoveNext() ) |
---|
| 1076 | { |
---|
| 1077 | number& nn = numberCollectionEnumerator.Current(); |
---|
| 1078 | nn = nlNeg(nn, cf); |
---|
| 1079 | } |
---|
| 1080 | } |
---|
| 1081 | return; |
---|
| 1082 | } |
---|
| 1083 | */ |
---|
| 1084 | |
---|
| 1085 | } |
---|
| 1086 | |
---|
| 1087 | // part3: all coeffs = all coeffs / cand |
---|
| 1088 | if (!lc_is_pos) |
---|
| 1089 | cand = p_Neg(cand, R); |
---|
| 1090 | |
---|
| 1091 | c = (number)cand; naTest(c); |
---|
| 1092 | |
---|
| 1093 | poly cInverse = (poly)naInvers(c, cf); |
---|
| 1094 | assume(cInverse != NULL); // c is non-zero divisor!? |
---|
| 1095 | |
---|
| 1096 | |
---|
| 1097 | numberCollectionEnumerator.Reset(); |
---|
| 1098 | |
---|
| 1099 | |
---|
| 1100 | while (numberCollectionEnumerator.MoveNext() ) |
---|
| 1101 | { |
---|
| 1102 | number& n = numberCollectionEnumerator.Current(); |
---|
| 1103 | |
---|
| 1104 | assume( length > 0 ); |
---|
| 1105 | |
---|
| 1106 | if( --length > 0 ) |
---|
| 1107 | { |
---|
| 1108 | assume( cInverse != NULL ); |
---|
| 1109 | n = (number) p_Mult_q(p_Copy(cInverse, R), (poly)n, R); |
---|
| 1110 | } |
---|
| 1111 | else |
---|
| 1112 | { |
---|
| 1113 | n = (number) p_Mult_q(cInverse, (poly)n, R); |
---|
| 1114 | cInverse = NULL; |
---|
| 1115 | assume(length == 0); |
---|
| 1116 | } |
---|
| 1117 | |
---|
| 1118 | definiteReduce((poly &)n, naMinpoly, cf); |
---|
| 1119 | } |
---|
| 1120 | |
---|
| 1121 | assume(length == 0); |
---|
| 1122 | assume(cInverse == NULL); // p_Delete(&cInverse, R); |
---|
| 1123 | |
---|
| 1124 | // Quick and dirty fix for constant content clearing... !? |
---|
| 1125 | CRecursivePolyCoeffsEnumerator<NAConverter> itr(numberCollectionEnumerator); // recursively treat the numbers as polys! |
---|
| 1126 | |
---|
| 1127 | number cc; |
---|
| 1128 | |
---|
| 1129 | extern void nlClearContentNoPositiveLead(ICoeffsEnumerator&, number&, const coeffs); |
---|
| 1130 | |
---|
| 1131 | nlClearContentNoPositiveLead(itr, cc, Q); // TODO: get rid of (-LC) normalization!? |
---|
| 1132 | |
---|
| 1133 | // over alg. ext. of Q // takes over the input number |
---|
| 1134 | c = (number) p_Mult_nn( (poly)c, cc, R); |
---|
| 1135 | // p_Mult_q(p_NSet(cc, R), , R); |
---|
| 1136 | |
---|
| 1137 | n_Delete(&cc, Q); |
---|
| 1138 | |
---|
| 1139 | // TODO: the above is not enough! need GCD's of polynomial coeffs...! |
---|
| 1140 | /* |
---|
| 1141 | // old and wrong part of p_Content |
---|
| 1142 | if (rField_is_Q_a(r) && !CLEARENUMERATORS) // should not be used anymore if CLEARENUMERATORS is 1 |
---|
| 1143 | { |
---|
| 1144 | // we only need special handling for alg. ext. |
---|
| 1145 | if (getCoeffType(r->cf)==n_algExt) |
---|
| 1146 | { |
---|
| 1147 | number hzz = n_Init(1, r->cf->extRing->cf); |
---|
| 1148 | p=ph; |
---|
| 1149 | while (p!=NULL) |
---|
| 1150 | { // each monom: coeff in Q_a |
---|
| 1151 | poly c_n_n=(poly)pGetCoeff(p); |
---|
| 1152 | poly c_n=c_n_n; |
---|
| 1153 | while (c_n!=NULL) |
---|
| 1154 | { // each monom: coeff in Q |
---|
| 1155 | d=n_Lcm(hzz,pGetCoeff(c_n),r->cf->extRing->cf); |
---|
| 1156 | n_Delete(&hzz,r->cf->extRing->cf); |
---|
| 1157 | hzz=d; |
---|
| 1158 | pIter(c_n); |
---|
| 1159 | } |
---|
| 1160 | pIter(p); |
---|
| 1161 | } |
---|
| 1162 | // hzz contains the 1/lcm of all denominators in c_n_n |
---|
| 1163 | h=n_Invers(hzz,r->cf->extRing->cf); |
---|
| 1164 | n_Delete(&hzz,r->cf->extRing->cf); |
---|
| 1165 | n_Normalize(h,r->cf->extRing->cf); |
---|
| 1166 | if(!n_IsOne(h,r->cf->extRing->cf)) |
---|
| 1167 | { |
---|
| 1168 | p=ph; |
---|
| 1169 | while (p!=NULL) |
---|
| 1170 | { // each monom: coeff in Q_a |
---|
| 1171 | poly c_n=(poly)pGetCoeff(p); |
---|
| 1172 | while (c_n!=NULL) |
---|
| 1173 | { // each monom: coeff in Q |
---|
| 1174 | d=n_Mult(h,pGetCoeff(c_n),r->cf->extRing->cf); |
---|
| 1175 | n_Normalize(d,r->cf->extRing->cf); |
---|
| 1176 | n_Delete(&pGetCoeff(c_n),r->cf->extRing->cf); |
---|
| 1177 | pGetCoeff(c_n)=d; |
---|
| 1178 | pIter(c_n); |
---|
| 1179 | } |
---|
| 1180 | pIter(p); |
---|
| 1181 | } |
---|
| 1182 | } |
---|
| 1183 | n_Delete(&h,r->cf->extRing->cf); |
---|
| 1184 | } |
---|
| 1185 | } |
---|
| 1186 | */ |
---|
| 1187 | |
---|
| 1188 | |
---|
| 1189 | // c = n_Init(1, cf); assume(FALSE); // TODO: NOT YET IMPLEMENTED!!! |
---|
| 1190 | } |
---|
| 1191 | |
---|
| 1192 | |
---|
| 1193 | static void naClearDenominators(ICoeffsEnumerator& numberCollectionEnumerator, number& c, const coeffs cf) |
---|
| 1194 | { |
---|
| 1195 | assume(cf != NULL); |
---|
| 1196 | assume(getCoeffType(cf) == ID); |
---|
| 1197 | assume(nCoeff_is_Q_algext(cf)); // only over (Q[a]/m(a)), while the default impl. is used over Zp[a]/m(a) ! |
---|
| 1198 | |
---|
| 1199 | assume(cf->extRing != NULL); |
---|
| 1200 | const coeffs Q = cf->extRing->cf; |
---|
| 1201 | assume(Q != NULL); |
---|
| 1202 | assume(nCoeff_is_Q(Q)); |
---|
| 1203 | number n; |
---|
| 1204 | CRecursivePolyCoeffsEnumerator<NAConverter> itr(numberCollectionEnumerator); // recursively treat the numbers as polys! |
---|
| 1205 | |
---|
| 1206 | extern void nlClearDenominatorsNoPositiveLead(ICoeffsEnumerator&, number&, const coeffs); |
---|
| 1207 | |
---|
| 1208 | nlClearDenominatorsNoPositiveLead(itr, n, Q); // this should probably be fine... |
---|
| 1209 | c = (number)p_NSet(n, cf->extRing); // over alg. ext. of Q // takes over the input number |
---|
[de88371] | 1210 | } |
---|
| 1211 | |
---|
| 1212 | |
---|
[ba2359] | 1213 | BOOLEAN naInitChar(coeffs cf, void * infoStruct) |
---|
[3c0498] | 1214 | { |
---|
[a55ef0] | 1215 | assume( infoStruct != NULL ); |
---|
| 1216 | |
---|
[6ccdd3a] | 1217 | AlgExtInfo *e = (AlgExtInfo *)infoStruct; |
---|
| 1218 | /// first check whether cf->extRing != NULL and delete old ring??? |
---|
| 1219 | |
---|
[ec5ec8] | 1220 | assume(e->r != NULL); // extRing; |
---|
| 1221 | assume(e->r->cf != NULL); // extRing->cf; |
---|
| 1222 | |
---|
[7e9f12] | 1223 | assume((e->r->qideal != NULL) && // minideal has one |
---|
| 1224 | (IDELEMS(e->r->qideal) == 1) && // non-zero generator |
---|
| 1225 | (e->r->qideal->m[0] != NULL) ); // at m[0]; |
---|
[ec5ec8] | 1226 | |
---|
| 1227 | assume( cf != NULL ); |
---|
[1f414c8] | 1228 | assume(getCoeffType(cf) == ID); // coeff type; |
---|
[7fee876] | 1229 | |
---|
[7e9f12] | 1230 | e->r->ref ++; // increase the ref.counter for the ground poly. ring! |
---|
| 1231 | const ring R = e->r; // no copy! |
---|
| 1232 | assume( R->qideal == e->r->qideal ); |
---|
| 1233 | cf->extRing = R; |
---|
[ec5ec8] | 1234 | |
---|
[73a9ffb] | 1235 | /* propagate characteristic up so that it becomes |
---|
| 1236 | directly accessible in cf: */ |
---|
[7fee876] | 1237 | cf->ch = R->cf->ch; |
---|
| 1238 | |
---|
[fba6f18] | 1239 | #ifdef LDEBUG |
---|
[c28ecf] | 1240 | p_Test((poly)naMinpoly, naRing); |
---|
[fba6f18] | 1241 | #endif |
---|
[3c0498] | 1242 | |
---|
[fba6f18] | 1243 | cf->cfGreaterZero = naGreaterZero; |
---|
| 1244 | cf->cfGreater = naGreater; |
---|
| 1245 | cf->cfEqual = naEqual; |
---|
| 1246 | cf->cfIsZero = naIsZero; |
---|
| 1247 | cf->cfIsOne = naIsOne; |
---|
| 1248 | cf->cfIsMOne = naIsMOne; |
---|
| 1249 | cf->cfInit = naInit; |
---|
[3c0498] | 1250 | cf->cfInit_bigint = naInit_bigint; |
---|
[fba6f18] | 1251 | cf->cfInt = naInt; |
---|
| 1252 | cf->cfNeg = naNeg; |
---|
[ba2359] | 1253 | cf->cfAdd = naAdd; |
---|
[fba6f18] | 1254 | cf->cfSub = naSub; |
---|
[ba2359] | 1255 | cf->cfMult = naMult; |
---|
| 1256 | cf->cfDiv = naDiv; |
---|
| 1257 | cf->cfExactDiv = naDiv; |
---|
| 1258 | cf->cfPower = naPower; |
---|
| 1259 | cf->cfCopy = naCopy; |
---|
[ce1f78] | 1260 | |
---|
| 1261 | cf->cfWriteLong = naWriteLong; |
---|
| 1262 | |
---|
| 1263 | if( rCanShortOut(naRing) ) |
---|
| 1264 | cf->cfWriteShort = naWriteShort; |
---|
| 1265 | else |
---|
| 1266 | cf->cfWriteShort = naWriteLong; |
---|
[3c0498] | 1267 | |
---|
[ba2359] | 1268 | cf->cfRead = naRead; |
---|
[fba6f18] | 1269 | cf->cfDelete = naDelete; |
---|
| 1270 | cf->cfSetMap = naSetMap; |
---|
| 1271 | cf->cfGetDenom = naGetDenom; |
---|
[0fb5991] | 1272 | cf->cfGetNumerator = naGetNumerator; |
---|
[ba2359] | 1273 | cf->cfRePart = naCopy; |
---|
[fba6f18] | 1274 | cf->cfImPart = naImPart; |
---|
| 1275 | cf->cfCoeffWrite = naCoeffWrite; |
---|
[146c603] | 1276 | cf->cfNormalize = naNormalize; |
---|
[dbcf787] | 1277 | #ifdef LDEBUG |
---|
[fba6f18] | 1278 | cf->cfDBTest = naDBTest; |
---|
[dbcf787] | 1279 | #endif |
---|
[ba2359] | 1280 | cf->cfGcd = naGcd; |
---|
[36ef6e0] | 1281 | //cf->cfLcm = naLcm; |
---|
[ba2359] | 1282 | cf->cfSize = naSize; |
---|
| 1283 | cf->nCoeffIsEqual = naCoeffIsEqual; |
---|
| 1284 | cf->cfInvers = naInvers; |
---|
[dc79bd] | 1285 | cf->cfIntDiv = naDiv; // ??? |
---|
[146c603] | 1286 | #ifdef HAVE_FACTORY |
---|
| 1287 | cf->convFactoryNSingN=naConvFactoryNSingN; |
---|
| 1288 | cf->convSingNFactoryN=naConvSingNFactoryN; |
---|
| 1289 | #endif |
---|
[48a41a] | 1290 | cf->cfParDeg = naParDeg; |
---|
[3c0498] | 1291 | |
---|
[7fee876] | 1292 | cf->iNumberOfParameters = rVar(R); |
---|
| 1293 | cf->pParameterNames = R->names; |
---|
| 1294 | cf->cfParameter = naParameter; |
---|
[3c0498] | 1295 | |
---|
[de88371] | 1296 | if( nCoeff_is_Q(R->cf) ) |
---|
[dc79bd] | 1297 | { |
---|
[de88371] | 1298 | cf->cfClearContent = naClearContent; |
---|
[dc79bd] | 1299 | cf->cfClearDenominators = naClearDenominators; |
---|
| 1300 | } |
---|
[de88371] | 1301 | |
---|
[7fee876] | 1302 | return FALSE; |
---|
[e82417] | 1303 | } |
---|