[ad3c3ff] | 1 | /*****************************************************************************\ |
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[806c18] | 2 | * Computer Algebra System SINGULAR |
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[ad3c3ff] | 3 | \*****************************************************************************/ |
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| 4 | /** @file facHensel.h |
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| 5 | * |
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[806c18] | 6 | * This file defines functions for Hensel lifting and modular multiplication. |
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| 7 | * |
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| 8 | * ABSTRACT: function are used for bi- and multivariate factorization over |
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[ad3c3ff] | 9 | * finite fields |
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| 10 | * |
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| 11 | * @author Martin Lee |
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| 12 | * |
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| 13 | * @internal @version \$Id$ |
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| 14 | * |
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| 15 | **/ |
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| 16 | /*****************************************************************************/ |
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| 17 | |
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| 18 | #ifndef FAC_HENSEL_H |
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| 19 | #define FAC_HENSEL_H |
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| 20 | |
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[e4fe2b] | 21 | // #include "config.h" |
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[650f2d8] | 22 | #include "cf_assert.h" |
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[ad3c3ff] | 23 | #include "debug.h" |
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| 24 | #include "timing.h" |
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| 25 | |
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| 26 | #include "canonicalform.h" |
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[806c18] | 27 | #include "cf_iter.h" |
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[6db552] | 28 | #include "templates/ftmpl_functions.h" |
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[ad3c3ff] | 29 | #include "algext.h" |
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| 30 | |
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[615ca8] | 31 | #ifdef HAVE_NTL |
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[806c18] | 32 | /// multiplication of univariate polys over a finite field using NTL, if we are |
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[ad3c3ff] | 33 | /// in GF factory's default multiplication is used. |
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| 34 | /// |
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| 35 | /// @return @a mulNTL returns F*G |
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| 36 | CanonicalForm |
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[806c18] | 37 | mulNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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[ad3c3ff] | 38 | const CanonicalForm& G ///< [in] a univariate poly |
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| 39 | ); |
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| 40 | |
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[806c18] | 41 | /// mod of univariate polys over a finite field using NTL, if we are |
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[ad3c3ff] | 42 | /// in GF factory's default mod is used. |
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| 43 | /// |
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| 44 | /// @return @a modNTL returns F mod G |
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| 45 | CanonicalForm |
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[806c18] | 46 | modNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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[ad3c3ff] | 47 | const CanonicalForm& G ///< [in] a univariate poly |
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| 48 | ); |
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| 49 | |
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[806c18] | 50 | /// division of univariate polys over a finite field using NTL, if we are |
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[ad3c3ff] | 51 | /// in GF factory's default division is used. |
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| 52 | /// |
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| 53 | /// @return @a divNTL returns F/G |
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| 54 | CanonicalForm |
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[806c18] | 55 | divNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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[ad3c3ff] | 56 | const CanonicalForm& G ///< [in] a univariate poly |
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| 57 | ); |
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| 58 | |
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[806c18] | 59 | /*/// division with remainder of univariate polys over a finite field using NTL, |
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[ad3c3ff] | 60 | /// if we are in GF factory's default division with remainder is used. |
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| 61 | void |
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[806c18] | 62 | divremNTL (const CanonicalForm& F, ///< [in] a univariate poly |
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[ad3c3ff] | 63 | const CanonicalForm& G, ///< [in] a univariate poly |
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| 64 | CanonicalForm& Q, ///< [in,out] dividend |
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| 65 | CanonicalForm& R ///< [in,out] remainder |
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| 66 | );*/ |
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| 67 | |
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| 68 | /// division with remainder of @a F by |
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[806c18] | 69 | /// @a G wrt Variable (1) modulo @a M. |
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| 70 | /// |
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[ad3c3ff] | 71 | /// @return @a Q returns the dividend, @a R returns the remainder. |
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| 72 | /// @sa divrem() |
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[806c18] | 73 | void divrem2 (const CanonicalForm& F, ///< [in] bivariate, compressed polynomial |
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[ad3c3ff] | 74 | const CanonicalForm& G, ///< [in] bivariate, compressed polynomial |
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| 75 | CanonicalForm& Q, ///< [in,out] dividend |
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[806c18] | 76 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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[ad3c3ff] | 77 | ///< degree (G, 1) |
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| 78 | const CanonicalForm& M ///< [in] power of Variable (2) |
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| 79 | ); |
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| 80 | |
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| 81 | /// division with remainder of @a F by |
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| 82 | /// @a G wrt Variable (1) modulo @a MOD. |
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| 83 | /// |
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| 84 | /// @sa divrem2() |
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| 85 | void divrem ( |
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| 86 | const CanonicalForm& F, ///< [in] multivariate, compressed polynomial |
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| 87 | const CanonicalForm& G, ///< [in] multivariate, compressed polynomial |
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| 88 | CanonicalForm& Q, ///< [in,out] dividend |
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[806c18] | 89 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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[ad3c3ff] | 90 | ///< degree (G, 1) |
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| 91 | const CFList& MOD ///< [in] only contains powers of |
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| 92 | ///< Variables of level higher than 1 |
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| 93 | ); |
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| 94 | |
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[c9b1d66] | 95 | |
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| 96 | /// division with remainder of @a F by |
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| 97 | /// @a G wrt Variable (1) modulo @a M using Newton inversion |
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| 98 | /// |
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| 99 | /// @return @a Q returns the dividend, @a R returns the remainder. |
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| 100 | /// @sa divrem2(), newtonDiv() |
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| 101 | void |
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| 102 | newtonDivrem (const CanonicalForm& F, ///< [in] bivariate, compressed polynomial |
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| 103 | const CanonicalForm& G, ///< [in] bivariate, compressed polynomial |
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| 104 | ///< which is monic in Variable (1) |
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| 105 | CanonicalForm& Q, ///< [in,out] dividend |
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| 106 | CanonicalForm& R, ///< [in,out] remainder, degree (R, 1) < |
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| 107 | ///< degree (G, 1) |
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| 108 | const CanonicalForm& M ///< [in] power of Variable (2) |
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| 109 | ); |
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| 110 | |
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| 111 | /// division of @a F by |
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| 112 | /// @a G wrt Variable (1) modulo @a M using Newton inversion |
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| 113 | /// |
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| 114 | /// @return @a newtonDiv returns the dividend |
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| 115 | /// @sa divrem2(), newtonDivrem() |
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| 116 | CanonicalForm |
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| 117 | newtonDiv (const CanonicalForm& F, ///< [in] bivariate, compressed polynomial |
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| 118 | const CanonicalForm& G, ///< [in] bivariate, compressed polynomial |
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| 119 | ///< which is monic in Variable (1) |
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| 120 | const CanonicalForm& M ///< [in] power of Variable (2) |
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| 121 | ); |
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| 122 | |
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[ad3c3ff] | 123 | /// reduce @a F modulo elements in @a M. |
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| 124 | /// |
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| 125 | /// @return @a mod returns @a F modulo @a M |
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| 126 | CanonicalForm mod (const CanonicalForm& F, ///< [in] compressed polynomial |
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[806c18] | 127 | const CFList& M ///< [in] list containing only |
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[ad3c3ff] | 128 | ///< univariate polynomials |
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| 129 | ); |
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| 130 | |
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| 131 | /// Karatsuba style modular multiplication for bivariate polynomials. |
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| 132 | /// |
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| 133 | /// @return @a mulMod2 returns @a A * @a B mod @a M. |
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[806c18] | 134 | CanonicalForm |
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[ad3c3ff] | 135 | mulMod2 (const CanonicalForm& A, ///< [in] bivariate, compressed polynomial |
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| 136 | const CanonicalForm& B, ///< [in] bivariate, compressed polynomial |
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| 137 | const CanonicalForm& M ///< [in] power of Variable (2) |
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| 138 | ); |
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| 139 | |
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| 140 | /// Karatsuba style modular multiplication for multivariate polynomials. |
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| 141 | /// |
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| 142 | /// @return @a mulMod2 returns @a A * @a B mod @a MOD. |
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[806c18] | 143 | CanonicalForm |
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[ad3c3ff] | 144 | mulMod (const CanonicalForm& A, ///< [in] multivariate, compressed polynomial |
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| 145 | const CanonicalForm& B, ///< [in] multivariate, compressed polynomial |
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| 146 | const CFList& MOD ///< [in] only contains powers of |
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| 147 | ///< Variables of level higher than 1 |
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| 148 | ); |
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| 149 | |
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| 150 | /// product of all elements in @a L modulo @a M via divide-and-conquer. |
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| 151 | /// |
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| 152 | /// @return @a prodMod returns product of all elements in @a L modulo @a M. |
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[806c18] | 153 | CanonicalForm |
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| 154 | prodMod (const CFList& L, ///< [in] contains only bivariate, compressed |
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[ad3c3ff] | 155 | ///< polynomials |
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| 156 | const CanonicalForm& M ///< [in] power of Variable (2) |
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| 157 | ); |
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| 158 | |
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| 159 | /// product of all elements in @a L modulo @a M via divide-and-conquer. |
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| 160 | /// |
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| 161 | /// @return @a prodMod returns product of all elements in @a L modulo @a M. |
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[806c18] | 162 | CanonicalForm |
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| 163 | prodMod (const CFList& L, ///< [in] contains multivariate, compressed |
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[ad3c3ff] | 164 | ///< polynomials |
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| 165 | const CFList& M ///< [in] contains only powers of Variables |
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| 166 | ); |
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[806c18] | 167 | |
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[ad3c3ff] | 168 | /// sort a list of polynomials by their degree in @a x. |
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| 169 | /// |
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[806c18] | 170 | void sortList (CFList& list, ///< [in, out] list of polys, sorted list |
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| 171 | const Variable& x ///< [in] some Variable |
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[ad3c3ff] | 172 | ); |
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| 173 | |
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| 174 | /// Hensel lift from univariate to bivariate. |
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[806c18] | 175 | /// |
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[ad3c3ff] | 176 | /// @sa henselLiftResume12(), henselLift23(), henselLiftResume(), henselLift() |
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[806c18] | 177 | void |
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[ad3c3ff] | 178 | henselLift12 (const CanonicalForm& F, ///< [in] compressed, bivariate poly |
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[806c18] | 179 | CFList& factors, ///< [in, out] monic univariate factors of |
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[ad3c3ff] | 180 | ///< F, including leading coefficient as |
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[806c18] | 181 | ///< first element. Returns monic lifted |
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| 182 | ///< factors without the leading |
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[ad3c3ff] | 183 | ///< coefficient |
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| 184 | int l, ///< [in] lifting precision |
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| 185 | CFArray& Pi, ///< [in,out] stores intermediate results |
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| 186 | CFList& diophant, ///< [in,out] result of diophantine() |
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[e368746] | 187 | CFMatrix& M, ///< [in,out] stores intermediate results |
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| 188 | bool sort= true ///< [in] sort factors by degree in |
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| 189 | ///< Variable(1) |
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[ad3c3ff] | 190 | ); |
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| 191 | |
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[806c18] | 192 | /// resume Hensel lift from univariate to bivariate. Assumes factors are lifted |
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[ad3c3ff] | 193 | /// to precision Variable (2)^start and lifts them to precision Variable (2)^end |
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[806c18] | 194 | /// |
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[ad3c3ff] | 195 | /// @sa henselLift12(), henselLift23(), henselLiftResume(), henselLift() |
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[806c18] | 196 | void |
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[ad3c3ff] | 197 | henselLiftResume12 (const CanonicalForm& F, ///< [in] compressed, bivariate poly |
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[806c18] | 198 | CFList& factors, ///< [in,out] monic factors of F, |
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| 199 | ///< lifted to precision start, |
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| 200 | ///< including leading coefficient |
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| 201 | ///< as first element. Returns monic |
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| 202 | ///< lifted factors without the |
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[ad3c3ff] | 203 | ///< leading coefficient |
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| 204 | int start, ///< [in] starting precision |
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| 205 | int end, ///< [in] end precision |
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| 206 | CFArray& Pi, ///< [in,out] stores intermediate |
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| 207 | ///< results |
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| 208 | const CFList& diophant, ///< [in] result of diophantine |
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| 209 | CFMatrix& M ///< [in,out] stores intermediate |
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| 210 | ///< results |
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| 211 | ); |
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| 212 | |
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| 213 | /// Hensel lifting from bivariate to trivariate. |
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| 214 | /// |
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| 215 | /// @return @a henselLift23 returns a list of polynomials lifted to precision |
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| 216 | /// Variable (3)^l[1] |
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| 217 | /// @sa henselLift12(), henselLiftResume12(), henselLiftResume(), henselLift() |
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[806c18] | 218 | CFList |
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[ad3c3ff] | 219 | henselLift23 (const CFList& eval, ///< [in] contains compressed, bivariate |
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| 220 | ///< as first element and trivariate one as |
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[806c18] | 221 | ///< second element |
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| 222 | const CFList& factors, ///< [in] monic bivariate factors, |
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| 223 | ///< including leading coefficient |
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[ad3c3ff] | 224 | ///< as first element. |
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[806c18] | 225 | const int* l, ///< [in] l[0]: precision of bivariate |
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[ad3c3ff] | 226 | ///< lifting, l[1] as above |
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| 227 | CFList& diophant, ///< [in,out] returns the result of |
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| 228 | ///< biDiophantine() |
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| 229 | CFArray& Pi, ///< [in,out] stores intermediate results |
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| 230 | CFMatrix& M ///< [in,out] stores intermediate results |
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| 231 | ); |
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| 232 | |
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| 233 | /// resume Hensel lifting. |
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| 234 | /// |
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| 235 | /// @sa henselLift12(), henselLiftResume12(), henselLift23(), henselLift() |
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[806c18] | 236 | void |
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[ad3c3ff] | 237 | henselLiftResume ( |
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[806c18] | 238 | const CanonicalForm& F, ///< [in] compressed, multivariate poly |
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[ad3c3ff] | 239 | CFList& factors, ///< [in,out] monic multivariate factors |
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[806c18] | 240 | ///< lifted to precision F.mvar()^start, |
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| 241 | ///< including leading coefficient |
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[ad3c3ff] | 242 | ///< as first element. Returns factors |
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| 243 | ///< lifted to precision F.mvar()^end |
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[806c18] | 244 | int start, ///< [in] starting precision |
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[ad3c3ff] | 245 | int end, ///< [in] end precision |
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| 246 | CFArray& Pi, ///< [in,out] stores intermediate results |
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| 247 | const CFList& diophant, ///< [in] result of multiRecDiophantine() |
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| 248 | CFMatrix& M, ///< [in, out] stores intermediate results |
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| 249 | const CFList& MOD ///< [in] a list of powers of Variables |
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| 250 | ///< of level higher than 1 |
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| 251 | ); |
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| 252 | |
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| 253 | /// Hensel lifting |
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| 254 | /// |
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[806c18] | 255 | /// @return @a henselLift returns a list of polynomials lifted to |
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[ad3c3ff] | 256 | /// precision F.getLast().mvar()^l_new |
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| 257 | /// @sa henselLift12(), henselLiftResume12(), henselLift23(), henselLiftResume() |
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| 258 | CFList |
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| 259 | henselLift (const CFList& F, ///< [in] two compressed, multivariate |
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| 260 | ///< polys F and G |
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[806c18] | 261 | const CFList& factors, ///< [in] monic multivariate factors |
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| 262 | ///< including leading coefficient |
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| 263 | ///< as first element. |
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[ad3c3ff] | 264 | const CFList& MOD, ///< [in] a list of powers of Variables |
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| 265 | ///< of level higher than 1 |
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| 266 | CFList& diophant, ///< [in,out] result of multiRecDiophantine() |
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| 267 | CFArray& Pi, ///< [in,out] stores intermediate results |
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| 268 | CFMatrix& M, ///< [in,out] stores intermediate results |
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[806c18] | 269 | const int lOld, ///< [in] lifting precision of F.mvar() |
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[ad3c3ff] | 270 | const int lNew ///< [in] lifting precision of G.mvar() |
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| 271 | ); |
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| 272 | |
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[806c18] | 273 | /// Hensel lifting from bivariate to multivariate |
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[ad3c3ff] | 274 | /// |
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[806c18] | 275 | /// @return @a henselLift returns a list of lifted factors |
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| 276 | /// @sa henselLift12(), henselLiftResume12(), henselLift23(), henselLiftResume() |
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[ad3c3ff] | 277 | CFList |
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[806c18] | 278 | henselLift (const CFList& eval, ///< [in] a list of polynomials the last |
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| 279 | ///< element is a compressed multivariate |
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| 280 | ///< poly, last but one element equals the |
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[ad3c3ff] | 281 | ///< last elements modulo its main variable |
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[806c18] | 282 | ///< and so on. The first element is a |
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[ad3c3ff] | 283 | ///< compressed bivariate poly. |
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[806c18] | 284 | const CFList& factors, ///< [in] bivariate factors, including |
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[ad3c3ff] | 285 | ///< leading coefficient |
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| 286 | const int* l, ///< [in] lifting bounds |
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[e368746] | 287 | const int lLength, ///< [in] length of l |
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| 288 | bool sort= true ///< [in] sort factors by degree in |
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| 289 | ///< Variable(1) |
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[ad3c3ff] | 290 | ); |
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| 291 | |
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[08daea] | 292 | /// two factor Hensel lifting from univariate to bivariate, factors need not to |
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| 293 | /// be monic |
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| 294 | void |
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| 295 | henselLift122 (const CanonicalForm& F,///< [in] a bivariate poly |
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| 296 | CFList& factors, ///< [in, out] a list of univariate polys |
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| 297 | ///< lifted factors |
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| 298 | int l, ///< [in] lift bound |
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| 299 | CFArray& Pi, ///< [in, out] stores intermediate results |
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| 300 | CFList& diophant, ///< [in, out] result of diophantine |
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| 301 | CFMatrix& M, ///< [in, out] stores intermediate results |
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| 302 | const CFArray& LCs, ///< [in] leading coefficients |
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| 303 | bool sort ///< [in] if true factors are sorted by |
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| 304 | ///< their degree |
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| 305 | ); |
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| 306 | |
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| 307 | /// two factor Hensel lifting from bivariate to multivariate, factors need not |
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| 308 | /// to be monic |
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| 309 | /// |
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| 310 | /// @return @a henselLift122 returns a list of lifted factors |
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| 311 | CFList |
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| 312 | henselLift2 (const CFList& eval, ///< [in] a list of polynomials the last |
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| 313 | ///< element is a compressed multivariate |
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| 314 | ///< poly, last but one element equals the |
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| 315 | ///< last elements modulo its main variable |
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| 316 | ///< and so on. The first element is a |
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| 317 | ///< compressed bivariate poly. |
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| 318 | const CFList& factors,///< [in] bivariate factors |
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| 319 | int* l, ///< [in] lift bounds |
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| 320 | const int lLength, ///< [in] length of l |
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| 321 | bool sort, ///< [in] if true factors are sorted by |
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| 322 | ///< their degree in Variable(1) |
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| 323 | const CFList& LCs1, ///< [in] a list of evaluated LC of first |
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| 324 | ///< factor |
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| 325 | const CFList& LCs2, ///< [in] a list of evaluated LC of second |
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| 326 | ///< factor |
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| 327 | const CFArray& Pi, ///< [in] intermediate result |
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[e368746] | 328 | const CFList& diophant,///< [in] result of diophantine |
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| 329 | bool& noOneToOne ///< [in,out] check for one to one |
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| 330 | ///< correspondence |
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[08daea] | 331 | ); |
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| 332 | |
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[c1b9927] | 333 | /// Hensel lifting of non monic factors, needs correct leading coefficients of |
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[e368746] | 334 | /// factors and a one to one corresponds between bivariate and multivariate |
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| 335 | /// factors to succeed |
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| 336 | /// |
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| 337 | /// @return @a nonMonicHenselLift returns a list of lifted factors |
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[c1b9927] | 338 | /// such that prod (factors) == eval.getLast() if there is a one to one |
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[e368746] | 339 | /// correspondence |
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| 340 | CFList |
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| 341 | nonMonicHenselLift (const CFList& eval, ///< [in] a list of polys the last |
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[c1b9927] | 342 | ///< element is a compressed |
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[e368746] | 343 | ///< multivariate poly, last but one |
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| 344 | ///< element equals the last elements |
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[c1b9927] | 345 | ///< modulo its main variable and so |
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| 346 | ///< on. The first element is a |
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[e368746] | 347 | ///< compressed poly in 3 variables |
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| 348 | const CFList& factors, ///< [in] a list of bivariate factors |
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| 349 | CFList* const& LCs, ///< [in] leading coefficients, |
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[c1b9927] | 350 | ///< evaluated in the same way as |
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[e368746] | 351 | ///< eval |
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| 352 | CFList& diophant, ///< [in, out] solution of univariate |
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[c1b9927] | 353 | ///< diophantine equation |
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[e368746] | 354 | CFArray& Pi, ///< [in, out] buffer intermediate |
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| 355 | ///< results |
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| 356 | int* liftBound, ///< [in] lifting bounds |
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| 357 | int length, ///< [in] length of lifting bounds |
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| 358 | bool& noOneToOne ///< [in, out] check for one to one |
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| 359 | ///< correspondence |
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| 360 | ); |
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[615ca8] | 361 | #endif /* HAVE_NTL */ |
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[ad3c3ff] | 362 | #endif |
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| 363 | /* FAC_HENSEL_H */ |
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| 364 | |
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