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