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