1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file facHensel.cc |
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5 | * |
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6 | * This file implements functions to lift factors via Hensel lifting. |
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7 | * |
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8 | * ABSTRACT: Hensel lifting is described in "Efficient Multivariate |
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9 | * Factorization over Finite Fields" by L. Bernardin & M. Monagon and |
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10 | * "Algorithms for Computer Algebra" by Geddes, Czapor, Labahn |
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11 | * |
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12 | * @author Martin Lee |
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13 | * |
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14 | **/ |
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15 | /*****************************************************************************/ |
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16 | |
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17 | #ifdef HAVE_CONFIG_H |
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18 | #include "config.h" |
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19 | #endif /* HAVE_CONFIG_H */ |
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20 | |
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21 | #include "cf_assert.h" |
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22 | #include "debug.h" |
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23 | #include "timing.h" |
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24 | |
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25 | #include "algext.h" |
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26 | #include "facHensel.h" |
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27 | #include "facMul.h" |
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28 | #include "fac_util.h" |
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29 | #include "cf_algorithm.h" |
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30 | #include "cf_primes.h" |
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31 | #include "facBivar.h" |
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32 | #include "facNTLzzpEXGCD.h" |
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33 | |
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34 | #ifdef HAVE_NTL |
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35 | #include <NTL/lzz_pEX.h> |
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36 | #include "NTLconvert.h" |
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37 | |
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38 | #ifdef HAVE_FLINT |
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39 | #include "FLINTconvert.h" |
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40 | #endif |
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41 | |
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42 | TIMING_DEFINE_PRINT (diotime) |
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43 | TIMING_DEFINE_PRINT (product1) |
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44 | TIMING_DEFINE_PRINT (product2) |
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45 | TIMING_DEFINE_PRINT (hensel23) |
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46 | TIMING_DEFINE_PRINT (hensel) |
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47 | |
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48 | #if (!(HAVE_FLINT && __FLINT_VERSION_MINOR >= 4)) |
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49 | static |
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50 | CFList productsNTL (const CFList& factors, const CanonicalForm& M) |
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51 | { |
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52 | if (fac_NTL_char != getCharacteristic()) |
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53 | { |
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54 | fac_NTL_char= getCharacteristic(); |
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55 | zz_p::init (getCharacteristic()); |
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56 | } |
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57 | zz_pX NTLMipo= convertFacCF2NTLzzpX (M); |
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58 | zz_pE::init (NTLMipo); |
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59 | zz_pEX prod; |
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60 | vec_zz_pEX v; |
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61 | v.SetLength (factors.length()); |
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62 | int j= 0; |
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63 | for (CFListIterator i= factors; i.hasItem(); i++, j++) |
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64 | { |
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65 | if (i.getItem().inCoeffDomain()) |
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66 | v[j]= to_zz_pEX (to_zz_pE (convertFacCF2NTLzzpX (i.getItem()))); |
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67 | else |
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68 | v[j]= convertFacCF2NTLzz_pEX (i.getItem(), NTLMipo); |
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69 | } |
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70 | CFList result; |
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71 | Variable x= Variable (1); |
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72 | for (int j= 0; j < factors.length(); j++) |
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73 | { |
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74 | int k= 0; |
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75 | set(prod); |
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76 | for (int i= 0; i < factors.length(); i++, k++) |
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77 | { |
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78 | if (k == j) |
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79 | continue; |
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80 | prod *= v[i]; |
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81 | } |
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82 | result.append (convertNTLzz_pEX2CF (prod, x, M.mvar())); |
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83 | } |
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84 | return result; |
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85 | } |
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86 | #endif |
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87 | |
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88 | #if (HAVE_FLINT && __FLINT_VERSION_MINOR >= 4) |
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89 | static |
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90 | CFList productsFLINT (const CFList& factors, const CanonicalForm& M) |
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91 | { |
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92 | nmod_poly_t FLINTmipo; |
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93 | fq_nmod_ctx_t fq_con; |
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94 | fq_nmod_poly_t prod; |
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95 | fq_nmod_t buf; |
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96 | |
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97 | nmod_poly_init (FLINTmipo, getCharacteristic()); |
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98 | convertFacCF2nmod_poly_t (FLINTmipo, M); |
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99 | |
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100 | fq_nmod_ctx_init_modulus (fq_con, FLINTmipo, "Z"); |
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101 | |
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102 | fq_nmod_poly_t * vec=new fq_nmod_poly_t [factors.length()]; |
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103 | |
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104 | int j= 0; |
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105 | |
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106 | for (CFListIterator i= factors; i.hasItem(); i++, j++) |
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107 | { |
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108 | if (i.getItem().inCoeffDomain()) |
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109 | { |
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110 | fq_nmod_poly_init (vec[j], fq_con); |
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111 | fq_nmod_init2 (buf, fq_con); |
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112 | convertFacCF2Fq_nmod_t (buf, i.getItem(), fq_con); |
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113 | fq_nmod_poly_set_coeff (vec[j], 0, buf, fq_con); |
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114 | fq_nmod_clear (buf, fq_con); |
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115 | } |
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116 | else |
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117 | convertFacCF2Fq_nmod_poly_t (vec[j], i.getItem(), fq_con); |
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118 | } |
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119 | |
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120 | CFList result; |
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121 | Variable x= Variable (1); |
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122 | fq_nmod_poly_init (prod, fq_con); |
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123 | for (j= 0; j < factors.length(); j++) |
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124 | { |
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125 | int k= 0; |
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126 | fq_nmod_poly_one (prod, fq_con); |
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127 | for (int i= 0; i < factors.length(); i++, k++) |
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128 | { |
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129 | if (k == j) |
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130 | continue; |
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131 | fq_nmod_poly_mul (prod, prod, vec[i], fq_con); |
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132 | } |
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133 | result.append (convertFq_nmod_poly_t2FacCF (prod, x, M.mvar(), fq_con)); |
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134 | } |
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135 | for (j= 0; j < factors.length(); j++) |
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136 | fq_nmod_poly_clear (vec[j], fq_con); |
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137 | |
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138 | nmod_poly_clear (FLINTmipo); |
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139 | fq_nmod_poly_clear (prod, fq_con); |
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140 | fq_nmod_ctx_clear (fq_con); |
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141 | delete [] vec; |
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142 | return result; |
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143 | } |
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144 | #endif |
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145 | |
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146 | static |
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147 | void tryDiophantine (CFList& result, const CanonicalForm& F, |
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148 | const CFList& factors, const CanonicalForm& M, bool& fail) |
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149 | { |
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150 | ASSERT (M.isUnivariate(), "expected univariate poly"); |
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151 | |
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152 | CFList bufFactors= factors; |
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153 | bufFactors.removeFirst(); |
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154 | bufFactors.insert (factors.getFirst () (0,2)); |
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155 | CanonicalForm inv, leadingCoeff= Lc (F); |
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156 | CFListIterator i= bufFactors; |
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157 | if (bufFactors.getFirst().inCoeffDomain()) |
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158 | { |
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159 | if (i.hasItem()) |
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160 | i++; |
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161 | } |
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162 | for (; i.hasItem(); i++) |
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163 | { |
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164 | tryInvert (Lc (i.getItem()), M, inv ,fail); |
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165 | if (fail) |
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166 | return; |
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167 | i.getItem()= reduce (i.getItem()*inv, M); |
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168 | } |
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169 | #if (HAVE_FLINT && __FLINT_VERSION_MINOR >= 4) |
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170 | bufFactors= productsFLINT (bufFactors, M); |
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171 | #else |
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172 | bufFactors= productsNTL (bufFactors, M); |
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173 | #endif |
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174 | |
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175 | CanonicalForm buf1, buf2, buf3, S, T; |
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176 | i= bufFactors; |
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177 | if (i.hasItem()) |
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178 | i++; |
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179 | buf1= bufFactors.getFirst(); |
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180 | buf2= i.getItem(); |
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181 | #ifdef HAVE_NTL |
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182 | Variable x= Variable (1); |
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183 | if (fac_NTL_char != getCharacteristic()) |
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184 | { |
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185 | fac_NTL_char= getCharacteristic(); |
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186 | zz_p::init (getCharacteristic()); |
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187 | } |
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188 | zz_pX NTLMipo= convertFacCF2NTLzzpX (M); |
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189 | zz_pE::init (NTLMipo); |
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190 | zz_pEX NTLbuf1, NTLbuf2, NTLbuf3, NTLS, NTLT; |
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191 | NTLbuf1= convertFacCF2NTLzz_pEX (buf1, NTLMipo); |
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192 | NTLbuf2= convertFacCF2NTLzz_pEX (buf2, NTLMipo); |
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193 | tryNTLXGCD (NTLbuf3, NTLS, NTLT, NTLbuf1, NTLbuf2, fail); |
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194 | if (fail) |
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195 | return; |
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196 | S= convertNTLzz_pEX2CF (NTLS, x, M.mvar()); |
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197 | T= convertNTLzz_pEX2CF (NTLT, x, M.mvar()); |
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198 | #else |
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199 | tryExtgcd (buf1, buf2, M, buf3, S, T, fail); |
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200 | if (fail) |
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201 | return; |
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202 | #endif |
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203 | result.append (S); |
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204 | result.append (T); |
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205 | if (i.hasItem()) |
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206 | i++; |
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207 | for (; i.hasItem(); i++) |
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208 | { |
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209 | #ifdef HAVE_NTL |
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210 | NTLbuf1= convertFacCF2NTLzz_pEX (i.getItem(), NTLMipo); |
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211 | tryNTLXGCD (NTLbuf3, NTLS, NTLT, NTLbuf3, NTLbuf1, fail); |
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212 | if (fail) |
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213 | return; |
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214 | S= convertNTLzz_pEX2CF (NTLS, x, M.mvar()); |
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215 | T= convertNTLzz_pEX2CF (NTLT, x, M.mvar()); |
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216 | #else |
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217 | buf1= i.getItem(); |
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218 | tryExtgcd (buf3, buf1, M, buf3, S, T, fail); |
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219 | if (fail) |
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220 | return; |
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221 | #endif |
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222 | CFListIterator k= factors; |
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223 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
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224 | { |
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225 | j.getItem() *= S; |
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226 | j.getItem()= mod (j.getItem(), k.getItem()); |
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227 | j.getItem()= reduce (j.getItem(), M); |
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228 | } |
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229 | result.append (T); |
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230 | } |
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231 | } |
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232 | |
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233 | static |
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234 | CFList mapinto (const CFList& L) |
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235 | { |
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236 | CFList result; |
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237 | for (CFListIterator i= L; i.hasItem(); i++) |
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238 | result.append (mapinto (i.getItem())); |
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239 | return result; |
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240 | } |
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241 | |
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242 | static |
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243 | int mod (const CFList& L, const CanonicalForm& p) |
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244 | { |
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245 | for (CFListIterator i= L; i.hasItem(); i++) |
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246 | { |
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247 | if (mod (i.getItem(), p) == 0) |
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248 | return 0; |
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249 | } |
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250 | return 1; |
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251 | } |
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252 | |
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253 | |
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254 | static void |
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255 | chineseRemainder (const CFList & x1, const CanonicalForm & q1, |
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256 | const CFList & x2, const CanonicalForm & q2, |
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257 | CFList & xnew, CanonicalForm & qnew) |
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258 | { |
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259 | ASSERT (x1.length() == x2.length(), "expected lists of equal length"); |
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260 | CanonicalForm tmp1, tmp2; |
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261 | CFListIterator j= x2; |
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262 | for (CFListIterator i= x1; i.hasItem() && j.hasItem(); i++, j++) |
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263 | { |
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264 | chineseRemainder (i.getItem(), q1, j.getItem(), q2, tmp1, tmp2); |
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265 | xnew.append (tmp1); |
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266 | } |
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267 | qnew= tmp2; |
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268 | } |
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269 | |
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270 | static |
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271 | CFList Farey (const CFList& L, const CanonicalForm& q) |
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272 | { |
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273 | CFList result; |
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274 | for (CFListIterator i= L; i.hasItem(); i++) |
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275 | result.append (Farey (i.getItem(), q)); |
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276 | return result; |
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277 | } |
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278 | |
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279 | static |
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280 | CFList replacevar (const CFList& L, const Variable& a, const Variable& b) |
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281 | { |
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282 | CFList result; |
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283 | for (CFListIterator i= L; i.hasItem(); i++) |
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284 | result.append (replacevar (i.getItem(), a, b)); |
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285 | return result; |
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286 | } |
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287 | |
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288 | CFList |
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289 | modularDiophant (const CanonicalForm& f, const CFList& factors, |
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290 | const CanonicalForm& M) |
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291 | { |
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292 | bool save_rat=!isOn (SW_RATIONAL); |
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293 | On (SW_RATIONAL); |
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294 | CanonicalForm F= f*bCommonDen (f); |
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295 | CFList products= factors; |
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296 | for (CFListIterator i= products; i.hasItem(); i++) |
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297 | { |
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298 | if (products.getFirst().level() == 1) |
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299 | i.getItem() /= Lc (i.getItem()); |
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300 | i.getItem() *= bCommonDen (i.getItem()); |
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301 | } |
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302 | if (products.getFirst().level() == 1) |
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303 | products.insert (Lc (F)); |
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304 | CanonicalForm bound= maxNorm (F); |
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305 | CFList leadingCoeffs; |
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306 | leadingCoeffs.append (lc (F)); |
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307 | CanonicalForm dummy; |
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308 | for (CFListIterator i= products; i.hasItem(); i++) |
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309 | { |
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310 | leadingCoeffs.append (lc (i.getItem())); |
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311 | dummy= maxNorm (i.getItem()); |
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312 | bound= (dummy > bound) ? dummy : bound; |
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313 | } |
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314 | bound *= maxNorm (Lc (F))*maxNorm (Lc(F))*bound; |
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315 | bound *= bound*bound; |
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316 | bound= power (bound, degree (M)); |
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317 | bound *= power (CanonicalForm (2),degree (f)); |
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318 | CanonicalForm bufBound= bound; |
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319 | int i = cf_getNumBigPrimes() - 1; |
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320 | int p; |
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321 | CFList resultModP, result, newResult; |
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322 | CanonicalForm q (0), newQ; |
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323 | bool fail= false; |
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324 | Variable a= M.mvar(); |
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325 | Variable b= Variable (2); |
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326 | setReduce (M.mvar(), false); |
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327 | CanonicalForm mipo= bCommonDen (M)*M; |
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328 | Off (SW_RATIONAL); |
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329 | CanonicalForm modMipo; |
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330 | leadingCoeffs.append (lc (mipo)); |
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331 | CFList tmp1, tmp2; |
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332 | bool equal= false; |
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333 | int count= 0; |
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334 | do |
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335 | { |
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336 | p = cf_getBigPrime( i ); |
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337 | i--; |
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338 | while ( i >= 0 && mod( leadingCoeffs, p ) == 0) |
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339 | { |
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340 | p = cf_getBigPrime( i ); |
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341 | i--; |
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342 | } |
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343 | |
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344 | ASSERT (i >= 0, "ran out of primes"); //sic |
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345 | |
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346 | setCharacteristic (p); |
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347 | modMipo= mapinto (mipo); |
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348 | modMipo /= lc (modMipo); |
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349 | resultModP= CFList(); |
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350 | tryDiophantine (resultModP, mapinto (F), mapinto (products), modMipo, fail); |
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351 | setCharacteristic (0); |
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352 | if (fail) |
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353 | { |
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354 | fail= false; |
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355 | continue; |
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356 | } |
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357 | |
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358 | if ( q.isZero() ) |
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359 | { |
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360 | result= replacevar (mapinto(resultModP), a, b); |
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361 | q= p; |
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362 | } |
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363 | else |
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364 | { |
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365 | result= replacevar (result, a, b); |
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366 | newResult= CFList(); |
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367 | chineseRemainder( result, q, replacevar (mapinto (resultModP), a, b), |
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368 | p, newResult, newQ ); |
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369 | q= newQ; |
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370 | result= newResult; |
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371 | if (newQ > bound) |
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372 | { |
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373 | count++; |
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374 | tmp1= replacevar (Farey (result, q), b, a); |
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375 | if (tmp2.isEmpty()) |
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376 | tmp2= tmp1; |
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377 | else |
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378 | { |
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379 | equal= true; |
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380 | CFListIterator k= tmp1; |
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381 | for (CFListIterator j= tmp2; j.hasItem(); j++, k++) |
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382 | { |
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383 | if (j.getItem() != k.getItem()) |
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384 | equal= false; |
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385 | } |
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386 | if (!equal) |
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387 | tmp2= tmp1; |
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388 | } |
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389 | if (count > 2) |
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390 | { |
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391 | bound *= bufBound; |
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392 | equal= false; |
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393 | count= 0; |
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394 | } |
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395 | } |
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396 | if (newQ > bound && equal) |
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397 | { |
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398 | On( SW_RATIONAL ); |
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399 | CFList bufResult= result; |
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400 | result= tmp2; |
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401 | setReduce (M.mvar(), true); |
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402 | if (factors.getFirst().level() == 1) |
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403 | { |
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404 | result.removeFirst(); |
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405 | CFListIterator j= factors; |
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406 | CanonicalForm denf= bCommonDen (f); |
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407 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
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408 | i.getItem() *= Lc (j.getItem())*denf; |
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409 | } |
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410 | if (factors.getFirst().level() != 1 && |
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411 | !bCommonDen (factors.getFirst()).isOne()) |
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412 | { |
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413 | CanonicalForm denFirst= bCommonDen (factors.getFirst()); |
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414 | for (CFListIterator i= result; i.hasItem(); i++) |
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415 | i.getItem() *= denFirst; |
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416 | } |
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417 | |
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418 | CanonicalForm test= 0; |
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419 | CFListIterator jj= factors; |
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420 | for (CFListIterator ii= result; ii.hasItem(); ii++, jj++) |
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421 | test += ii.getItem()*(f/jj.getItem()); |
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422 | if (!test.isOne()) |
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423 | { |
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424 | bound *= bufBound; |
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425 | equal= false; |
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426 | count= 0; |
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427 | setReduce (M.mvar(), false); |
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428 | result= bufResult; |
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429 | Off (SW_RATIONAL); |
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430 | } |
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431 | else |
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432 | break; |
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433 | } |
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434 | } |
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435 | } while (1); |
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436 | if (save_rat) Off(SW_RATIONAL); |
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437 | return result; |
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438 | } |
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439 | |
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440 | void sortList (CFList& list, const Variable& x) |
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441 | { |
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442 | int l= 1; |
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443 | int k= 1; |
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444 | CanonicalForm buf; |
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445 | CFListIterator m; |
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446 | for (CFListIterator i= list; l <= list.length(); i++, l++) |
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447 | { |
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448 | for (CFListIterator j= list; k <= list.length() - l; k++) |
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449 | { |
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450 | m= j; |
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451 | m++; |
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452 | if (degree (j.getItem(), x) > degree (m.getItem(), x)) |
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453 | { |
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454 | buf= m.getItem(); |
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455 | m.getItem()= j.getItem(); |
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456 | j.getItem()= buf; |
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457 | j++; |
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458 | j.getItem()= m.getItem(); |
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459 | } |
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460 | else |
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461 | j++; |
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462 | } |
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463 | k= 1; |
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464 | } |
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465 | } |
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466 | |
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467 | CFList |
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468 | diophantine (const CanonicalForm& F, const CFList& factors); |
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469 | |
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470 | CFList |
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471 | diophantineHensel (const CanonicalForm & F, const CFList& factors, |
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472 | const modpk& b) |
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473 | { |
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474 | int p= b.getp(); |
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475 | setCharacteristic (p); |
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476 | CFList recResult= diophantine (mapinto (F), mapinto (factors)); |
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477 | setCharacteristic (0); |
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478 | recResult= mapinto (recResult); |
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479 | CanonicalForm e= 1; |
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480 | CFList L; |
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481 | CFArray bufFactors= CFArray (factors.length()); |
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482 | int k= 0; |
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483 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
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484 | { |
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485 | if (k == 0) |
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486 | bufFactors[k]= i.getItem() (0); |
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487 | else |
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488 | bufFactors [k]= i.getItem(); |
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489 | } |
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490 | CanonicalForm tmp, quot; |
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491 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
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492 | { |
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493 | tmp= 1; |
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494 | for (int l= 0; l < factors.length(); l++) |
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495 | { |
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496 | if (l == k) |
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497 | continue; |
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498 | else |
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499 | { |
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500 | tmp= mulNTL (tmp, bufFactors[l]); |
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501 | } |
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502 | } |
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503 | L.append (tmp); |
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504 | } |
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505 | |
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506 | setCharacteristic (p); |
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507 | for (k= 0; k < factors.length(); k++) |
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508 | bufFactors [k]= bufFactors[k].mapinto(); |
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509 | setCharacteristic(0); |
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510 | |
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511 | CFListIterator j= L; |
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512 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
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513 | e= b (e - mulNTL (i.getItem(),j.getItem(), b)); |
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514 | |
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515 | if (e.isZero()) |
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516 | return recResult; |
---|
517 | CanonicalForm coeffE; |
---|
518 | CFList s; |
---|
519 | CFList result= recResult; |
---|
520 | setCharacteristic (p); |
---|
521 | recResult= mapinto (recResult); |
---|
522 | setCharacteristic (0); |
---|
523 | CanonicalForm g; |
---|
524 | CanonicalForm modulus= p; |
---|
525 | int d= b.getk(); |
---|
526 | modpk b2; |
---|
527 | for (int i= 1; i < d; i++) |
---|
528 | { |
---|
529 | coeffE= div (e, modulus); |
---|
530 | setCharacteristic (p); |
---|
531 | coeffE= coeffE.mapinto(); |
---|
532 | setCharacteristic (0); |
---|
533 | b2= modpk (p, d - i); |
---|
534 | if (!coeffE.isZero()) |
---|
535 | { |
---|
536 | CFListIterator k= result; |
---|
537 | CFListIterator l= L; |
---|
538 | int ii= 0; |
---|
539 | j= recResult; |
---|
540 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
541 | { |
---|
542 | setCharacteristic (p); |
---|
543 | g= modNTL (coeffE, bufFactors[ii]); |
---|
544 | g= mulNTL (g, j.getItem()); |
---|
545 | g= modNTL (g, bufFactors[ii]); |
---|
546 | setCharacteristic (0); |
---|
547 | k.getItem() += g.mapinto()*modulus; |
---|
548 | e -= mulNTL (g.mapinto(), b2 (l.getItem()), b2)*modulus; |
---|
549 | e= b(e); |
---|
550 | } |
---|
551 | } |
---|
552 | modulus *= p; |
---|
553 | if (e.isZero()) |
---|
554 | break; |
---|
555 | } |
---|
556 | |
---|
557 | return result; |
---|
558 | } |
---|
559 | |
---|
560 | /// solve \f$ 1=\sum_{i=1}^n{\delta_{i} \prod_{j\neq i}{f_j}} \f$ mod \f$p^k\f$ |
---|
561 | /// over Q(alpha) by p-adic lifting |
---|
562 | CFList |
---|
563 | diophantineHenselQa (const CanonicalForm & F, const CanonicalForm& G, |
---|
564 | const CFList& factors, modpk& b, const Variable& alpha) |
---|
565 | { |
---|
566 | bool fail= false; |
---|
567 | CFList recResult; |
---|
568 | CanonicalForm modMipo, mipo; |
---|
569 | //here SW_RATIONAL is off |
---|
570 | On (SW_RATIONAL); |
---|
571 | mipo= getMipo (alpha); |
---|
572 | bool mipoHasDen= false; |
---|
573 | if (!bCommonDen (mipo).isOne()) |
---|
574 | { |
---|
575 | mipo *= bCommonDen (mipo); |
---|
576 | mipoHasDen= true; |
---|
577 | } |
---|
578 | Off (SW_RATIONAL); |
---|
579 | int p= b.getp(); |
---|
580 | setCharacteristic (p); |
---|
581 | setReduce (alpha, false); |
---|
582 | while (1) |
---|
583 | { |
---|
584 | setCharacteristic (p); |
---|
585 | modMipo= mapinto (mipo); |
---|
586 | modMipo /= lc (modMipo); |
---|
587 | tryDiophantine (recResult, mapinto (F), mapinto (factors), modMipo, fail); |
---|
588 | if (fail) |
---|
589 | { |
---|
590 | int i= 0; |
---|
591 | while (cf_getBigPrime (i) < p) |
---|
592 | i++; |
---|
593 | findGoodPrime (F, i); |
---|
594 | findGoodPrime (G, i); |
---|
595 | p=cf_getBigPrime(i); |
---|
596 | b = coeffBound( G, p, mipo ); |
---|
597 | modpk bb= coeffBound (F, p, mipo ); |
---|
598 | if (bb.getk() > b.getk() ) b=bb; |
---|
599 | fail= false; |
---|
600 | } |
---|
601 | else |
---|
602 | break; |
---|
603 | } |
---|
604 | setCharacteristic (0); |
---|
605 | recResult= mapinto (recResult); |
---|
606 | setReduce (alpha, true); |
---|
607 | CanonicalForm e= 1; |
---|
608 | CFList L; |
---|
609 | CFArray bufFactors= CFArray (factors.length()); |
---|
610 | int k= 0; |
---|
611 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
612 | { |
---|
613 | if (k == 0) |
---|
614 | bufFactors[k]= i.getItem() (0); |
---|
615 | else |
---|
616 | bufFactors [k]= i.getItem(); |
---|
617 | } |
---|
618 | CanonicalForm tmp; |
---|
619 | On (SW_RATIONAL); |
---|
620 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
621 | { |
---|
622 | tmp= 1; |
---|
623 | for (int l= 0; l < factors.length(); l++) |
---|
624 | { |
---|
625 | if (l == k) |
---|
626 | continue; |
---|
627 | else |
---|
628 | tmp= mulNTL (tmp, bufFactors[l]); |
---|
629 | } |
---|
630 | L.append (tmp*bCommonDen(tmp)); |
---|
631 | } |
---|
632 | |
---|
633 | Variable gamma; |
---|
634 | CanonicalForm den; |
---|
635 | if (mipoHasDen) |
---|
636 | { |
---|
637 | modMipo= getMipo (alpha); |
---|
638 | den= bCommonDen (modMipo); |
---|
639 | modMipo *= den; |
---|
640 | Off (SW_RATIONAL); |
---|
641 | setReduce (alpha, false); |
---|
642 | gamma= rootOf (b (modMipo*b.inverse (den))); |
---|
643 | setReduce (alpha, true); |
---|
644 | } |
---|
645 | |
---|
646 | setCharacteristic (p); |
---|
647 | Variable beta; |
---|
648 | Off (SW_RATIONAL); |
---|
649 | setReduce (alpha, false); |
---|
650 | modMipo= modMipo.mapinto(); |
---|
651 | modMipo /= lc (modMipo); |
---|
652 | beta= rootOf (modMipo); |
---|
653 | setReduce (alpha, true); |
---|
654 | |
---|
655 | setReduce (alpha, false); |
---|
656 | for (k= 0; k < factors.length(); k++) |
---|
657 | { |
---|
658 | bufFactors [k]= bufFactors[k].mapinto(); |
---|
659 | bufFactors [k]= replacevar (bufFactors[k], alpha, beta); |
---|
660 | } |
---|
661 | setReduce (alpha, true); |
---|
662 | setCharacteristic(0); |
---|
663 | |
---|
664 | CFListIterator j= L; |
---|
665 | for (;j.hasItem(); j++) |
---|
666 | { |
---|
667 | if (mipoHasDen) |
---|
668 | j.getItem()= replacevar (b(j.getItem()*b.inverse(lc(j.getItem()))), |
---|
669 | alpha, gamma); |
---|
670 | else |
---|
671 | j.getItem()= b(j.getItem()*b.inverse(lc(j.getItem()))); |
---|
672 | } |
---|
673 | j= L; |
---|
674 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
675 | { |
---|
676 | if (mipoHasDen) |
---|
677 | e= b (e - mulNTL (replacevar (i.getItem(), alpha, gamma),j.getItem(), b)); |
---|
678 | else |
---|
679 | e= b (e - mulNTL (i.getItem(), j.getItem(), b)); |
---|
680 | } |
---|
681 | |
---|
682 | if (e.isZero()) |
---|
683 | { |
---|
684 | if (mipoHasDen) |
---|
685 | { |
---|
686 | for (CFListIterator i= recResult; i.hasItem(); i++) |
---|
687 | i.getItem()= replacevar (i.getItem(), alpha, gamma); |
---|
688 | } |
---|
689 | return recResult; |
---|
690 | } |
---|
691 | CanonicalForm coeffE; |
---|
692 | CFList result= recResult; |
---|
693 | if (mipoHasDen) |
---|
694 | { |
---|
695 | for (CFListIterator i= result; i.hasItem(); i++) |
---|
696 | i.getItem()= replacevar (i.getItem(), alpha, gamma); |
---|
697 | } |
---|
698 | setCharacteristic (p); |
---|
699 | setReduce (alpha, false); |
---|
700 | recResult= mapinto (recResult); |
---|
701 | setReduce (alpha, true); |
---|
702 | |
---|
703 | for (CFListIterator i= recResult; i.hasItem(); i++) |
---|
704 | i.getItem()= replacevar (i.getItem(), alpha, beta); |
---|
705 | |
---|
706 | setCharacteristic (0); |
---|
707 | CanonicalForm g; |
---|
708 | CanonicalForm modulus= p; |
---|
709 | int d= b.getk(); |
---|
710 | modpk b2; |
---|
711 | for (int i= 1; i < d; i++) |
---|
712 | { |
---|
713 | coeffE= div (e, modulus); |
---|
714 | setCharacteristic (p); |
---|
715 | if (mipoHasDen) |
---|
716 | setReduce (gamma, false); |
---|
717 | else |
---|
718 | setReduce (alpha, false); |
---|
719 | coeffE= coeffE.mapinto(); |
---|
720 | if (mipoHasDen) |
---|
721 | setReduce (gamma, true); |
---|
722 | else |
---|
723 | setReduce (alpha, true); |
---|
724 | if (mipoHasDen) |
---|
725 | coeffE= replacevar (coeffE, gamma, beta); |
---|
726 | else |
---|
727 | coeffE= replacevar (coeffE, alpha, beta); |
---|
728 | setCharacteristic (0); |
---|
729 | b2= modpk (p, d - i); |
---|
730 | if (!coeffE.isZero()) |
---|
731 | { |
---|
732 | CFListIterator k= result; |
---|
733 | CFListIterator l= L; |
---|
734 | int ii= 0; |
---|
735 | j= recResult; |
---|
736 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
737 | { |
---|
738 | setCharacteristic (p); |
---|
739 | g= modNTL (coeffE, bufFactors[ii]); |
---|
740 | g= mulNTL (g, j.getItem()); |
---|
741 | g= modNTL (g, bufFactors[ii]); |
---|
742 | setCharacteristic (0); |
---|
743 | if (mipoHasDen) |
---|
744 | { |
---|
745 | setReduce (beta, false); |
---|
746 | k.getItem() += replacevar (g.mapinto()*modulus, beta, gamma); |
---|
747 | e -= mulNTL (replacevar (g.mapinto(), beta, gamma), |
---|
748 | b2 (l.getItem()), b2)*modulus; |
---|
749 | setReduce (beta, true); |
---|
750 | } |
---|
751 | else |
---|
752 | { |
---|
753 | setReduce (beta, false); |
---|
754 | k.getItem() += replacevar (g.mapinto()*modulus, beta, alpha); |
---|
755 | e -= mulNTL (replacevar (g.mapinto(), beta, alpha), |
---|
756 | b2 (l.getItem()), b2)*modulus; |
---|
757 | setReduce (beta, true); |
---|
758 | } |
---|
759 | e= b(e); |
---|
760 | } |
---|
761 | } |
---|
762 | modulus *= p; |
---|
763 | if (e.isZero()) |
---|
764 | break; |
---|
765 | } |
---|
766 | |
---|
767 | return result; |
---|
768 | } |
---|
769 | |
---|
770 | |
---|
771 | /// solve \f$ 1=\sum_{i=1}^n{\delta_{i} \prod_{j\neq i}{f_j}} \f$ mod \f$p^k\f$ |
---|
772 | /// over Q(alpha) by first computing mod \f$p\f$ and if no zero divisor occured |
---|
773 | /// compute it mod \f$p^k\f$ |
---|
774 | CFList |
---|
775 | diophantineQa (const CanonicalForm& F, const CanonicalForm& G, |
---|
776 | const CFList& factors, modpk& b, const Variable& alpha) |
---|
777 | { |
---|
778 | bool fail= false; |
---|
779 | CFList recResult; |
---|
780 | CanonicalForm modMipo, mipo; |
---|
781 | //here SW_RATIONAL is off |
---|
782 | On (SW_RATIONAL); |
---|
783 | mipo= getMipo (alpha); |
---|
784 | bool mipoHasDen= false; |
---|
785 | if (!bCommonDen (mipo).isOne()) |
---|
786 | { |
---|
787 | mipo *= bCommonDen (mipo); |
---|
788 | mipoHasDen= true; |
---|
789 | } |
---|
790 | Off (SW_RATIONAL); |
---|
791 | int p= b.getp(); |
---|
792 | setCharacteristic (p); |
---|
793 | setReduce (alpha, false); |
---|
794 | while (1) |
---|
795 | { |
---|
796 | setCharacteristic (p); |
---|
797 | modMipo= mapinto (mipo); |
---|
798 | modMipo /= lc (modMipo); |
---|
799 | tryDiophantine (recResult, mapinto (F), mapinto (factors), modMipo, fail); |
---|
800 | if (fail) |
---|
801 | { |
---|
802 | int i= 0; |
---|
803 | while (cf_getBigPrime (i) < p) |
---|
804 | i++; |
---|
805 | findGoodPrime (F, i); |
---|
806 | findGoodPrime (G, i); |
---|
807 | p=cf_getBigPrime(i); |
---|
808 | b = coeffBound( G, p, mipo ); |
---|
809 | modpk bb= coeffBound (F, p, mipo ); |
---|
810 | if (bb.getk() > b.getk() ) b=bb; |
---|
811 | fail= false; |
---|
812 | } |
---|
813 | else |
---|
814 | break; |
---|
815 | } |
---|
816 | setReduce (alpha, true); |
---|
817 | setCharacteristic (0); |
---|
818 | |
---|
819 | Variable gamma= alpha; |
---|
820 | CanonicalForm den; |
---|
821 | if (mipoHasDen) |
---|
822 | { |
---|
823 | On (SW_RATIONAL); |
---|
824 | modMipo= getMipo (alpha); |
---|
825 | den= bCommonDen (modMipo); |
---|
826 | modMipo *= den; |
---|
827 | Off (SW_RATIONAL); |
---|
828 | setReduce (alpha, false); |
---|
829 | gamma= rootOf (b (modMipo*b.inverse (den))); |
---|
830 | setReduce (alpha, true); |
---|
831 | } |
---|
832 | |
---|
833 | Variable x= Variable (1); |
---|
834 | CanonicalForm buf1, buf2, buf3, S; |
---|
835 | CFList bufFactors= factors; |
---|
836 | CFListIterator i= bufFactors; |
---|
837 | if (mipoHasDen) |
---|
838 | { |
---|
839 | for (; i.hasItem(); i++) |
---|
840 | i.getItem()= replacevar (i.getItem(), alpha, gamma); |
---|
841 | } |
---|
842 | i= bufFactors; |
---|
843 | CFList result; |
---|
844 | if (i.hasItem()) |
---|
845 | i++; |
---|
846 | buf1= 0; |
---|
847 | CanonicalForm Freplaced; |
---|
848 | if (mipoHasDen) |
---|
849 | { |
---|
850 | Freplaced= replacevar (F, alpha, gamma); |
---|
851 | buf2= divNTL (Freplaced, replacevar (i.getItem(), alpha, gamma), b); |
---|
852 | } |
---|
853 | else |
---|
854 | buf2= divNTL (F, i.getItem(), b); |
---|
855 | ZZ_p::init (convertFacCF2NTLZZ (b.getpk())); |
---|
856 | ZZ_pX NTLmipo= to_ZZ_pX (convertFacCF2NTLZZX (getMipo (gamma))); |
---|
857 | ZZ_pE::init (NTLmipo); |
---|
858 | ZZ_pEX NTLS, NTLT, NTLbuf3; |
---|
859 | ZZ_pEX NTLbuf1= convertFacCF2NTLZZ_pEX (buf1, NTLmipo); |
---|
860 | ZZ_pEX NTLbuf2= convertFacCF2NTLZZ_pEX (buf2, NTLmipo); |
---|
861 | XGCD (NTLbuf3, NTLS, NTLT, NTLbuf1, NTLbuf2); |
---|
862 | result.append (b (convertNTLZZ_pEX2CF (NTLS, x, gamma))); |
---|
863 | result.append (b (convertNTLZZ_pEX2CF (NTLT, x, gamma))); |
---|
864 | if (i.hasItem()) |
---|
865 | i++; |
---|
866 | for (; i.hasItem(); i++) |
---|
867 | { |
---|
868 | if (mipoHasDen) |
---|
869 | buf1= divNTL (Freplaced, i.getItem(), b); |
---|
870 | else |
---|
871 | buf1= divNTL (F, i.getItem(), b); |
---|
872 | XGCD (NTLbuf3, NTLS, NTLT, NTLbuf3, convertFacCF2NTLZZ_pEX (buf1, NTLmipo)); |
---|
873 | CFListIterator k= bufFactors; |
---|
874 | S= convertNTLZZ_pEX2CF (NTLS, x, gamma); |
---|
875 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
876 | { |
---|
877 | j.getItem()= mulNTL (j.getItem(), S, b); |
---|
878 | j.getItem()= modNTL (j.getItem(), k.getItem(), b); |
---|
879 | } |
---|
880 | result.append (b (convertNTLZZ_pEX2CF (NTLT, x, gamma))); |
---|
881 | } |
---|
882 | return result; |
---|
883 | } |
---|
884 | |
---|
885 | CFList |
---|
886 | diophantine (const CanonicalForm& F, const CanonicalForm& G, |
---|
887 | const CFList& factors, modpk& b) |
---|
888 | { |
---|
889 | if (getCharacteristic() == 0) |
---|
890 | { |
---|
891 | Variable v; |
---|
892 | bool hasAlgVar= hasFirstAlgVar (F, v); |
---|
893 | for (CFListIterator i= factors; i.hasItem() && !hasAlgVar; i++) |
---|
894 | hasAlgVar= hasFirstAlgVar (i.getItem(), v); |
---|
895 | if (hasAlgVar) |
---|
896 | { |
---|
897 | if (b.getp() != 0) |
---|
898 | { |
---|
899 | CFList result= diophantineQa (F, G, factors, b, v); |
---|
900 | return result; |
---|
901 | } |
---|
902 | CFList result= modularDiophant (F, factors, getMipo (v)); |
---|
903 | return result; |
---|
904 | } |
---|
905 | if (b.getp() != 0) |
---|
906 | return diophantineHensel (F, factors, b); |
---|
907 | } |
---|
908 | |
---|
909 | CanonicalForm buf1, buf2, buf3, S, T; |
---|
910 | CFListIterator i= factors; |
---|
911 | CFList result; |
---|
912 | if (i.hasItem()) |
---|
913 | i++; |
---|
914 | buf1= F/factors.getFirst(); |
---|
915 | buf2= divNTL (F, i.getItem()); |
---|
916 | buf3= extgcd (buf1, buf2, S, T); |
---|
917 | result.append (S); |
---|
918 | result.append (T); |
---|
919 | if (i.hasItem()) |
---|
920 | i++; |
---|
921 | for (; i.hasItem(); i++) |
---|
922 | { |
---|
923 | buf1= divNTL (F, i.getItem()); |
---|
924 | buf3= extgcd (buf3, buf1, S, T); |
---|
925 | CFListIterator k= factors; |
---|
926 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
927 | { |
---|
928 | j.getItem()= mulNTL (j.getItem(), S); |
---|
929 | j.getItem()= modNTL (j.getItem(), k.getItem()); |
---|
930 | } |
---|
931 | result.append (T); |
---|
932 | } |
---|
933 | return result; |
---|
934 | } |
---|
935 | |
---|
936 | CFList |
---|
937 | diophantine (const CanonicalForm& F, const CFList& factors) |
---|
938 | { |
---|
939 | modpk b= modpk(); |
---|
940 | return diophantine (F, 1, factors, b); |
---|
941 | } |
---|
942 | |
---|
943 | void |
---|
944 | henselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
945 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
946 | CFArray& Pi, int j, const modpk& b) |
---|
947 | { |
---|
948 | CanonicalForm E; |
---|
949 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
950 | Variable x= F.mvar(); |
---|
951 | // compute the error |
---|
952 | if (j == 1) |
---|
953 | E= F[j]; |
---|
954 | else |
---|
955 | { |
---|
956 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
957 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
958 | else |
---|
959 | E= F[j]; |
---|
960 | } |
---|
961 | |
---|
962 | if (b.getp() != 0) |
---|
963 | E= b(E); |
---|
964 | CFArray buf= CFArray (diophant.length()); |
---|
965 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
966 | int k= 0; |
---|
967 | CanonicalForm remainder; |
---|
968 | // actual lifting |
---|
969 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
970 | { |
---|
971 | if (degree (bufFactors[k], x) > 0) |
---|
972 | { |
---|
973 | if (k > 0) |
---|
974 | remainder= modNTL (E, bufFactors[k] [0], b); //TODO precompute inverses of bufFactors[k][0] |
---|
975 | else |
---|
976 | remainder= E; |
---|
977 | } |
---|
978 | else |
---|
979 | remainder= modNTL (E, bufFactors[k], b); |
---|
980 | |
---|
981 | buf[k]= mulNTL (i.getItem(), remainder, b); |
---|
982 | if (degree (bufFactors[k], x) > 0) |
---|
983 | buf[k]= modNTL (buf[k], bufFactors[k] [0], b); |
---|
984 | else |
---|
985 | buf[k]= modNTL (buf[k], bufFactors[k], b); |
---|
986 | } |
---|
987 | for (k= 1; k < factors.length(); k++) |
---|
988 | { |
---|
989 | bufFactors[k] += xToJ*buf[k]; |
---|
990 | if (b.getp() != 0) |
---|
991 | bufFactors[k]= b(bufFactors[k]); |
---|
992 | } |
---|
993 | |
---|
994 | // update Pi [0] |
---|
995 | int degBuf0= degree (bufFactors[0], x); |
---|
996 | int degBuf1= degree (bufFactors[1], x); |
---|
997 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
998 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j], b); |
---|
999 | CanonicalForm uIZeroJ; |
---|
1000 | |
---|
1001 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1002 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1003 | (bufFactors[1] [0] + buf[1]), b) - M(1, 1) - M(j + 1, 1); |
---|
1004 | else if (degBuf0 > 0) |
---|
1005 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1], b); |
---|
1006 | else if (degBuf1 > 0) |
---|
1007 | uIZeroJ= mulNTL (bufFactors[0], buf[1], b); |
---|
1008 | else |
---|
1009 | uIZeroJ= 0; |
---|
1010 | if (b.getp() != 0) |
---|
1011 | uIZeroJ= b (uIZeroJ); |
---|
1012 | Pi [0] += xToJ*uIZeroJ; |
---|
1013 | if (b.getp() != 0) |
---|
1014 | Pi [0]= b (Pi[0]); |
---|
1015 | |
---|
1016 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1017 | for (k= 0; k < factors.length() - 1; k++) |
---|
1018 | tmp[k]= 0; |
---|
1019 | CFIterator one, two; |
---|
1020 | one= bufFactors [0]; |
---|
1021 | two= bufFactors [1]; |
---|
1022 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1023 | { |
---|
1024 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1025 | { |
---|
1026 | if (k != j - k + 1) |
---|
1027 | { |
---|
1028 | if ((one.hasTerms() && one.exp() == j - k + 1) |
---|
1029 | && (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1030 | { |
---|
1031 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), (bufFactors[1][k]+ |
---|
1032 | two.coeff()), b) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
1033 | one++; |
---|
1034 | two++; |
---|
1035 | } |
---|
1036 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1037 | { |
---|
1038 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), bufFactors[1][k], b) |
---|
1039 | - M (k + 1, 1); |
---|
1040 | one++; |
---|
1041 | } |
---|
1042 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1043 | { |
---|
1044 | tmp[0] += mulNTL (bufFactors[0][k], (bufFactors[1][k]+two.coeff()), b) |
---|
1045 | - M (k + 1, 1); |
---|
1046 | two++; |
---|
1047 | } |
---|
1048 | } |
---|
1049 | else |
---|
1050 | { |
---|
1051 | tmp[0] += M (k + 1, 1); |
---|
1052 | } |
---|
1053 | } |
---|
1054 | } |
---|
1055 | if (b.getp() != 0) |
---|
1056 | tmp[0]= b (tmp[0]); |
---|
1057 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1058 | |
---|
1059 | // update Pi [l] |
---|
1060 | int degPi, degBuf; |
---|
1061 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1062 | { |
---|
1063 | degPi= degree (Pi [l - 1], x); |
---|
1064 | degBuf= degree (bufFactors[l + 1], x); |
---|
1065 | if (degPi > 0 && degBuf > 0) |
---|
1066 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j], b); |
---|
1067 | if (j == 1) |
---|
1068 | { |
---|
1069 | if (degPi > 0 && degBuf > 0) |
---|
1070 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [0] + Pi [l - 1] [j], |
---|
1071 | bufFactors[l + 1] [0] + buf[l + 1], b) - M (j + 1, l +1) - |
---|
1072 | M (1, l + 1)); |
---|
1073 | else if (degPi > 0) |
---|
1074 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [j], bufFactors[l + 1], b)); |
---|
1075 | else if (degBuf > 0) |
---|
1076 | Pi [l] += xToJ*(mulNTL (Pi [l - 1], buf[l + 1], b)); |
---|
1077 | } |
---|
1078 | else |
---|
1079 | { |
---|
1080 | if (degPi > 0 && degBuf > 0) |
---|
1081 | { |
---|
1082 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0], b); |
---|
1083 | uIZeroJ += mulNTL (Pi [l - 1] [0], buf [l + 1], b); |
---|
1084 | } |
---|
1085 | else if (degPi > 0) |
---|
1086 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1], b); |
---|
1087 | else if (degBuf > 0) |
---|
1088 | { |
---|
1089 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0], b); |
---|
1090 | uIZeroJ += mulNTL (Pi [l - 1], buf[l + 1], b); |
---|
1091 | } |
---|
1092 | Pi[l] += xToJ*uIZeroJ; |
---|
1093 | } |
---|
1094 | one= bufFactors [l + 1]; |
---|
1095 | two= Pi [l - 1]; |
---|
1096 | if (two.hasTerms() && two.exp() == j + 1) |
---|
1097 | { |
---|
1098 | if (degBuf > 0 && degPi > 0) |
---|
1099 | { |
---|
1100 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1][0], b); |
---|
1101 | two++; |
---|
1102 | } |
---|
1103 | else if (degPi > 0) |
---|
1104 | { |
---|
1105 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1], b); |
---|
1106 | two++; |
---|
1107 | } |
---|
1108 | } |
---|
1109 | if (degBuf > 0 && degPi > 0) |
---|
1110 | { |
---|
1111 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1112 | { |
---|
1113 | if (k != j - k + 1) |
---|
1114 | { |
---|
1115 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1116 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1117 | { |
---|
1118 | tmp[l] += mulNTL ((bufFactors[l+1][k] + one.coeff()), (Pi[l-1][k] + |
---|
1119 | two.coeff()),b) - M (k + 1, l + 1) - M (j - k + 2, l + 1); |
---|
1120 | one++; |
---|
1121 | two++; |
---|
1122 | } |
---|
1123 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1124 | { |
---|
1125 | tmp[l] += mulNTL ((bufFactors[l+1][k]+one.coeff()), Pi[l-1][k], b) - |
---|
1126 | M (k + 1, l + 1); |
---|
1127 | one++; |
---|
1128 | } |
---|
1129 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1130 | { |
---|
1131 | tmp[l] += mulNTL (bufFactors[l+1][k], (Pi[l-1][k] + two.coeff()), b) |
---|
1132 | - M (k + 1, l + 1); |
---|
1133 | two++; |
---|
1134 | } |
---|
1135 | } |
---|
1136 | else |
---|
1137 | tmp[l] += M (k + 1, l + 1); |
---|
1138 | } |
---|
1139 | } |
---|
1140 | if (b.getp() != 0) |
---|
1141 | tmp[l]= b (tmp[l]); |
---|
1142 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1143 | } |
---|
1144 | } |
---|
1145 | |
---|
1146 | void |
---|
1147 | henselLift12 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
1148 | CFList& diophant, CFMatrix& M, modpk& b, bool sort) |
---|
1149 | { |
---|
1150 | if (sort) |
---|
1151 | sortList (factors, Variable (1)); |
---|
1152 | Pi= CFArray (factors.length() - 1); |
---|
1153 | CFListIterator j= factors; |
---|
1154 | diophant= diophantine (F[0], F, factors, b); |
---|
1155 | CanonicalForm bufF= F; |
---|
1156 | if (getCharacteristic() == 0 && b.getp() != 0) |
---|
1157 | { |
---|
1158 | Variable v; |
---|
1159 | bool hasAlgVar= hasFirstAlgVar (F, v); |
---|
1160 | for (CFListIterator i= factors; i.hasItem() && !hasAlgVar; i++) |
---|
1161 | hasAlgVar= hasFirstAlgVar (i.getItem(), v); |
---|
1162 | Variable w; |
---|
1163 | bool hasAlgVar2= false; |
---|
1164 | for (CFListIterator i= diophant; i.hasItem() && !hasAlgVar2; i++) |
---|
1165 | hasAlgVar2= hasFirstAlgVar (i.getItem(), w); |
---|
1166 | if (hasAlgVar && hasAlgVar2 && v!=w) |
---|
1167 | { |
---|
1168 | bufF= replacevar (bufF, v, w); |
---|
1169 | for (CFListIterator i= factors; i.hasItem(); i++) |
---|
1170 | i.getItem()= replacevar (i.getItem(), v, w); |
---|
1171 | } |
---|
1172 | } |
---|
1173 | |
---|
1174 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
1175 | j++; |
---|
1176 | Pi [0]= mulNTL (j.getItem(), mod (factors.getFirst(), F.mvar()), b); |
---|
1177 | M (1, 1)= Pi [0]; |
---|
1178 | int i= 1; |
---|
1179 | if (j.hasItem()) |
---|
1180 | j++; |
---|
1181 | for (; j.hasItem(); j++, i++) |
---|
1182 | { |
---|
1183 | Pi [i]= mulNTL (Pi [i - 1], j.getItem(), b); |
---|
1184 | M (1, i + 1)= Pi [i]; |
---|
1185 | } |
---|
1186 | CFArray bufFactors= CFArray (factors.length()); |
---|
1187 | i= 0; |
---|
1188 | for (CFListIterator k= factors; k.hasItem(); i++, k++) |
---|
1189 | { |
---|
1190 | if (i == 0) |
---|
1191 | bufFactors[i]= mod (k.getItem(), F.mvar()); |
---|
1192 | else |
---|
1193 | bufFactors[i]= k.getItem(); |
---|
1194 | } |
---|
1195 | for (i= 1; i < l; i++) |
---|
1196 | henselStep12 (bufF, factors, bufFactors, diophant, M, Pi, i, b); |
---|
1197 | |
---|
1198 | CFListIterator k= factors; |
---|
1199 | for (i= 0; i < factors.length (); i++, k++) |
---|
1200 | k.getItem()= bufFactors[i]; |
---|
1201 | factors.removeFirst(); |
---|
1202 | } |
---|
1203 | |
---|
1204 | void |
---|
1205 | henselLift12 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
1206 | CFList& diophant, CFMatrix& M, bool sort) |
---|
1207 | { |
---|
1208 | modpk dummy= modpk(); |
---|
1209 | henselLift12 (F, factors, l, Pi, diophant, M, dummy, sort); |
---|
1210 | } |
---|
1211 | |
---|
1212 | void |
---|
1213 | henselLiftResume12 (const CanonicalForm& F, CFList& factors, int start, int |
---|
1214 | end, CFArray& Pi, const CFList& diophant, CFMatrix& M, |
---|
1215 | const modpk& b) |
---|
1216 | { |
---|
1217 | CFArray bufFactors= CFArray (factors.length()); |
---|
1218 | int i= 0; |
---|
1219 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
1220 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
1221 | { |
---|
1222 | if (i == 0) |
---|
1223 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
1224 | else |
---|
1225 | bufFactors[i]= k.getItem(); |
---|
1226 | } |
---|
1227 | for (i= start; i < end; i++) |
---|
1228 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i, b); |
---|
1229 | |
---|
1230 | CFListIterator k= factors; |
---|
1231 | for (i= 0; i < factors.length(); k++, i++) |
---|
1232 | k.getItem()= bufFactors [i]; |
---|
1233 | factors.removeFirst(); |
---|
1234 | return; |
---|
1235 | } |
---|
1236 | |
---|
1237 | CFList |
---|
1238 | biDiophantine (const CanonicalForm& F, const CFList& factors, int d) |
---|
1239 | { |
---|
1240 | Variable y= F.mvar(); |
---|
1241 | CFList result; |
---|
1242 | if (y.level() == 1) |
---|
1243 | { |
---|
1244 | result= diophantine (F, factors); |
---|
1245 | return result; |
---|
1246 | } |
---|
1247 | else |
---|
1248 | { |
---|
1249 | CFList buf= factors; |
---|
1250 | for (CFListIterator i= buf; i.hasItem(); i++) |
---|
1251 | i.getItem()= mod (i.getItem(), y); |
---|
1252 | CanonicalForm A= mod (F, y); |
---|
1253 | int bufD= 1; |
---|
1254 | CFList recResult= biDiophantine (A, buf, bufD); |
---|
1255 | CanonicalForm e= 1; |
---|
1256 | CFList p; |
---|
1257 | CFArray bufFactors= CFArray (factors.length()); |
---|
1258 | CanonicalForm yToD= power (y, d); |
---|
1259 | int k= 0; |
---|
1260 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1261 | { |
---|
1262 | bufFactors [k]= i.getItem(); |
---|
1263 | } |
---|
1264 | CanonicalForm b, quot; |
---|
1265 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
1266 | { |
---|
1267 | b= 1; |
---|
1268 | if (fdivides (bufFactors[k], F, quot)) |
---|
1269 | b= quot; |
---|
1270 | else |
---|
1271 | { |
---|
1272 | for (int l= 0; l < factors.length(); l++) |
---|
1273 | { |
---|
1274 | if (l == k) |
---|
1275 | continue; |
---|
1276 | else |
---|
1277 | { |
---|
1278 | b= mulMod2 (b, bufFactors[l], yToD); |
---|
1279 | } |
---|
1280 | } |
---|
1281 | } |
---|
1282 | p.append (b); |
---|
1283 | } |
---|
1284 | |
---|
1285 | CFListIterator j= p; |
---|
1286 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1287 | e -= i.getItem()*j.getItem(); |
---|
1288 | |
---|
1289 | if (e.isZero()) |
---|
1290 | return recResult; |
---|
1291 | CanonicalForm coeffE; |
---|
1292 | CFList s; |
---|
1293 | result= recResult; |
---|
1294 | CanonicalForm g; |
---|
1295 | for (int i= 1; i < d; i++) |
---|
1296 | { |
---|
1297 | if (degree (e, y) > 0) |
---|
1298 | coeffE= e[i]; |
---|
1299 | else |
---|
1300 | coeffE= 0; |
---|
1301 | if (!coeffE.isZero()) |
---|
1302 | { |
---|
1303 | CFListIterator k= result; |
---|
1304 | CFListIterator l= p; |
---|
1305 | int ii= 0; |
---|
1306 | j= recResult; |
---|
1307 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
1308 | { |
---|
1309 | g= coeffE*j.getItem(); |
---|
1310 | if (degree (bufFactors[ii], y) <= 0) |
---|
1311 | g= mod (g, bufFactors[ii]); |
---|
1312 | else |
---|
1313 | g= mod (g, bufFactors[ii][0]); |
---|
1314 | k.getItem() += g*power (y, i); |
---|
1315 | e -= mulMod2 (g*power(y, i), l.getItem(), yToD); |
---|
1316 | DEBOUTLN (cerr, "mod (e, power (y, i + 1))= " << |
---|
1317 | mod (e, power (y, i + 1))); |
---|
1318 | } |
---|
1319 | } |
---|
1320 | if (e.isZero()) |
---|
1321 | break; |
---|
1322 | } |
---|
1323 | |
---|
1324 | DEBOUTLN (cerr, "mod (e, y)= " << mod (e, y)); |
---|
1325 | |
---|
1326 | #ifdef DEBUGOUTPUT |
---|
1327 | CanonicalForm test= 0; |
---|
1328 | j= p; |
---|
1329 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
1330 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
1331 | DEBOUTLN (cerr, "test= " << test); |
---|
1332 | #endif |
---|
1333 | return result; |
---|
1334 | } |
---|
1335 | } |
---|
1336 | |
---|
1337 | CFList |
---|
1338 | multiRecDiophantine (const CanonicalForm& F, const CFList& factors, |
---|
1339 | const CFList& recResult, const CFList& M, int d) |
---|
1340 | { |
---|
1341 | Variable y= F.mvar(); |
---|
1342 | CFList result; |
---|
1343 | CFListIterator i; |
---|
1344 | CanonicalForm e= 1; |
---|
1345 | CFListIterator j= factors; |
---|
1346 | CFList p; |
---|
1347 | CFArray bufFactors= CFArray (factors.length()); |
---|
1348 | CanonicalForm yToD= power (y, d); |
---|
1349 | int k= 0; |
---|
1350 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1351 | bufFactors [k]= i.getItem(); |
---|
1352 | CanonicalForm b, quot; |
---|
1353 | CFList buf= M; |
---|
1354 | buf.removeLast(); |
---|
1355 | buf.append (yToD); |
---|
1356 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
1357 | { |
---|
1358 | b= 1; |
---|
1359 | if (fdivides (bufFactors[k], F, quot)) |
---|
1360 | b= quot; |
---|
1361 | else |
---|
1362 | { |
---|
1363 | for (int l= 0; l < factors.length(); l++) |
---|
1364 | { |
---|
1365 | if (l == k) |
---|
1366 | continue; |
---|
1367 | else |
---|
1368 | { |
---|
1369 | b= mulMod (b, bufFactors[l], buf); |
---|
1370 | } |
---|
1371 | } |
---|
1372 | } |
---|
1373 | p.append (b); |
---|
1374 | } |
---|
1375 | j= p; |
---|
1376 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1377 | e -= mulMod (i.getItem(), j.getItem(), M); |
---|
1378 | |
---|
1379 | if (e.isZero()) |
---|
1380 | return recResult; |
---|
1381 | CanonicalForm coeffE; |
---|
1382 | CFList s; |
---|
1383 | result= recResult; |
---|
1384 | CanonicalForm g; |
---|
1385 | for (int i= 1; i < d; i++) |
---|
1386 | { |
---|
1387 | if (degree (e, y) > 0) |
---|
1388 | coeffE= e[i]; |
---|
1389 | else |
---|
1390 | coeffE= 0; |
---|
1391 | if (!coeffE.isZero()) |
---|
1392 | { |
---|
1393 | CFListIterator k= result; |
---|
1394 | CFListIterator l= p; |
---|
1395 | j= recResult; |
---|
1396 | int ii= 0; |
---|
1397 | CanonicalForm dummy; |
---|
1398 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
1399 | { |
---|
1400 | g= mulMod (coeffE, j.getItem(), M); |
---|
1401 | if (degree (bufFactors[ii], y) <= 0) |
---|
1402 | divrem (g, mod (bufFactors[ii], Variable (y.level() - 1)), dummy, |
---|
1403 | g, M); |
---|
1404 | else |
---|
1405 | divrem (g, bufFactors[ii][0], dummy, g, M); |
---|
1406 | k.getItem() += g*power (y, i); |
---|
1407 | e -= mulMod (g*power (y, i), l.getItem(), M); |
---|
1408 | } |
---|
1409 | } |
---|
1410 | |
---|
1411 | if (e.isZero()) |
---|
1412 | break; |
---|
1413 | } |
---|
1414 | |
---|
1415 | #ifdef DEBUGOUTPUT |
---|
1416 | CanonicalForm test= 0; |
---|
1417 | j= p; |
---|
1418 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
1419 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
1420 | DEBOUTLN (cerr, "test in multiRecDiophantine= " << test); |
---|
1421 | #endif |
---|
1422 | return result; |
---|
1423 | } |
---|
1424 | |
---|
1425 | static |
---|
1426 | void |
---|
1427 | henselStep (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
1428 | const CFList& diophant, CFMatrix& M, CFArray& Pi, int j, |
---|
1429 | const CFList& MOD) |
---|
1430 | { |
---|
1431 | CanonicalForm E; |
---|
1432 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
1433 | Variable x= F.mvar(); |
---|
1434 | // compute the error |
---|
1435 | if (j == 1) |
---|
1436 | { |
---|
1437 | E= F[j]; |
---|
1438 | #ifdef DEBUGOUTPUT |
---|
1439 | CanonicalForm test= 1; |
---|
1440 | for (int i= 0; i < factors.length(); i++) |
---|
1441 | { |
---|
1442 | if (i == 0) |
---|
1443 | test= mulMod (test, mod (bufFactors [i], xToJ), MOD); |
---|
1444 | else |
---|
1445 | test= mulMod (test, bufFactors[i], MOD); |
---|
1446 | } |
---|
1447 | CanonicalForm test2= mod (F-test, xToJ); |
---|
1448 | |
---|
1449 | test2= mod (test2, MOD); |
---|
1450 | DEBOUTLN (cerr, "test in henselStep= " << test2); |
---|
1451 | #endif |
---|
1452 | } |
---|
1453 | else |
---|
1454 | { |
---|
1455 | #ifdef DEBUGOUTPUT |
---|
1456 | CanonicalForm test= 1; |
---|
1457 | for (int i= 0; i < factors.length(); i++) |
---|
1458 | { |
---|
1459 | if (i == 0) |
---|
1460 | test *= mod (bufFactors [i], power (x, j)); |
---|
1461 | else |
---|
1462 | test *= bufFactors[i]; |
---|
1463 | } |
---|
1464 | test= mod (test, power (x, j)); |
---|
1465 | test= mod (test, MOD); |
---|
1466 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
1467 | DEBOUTLN (cerr, "test in henselStep= " << test2); |
---|
1468 | #endif |
---|
1469 | |
---|
1470 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1471 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1472 | else |
---|
1473 | E= F[j]; |
---|
1474 | } |
---|
1475 | |
---|
1476 | CFArray buf= CFArray (diophant.length()); |
---|
1477 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
1478 | int k= 0; |
---|
1479 | // actual lifting |
---|
1480 | CanonicalForm dummy, rest1; |
---|
1481 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1482 | { |
---|
1483 | if (degree (bufFactors[k], x) > 0) |
---|
1484 | { |
---|
1485 | if (k > 0) |
---|
1486 | divrem (E, bufFactors[k] [0], dummy, rest1, MOD); |
---|
1487 | else |
---|
1488 | rest1= E; |
---|
1489 | } |
---|
1490 | else |
---|
1491 | divrem (E, bufFactors[k], dummy, rest1, MOD); |
---|
1492 | |
---|
1493 | buf[k]= mulMod (i.getItem(), rest1, MOD); |
---|
1494 | |
---|
1495 | if (degree (bufFactors[k], x) > 0) |
---|
1496 | divrem (buf[k], bufFactors[k] [0], dummy, buf[k], MOD); |
---|
1497 | else |
---|
1498 | divrem (buf[k], bufFactors[k], dummy, buf[k], MOD); |
---|
1499 | } |
---|
1500 | for (k= 1; k < factors.length(); k++) |
---|
1501 | bufFactors[k] += xToJ*buf[k]; |
---|
1502 | |
---|
1503 | // update Pi [0] |
---|
1504 | int degBuf0= degree (bufFactors[0], x); |
---|
1505 | int degBuf1= degree (bufFactors[1], x); |
---|
1506 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1507 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
1508 | CanonicalForm uIZeroJ; |
---|
1509 | |
---|
1510 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1511 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1512 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
1513 | else if (degBuf0 > 0) |
---|
1514 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
1515 | else if (degBuf1 > 0) |
---|
1516 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
1517 | else |
---|
1518 | uIZeroJ= 0; |
---|
1519 | Pi [0] += xToJ*uIZeroJ; |
---|
1520 | |
---|
1521 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1522 | for (k= 0; k < factors.length() - 1; k++) |
---|
1523 | tmp[k]= 0; |
---|
1524 | CFIterator one, two; |
---|
1525 | one= bufFactors [0]; |
---|
1526 | two= bufFactors [1]; |
---|
1527 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1528 | { |
---|
1529 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1530 | { |
---|
1531 | if (k != j - k + 1) |
---|
1532 | { |
---|
1533 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1534 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1535 | { |
---|
1536 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
1537 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
1538 | M (j - k + 2, 1); |
---|
1539 | one++; |
---|
1540 | two++; |
---|
1541 | } |
---|
1542 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1543 | { |
---|
1544 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
1545 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
1546 | one++; |
---|
1547 | } |
---|
1548 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1549 | { |
---|
1550 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
1551 | two.coeff()), MOD) - M (k + 1, 1); |
---|
1552 | two++; |
---|
1553 | } |
---|
1554 | } |
---|
1555 | else |
---|
1556 | { |
---|
1557 | tmp[0] += M (k + 1, 1); |
---|
1558 | } |
---|
1559 | } |
---|
1560 | } |
---|
1561 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1562 | |
---|
1563 | // update Pi [l] |
---|
1564 | int degPi, degBuf; |
---|
1565 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1566 | { |
---|
1567 | degPi= degree (Pi [l - 1], x); |
---|
1568 | degBuf= degree (bufFactors[l + 1], x); |
---|
1569 | if (degPi > 0 && degBuf > 0) |
---|
1570 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
1571 | if (j == 1) |
---|
1572 | { |
---|
1573 | if (degPi > 0 && degBuf > 0) |
---|
1574 | Pi [l] += xToJ*(mulMod ((Pi [l - 1] [0] + Pi [l - 1] [j]), |
---|
1575 | (bufFactors[l + 1] [0] + buf[l + 1]), MOD) - M (j + 1, l +1)- |
---|
1576 | M (1, l + 1)); |
---|
1577 | else if (degPi > 0) |
---|
1578 | Pi [l] += xToJ*(mulMod (Pi [l - 1] [j], bufFactors[l + 1], MOD)); |
---|
1579 | else if (degBuf > 0) |
---|
1580 | Pi [l] += xToJ*(mulMod (Pi [l - 1], buf[l + 1], MOD)); |
---|
1581 | } |
---|
1582 | else |
---|
1583 | { |
---|
1584 | if (degPi > 0 && degBuf > 0) |
---|
1585 | { |
---|
1586 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
1587 | uIZeroJ += mulMod (Pi [l - 1] [0], buf [l + 1], MOD); |
---|
1588 | } |
---|
1589 | else if (degPi > 0) |
---|
1590 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1], MOD); |
---|
1591 | else if (degBuf > 0) |
---|
1592 | { |
---|
1593 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
1594 | uIZeroJ += mulMod (Pi [l - 1], buf[l + 1], MOD); |
---|
1595 | } |
---|
1596 | Pi[l] += xToJ*uIZeroJ; |
---|
1597 | } |
---|
1598 | one= bufFactors [l + 1]; |
---|
1599 | two= Pi [l - 1]; |
---|
1600 | if (two.hasTerms() && two.exp() == j + 1) |
---|
1601 | { |
---|
1602 | if (degBuf > 0 && degPi > 0) |
---|
1603 | { |
---|
1604 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1][0], MOD); |
---|
1605 | two++; |
---|
1606 | } |
---|
1607 | else if (degPi > 0) |
---|
1608 | { |
---|
1609 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1], MOD); |
---|
1610 | two++; |
---|
1611 | } |
---|
1612 | } |
---|
1613 | if (degBuf > 0 && degPi > 0) |
---|
1614 | { |
---|
1615 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1616 | { |
---|
1617 | if (k != j - k + 1) |
---|
1618 | { |
---|
1619 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1620 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1621 | { |
---|
1622 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1623 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
1624 | M (j - k + 2, l + 1); |
---|
1625 | one++; |
---|
1626 | two++; |
---|
1627 | } |
---|
1628 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1629 | { |
---|
1630 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1631 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
1632 | one++; |
---|
1633 | } |
---|
1634 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1635 | { |
---|
1636 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
1637 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
1638 | two++; |
---|
1639 | } |
---|
1640 | } |
---|
1641 | else |
---|
1642 | tmp[l] += M (k + 1, l + 1); |
---|
1643 | } |
---|
1644 | } |
---|
1645 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1646 | } |
---|
1647 | |
---|
1648 | return; |
---|
1649 | } |
---|
1650 | |
---|
1651 | CFList |
---|
1652 | henselLift23 (const CFList& eval, const CFList& factors, int* l, CFList& |
---|
1653 | diophant, CFArray& Pi, CFMatrix& M) |
---|
1654 | { |
---|
1655 | CFList buf= factors; |
---|
1656 | int k= 0; |
---|
1657 | int liftBoundBivar= l[k]; |
---|
1658 | diophant= biDiophantine (eval.getFirst(), buf, liftBoundBivar); |
---|
1659 | CFList MOD; |
---|
1660 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
1661 | CFArray bufFactors= CFArray (factors.length()); |
---|
1662 | k= 0; |
---|
1663 | CFListIterator j= eval; |
---|
1664 | j++; |
---|
1665 | buf.removeFirst(); |
---|
1666 | buf.insert (LC (j.getItem(), 1)); |
---|
1667 | for (CFListIterator i= buf; i.hasItem(); i++, k++) |
---|
1668 | bufFactors[k]= i.getItem(); |
---|
1669 | Pi= CFArray (factors.length() - 1); |
---|
1670 | CFListIterator i= buf; |
---|
1671 | i++; |
---|
1672 | Variable y= j.getItem().mvar(); |
---|
1673 | Pi [0]= mulMod (i.getItem(), mod (buf.getFirst(), y), MOD); |
---|
1674 | M (1, 1)= Pi [0]; |
---|
1675 | k= 1; |
---|
1676 | if (i.hasItem()) |
---|
1677 | i++; |
---|
1678 | for (; i.hasItem(); i++, k++) |
---|
1679 | { |
---|
1680 | Pi [k]= mulMod (Pi [k - 1], i.getItem(), MOD); |
---|
1681 | M (1, k + 1)= Pi [k]; |
---|
1682 | } |
---|
1683 | |
---|
1684 | for (int d= 1; d < l[1]; d++) |
---|
1685 | henselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
1686 | CFList result; |
---|
1687 | for (k= 1; k < factors.length(); k++) |
---|
1688 | result.append (bufFactors[k]); |
---|
1689 | return result; |
---|
1690 | } |
---|
1691 | |
---|
1692 | void |
---|
1693 | henselLiftResume (const CanonicalForm& F, CFList& factors, int start, int end, |
---|
1694 | CFArray& Pi, const CFList& diophant, CFMatrix& M, |
---|
1695 | const CFList& MOD) |
---|
1696 | { |
---|
1697 | CFArray bufFactors= CFArray (factors.length()); |
---|
1698 | int i= 0; |
---|
1699 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
1700 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
1701 | { |
---|
1702 | if (i == 0) |
---|
1703 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
1704 | else |
---|
1705 | bufFactors[i]= k.getItem(); |
---|
1706 | } |
---|
1707 | for (i= start; i < end; i++) |
---|
1708 | henselStep (F, factors, bufFactors, diophant, M, Pi, i, MOD); |
---|
1709 | |
---|
1710 | CFListIterator k= factors; |
---|
1711 | for (i= 0; i < factors.length(); k++, i++) |
---|
1712 | k.getItem()= bufFactors [i]; |
---|
1713 | factors.removeFirst(); |
---|
1714 | return; |
---|
1715 | } |
---|
1716 | |
---|
1717 | CFList |
---|
1718 | henselLift (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
1719 | diophant, CFArray& Pi, CFMatrix& M, int lOld, int lNew) |
---|
1720 | { |
---|
1721 | diophant= multiRecDiophantine (F.getFirst(), factors, diophant, MOD, lOld); |
---|
1722 | int k= 0; |
---|
1723 | CFArray bufFactors= CFArray (factors.length()); |
---|
1724 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1725 | { |
---|
1726 | if (k == 0) |
---|
1727 | bufFactors[k]= LC (F.getLast(), 1); |
---|
1728 | else |
---|
1729 | bufFactors[k]= i.getItem(); |
---|
1730 | } |
---|
1731 | CFList buf= factors; |
---|
1732 | buf.removeFirst(); |
---|
1733 | buf.insert (LC (F.getLast(), 1)); |
---|
1734 | CFListIterator i= buf; |
---|
1735 | i++; |
---|
1736 | Variable y= F.getLast().mvar(); |
---|
1737 | Variable x= F.getFirst().mvar(); |
---|
1738 | CanonicalForm xToLOld= power (x, lOld); |
---|
1739 | Pi [0]= mod (Pi[0], xToLOld); |
---|
1740 | M (1, 1)= Pi [0]; |
---|
1741 | k= 1; |
---|
1742 | if (i.hasItem()) |
---|
1743 | i++; |
---|
1744 | for (; i.hasItem(); i++, k++) |
---|
1745 | { |
---|
1746 | Pi [k]= mod (Pi [k], xToLOld); |
---|
1747 | M (1, k + 1)= Pi [k]; |
---|
1748 | } |
---|
1749 | |
---|
1750 | for (int d= 1; d < lNew; d++) |
---|
1751 | henselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
1752 | CFList result; |
---|
1753 | for (k= 1; k < factors.length(); k++) |
---|
1754 | result.append (bufFactors[k]); |
---|
1755 | return result; |
---|
1756 | } |
---|
1757 | |
---|
1758 | CFList |
---|
1759 | henselLift (const CFList& eval, const CFList& factors, int* l, int lLength, |
---|
1760 | bool sort) |
---|
1761 | { |
---|
1762 | CFList diophant; |
---|
1763 | CFList buf= factors; |
---|
1764 | buf.insert (LC (eval.getFirst(), 1)); |
---|
1765 | if (sort) |
---|
1766 | sortList (buf, Variable (1)); |
---|
1767 | CFArray Pi; |
---|
1768 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
1769 | CFList result= henselLift23 (eval, buf, l, diophant, Pi, M); |
---|
1770 | if (eval.length() == 2) |
---|
1771 | return result; |
---|
1772 | CFList MOD; |
---|
1773 | for (int i= 0; i < 2; i++) |
---|
1774 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1775 | CFListIterator j= eval; |
---|
1776 | j++; |
---|
1777 | CFList bufEval; |
---|
1778 | bufEval.append (j.getItem()); |
---|
1779 | j++; |
---|
1780 | |
---|
1781 | for (int i= 2; i < lLength && j.hasItem(); i++, j++) |
---|
1782 | { |
---|
1783 | result.insert (LC (bufEval.getFirst(), 1)); |
---|
1784 | bufEval.append (j.getItem()); |
---|
1785 | M= CFMatrix (l[i], factors.length()); |
---|
1786 | result= henselLift (bufEval, result, MOD, diophant, Pi, M, l[i - 1], l[i]); |
---|
1787 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1788 | bufEval.removeFirst(); |
---|
1789 | } |
---|
1790 | return result; |
---|
1791 | } |
---|
1792 | |
---|
1793 | // nonmonic |
---|
1794 | |
---|
1795 | void |
---|
1796 | nonMonicHenselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
1797 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
1798 | CFArray& Pi, int j, const CFArray& /*LCs*/) |
---|
1799 | { |
---|
1800 | Variable x= F.mvar(); |
---|
1801 | CanonicalForm xToJ= power (x, j); |
---|
1802 | |
---|
1803 | CanonicalForm E; |
---|
1804 | // compute the error |
---|
1805 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1806 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1807 | else |
---|
1808 | E= F[j]; |
---|
1809 | |
---|
1810 | CFArray buf= CFArray (diophant.length()); |
---|
1811 | |
---|
1812 | int k= 0; |
---|
1813 | CanonicalForm remainder; |
---|
1814 | // actual lifting |
---|
1815 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1816 | { |
---|
1817 | if (degree (bufFactors[k], x) > 0) |
---|
1818 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
1819 | else |
---|
1820 | remainder= modNTL (E, bufFactors[k]); |
---|
1821 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
1822 | if (degree (bufFactors[k], x) > 0) |
---|
1823 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
1824 | else |
---|
1825 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
1826 | } |
---|
1827 | |
---|
1828 | for (k= 0; k < factors.length(); k++) |
---|
1829 | bufFactors[k] += xToJ*buf[k]; |
---|
1830 | |
---|
1831 | // update Pi [0] |
---|
1832 | int degBuf0= degree (bufFactors[0], x); |
---|
1833 | int degBuf1= degree (bufFactors[1], x); |
---|
1834 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1835 | { |
---|
1836 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
1837 | if (j + 2 <= M.rows()) |
---|
1838 | M (j + 2, 1)= mulNTL (bufFactors[0] [j + 1], bufFactors[1] [j + 1]); |
---|
1839 | } |
---|
1840 | else |
---|
1841 | M (j + 1, 1)= 0; |
---|
1842 | |
---|
1843 | CanonicalForm uIZeroJ; |
---|
1844 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1845 | uIZeroJ= mulNTL(bufFactors[0][0], buf[1]) + |
---|
1846 | mulNTL (bufFactors[1][0], buf[0]); |
---|
1847 | else if (degBuf0 > 0) |
---|
1848 | uIZeroJ= mulNTL (buf[0], bufFactors[1]) + |
---|
1849 | mulNTL (buf[1], bufFactors[0][0]); |
---|
1850 | else if (degBuf1 > 0) |
---|
1851 | uIZeroJ= mulNTL (bufFactors[0], buf[1]) + |
---|
1852 | mulNTL (buf[0], bufFactors[1][0]); |
---|
1853 | else |
---|
1854 | uIZeroJ= mulNTL (bufFactors[0], buf[1]) + |
---|
1855 | mulNTL (buf[0], bufFactors[1]); |
---|
1856 | |
---|
1857 | Pi [0] += xToJ*uIZeroJ; |
---|
1858 | |
---|
1859 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1860 | for (k= 0; k < factors.length() - 1; k++) |
---|
1861 | tmp[k]= 0; |
---|
1862 | CFIterator one, two; |
---|
1863 | one= bufFactors [0]; |
---|
1864 | two= bufFactors [1]; |
---|
1865 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1866 | { |
---|
1867 | while (one.hasTerms() && one.exp() > j) one++; |
---|
1868 | while (two.hasTerms() && two.exp() > j) two++; |
---|
1869 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1870 | { |
---|
1871 | if (k != j - k + 1) |
---|
1872 | { |
---|
1873 | if ((one.hasTerms() && one.exp() == j - k + 1) && + |
---|
1874 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1875 | { |
---|
1876 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()),(bufFactors[1][k] + |
---|
1877 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
1878 | one++; |
---|
1879 | two++; |
---|
1880 | } |
---|
1881 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1882 | { |
---|
1883 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), bufFactors[1] [k]) - |
---|
1884 | M (k + 1, 1); |
---|
1885 | one++; |
---|
1886 | } |
---|
1887 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1888 | { |
---|
1889 | tmp[0] += mulNTL (bufFactors[0][k],(bufFactors[1][k] + two.coeff())) - |
---|
1890 | M (k + 1, 1); |
---|
1891 | two++; |
---|
1892 | } |
---|
1893 | } |
---|
1894 | else |
---|
1895 | tmp[0] += M (k + 1, 1); |
---|
1896 | } |
---|
1897 | } |
---|
1898 | |
---|
1899 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
1900 | { |
---|
1901 | if (j + 2 <= M.rows()) |
---|
1902 | tmp [0] += mulNTL ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
1903 | (bufFactors [1] [j + 1] + bufFactors [1] [0])) |
---|
1904 | - M(1,1) - M (j + 2,1); |
---|
1905 | } |
---|
1906 | else if (degBuf0 >= j + 1) |
---|
1907 | { |
---|
1908 | if (degBuf1 > 0) |
---|
1909 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1] [0]); |
---|
1910 | else |
---|
1911 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1]); |
---|
1912 | } |
---|
1913 | else if (degBuf1 >= j + 1) |
---|
1914 | { |
---|
1915 | if (degBuf0 > 0) |
---|
1916 | tmp[0] += mulNTL (bufFactors [0] [0], bufFactors [1] [j + 1]); |
---|
1917 | else |
---|
1918 | tmp[0] += mulNTL (bufFactors [0], bufFactors [1] [j + 1]); |
---|
1919 | } |
---|
1920 | |
---|
1921 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1922 | |
---|
1923 | int degPi, degBuf; |
---|
1924 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1925 | { |
---|
1926 | degPi= degree (Pi [l - 1], x); |
---|
1927 | degBuf= degree (bufFactors[l + 1], x); |
---|
1928 | if (degPi > 0 && degBuf > 0) |
---|
1929 | { |
---|
1930 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
1931 | if (j + 2 <= M.rows()) |
---|
1932 | M (j + 2, l + 1)= mulNTL (Pi [l - 1][j + 1], bufFactors[l + 1] [j + 1]); |
---|
1933 | } |
---|
1934 | else |
---|
1935 | M (j + 1, l + 1)= 0; |
---|
1936 | |
---|
1937 | if (degPi > 0 && degBuf > 0) |
---|
1938 | uIZeroJ= mulNTL (Pi[l - 1] [0], buf[l + 1]) + |
---|
1939 | mulNTL (uIZeroJ, bufFactors[l+1] [0]); |
---|
1940 | else if (degPi > 0) |
---|
1941 | uIZeroJ= mulNTL (uIZeroJ, bufFactors[l + 1]) + |
---|
1942 | mulNTL (Pi[l - 1][0], buf[l + 1]); |
---|
1943 | else if (degBuf > 0) |
---|
1944 | uIZeroJ= mulNTL (uIZeroJ, bufFactors[l + 1][0]) + |
---|
1945 | mulNTL (Pi[l - 1], buf[l + 1]); |
---|
1946 | else |
---|
1947 | uIZeroJ= mulNTL (uIZeroJ, bufFactors[l + 1]) + |
---|
1948 | mulNTL (Pi[l - 1], buf[l + 1]); |
---|
1949 | |
---|
1950 | Pi [l] += xToJ*uIZeroJ; |
---|
1951 | |
---|
1952 | one= bufFactors [l + 1]; |
---|
1953 | two= Pi [l - 1]; |
---|
1954 | if (degBuf > 0 && degPi > 0) |
---|
1955 | { |
---|
1956 | while (one.hasTerms() && one.exp() > j) one++; |
---|
1957 | while (two.hasTerms() && two.exp() > j) two++; |
---|
1958 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1959 | { |
---|
1960 | if (k != j - k + 1) |
---|
1961 | { |
---|
1962 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1963 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1964 | { |
---|
1965 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1966 | (Pi[l - 1] [k] + two.coeff())) - M (k + 1, l + 1) - |
---|
1967 | M (j - k + 2, l + 1); |
---|
1968 | one++; |
---|
1969 | two++; |
---|
1970 | } |
---|
1971 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1972 | { |
---|
1973 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1974 | Pi[l - 1] [k]) - M (k + 1, l + 1); |
---|
1975 | one++; |
---|
1976 | } |
---|
1977 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1978 | { |
---|
1979 | tmp[l] += mulNTL (bufFactors[l + 1] [k], |
---|
1980 | (Pi[l - 1] [k] + two.coeff())) - M (k + 1, l + 1); |
---|
1981 | two++; |
---|
1982 | } |
---|
1983 | } |
---|
1984 | else |
---|
1985 | tmp[l] += M (k + 1, l + 1); |
---|
1986 | } |
---|
1987 | } |
---|
1988 | |
---|
1989 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
1990 | { |
---|
1991 | if (j + 2 <= M.rows()) |
---|
1992 | tmp [l] += mulNTL ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
1993 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
1994 | ) - M(1,l+1) - M (j + 2,l+1); |
---|
1995 | } |
---|
1996 | else if (degPi >= j + 1) |
---|
1997 | { |
---|
1998 | if (degBuf > 0) |
---|
1999 | tmp[l] += mulNTL (Pi [l - 1] [j+1], bufFactors [l + 1] [0]); |
---|
2000 | else |
---|
2001 | tmp[l] += mulNTL (Pi [l - 1] [j+1], bufFactors [l + 1]); |
---|
2002 | } |
---|
2003 | else if (degBuf >= j + 1) |
---|
2004 | { |
---|
2005 | if (degPi > 0) |
---|
2006 | tmp[l] += mulNTL (Pi [l - 1] [0], bufFactors [l + 1] [j + 1]); |
---|
2007 | else |
---|
2008 | tmp[l] += mulNTL (Pi [l - 1], bufFactors [l + 1] [j + 1]); |
---|
2009 | } |
---|
2010 | |
---|
2011 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2012 | } |
---|
2013 | return; |
---|
2014 | } |
---|
2015 | |
---|
2016 | void |
---|
2017 | nonMonicHenselLift12 (const CanonicalForm& F, CFList& factors, int l, |
---|
2018 | CFArray& Pi, CFList& diophant, CFMatrix& M, |
---|
2019 | const CFArray& LCs, bool sort) |
---|
2020 | { |
---|
2021 | if (sort) |
---|
2022 | sortList (factors, Variable (1)); |
---|
2023 | Pi= CFArray (factors.length() - 2); |
---|
2024 | CFList bufFactors2= factors; |
---|
2025 | bufFactors2.removeFirst(); |
---|
2026 | diophant= diophantine (F[0], bufFactors2); |
---|
2027 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
2028 | |
---|
2029 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
2030 | int i= 0; |
---|
2031 | for (CFListIterator k= bufFactors2; k.hasItem(); i++, k++) |
---|
2032 | bufFactors[i]= replaceLc (k.getItem(), LCs [i]); |
---|
2033 | |
---|
2034 | Variable x= F.mvar(); |
---|
2035 | if (degree (bufFactors[0], x) > 0 && degree (bufFactors [1], x) > 0) |
---|
2036 | { |
---|
2037 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1] [0]); |
---|
2038 | Pi [0]= M (1, 1) + (mulNTL (bufFactors [0] [1], bufFactors[1] [0]) + |
---|
2039 | mulNTL (bufFactors [0] [0], bufFactors [1] [1]))*x; |
---|
2040 | } |
---|
2041 | else if (degree (bufFactors[0], x) > 0) |
---|
2042 | { |
---|
2043 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1]); |
---|
2044 | Pi [0]= M (1, 1) + |
---|
2045 | mulNTL (bufFactors [0] [1], bufFactors[1])*x; |
---|
2046 | } |
---|
2047 | else if (degree (bufFactors[1], x) > 0) |
---|
2048 | { |
---|
2049 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1] [0]); |
---|
2050 | Pi [0]= M (1, 1) + |
---|
2051 | mulNTL (bufFactors [0], bufFactors[1] [1])*x; |
---|
2052 | } |
---|
2053 | else |
---|
2054 | { |
---|
2055 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1]); |
---|
2056 | Pi [0]= M (1, 1); |
---|
2057 | } |
---|
2058 | |
---|
2059 | for (i= 1; i < Pi.size(); i++) |
---|
2060 | { |
---|
2061 | if (degree (Pi[i-1], x) > 0 && degree (bufFactors [i+1], x) > 0) |
---|
2062 | { |
---|
2063 | M (1,i+1)= mulNTL (Pi[i-1] [0], bufFactors[i+1] [0]); |
---|
2064 | Pi [i]= M (1,i+1) + (mulNTL (Pi[i-1] [1], bufFactors[i+1] [0]) + |
---|
2065 | mulNTL (Pi[i-1] [0], bufFactors [i+1] [1]))*x; |
---|
2066 | } |
---|
2067 | else if (degree (Pi[i-1], x) > 0) |
---|
2068 | { |
---|
2069 | M (1,i+1)= mulNTL (Pi[i-1] [0], bufFactors [i+1]); |
---|
2070 | Pi [i]= M(1,i+1) + mulNTL (Pi[i-1] [1], bufFactors[i+1])*x; |
---|
2071 | } |
---|
2072 | else if (degree (bufFactors[i+1], x) > 0) |
---|
2073 | { |
---|
2074 | M (1,i+1)= mulNTL (Pi[i-1], bufFactors [i+1] [0]); |
---|
2075 | Pi [i]= M (1,i+1) + mulNTL (Pi[i-1], bufFactors[i+1] [1])*x; |
---|
2076 | } |
---|
2077 | else |
---|
2078 | { |
---|
2079 | M (1,i+1)= mulNTL (Pi [i-1], bufFactors [i+1]); |
---|
2080 | Pi [i]= M (1,i+1); |
---|
2081 | } |
---|
2082 | } |
---|
2083 | |
---|
2084 | for (i= 1; i < l; i++) |
---|
2085 | nonMonicHenselStep12 (F, bufFactors2, bufFactors, diophant, M, Pi, i, LCs); |
---|
2086 | |
---|
2087 | factors= CFList(); |
---|
2088 | for (i= 0; i < bufFactors.size(); i++) |
---|
2089 | factors.append (bufFactors[i]); |
---|
2090 | return; |
---|
2091 | } |
---|
2092 | |
---|
2093 | |
---|
2094 | /// solve \f$ E=\sum_{i= 1}^r{\sigma_{i}\prod_{j=1, j\neq i}^{r}{f_{i}}}\f$ |
---|
2095 | /// mod M, @a products contains \f$ \prod_{j=1, j\neq i}^{r}{f_{j}} \f$ |
---|
2096 | CFList |
---|
2097 | diophantine (const CFList& recResult, const CFList& factors, |
---|
2098 | const CFList& products, const CFList& M, const CanonicalForm& E, |
---|
2099 | bool& bad) |
---|
2100 | { |
---|
2101 | if (M.isEmpty()) |
---|
2102 | { |
---|
2103 | CFList result; |
---|
2104 | CFListIterator j= factors; |
---|
2105 | CanonicalForm buf; |
---|
2106 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
2107 | { |
---|
2108 | ASSERT (E.isUnivariate() || E.inCoeffDomain(), |
---|
2109 | "constant or univariate poly expected"); |
---|
2110 | ASSERT (i.getItem().isUnivariate() || i.getItem().inCoeffDomain(), |
---|
2111 | "constant or univariate poly expected"); |
---|
2112 | ASSERT (j.getItem().isUnivariate() || j.getItem().inCoeffDomain(), |
---|
2113 | "constant or univariate poly expected"); |
---|
2114 | buf= mulNTL (E, i.getItem()); |
---|
2115 | result.append (modNTL (buf, j.getItem())); |
---|
2116 | } |
---|
2117 | return result; |
---|
2118 | } |
---|
2119 | Variable y= M.getLast().mvar(); |
---|
2120 | CFList bufFactors= factors; |
---|
2121 | for (CFListIterator i= bufFactors; i.hasItem(); i++) |
---|
2122 | i.getItem()= mod (i.getItem(), y); |
---|
2123 | CFList bufProducts= products; |
---|
2124 | for (CFListIterator i= bufProducts; i.hasItem(); i++) |
---|
2125 | i.getItem()= mod (i.getItem(), y); |
---|
2126 | CFList buf= M; |
---|
2127 | buf.removeLast(); |
---|
2128 | CanonicalForm bufE= mod (E, y); |
---|
2129 | CFList recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
2130 | bufE, bad); |
---|
2131 | |
---|
2132 | if (bad) |
---|
2133 | return CFList(); |
---|
2134 | |
---|
2135 | CanonicalForm e= E; |
---|
2136 | CFListIterator j= products; |
---|
2137 | for (CFListIterator i= recDiophantine; i.hasItem(); i++, j++) |
---|
2138 | e -= j.getItem()*i.getItem(); |
---|
2139 | |
---|
2140 | CFList result= recDiophantine; |
---|
2141 | int d= degree (M.getLast()); |
---|
2142 | CanonicalForm coeffE; |
---|
2143 | for (int i= 1; i < d; i++) |
---|
2144 | { |
---|
2145 | if (degree (e, y) > 0) |
---|
2146 | coeffE= e[i]; |
---|
2147 | else |
---|
2148 | coeffE= 0; |
---|
2149 | if (!coeffE.isZero()) |
---|
2150 | { |
---|
2151 | CFListIterator k= result; |
---|
2152 | recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
2153 | coeffE, bad); |
---|
2154 | if (bad) |
---|
2155 | return CFList(); |
---|
2156 | CFListIterator l= products; |
---|
2157 | for (j= recDiophantine; j.hasItem(); j++, k++, l++) |
---|
2158 | { |
---|
2159 | k.getItem() += j.getItem()*power (y, i); |
---|
2160 | e -= l.getItem()*(j.getItem()*power (y, i)); |
---|
2161 | } |
---|
2162 | } |
---|
2163 | if (e.isZero()) |
---|
2164 | break; |
---|
2165 | } |
---|
2166 | if (!e.isZero()) |
---|
2167 | { |
---|
2168 | bad= true; |
---|
2169 | return CFList(); |
---|
2170 | } |
---|
2171 | return result; |
---|
2172 | } |
---|
2173 | |
---|
2174 | void |
---|
2175 | nonMonicHenselStep (const CanonicalForm& F, const CFList& factors, |
---|
2176 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
2177 | CFArray& Pi, const CFList& products, int j, |
---|
2178 | const CFList& MOD, bool& noOneToOne) |
---|
2179 | { |
---|
2180 | CanonicalForm E; |
---|
2181 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
2182 | Variable x= F.mvar(); |
---|
2183 | |
---|
2184 | // compute the error |
---|
2185 | #ifdef DEBUGOUTPUT |
---|
2186 | CanonicalForm test= 1; |
---|
2187 | for (int i= 0; i < factors.length(); i++) |
---|
2188 | { |
---|
2189 | if (i == 0) |
---|
2190 | test *= mod (bufFactors [i], power (x, j)); |
---|
2191 | else |
---|
2192 | test *= bufFactors[i]; |
---|
2193 | } |
---|
2194 | test= mod (test, power (x, j)); |
---|
2195 | test= mod (test, MOD); |
---|
2196 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
2197 | DEBOUTLN (cerr, "test in nonMonicHenselStep= " << test2); |
---|
2198 | #endif |
---|
2199 | |
---|
2200 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
2201 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
2202 | else |
---|
2203 | E= F[j]; |
---|
2204 | |
---|
2205 | CFArray buf= CFArray (diophant.length()); |
---|
2206 | |
---|
2207 | // actual lifting |
---|
2208 | TIMING_START (diotime); |
---|
2209 | CFList diophantine2= diophantine (diophant, factors, products, MOD, E, |
---|
2210 | noOneToOne); |
---|
2211 | |
---|
2212 | if (noOneToOne) |
---|
2213 | return; |
---|
2214 | |
---|
2215 | int k= 0; |
---|
2216 | for (CFListIterator i= diophantine2; k < factors.length(); k++, i++) |
---|
2217 | { |
---|
2218 | buf[k]= i.getItem(); |
---|
2219 | bufFactors[k] += xToJ*i.getItem(); |
---|
2220 | } |
---|
2221 | TIMING_END_AND_PRINT (diotime, "time for dio: "); |
---|
2222 | |
---|
2223 | // update Pi [0] |
---|
2224 | TIMING_START (product2); |
---|
2225 | int degBuf0= degree (bufFactors[0], x); |
---|
2226 | int degBuf1= degree (bufFactors[1], x); |
---|
2227 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2228 | { |
---|
2229 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
2230 | if (j + 2 <= M.rows()) |
---|
2231 | M (j + 2, 1)= mulMod (bufFactors[0] [j + 1], bufFactors[1] [j + 1], MOD); |
---|
2232 | } |
---|
2233 | else |
---|
2234 | M (j + 1, 1)= 0; |
---|
2235 | |
---|
2236 | CanonicalForm uIZeroJ; |
---|
2237 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2238 | uIZeroJ= mulMod (bufFactors[0] [0], buf[1], MOD) + |
---|
2239 | mulMod (bufFactors[1] [0], buf[0], MOD); |
---|
2240 | else if (degBuf0 > 0) |
---|
2241 | uIZeroJ= mulMod (buf[0], bufFactors[1], MOD) + |
---|
2242 | mulMod (buf[1], bufFactors[0][0], MOD); |
---|
2243 | else if (degBuf1 > 0) |
---|
2244 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD) + |
---|
2245 | mulMod (buf[0], bufFactors[1][0], MOD); |
---|
2246 | else |
---|
2247 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD) + |
---|
2248 | mulMod (buf[0], bufFactors[1], MOD); |
---|
2249 | Pi [0] += xToJ*uIZeroJ; |
---|
2250 | |
---|
2251 | CFArray tmp= CFArray (factors.length() - 1); |
---|
2252 | for (k= 0; k < factors.length() - 1; k++) |
---|
2253 | tmp[k]= 0; |
---|
2254 | CFIterator one, two; |
---|
2255 | one= bufFactors [0]; |
---|
2256 | two= bufFactors [1]; |
---|
2257 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2258 | { |
---|
2259 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2260 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2261 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2262 | { |
---|
2263 | if (k != j - k + 1) |
---|
2264 | { |
---|
2265 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2266 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2267 | { |
---|
2268 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2269 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
2270 | M (j - k + 2, 1); |
---|
2271 | one++; |
---|
2272 | two++; |
---|
2273 | } |
---|
2274 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2275 | { |
---|
2276 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2277 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
2278 | one++; |
---|
2279 | } |
---|
2280 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2281 | { |
---|
2282 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
2283 | two.coeff()), MOD) - M (k + 1, 1); |
---|
2284 | two++; |
---|
2285 | } |
---|
2286 | } |
---|
2287 | else |
---|
2288 | { |
---|
2289 | tmp[0] += M (k + 1, 1); |
---|
2290 | } |
---|
2291 | } |
---|
2292 | } |
---|
2293 | |
---|
2294 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
2295 | { |
---|
2296 | if (j + 2 <= M.rows()) |
---|
2297 | tmp [0] += mulMod ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
2298 | (bufFactors [1] [j + 1] + bufFactors [1] [0]), MOD) |
---|
2299 | - M(1,1) - M (j + 2,1); |
---|
2300 | } |
---|
2301 | else if (degBuf0 >= j + 1) |
---|
2302 | { |
---|
2303 | if (degBuf1 > 0) |
---|
2304 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1] [0], MOD); |
---|
2305 | else |
---|
2306 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1], MOD); |
---|
2307 | } |
---|
2308 | else if (degBuf1 >= j + 1) |
---|
2309 | { |
---|
2310 | if (degBuf0 > 0) |
---|
2311 | tmp[0] += mulMod (bufFactors [0] [0], bufFactors [1] [j + 1], MOD); |
---|
2312 | else |
---|
2313 | tmp[0] += mulMod (bufFactors [0], bufFactors [1] [j + 1], MOD); |
---|
2314 | } |
---|
2315 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
2316 | |
---|
2317 | // update Pi [l] |
---|
2318 | int degPi, degBuf; |
---|
2319 | for (int l= 1; l < factors.length() - 1; l++) |
---|
2320 | { |
---|
2321 | degPi= degree (Pi [l - 1], x); |
---|
2322 | degBuf= degree (bufFactors[l + 1], x); |
---|
2323 | if (degPi > 0 && degBuf > 0) |
---|
2324 | { |
---|
2325 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
2326 | if (j + 2 <= M.rows()) |
---|
2327 | M (j + 2, l + 1)= mulMod (Pi [l - 1] [j + 1], bufFactors[l + 1] [j + 1], |
---|
2328 | MOD); |
---|
2329 | } |
---|
2330 | else |
---|
2331 | M (j + 1, l + 1)= 0; |
---|
2332 | |
---|
2333 | if (degPi > 0 && degBuf > 0) |
---|
2334 | uIZeroJ= mulMod (Pi[l - 1] [0], buf[l + 1], MOD) + |
---|
2335 | mulMod (uIZeroJ, bufFactors[l + 1] [0], MOD); |
---|
2336 | else if (degPi > 0) |
---|
2337 | uIZeroJ= mulMod (uIZeroJ, bufFactors[l + 1], MOD) + |
---|
2338 | mulMod (Pi[l - 1][0], buf[l + 1], MOD); |
---|
2339 | else if (degBuf > 0) |
---|
2340 | uIZeroJ= mulMod (Pi[l - 1], buf[l + 1], MOD) + |
---|
2341 | mulMod (uIZeroJ, bufFactors[l + 1][0], MOD); |
---|
2342 | else |
---|
2343 | uIZeroJ= mulMod (Pi[l - 1], buf[l + 1], MOD) + |
---|
2344 | mulMod (uIZeroJ, bufFactors[l + 1], MOD); |
---|
2345 | |
---|
2346 | Pi [l] += xToJ*uIZeroJ; |
---|
2347 | |
---|
2348 | one= bufFactors [l + 1]; |
---|
2349 | two= Pi [l - 1]; |
---|
2350 | if (degBuf > 0 && degPi > 0) |
---|
2351 | { |
---|
2352 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2353 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2354 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2355 | { |
---|
2356 | if (k != j - k + 1) |
---|
2357 | { |
---|
2358 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2359 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2360 | { |
---|
2361 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2362 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
2363 | M (j - k + 2, l + 1); |
---|
2364 | one++; |
---|
2365 | two++; |
---|
2366 | } |
---|
2367 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2368 | { |
---|
2369 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2370 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
2371 | one++; |
---|
2372 | } |
---|
2373 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2374 | { |
---|
2375 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
2376 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
2377 | two++; |
---|
2378 | } |
---|
2379 | } |
---|
2380 | else |
---|
2381 | tmp[l] += M (k + 1, l + 1); |
---|
2382 | } |
---|
2383 | } |
---|
2384 | |
---|
2385 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
2386 | { |
---|
2387 | if (j + 2 <= M.rows()) |
---|
2388 | tmp [l] += mulMod ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
2389 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
2390 | , MOD) - M(1,l+1) - M (j + 2,l+1); |
---|
2391 | } |
---|
2392 | else if (degPi >= j + 1) |
---|
2393 | { |
---|
2394 | if (degBuf > 0) |
---|
2395 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1] [0], MOD); |
---|
2396 | else |
---|
2397 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1], MOD); |
---|
2398 | } |
---|
2399 | else if (degBuf >= j + 1) |
---|
2400 | { |
---|
2401 | if (degPi > 0) |
---|
2402 | tmp[l] += mulMod (Pi [l - 1] [0], bufFactors [l + 1] [j + 1], MOD); |
---|
2403 | else |
---|
2404 | tmp[l] += mulMod (Pi [l - 1], bufFactors [l + 1] [j + 1], MOD); |
---|
2405 | } |
---|
2406 | |
---|
2407 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2408 | } |
---|
2409 | TIMING_END_AND_PRINT (product2, "time for product in hensel step: "); |
---|
2410 | return; |
---|
2411 | } |
---|
2412 | |
---|
2413 | // wrt. Variable (1) |
---|
2414 | CanonicalForm replaceLC (const CanonicalForm& F, const CanonicalForm& c) |
---|
2415 | { |
---|
2416 | if (degree (F, 1) <= 0) |
---|
2417 | return c; |
---|
2418 | else |
---|
2419 | { |
---|
2420 | CanonicalForm result= swapvar (F, Variable (F.level() + 1), Variable (1)); |
---|
2421 | result += (swapvar (c, Variable (F.level() + 1), Variable (1)) |
---|
2422 | - LC (result))*power (result.mvar(), degree (result)); |
---|
2423 | return swapvar (result, Variable (F.level() + 1), Variable (1)); |
---|
2424 | } |
---|
2425 | } |
---|
2426 | |
---|
2427 | CFList |
---|
2428 | nonMonicHenselLift232(const CFList& eval, const CFList& factors, int* l, CFList& |
---|
2429 | diophant, CFArray& Pi, CFMatrix& M, const CFList& LCs1, |
---|
2430 | const CFList& LCs2, bool& bad) |
---|
2431 | { |
---|
2432 | CFList buf= factors; |
---|
2433 | int k= 0; |
---|
2434 | int liftBoundBivar= l[k]; |
---|
2435 | CFList bufbuf= factors; |
---|
2436 | Variable v= Variable (2); |
---|
2437 | |
---|
2438 | CFList MOD; |
---|
2439 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
2440 | CFArray bufFactors= CFArray (factors.length()); |
---|
2441 | k= 0; |
---|
2442 | CFListIterator j= eval; |
---|
2443 | j++; |
---|
2444 | CFListIterator iter1= LCs1; |
---|
2445 | CFListIterator iter2= LCs2; |
---|
2446 | iter1++; |
---|
2447 | iter2++; |
---|
2448 | bufFactors[0]= replaceLC (buf.getFirst(), iter1.getItem()); |
---|
2449 | bufFactors[1]= replaceLC (buf.getLast(), iter2.getItem()); |
---|
2450 | |
---|
2451 | CFListIterator i= buf; |
---|
2452 | i++; |
---|
2453 | Variable y= j.getItem().mvar(); |
---|
2454 | if (y.level() != 3) |
---|
2455 | y= Variable (3); |
---|
2456 | |
---|
2457 | Pi[0]= mod (Pi[0], power (v, liftBoundBivar)); |
---|
2458 | M (1, 1)= Pi[0]; |
---|
2459 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
2460 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
2461 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
2462 | else if (degree (bufFactors[0], y) > 0) |
---|
2463 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
2464 | else if (degree (bufFactors[1], y) > 0) |
---|
2465 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
2466 | |
---|
2467 | CFList products; |
---|
2468 | for (int i= 0; i < bufFactors.size(); i++) |
---|
2469 | { |
---|
2470 | if (degree (bufFactors[i], y) > 0) |
---|
2471 | products.append (eval.getFirst()/bufFactors[i] [0]); |
---|
2472 | else |
---|
2473 | products.append (eval.getFirst()/bufFactors[i]); |
---|
2474 | } |
---|
2475 | |
---|
2476 | for (int d= 1; d < l[1]; d++) |
---|
2477 | { |
---|
2478 | nonMonicHenselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, products, |
---|
2479 | d, MOD, bad); |
---|
2480 | if (bad) |
---|
2481 | return CFList(); |
---|
2482 | } |
---|
2483 | CFList result; |
---|
2484 | for (k= 0; k < factors.length(); k++) |
---|
2485 | result.append (bufFactors[k]); |
---|
2486 | return result; |
---|
2487 | } |
---|
2488 | |
---|
2489 | |
---|
2490 | CFList |
---|
2491 | nonMonicHenselLift2 (const CFList& F, const CFList& factors, const CFList& MOD, |
---|
2492 | CFList& diophant, CFArray& Pi, CFMatrix& M, int lOld, |
---|
2493 | int& lNew, const CFList& LCs1, const CFList& LCs2, bool& bad |
---|
2494 | ) |
---|
2495 | { |
---|
2496 | int k= 0; |
---|
2497 | CFArray bufFactors= CFArray (factors.length()); |
---|
2498 | bufFactors[0]= replaceLC (factors.getFirst(), LCs1.getLast()); |
---|
2499 | bufFactors[1]= replaceLC (factors.getLast(), LCs2.getLast()); |
---|
2500 | CFList buf= factors; |
---|
2501 | Variable y= F.getLast().mvar(); |
---|
2502 | Variable x= F.getFirst().mvar(); |
---|
2503 | CanonicalForm xToLOld= power (x, lOld); |
---|
2504 | Pi [0]= mod (Pi[0], xToLOld); |
---|
2505 | M (1, 1)= Pi [0]; |
---|
2506 | |
---|
2507 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
2508 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
2509 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
2510 | else if (degree (bufFactors[0], y) > 0) |
---|
2511 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
2512 | else if (degree (bufFactors[1], y) > 0) |
---|
2513 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
2514 | |
---|
2515 | CFList products; |
---|
2516 | CanonicalForm quot; |
---|
2517 | for (int i= 0; i < bufFactors.size(); i++) |
---|
2518 | { |
---|
2519 | if (degree (bufFactors[i], y) > 0) |
---|
2520 | { |
---|
2521 | if (!fdivides (bufFactors[i] [0], F.getFirst(), quot)) |
---|
2522 | { |
---|
2523 | bad= true; |
---|
2524 | return CFList(); |
---|
2525 | } |
---|
2526 | products.append (quot); |
---|
2527 | } |
---|
2528 | else |
---|
2529 | { |
---|
2530 | if (!fdivides (bufFactors[i], F.getFirst(), quot)) |
---|
2531 | { |
---|
2532 | bad= true; |
---|
2533 | return CFList(); |
---|
2534 | } |
---|
2535 | products.append (quot); |
---|
2536 | } |
---|
2537 | } |
---|
2538 | |
---|
2539 | for (int d= 1; d < lNew; d++) |
---|
2540 | { |
---|
2541 | nonMonicHenselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, products, |
---|
2542 | d, MOD, bad); |
---|
2543 | if (bad) |
---|
2544 | return CFList(); |
---|
2545 | } |
---|
2546 | |
---|
2547 | CFList result; |
---|
2548 | for (k= 0; k < factors.length(); k++) |
---|
2549 | result.append (bufFactors[k]); |
---|
2550 | return result; |
---|
2551 | } |
---|
2552 | |
---|
2553 | CFList |
---|
2554 | nonMonicHenselLift2 (const CFList& eval, const CFList& factors, int* l, int |
---|
2555 | lLength, bool sort, const CFList& LCs1, const CFList& LCs2, |
---|
2556 | const CFArray& Pi, const CFList& diophant, bool& bad) |
---|
2557 | { |
---|
2558 | CFList bufDiophant= diophant; |
---|
2559 | CFList buf= factors; |
---|
2560 | if (sort) |
---|
2561 | sortList (buf, Variable (1)); |
---|
2562 | CFArray bufPi= Pi; |
---|
2563 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
2564 | CFList result= |
---|
2565 | nonMonicHenselLift232(eval, buf, l, bufDiophant, bufPi, M, LCs1, LCs2, bad); |
---|
2566 | if (bad) |
---|
2567 | return CFList(); |
---|
2568 | |
---|
2569 | if (eval.length() == 2) |
---|
2570 | return result; |
---|
2571 | CFList MOD; |
---|
2572 | for (int i= 0; i < 2; i++) |
---|
2573 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2574 | CFListIterator j= eval; |
---|
2575 | j++; |
---|
2576 | CFList bufEval; |
---|
2577 | bufEval.append (j.getItem()); |
---|
2578 | j++; |
---|
2579 | CFListIterator jj= LCs1; |
---|
2580 | CFListIterator jjj= LCs2; |
---|
2581 | CFList bufLCs1, bufLCs2; |
---|
2582 | jj++, jjj++; |
---|
2583 | bufLCs1.append (jj.getItem()); |
---|
2584 | bufLCs2.append (jjj.getItem()); |
---|
2585 | jj++, jjj++; |
---|
2586 | |
---|
2587 | for (int i= 2; i < lLength && j.hasItem(); i++, j++, jj++, jjj++) |
---|
2588 | { |
---|
2589 | bufEval.append (j.getItem()); |
---|
2590 | bufLCs1.append (jj.getItem()); |
---|
2591 | bufLCs2.append (jjj.getItem()); |
---|
2592 | M= CFMatrix (l[i], factors.length()); |
---|
2593 | result= nonMonicHenselLift2 (bufEval, result, MOD, bufDiophant, bufPi, M, |
---|
2594 | l[i - 1], l[i], bufLCs1, bufLCs2, bad); |
---|
2595 | if (bad) |
---|
2596 | return CFList(); |
---|
2597 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2598 | bufEval.removeFirst(); |
---|
2599 | bufLCs1.removeFirst(); |
---|
2600 | bufLCs2.removeFirst(); |
---|
2601 | } |
---|
2602 | return result; |
---|
2603 | } |
---|
2604 | |
---|
2605 | CFList |
---|
2606 | nonMonicHenselLift23 (const CanonicalForm& F, const CFList& factors, const |
---|
2607 | CFList& LCs, CFList& diophant, CFArray& Pi, int liftBound, |
---|
2608 | int bivarLiftBound, bool& bad) |
---|
2609 | { |
---|
2610 | CFList bufFactors2= factors; |
---|
2611 | |
---|
2612 | Variable y= Variable (2); |
---|
2613 | for (CFListIterator i= bufFactors2; i.hasItem(); i++) |
---|
2614 | i.getItem()= mod (i.getItem(), y); |
---|
2615 | |
---|
2616 | CanonicalForm bufF= F; |
---|
2617 | bufF= mod (bufF, y); |
---|
2618 | bufF= mod (bufF, Variable (3)); |
---|
2619 | |
---|
2620 | TIMING_START (diotime); |
---|
2621 | diophant= diophantine (bufF, bufFactors2); |
---|
2622 | TIMING_END_AND_PRINT (diotime, "time for dio in 23: "); |
---|
2623 | |
---|
2624 | CFMatrix M= CFMatrix (liftBound, bufFactors2.length() - 1); |
---|
2625 | |
---|
2626 | Pi= CFArray (bufFactors2.length() - 1); |
---|
2627 | |
---|
2628 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
2629 | CFListIterator j= LCs; |
---|
2630 | int i= 0; |
---|
2631 | for (CFListIterator k= factors; k.hasItem(); j++, k++, i++) |
---|
2632 | bufFactors[i]= replaceLC (k.getItem(), j.getItem()); |
---|
2633 | |
---|
2634 | //initialise Pi |
---|
2635 | Variable v= Variable (3); |
---|
2636 | CanonicalForm yToL= power (y, bivarLiftBound); |
---|
2637 | if (degree (bufFactors[0], v) > 0 && degree (bufFactors [1], v) > 0) |
---|
2638 | { |
---|
2639 | M (1, 1)= mulMod2 (bufFactors [0] [0], bufFactors[1] [0], yToL); |
---|
2640 | Pi [0]= M (1,1) + (mulMod2 (bufFactors [0] [1], bufFactors[1] [0], yToL) + |
---|
2641 | mulMod2 (bufFactors [0] [0], bufFactors [1] [1], yToL))*v; |
---|
2642 | } |
---|
2643 | else if (degree (bufFactors[0], v) > 0) |
---|
2644 | { |
---|
2645 | M (1,1)= mulMod2 (bufFactors [0] [0], bufFactors [1], yToL); |
---|
2646 | Pi [0]= M(1,1) + mulMod2 (bufFactors [0] [1], bufFactors[1], yToL)*v; |
---|
2647 | } |
---|
2648 | else if (degree (bufFactors[1], v) > 0) |
---|
2649 | { |
---|
2650 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1] [0], yToL); |
---|
2651 | Pi [0]= M (1,1) + mulMod2 (bufFactors [0], bufFactors[1] [1], yToL)*v; |
---|
2652 | } |
---|
2653 | else |
---|
2654 | { |
---|
2655 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1], yToL); |
---|
2656 | Pi [0]= M (1,1); |
---|
2657 | } |
---|
2658 | |
---|
2659 | for (i= 1; i < Pi.size(); i++) |
---|
2660 | { |
---|
2661 | if (degree (Pi[i-1], v) > 0 && degree (bufFactors [i+1], v) > 0) |
---|
2662 | { |
---|
2663 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors[i+1] [0], yToL); |
---|
2664 | Pi [i]= M (1,i+1) + (mulMod2 (Pi[i-1] [1], bufFactors[i+1] [0], yToL) + |
---|
2665 | mulMod2 (Pi[i-1] [0], bufFactors [i+1] [1], yToL))*v; |
---|
2666 | } |
---|
2667 | else if (degree (Pi[i-1], v) > 0) |
---|
2668 | { |
---|
2669 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors [i+1], yToL); |
---|
2670 | Pi [i]= M(1,i+1) + mulMod2 (Pi[i-1] [1], bufFactors[i+1], yToL)*v; |
---|
2671 | } |
---|
2672 | else if (degree (bufFactors[i+1], v) > 0) |
---|
2673 | { |
---|
2674 | M (1,i+1)= mulMod2 (Pi[i-1], bufFactors [i+1] [0], yToL); |
---|
2675 | Pi [i]= M (1,i+1) + mulMod2 (Pi[i-1], bufFactors[i+1] [1], yToL)*v; |
---|
2676 | } |
---|
2677 | else |
---|
2678 | { |
---|
2679 | M (1,i+1)= mulMod2 (Pi [i-1], bufFactors [i+1], yToL); |
---|
2680 | Pi [i]= M (1,i+1); |
---|
2681 | } |
---|
2682 | } |
---|
2683 | |
---|
2684 | CFList products; |
---|
2685 | bufF= mod (F, Variable (3)); |
---|
2686 | TIMING_START (product1); |
---|
2687 | for (CFListIterator k= factors; k.hasItem(); k++) |
---|
2688 | products.append (bufF/k.getItem()); |
---|
2689 | TIMING_END_AND_PRINT (product1, "time for product1 in 23: "); |
---|
2690 | |
---|
2691 | CFList MOD= CFList (power (v, liftBound)); |
---|
2692 | MOD.insert (yToL); |
---|
2693 | for (int d= 1; d < liftBound; d++) |
---|
2694 | { |
---|
2695 | nonMonicHenselStep (F, factors, bufFactors, diophant, M, Pi, products, d, |
---|
2696 | MOD, bad); |
---|
2697 | if (bad) |
---|
2698 | return CFList(); |
---|
2699 | } |
---|
2700 | |
---|
2701 | CFList result; |
---|
2702 | for (i= 0; i < factors.length(); i++) |
---|
2703 | result.append (bufFactors[i]); |
---|
2704 | return result; |
---|
2705 | } |
---|
2706 | |
---|
2707 | CFList |
---|
2708 | nonMonicHenselLift (const CFList& F, const CFList& factors, const CFList& LCs, |
---|
2709 | CFList& diophant, CFArray& Pi, CFMatrix& M, int lOld, |
---|
2710 | int& lNew, const CFList& MOD, bool& noOneToOne |
---|
2711 | ) |
---|
2712 | { |
---|
2713 | |
---|
2714 | int k= 0; |
---|
2715 | CFArray bufFactors= CFArray (factors.length()); |
---|
2716 | CFListIterator j= LCs; |
---|
2717 | for (CFListIterator i= factors; i.hasItem(); i++, j++, k++) |
---|
2718 | bufFactors [k]= replaceLC (i.getItem(), j.getItem()); |
---|
2719 | |
---|
2720 | Variable y= F.getLast().mvar(); |
---|
2721 | Variable x= F.getFirst().mvar(); |
---|
2722 | CanonicalForm xToLOld= power (x, lOld); |
---|
2723 | |
---|
2724 | Pi [0]= mod (Pi[0], xToLOld); |
---|
2725 | M (1, 1)= Pi [0]; |
---|
2726 | |
---|
2727 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
2728 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
2729 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
2730 | else if (degree (bufFactors[0], y) > 0) |
---|
2731 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
2732 | else if (degree (bufFactors[1], y) > 0) |
---|
2733 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
2734 | |
---|
2735 | for (int i= 1; i < Pi.size(); i++) |
---|
2736 | { |
---|
2737 | Pi [i]= mod (Pi [i], xToLOld); |
---|
2738 | M (1, i + 1)= Pi [i]; |
---|
2739 | |
---|
2740 | if (degree (Pi[i-1], y) > 0 && degree (bufFactors [i+1], y) > 0) |
---|
2741 | Pi [i] += (mulMod (Pi[i-1] [1], bufFactors[i+1] [0], MOD) + |
---|
2742 | mulMod (Pi[i-1] [0], bufFactors [i+1] [1], MOD))*y; |
---|
2743 | else if (degree (Pi[i-1], y) > 0) |
---|
2744 | Pi [i] += mulMod (Pi[i-1] [1], bufFactors[i+1], MOD)*y; |
---|
2745 | else if (degree (bufFactors[i+1], y) > 0) |
---|
2746 | Pi [i] += mulMod (Pi[i-1], bufFactors[i+1] [1], MOD)*y; |
---|
2747 | } |
---|
2748 | |
---|
2749 | CFList products; |
---|
2750 | CanonicalForm quot, bufF= F.getFirst(); |
---|
2751 | |
---|
2752 | TIMING_START (product1); |
---|
2753 | for (int i= 0; i < bufFactors.size(); i++) |
---|
2754 | { |
---|
2755 | if (degree (bufFactors[i], y) > 0) |
---|
2756 | { |
---|
2757 | if (!fdivides (bufFactors[i] [0], bufF, quot)) |
---|
2758 | { |
---|
2759 | noOneToOne= true; |
---|
2760 | return factors; |
---|
2761 | } |
---|
2762 | products.append (quot); |
---|
2763 | } |
---|
2764 | else |
---|
2765 | { |
---|
2766 | if (!fdivides (bufFactors[i], bufF, quot)) |
---|
2767 | { |
---|
2768 | noOneToOne= true; |
---|
2769 | return factors; |
---|
2770 | } |
---|
2771 | products.append (quot); |
---|
2772 | } |
---|
2773 | } |
---|
2774 | TIMING_END_AND_PRINT (product1, "time for product1 in further hensel:" ); |
---|
2775 | |
---|
2776 | for (int d= 1; d < lNew; d++) |
---|
2777 | { |
---|
2778 | nonMonicHenselStep (F.getLast(), factors, bufFactors, diophant, M, Pi, |
---|
2779 | products, d, MOD, noOneToOne); |
---|
2780 | if (noOneToOne) |
---|
2781 | return CFList(); |
---|
2782 | } |
---|
2783 | |
---|
2784 | CFList result; |
---|
2785 | for (k= 0; k < factors.length(); k++) |
---|
2786 | result.append (bufFactors[k]); |
---|
2787 | return result; |
---|
2788 | } |
---|
2789 | |
---|
2790 | CFList |
---|
2791 | nonMonicHenselLift (const CFList& eval, const CFList& factors, |
---|
2792 | CFList* const& LCs, CFList& diophant, CFArray& Pi, |
---|
2793 | int* liftBound, int length, bool& noOneToOne |
---|
2794 | ) |
---|
2795 | { |
---|
2796 | CFList bufDiophant= diophant; |
---|
2797 | CFList buf= factors; |
---|
2798 | CFArray bufPi= Pi; |
---|
2799 | CFMatrix M= CFMatrix (liftBound[1], factors.length() - 1); |
---|
2800 | int k= 0; |
---|
2801 | |
---|
2802 | TIMING_START (hensel23); |
---|
2803 | CFList result= |
---|
2804 | nonMonicHenselLift23 (eval.getFirst(), factors, LCs [0], diophant, bufPi, |
---|
2805 | liftBound[1], liftBound[0], noOneToOne); |
---|
2806 | TIMING_END_AND_PRINT (hensel23, "time for 23: "); |
---|
2807 | |
---|
2808 | if (noOneToOne) |
---|
2809 | return CFList(); |
---|
2810 | |
---|
2811 | if (eval.length() == 1) |
---|
2812 | return result; |
---|
2813 | |
---|
2814 | k++; |
---|
2815 | CFList MOD; |
---|
2816 | for (int i= 0; i < 2; i++) |
---|
2817 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
2818 | |
---|
2819 | CFListIterator j= eval; |
---|
2820 | CFList bufEval; |
---|
2821 | bufEval.append (j.getItem()); |
---|
2822 | j++; |
---|
2823 | |
---|
2824 | for (int i= 2; i <= length && j.hasItem(); i++, j++, k++) |
---|
2825 | { |
---|
2826 | bufEval.append (j.getItem()); |
---|
2827 | M= CFMatrix (liftBound[i], factors.length() - 1); |
---|
2828 | TIMING_START (hensel); |
---|
2829 | result= nonMonicHenselLift (bufEval, result, LCs [i-1], diophant, bufPi, M, |
---|
2830 | liftBound[i-1], liftBound[i], MOD, noOneToOne); |
---|
2831 | TIMING_END_AND_PRINT (hensel, "time for further hensel: "); |
---|
2832 | if (noOneToOne) |
---|
2833 | return result; |
---|
2834 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
2835 | bufEval.removeFirst(); |
---|
2836 | } |
---|
2837 | |
---|
2838 | return result; |
---|
2839 | } |
---|
2840 | |
---|
2841 | #endif |
---|
2842 | /* HAVE_NTL */ |
---|
2843 | |
---|