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 and |
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7 | * functions for modular multiplication and division with remainder. |
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8 | * |
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9 | * ABSTRACT: Hensel lifting is described in "Efficient Multivariate |
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10 | * Factorization over Finite Fields" by L. Bernardin & M. Monagon. Division with |
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11 | * remainder is described in "Fast Recursive Division" by C. Burnikel and |
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12 | * J. Ziegler. Karatsuba multiplication is described in "Modern Computer |
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13 | * Algebra" by J. von zur Gathen and J. Gerhard. |
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14 | * |
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15 | * @author Martin Lee |
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16 | * |
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17 | * @internal @version \$Id$ |
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18 | * |
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19 | **/ |
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20 | /*****************************************************************************/ |
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21 | |
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22 | #include "config.h" |
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23 | |
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24 | #include "cf_assert.h" |
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25 | #include "debug.h" |
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26 | #include "timing.h" |
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27 | |
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28 | #include "facHensel.h" |
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29 | #include "cf_util.h" |
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30 | #include "fac_util.h" |
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31 | #include "cf_algorithm.h" |
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32 | #include "cf_primes.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 | tryExtgcd (buf1, buf2, M, buf3, S, T, fail); |
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104 | if (fail) |
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105 | return; |
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106 | result.append (S); |
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107 | result.append (T); |
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108 | if (i.hasItem()) |
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109 | i++; |
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110 | for (; i.hasItem(); i++) |
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111 | { |
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112 | buf1= i.getItem(); |
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113 | tryExtgcd (buf3, buf1, M, buf3, S, T, fail); |
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114 | if (fail) |
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115 | return; |
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116 | CFListIterator k= factors; |
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117 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
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118 | { |
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119 | j.getItem() *= S; |
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120 | j.getItem()= mod (j.getItem(), k.getItem()); |
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121 | j.getItem()= reduce (j.getItem(), M); |
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122 | } |
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123 | result.append (T); |
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124 | } |
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125 | } |
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126 | |
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127 | static inline |
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128 | CFList mapinto (const CFList& L) |
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129 | { |
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130 | CFList result; |
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131 | for (CFListIterator i= L; i.hasItem(); i++) |
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132 | result.append (mapinto (i.getItem())); |
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133 | return result; |
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134 | } |
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135 | |
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136 | static inline |
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137 | int mod (const CFList& L, const CanonicalForm& p) |
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138 | { |
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139 | for (CFListIterator i= L; i.hasItem(); i++) |
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140 | { |
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141 | if (mod (i.getItem(), p) == 0) |
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142 | return 0; |
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143 | } |
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144 | return 1; |
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145 | } |
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146 | |
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147 | |
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148 | static inline void |
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149 | chineseRemainder (const CFList & x1, const CanonicalForm & q1, |
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150 | const CFList & x2, const CanonicalForm & q2, |
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151 | CFList & xnew, CanonicalForm & qnew) |
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152 | { |
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153 | ASSERT (x1.length() == x2.length(), "expected lists of equal length"); |
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154 | CanonicalForm tmp1, tmp2; |
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155 | CFListIterator j= x2; |
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156 | for (CFListIterator i= x1; i.hasItem() && j.hasItem(); i++, j++) |
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157 | { |
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158 | chineseRemainder (i.getItem(), q1, j.getItem(), q2, tmp1, tmp2); |
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159 | xnew.append (tmp1); |
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160 | } |
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161 | qnew= tmp2; |
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162 | } |
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163 | |
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164 | static inline |
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165 | CFList Farey (const CFList& L, const CanonicalForm& q) |
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166 | { |
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167 | CFList result; |
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168 | for (CFListIterator i= L; i.hasItem(); i++) |
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169 | result.append (Farey (i.getItem(), q)); |
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170 | return result; |
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171 | } |
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172 | |
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173 | static inline |
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174 | CFList replacevar (const CFList& L, const Variable& a, const Variable& b) |
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175 | { |
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176 | CFList result; |
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177 | for (CFListIterator i= L; i.hasItem(); i++) |
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178 | result.append (replacevar (i.getItem(), a, b)); |
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179 | return result; |
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180 | } |
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181 | |
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182 | CFList |
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183 | modularDiophant (const CanonicalForm& f, const CFList& factors, |
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184 | const CanonicalForm& M) |
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185 | { |
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186 | bool save_rat=!isOn (SW_RATIONAL); |
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187 | On (SW_RATIONAL); |
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188 | CanonicalForm F= f*bCommonDen (f); |
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189 | CFList products= factors; |
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190 | for (CFListIterator i= products; i.hasItem(); i++) |
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191 | { |
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192 | if (products.getFirst().level() == 1) |
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193 | i.getItem() /= Lc (i.getItem()); |
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194 | i.getItem() *= bCommonDen (i.getItem()); |
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195 | } |
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196 | if (products.getFirst().level() == 1) |
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197 | products.insert (Lc (F)); |
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198 | CanonicalForm bound= maxNorm (F); |
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199 | CFList leadingCoeffs; |
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200 | leadingCoeffs.append (lc (F)); |
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201 | CanonicalForm dummy; |
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202 | for (CFListIterator i= products; i.hasItem(); i++) |
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203 | { |
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204 | leadingCoeffs.append (lc (i.getItem())); |
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205 | dummy= maxNorm (i.getItem()); |
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206 | bound= (dummy > bound) ? dummy : bound; |
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207 | } |
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208 | bound *= maxNorm (Lc (F))*maxNorm (Lc(F))*bound; |
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209 | bound *= bound*bound; |
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210 | bound= power (bound, degree (M)); |
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211 | bound *= power (CanonicalForm (2),degree (f)); |
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212 | CanonicalForm bufBound= bound; |
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213 | int i = cf_getNumBigPrimes() - 1; |
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214 | int p; |
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215 | CFList resultModP, result, newResult; |
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216 | CanonicalForm q (0), newQ; |
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217 | bool fail= false; |
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218 | Variable a= M.mvar(); |
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219 | Variable b= Variable (2); |
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220 | setReduce (M.mvar(), false); |
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221 | CanonicalForm mipo= bCommonDen (M)*M; |
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222 | Off (SW_RATIONAL); |
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223 | CanonicalForm modMipo; |
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224 | leadingCoeffs.append (lc (mipo)); |
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225 | CFList tmp1, tmp2; |
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226 | bool equal= false; |
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227 | int count= 0; |
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228 | do |
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229 | { |
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230 | p = cf_getBigPrime( i ); |
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231 | i--; |
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232 | while ( i >= 0 && mod( leadingCoeffs, p ) == 0) |
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233 | { |
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234 | p = cf_getBigPrime( i ); |
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235 | i--; |
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236 | } |
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237 | |
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238 | ASSERT (i >= 0, "ran out of primes"); //sic |
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239 | |
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240 | setCharacteristic (p); |
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241 | modMipo= mapinto (mipo); |
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242 | modMipo /= lc (modMipo); |
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243 | resultModP= CFList(); |
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244 | tryDiophantine (resultModP, mapinto (F), mapinto (products), modMipo, fail); |
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245 | setCharacteristic (0); |
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246 | if (fail) |
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247 | { |
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248 | fail= false; |
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249 | continue; |
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250 | } |
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251 | |
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252 | if ( q.isZero() ) |
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253 | { |
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254 | result= replacevar (mapinto(resultModP), a, b); |
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255 | q= p; |
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256 | } |
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257 | else |
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258 | { |
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259 | result= replacevar (result, a, b); |
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260 | newResult= CFList(); |
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261 | chineseRemainder( result, q, replacevar (mapinto (resultModP), a, b), |
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262 | p, newResult, newQ ); |
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263 | q= newQ; |
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264 | result= newResult; |
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265 | if (newQ > bound) |
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266 | { |
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267 | count++; |
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268 | tmp1= replacevar (Farey (result, q), b, a); |
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269 | if (tmp2.isEmpty()) |
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270 | tmp2= tmp1; |
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271 | else |
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272 | { |
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273 | equal= true; |
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274 | CFListIterator k= tmp1; |
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275 | for (CFListIterator j= tmp2; j.hasItem(); j++, k++) |
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276 | { |
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277 | if (j.getItem() != k.getItem()) |
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278 | equal= false; |
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279 | } |
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280 | if (!equal) |
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281 | tmp2= tmp1; |
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282 | } |
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283 | if (count > 2) |
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284 | { |
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285 | bound *= bufBound; |
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286 | equal= false; |
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287 | count= 0; |
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288 | } |
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289 | } |
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290 | if (newQ > bound && equal) |
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291 | { |
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292 | On( SW_RATIONAL ); |
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293 | CFList bufResult= result; |
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294 | result= tmp2; |
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295 | setReduce (M.mvar(), true); |
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296 | if (factors.getFirst().level() == 1) |
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297 | { |
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298 | result.removeFirst(); |
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299 | CFListIterator j= factors; |
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300 | CanonicalForm denf= bCommonDen (f); |
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301 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
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302 | i.getItem() *= Lc (j.getItem())*denf; |
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303 | } |
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304 | if (factors.getFirst().level() != 1 && |
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305 | !bCommonDen (factors.getFirst()).isOne()) |
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306 | { |
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307 | CanonicalForm denFirst= bCommonDen (factors.getFirst()); |
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308 | for (CFListIterator i= result; i.hasItem(); i++) |
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309 | i.getItem() *= denFirst; |
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310 | } |
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311 | |
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312 | CanonicalForm test= 0; |
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313 | CFListIterator jj= factors; |
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314 | for (CFListIterator ii= result; ii.hasItem(); ii++, jj++) |
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315 | test += ii.getItem()*(f/jj.getItem()); |
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316 | if (!test.isOne()) |
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317 | { |
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318 | bound *= bufBound; |
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319 | equal= false; |
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320 | count= 0; |
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321 | setReduce (M.mvar(), false); |
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322 | result= bufResult; |
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323 | Off (SW_RATIONAL); |
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324 | } |
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325 | else |
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326 | break; |
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327 | } |
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328 | } |
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329 | } while (1); |
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330 | if (save_rat) Off(SW_RATIONAL); |
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331 | return result; |
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332 | } |
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333 | |
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334 | CanonicalForm |
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335 | mulNTL (const CanonicalForm& F, const CanonicalForm& G) |
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336 | { |
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337 | if (F.inCoeffDomain() || G.inCoeffDomain() || getCharacteristic() == 0) |
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338 | return F*G; |
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339 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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340 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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341 | if (CFFactory::gettype() == GaloisFieldDomain) |
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342 | return F*G; |
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343 | zz_p::init (getCharacteristic()); |
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344 | Variable alpha; |
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345 | CanonicalForm result; |
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346 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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347 | { |
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348 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
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349 | zz_pE::init (NTLMipo); |
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350 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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351 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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352 | mul (NTLF, NTLF, NTLG); |
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353 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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354 | } |
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355 | else |
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356 | { |
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357 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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358 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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359 | mul (NTLF, NTLF, NTLG); |
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360 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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361 | } |
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362 | return result; |
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363 | } |
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364 | |
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365 | CanonicalForm |
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366 | modNTL (const CanonicalForm& F, const CanonicalForm& G) |
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367 | { |
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368 | if (F.inCoeffDomain() && G.isUnivariate()) |
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369 | return F; |
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370 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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371 | return mod (F, G); |
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372 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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373 | return mod (F,G); |
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374 | |
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375 | if (getCharacteristic() == 0) |
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376 | return mod (F, G); |
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377 | |
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378 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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379 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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380 | if (CFFactory::gettype() == GaloisFieldDomain) |
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381 | return mod (F, G); |
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382 | zz_p::init (getCharacteristic()); |
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383 | Variable alpha; |
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384 | CanonicalForm result; |
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385 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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386 | { |
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387 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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388 | zz_pE::init (NTLMipo); |
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389 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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390 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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391 | rem (NTLF, NTLF, NTLG); |
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392 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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393 | } |
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394 | else |
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395 | { |
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396 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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397 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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398 | rem (NTLF, NTLF, NTLG); |
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399 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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400 | } |
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401 | return result; |
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402 | } |
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403 | |
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404 | CanonicalForm |
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405 | divNTL (const CanonicalForm& F, const CanonicalForm& G) |
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406 | { |
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407 | if (F.inCoeffDomain() && G.isUnivariate()) |
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408 | return F; |
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409 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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410 | return div (F, G); |
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411 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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412 | return div (F,G); |
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413 | |
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414 | if (getCharacteristic() == 0) |
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415 | return div (F, G); |
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416 | |
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417 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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418 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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419 | if (CFFactory::gettype() == GaloisFieldDomain) |
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420 | return div (F, G); |
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421 | zz_p::init (getCharacteristic()); |
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422 | Variable alpha; |
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423 | CanonicalForm result; |
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424 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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425 | { |
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426 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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427 | zz_pE::init (NTLMipo); |
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428 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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429 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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430 | div (NTLF, NTLF, NTLG); |
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431 | result= convertNTLzz_pEX2CF(NTLF, F.mvar(), alpha); |
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432 | } |
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433 | else |
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434 | { |
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435 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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436 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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437 | div (NTLF, NTLF, NTLG); |
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438 | result= convertNTLzzpX2CF(NTLF, F.mvar()); |
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439 | } |
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440 | return result; |
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441 | } |
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442 | |
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443 | /* |
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444 | void |
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445 | divremNTL (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
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446 | CanonicalForm& R) |
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447 | { |
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448 | if (F.inCoeffDomain() && G.isUnivariate()) |
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449 | { |
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450 | R= F; |
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451 | Q= 0; |
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452 | } |
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453 | else if (F.inCoeffDomain() && G.inCoeffDomain()) |
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454 | { |
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455 | divrem (F, G, Q, R); |
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456 | return; |
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457 | } |
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458 | else if (F.isUnivariate() && G.inCoeffDomain()) |
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459 | { |
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460 | divrem (F, G, Q, R); |
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461 | return; |
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462 | } |
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463 | |
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464 | ASSERT (F.isUnivariate() && G.isUnivariate(), "expected univariate polys"); |
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465 | ASSERT (F.level() == G.level(), "expected polys of same level"); |
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466 | if (CFFactory::gettype() == GaloisFieldDomain) |
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467 | { |
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468 | divrem (F, G, Q, R); |
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469 | return; |
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470 | } |
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471 | zz_p::init (getCharacteristic()); |
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472 | Variable alpha; |
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473 | CanonicalForm result; |
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474 | if (hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha)) |
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475 | { |
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476 | zz_pX NTLMipo= convertFacCF2NTLzzpX(getMipo (alpha)); |
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477 | zz_pE::init (NTLMipo); |
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478 | zz_pEX NTLF= convertFacCF2NTLzz_pEX (F, NTLMipo); |
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479 | zz_pEX NTLG= convertFacCF2NTLzz_pEX (G, NTLMipo); |
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480 | zz_pEX NTLQ; |
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481 | zz_pEX NTLR; |
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482 | DivRem (NTLQ, NTLR, NTLF, NTLG); |
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483 | Q= convertNTLzz_pEX2CF(NTLQ, F.mvar(), alpha); |
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484 | R= convertNTLzz_pEX2CF(NTLR, F.mvar(), alpha); |
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485 | return; |
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486 | } |
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487 | else |
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488 | { |
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489 | zz_pX NTLF= convertFacCF2NTLzzpX (F); |
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490 | zz_pX NTLG= convertFacCF2NTLzzpX (G); |
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491 | zz_pX NTLQ; |
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492 | zz_pX NTLR; |
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493 | DivRem (NTLQ, NTLR, NTLF, NTLG); |
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494 | Q= convertNTLzzpX2CF(NTLQ, F.mvar()); |
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495 | R= convertNTLzzpX2CF(NTLR, F.mvar()); |
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496 | return; |
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497 | } |
---|
498 | }*/ |
---|
499 | |
---|
500 | CanonicalForm mod (const CanonicalForm& F, const CFList& M) |
---|
501 | { |
---|
502 | CanonicalForm A= F; |
---|
503 | for (CFListIterator i= M; i.hasItem(); i++) |
---|
504 | A= mod (A, i.getItem()); |
---|
505 | return A; |
---|
506 | } |
---|
507 | |
---|
508 | zz_pX kronSubFp (const CanonicalForm& A, int d) |
---|
509 | { |
---|
510 | int degAy= degree (A); |
---|
511 | zz_pX result; |
---|
512 | result.rep.SetLength (d*(degAy + 1)); |
---|
513 | |
---|
514 | zz_p *resultp; |
---|
515 | resultp= result.rep.elts(); |
---|
516 | zz_pX buf; |
---|
517 | zz_p *bufp; |
---|
518 | |
---|
519 | for (CFIterator i= A; i.hasTerms(); i++) |
---|
520 | { |
---|
521 | if (i.coeff().inCoeffDomain()) |
---|
522 | buf= convertFacCF2NTLzzpX (i.coeff()); |
---|
523 | else |
---|
524 | buf= convertFacCF2NTLzzpX (i.coeff()); |
---|
525 | |
---|
526 | int k= i.exp()*d; |
---|
527 | bufp= buf.rep.elts(); |
---|
528 | int bufRepLength= (int) buf.rep.length(); |
---|
529 | for (int j= 0; j < bufRepLength; j++) |
---|
530 | resultp [j + k]= bufp [j]; |
---|
531 | } |
---|
532 | result.normalize(); |
---|
533 | |
---|
534 | return result; |
---|
535 | } |
---|
536 | |
---|
537 | zz_pEX kronSub (const CanonicalForm& A, int d, const Variable& alpha) |
---|
538 | { |
---|
539 | int degAy= degree (A); |
---|
540 | zz_pEX result; |
---|
541 | result.rep.SetLength (d*(degAy + 1)); |
---|
542 | |
---|
543 | Variable v; |
---|
544 | zz_pE *resultp; |
---|
545 | resultp= result.rep.elts(); |
---|
546 | zz_pEX buf1; |
---|
547 | zz_pE *buf1p; |
---|
548 | zz_pX buf2; |
---|
549 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
---|
550 | |
---|
551 | for (CFIterator i= A; i.hasTerms(); i++) |
---|
552 | { |
---|
553 | if (i.coeff().inCoeffDomain()) |
---|
554 | { |
---|
555 | buf2= convertFacCF2NTLzzpX (i.coeff()); |
---|
556 | buf1= to_zz_pEX (to_zz_pE (buf2)); |
---|
557 | } |
---|
558 | else |
---|
559 | buf1= convertFacCF2NTLzz_pEX (i.coeff(), NTLMipo); |
---|
560 | |
---|
561 | int k= i.exp()*d; |
---|
562 | buf1p= buf1.rep.elts(); |
---|
563 | int buf1RepLength= (int) buf1.rep.length(); |
---|
564 | for (int j= 0; j < buf1RepLength; j++) |
---|
565 | resultp [j + k]= buf1p [j]; |
---|
566 | } |
---|
567 | result.normalize(); |
---|
568 | |
---|
569 | return result; |
---|
570 | } |
---|
571 | |
---|
572 | void |
---|
573 | kronSubRecipro (zz_pEX& subA1, zz_pEX& subA2,const CanonicalForm& A, int d, |
---|
574 | const Variable& alpha) |
---|
575 | { |
---|
576 | int degAy= degree (A); |
---|
577 | subA1.rep.SetLength ((long) d*(degAy + 2)); |
---|
578 | subA2.rep.SetLength ((long) d*(degAy + 2)); |
---|
579 | |
---|
580 | Variable v; |
---|
581 | zz_pE *subA1p; |
---|
582 | zz_pE *subA2p; |
---|
583 | subA1p= subA1.rep.elts(); |
---|
584 | subA2p= subA2.rep.elts(); |
---|
585 | zz_pEX buf; |
---|
586 | zz_pE *bufp; |
---|
587 | zz_pX buf2; |
---|
588 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
---|
589 | |
---|
590 | for (CFIterator i= A; i.hasTerms(); i++) |
---|
591 | { |
---|
592 | if (i.coeff().inCoeffDomain()) |
---|
593 | { |
---|
594 | buf2= convertFacCF2NTLzzpX (i.coeff()); |
---|
595 | buf= to_zz_pEX (to_zz_pE (buf2)); |
---|
596 | } |
---|
597 | else |
---|
598 | buf= convertFacCF2NTLzz_pEX (i.coeff(), NTLMipo); |
---|
599 | |
---|
600 | int k= i.exp()*d; |
---|
601 | int kk= (degAy - i.exp())*d; |
---|
602 | bufp= buf.rep.elts(); |
---|
603 | int bufRepLength= (int) buf.rep.length(); |
---|
604 | for (int j= 0; j < bufRepLength; j++) |
---|
605 | { |
---|
606 | subA1p [j + k] += bufp [j]; |
---|
607 | subA2p [j + kk] += bufp [j]; |
---|
608 | } |
---|
609 | } |
---|
610 | subA1.normalize(); |
---|
611 | subA2.normalize(); |
---|
612 | } |
---|
613 | |
---|
614 | void |
---|
615 | kronSubRecipro (zz_pX& subA1, zz_pX& subA2,const CanonicalForm& A, int d) |
---|
616 | { |
---|
617 | int degAy= degree (A); |
---|
618 | subA1.rep.SetLength ((long) d*(degAy + 2)); |
---|
619 | subA2.rep.SetLength ((long) d*(degAy + 2)); |
---|
620 | |
---|
621 | zz_p *subA1p; |
---|
622 | zz_p *subA2p; |
---|
623 | subA1p= subA1.rep.elts(); |
---|
624 | subA2p= subA2.rep.elts(); |
---|
625 | zz_pX buf; |
---|
626 | zz_p *bufp; |
---|
627 | |
---|
628 | for (CFIterator i= A; i.hasTerms(); i++) |
---|
629 | { |
---|
630 | buf= convertFacCF2NTLzzpX (i.coeff()); |
---|
631 | |
---|
632 | int k= i.exp()*d; |
---|
633 | int kk= (degAy - i.exp())*d; |
---|
634 | bufp= buf.rep.elts(); |
---|
635 | int bufRepLength= (int) buf.rep.length(); |
---|
636 | for (int j= 0; j < bufRepLength; j++) |
---|
637 | { |
---|
638 | subA1p [j + k] += bufp [j]; |
---|
639 | subA2p [j + kk] += bufp [j]; |
---|
640 | } |
---|
641 | } |
---|
642 | subA1.normalize(); |
---|
643 | subA2.normalize(); |
---|
644 | } |
---|
645 | |
---|
646 | CanonicalForm |
---|
647 | reverseSubst (const zz_pEX& F, const zz_pEX& G, int d, int k, |
---|
648 | const Variable& alpha) |
---|
649 | { |
---|
650 | Variable y= Variable (2); |
---|
651 | Variable x= Variable (1); |
---|
652 | |
---|
653 | zz_pEX f= F; |
---|
654 | zz_pEX g= G; |
---|
655 | int degf= deg(f); |
---|
656 | int degg= deg(g); |
---|
657 | |
---|
658 | zz_pEX buf1; |
---|
659 | zz_pEX buf2; |
---|
660 | zz_pEX buf3; |
---|
661 | |
---|
662 | zz_pE *buf1p; |
---|
663 | zz_pE *buf2p; |
---|
664 | zz_pE *buf3p; |
---|
665 | if (f.rep.length() < (long) d*(k+1)) //zero padding |
---|
666 | f.rep.SetLength ((long)d*(k+1)); |
---|
667 | |
---|
668 | zz_pE *gp= g.rep.elts(); |
---|
669 | zz_pE *fp= f.rep.elts(); |
---|
670 | CanonicalForm result= 0; |
---|
671 | int i= 0; |
---|
672 | int lf= 0; |
---|
673 | int lg= d*k; |
---|
674 | int degfSubLf= degf; |
---|
675 | int deggSubLg= degg-lg; |
---|
676 | int repLengthBuf2; |
---|
677 | int repLengthBuf1; |
---|
678 | zz_pE zzpEZero= zz_pE(); |
---|
679 | |
---|
680 | while (degf >= lf || lg >= 0) |
---|
681 | { |
---|
682 | if (degfSubLf >= d) |
---|
683 | repLengthBuf1= d; |
---|
684 | else if (degfSubLf < 0) |
---|
685 | repLengthBuf1= 0; |
---|
686 | else |
---|
687 | repLengthBuf1= degfSubLf + 1; |
---|
688 | buf1.rep.SetLength((long) repLengthBuf1); |
---|
689 | |
---|
690 | buf1p= buf1.rep.elts(); |
---|
691 | for (int ind= 0; ind < repLengthBuf1; ind++) |
---|
692 | buf1p [ind]= fp [ind + lf]; |
---|
693 | buf1.normalize(); |
---|
694 | |
---|
695 | repLengthBuf1= buf1.rep.length(); |
---|
696 | |
---|
697 | if (deggSubLg >= d - 1) |
---|
698 | repLengthBuf2= d - 1; |
---|
699 | else if (deggSubLg < 0) |
---|
700 | repLengthBuf2= 0; |
---|
701 | else |
---|
702 | repLengthBuf2= deggSubLg + 1; |
---|
703 | |
---|
704 | buf2.rep.SetLength ((long) repLengthBuf2); |
---|
705 | buf2p= buf2.rep.elts(); |
---|
706 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
707 | { |
---|
708 | buf2p [ind]= gp [ind + lg]; |
---|
709 | } |
---|
710 | buf2.normalize(); |
---|
711 | |
---|
712 | repLengthBuf2= buf2.rep.length(); |
---|
713 | |
---|
714 | buf3.rep.SetLength((long) repLengthBuf2 + d); |
---|
715 | buf3p= buf3.rep.elts(); |
---|
716 | buf2p= buf2.rep.elts(); |
---|
717 | buf1p= buf1.rep.elts(); |
---|
718 | for (int ind= 0; ind < repLengthBuf1; ind++) |
---|
719 | buf3p [ind]= buf1p [ind]; |
---|
720 | for (int ind= repLengthBuf1; ind < d; ind++) |
---|
721 | buf3p [ind]= zzpEZero; |
---|
722 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
723 | buf3p [ind + d]= buf2p [ind]; |
---|
724 | buf3.normalize(); |
---|
725 | |
---|
726 | result += convertNTLzz_pEX2CF (buf3, x, alpha)*power (y, i); |
---|
727 | i++; |
---|
728 | |
---|
729 | |
---|
730 | lf= i*d; |
---|
731 | degfSubLf= degf - lf; |
---|
732 | |
---|
733 | lg= d*(k-i); |
---|
734 | deggSubLg= degg - lg; |
---|
735 | |
---|
736 | buf1p= buf1.rep.elts(); |
---|
737 | |
---|
738 | if (lg >= 0 && deggSubLg > 0) |
---|
739 | { |
---|
740 | if (repLengthBuf2 > degfSubLf + 1) |
---|
741 | degfSubLf= repLengthBuf2 - 1; |
---|
742 | int tmp= tmin (repLengthBuf1, deggSubLg + 1); |
---|
743 | for (int ind= 0; ind < tmp; ind++) |
---|
744 | gp [ind + lg] -= buf1p [ind]; |
---|
745 | } |
---|
746 | |
---|
747 | if (lg < 0) |
---|
748 | break; |
---|
749 | |
---|
750 | buf2p= buf2.rep.elts(); |
---|
751 | if (degfSubLf >= 0) |
---|
752 | { |
---|
753 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
754 | fp [ind + lf] -= buf2p [ind]; |
---|
755 | } |
---|
756 | } |
---|
757 | |
---|
758 | return result; |
---|
759 | } |
---|
760 | |
---|
761 | CanonicalForm |
---|
762 | reverseSubst (const zz_pX& F, const zz_pX& G, int d, int k) |
---|
763 | { |
---|
764 | Variable y= Variable (2); |
---|
765 | Variable x= Variable (1); |
---|
766 | |
---|
767 | zz_pX f= F; |
---|
768 | zz_pX g= G; |
---|
769 | int degf= deg(f); |
---|
770 | int degg= deg(g); |
---|
771 | |
---|
772 | zz_pX buf1; |
---|
773 | zz_pX buf2; |
---|
774 | zz_pX buf3; |
---|
775 | |
---|
776 | zz_p *buf1p; |
---|
777 | zz_p *buf2p; |
---|
778 | zz_p *buf3p; |
---|
779 | |
---|
780 | if (f.rep.length() < (long) d*(k+1)) //zero padding |
---|
781 | f.rep.SetLength ((long)d*(k+1)); |
---|
782 | |
---|
783 | zz_p *gp= g.rep.elts(); |
---|
784 | zz_p *fp= f.rep.elts(); |
---|
785 | CanonicalForm result= 0; |
---|
786 | int i= 0; |
---|
787 | int lf= 0; |
---|
788 | int lg= d*k; |
---|
789 | int degfSubLf= degf; |
---|
790 | int deggSubLg= degg-lg; |
---|
791 | int repLengthBuf2; |
---|
792 | int repLengthBuf1; |
---|
793 | zz_p zzpZero= zz_p(); |
---|
794 | while (degf >= lf || lg >= 0) |
---|
795 | { |
---|
796 | if (degfSubLf >= d) |
---|
797 | repLengthBuf1= d; |
---|
798 | else if (degfSubLf < 0) |
---|
799 | repLengthBuf1= 0; |
---|
800 | else |
---|
801 | repLengthBuf1= degfSubLf + 1; |
---|
802 | buf1.rep.SetLength((long) repLengthBuf1); |
---|
803 | |
---|
804 | buf1p= buf1.rep.elts(); |
---|
805 | for (int ind= 0; ind < repLengthBuf1; ind++) |
---|
806 | buf1p [ind]= fp [ind + lf]; |
---|
807 | buf1.normalize(); |
---|
808 | |
---|
809 | repLengthBuf1= buf1.rep.length(); |
---|
810 | |
---|
811 | if (deggSubLg >= d - 1) |
---|
812 | repLengthBuf2= d - 1; |
---|
813 | else if (deggSubLg < 0) |
---|
814 | repLengthBuf2= 0; |
---|
815 | else |
---|
816 | repLengthBuf2= deggSubLg + 1; |
---|
817 | |
---|
818 | buf2.rep.SetLength ((long) repLengthBuf2); |
---|
819 | buf2p= buf2.rep.elts(); |
---|
820 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
821 | buf2p [ind]= gp [ind + lg]; |
---|
822 | |
---|
823 | buf2.normalize(); |
---|
824 | |
---|
825 | repLengthBuf2= buf2.rep.length(); |
---|
826 | |
---|
827 | |
---|
828 | buf3.rep.SetLength((long) repLengthBuf2 + d); |
---|
829 | buf3p= buf3.rep.elts(); |
---|
830 | buf2p= buf2.rep.elts(); |
---|
831 | buf1p= buf1.rep.elts(); |
---|
832 | for (int ind= 0; ind < repLengthBuf1; ind++) |
---|
833 | buf3p [ind]= buf1p [ind]; |
---|
834 | for (int ind= repLengthBuf1; ind < d; ind++) |
---|
835 | buf3p [ind]= zzpZero; |
---|
836 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
837 | buf3p [ind + d]= buf2p [ind]; |
---|
838 | buf3.normalize(); |
---|
839 | |
---|
840 | result += convertNTLzzpX2CF (buf3, x)*power (y, i); |
---|
841 | i++; |
---|
842 | |
---|
843 | |
---|
844 | lf= i*d; |
---|
845 | degfSubLf= degf - lf; |
---|
846 | |
---|
847 | lg= d*(k-i); |
---|
848 | deggSubLg= degg - lg; |
---|
849 | |
---|
850 | buf1p= buf1.rep.elts(); |
---|
851 | |
---|
852 | if (lg >= 0 && deggSubLg > 0) |
---|
853 | { |
---|
854 | if (repLengthBuf2 > degfSubLf + 1) |
---|
855 | degfSubLf= repLengthBuf2 - 1; |
---|
856 | int tmp= tmin (repLengthBuf1, deggSubLg + 1); |
---|
857 | for (int ind= 0; ind < tmp; ind++) |
---|
858 | gp [ind + lg] -= buf1p [ind]; |
---|
859 | } |
---|
860 | if (lg < 0) |
---|
861 | break; |
---|
862 | |
---|
863 | buf2p= buf2.rep.elts(); |
---|
864 | if (degfSubLf >= 0) |
---|
865 | { |
---|
866 | for (int ind= 0; ind < repLengthBuf2; ind++) |
---|
867 | fp [ind + lf] -= buf2p [ind]; |
---|
868 | } |
---|
869 | } |
---|
870 | |
---|
871 | return result; |
---|
872 | } |
---|
873 | |
---|
874 | CanonicalForm reverseSubst (const zz_pEX& F, int d, const Variable& alpha) |
---|
875 | { |
---|
876 | Variable y= Variable (2); |
---|
877 | Variable x= Variable (1); |
---|
878 | |
---|
879 | zz_pEX f= F; |
---|
880 | zz_pE *fp= f.rep.elts(); |
---|
881 | |
---|
882 | zz_pEX buf; |
---|
883 | zz_pE *bufp; |
---|
884 | CanonicalForm result= 0; |
---|
885 | int i= 0; |
---|
886 | int degf= deg(f); |
---|
887 | int k= 0; |
---|
888 | int degfSubK; |
---|
889 | int repLength; |
---|
890 | while (degf >= k) |
---|
891 | { |
---|
892 | degfSubK= degf - k; |
---|
893 | if (degfSubK >= d) |
---|
894 | repLength= d; |
---|
895 | else |
---|
896 | repLength= degfSubK + 1; |
---|
897 | |
---|
898 | buf.rep.SetLength ((long) repLength); |
---|
899 | bufp= buf.rep.elts(); |
---|
900 | for (int j= 0; j < repLength; j++) |
---|
901 | bufp [j]= fp [j + k]; |
---|
902 | buf.normalize(); |
---|
903 | |
---|
904 | result += convertNTLzz_pEX2CF (buf, x, alpha)*power (y, i); |
---|
905 | i++; |
---|
906 | k= d*i; |
---|
907 | } |
---|
908 | |
---|
909 | return result; |
---|
910 | } |
---|
911 | |
---|
912 | CanonicalForm reverseSubstFp (const zz_pX& F, int d) |
---|
913 | { |
---|
914 | Variable y= Variable (2); |
---|
915 | Variable x= Variable (1); |
---|
916 | |
---|
917 | zz_pX f= F; |
---|
918 | zz_p *fp= f.rep.elts(); |
---|
919 | |
---|
920 | zz_pX buf; |
---|
921 | zz_p *bufp; |
---|
922 | CanonicalForm result= 0; |
---|
923 | int i= 0; |
---|
924 | int degf= deg(f); |
---|
925 | int k= 0; |
---|
926 | int degfSubK; |
---|
927 | int repLength; |
---|
928 | while (degf >= k) |
---|
929 | { |
---|
930 | degfSubK= degf - k; |
---|
931 | if (degfSubK >= d) |
---|
932 | repLength= d; |
---|
933 | else |
---|
934 | repLength= degfSubK + 1; |
---|
935 | |
---|
936 | buf.rep.SetLength ((long) repLength); |
---|
937 | bufp= buf.rep.elts(); |
---|
938 | for (int j= 0; j < repLength; j++) |
---|
939 | bufp [j]= fp [j + k]; |
---|
940 | buf.normalize(); |
---|
941 | |
---|
942 | result += convertNTLzzpX2CF (buf, x)*power (y, i); |
---|
943 | i++; |
---|
944 | k= d*i; |
---|
945 | } |
---|
946 | |
---|
947 | return result; |
---|
948 | } |
---|
949 | |
---|
950 | // assumes input to be reduced mod M and to be an element of Fq not Fp |
---|
951 | CanonicalForm |
---|
952 | mulMod2NTLFpReci (const CanonicalForm& F, const CanonicalForm& G, const |
---|
953 | CanonicalForm& M) |
---|
954 | { |
---|
955 | int d1= degree (F, 1) + degree (G, 1) + 1; |
---|
956 | d1 /= 2; |
---|
957 | d1 += 1; |
---|
958 | |
---|
959 | zz_pX F1, F2; |
---|
960 | kronSubRecipro (F1, F2, F, d1); |
---|
961 | zz_pX G1, G2; |
---|
962 | kronSubRecipro (G1, G2, G, d1); |
---|
963 | |
---|
964 | int k= d1*degree (M); |
---|
965 | MulTrunc (F1, F1, G1, (long) k); |
---|
966 | |
---|
967 | int degtailF= degree (tailcoeff (F), 1); |
---|
968 | int degtailG= degree (tailcoeff (G), 1); |
---|
969 | int taildegF= taildegree (F); |
---|
970 | int taildegG= taildegree (G); |
---|
971 | int b= k + degtailF + degtailG - d1*(2+taildegF+taildegG); |
---|
972 | |
---|
973 | reverse (F2, F2); |
---|
974 | reverse (G2, G2); |
---|
975 | MulTrunc (F2, F2, G2, b + 1); |
---|
976 | reverse (F2, F2, b); |
---|
977 | |
---|
978 | int d2= tmax (deg (F2)/d1, deg (F1)/d1); |
---|
979 | return reverseSubst (F1, F2, d1, d2); |
---|
980 | } |
---|
981 | |
---|
982 | //Kronecker substitution |
---|
983 | CanonicalForm |
---|
984 | mulMod2NTLFp (const CanonicalForm& F, const CanonicalForm& G, const |
---|
985 | CanonicalForm& M) |
---|
986 | { |
---|
987 | CanonicalForm A= F; |
---|
988 | CanonicalForm B= G; |
---|
989 | |
---|
990 | int degAx= degree (A, 1); |
---|
991 | int degAy= degree (A, 2); |
---|
992 | int degBx= degree (B, 1); |
---|
993 | int degBy= degree (B, 2); |
---|
994 | int d1= degAx + 1 + degBx; |
---|
995 | int d2= tmax (degAy, degBy); |
---|
996 | |
---|
997 | if (d1 > 128 && d2 > 160 && (degAy == degBy) && (2*degAy > degree (M))) |
---|
998 | return mulMod2NTLFpReci (A, B, M); |
---|
999 | |
---|
1000 | zz_pX NTLA= kronSubFp (A, d1); |
---|
1001 | zz_pX NTLB= kronSubFp (B, d1); |
---|
1002 | |
---|
1003 | int k= d1*degree (M); |
---|
1004 | MulTrunc (NTLA, NTLA, NTLB, (long) k); |
---|
1005 | |
---|
1006 | A= reverseSubstFp (NTLA, d1); |
---|
1007 | |
---|
1008 | return A; |
---|
1009 | } |
---|
1010 | |
---|
1011 | // assumes input to be reduced mod M and to be an element of Fq not Fp |
---|
1012 | CanonicalForm |
---|
1013 | mulMod2NTLFqReci (const CanonicalForm& F, const CanonicalForm& G, const |
---|
1014 | CanonicalForm& M, const Variable& alpha) |
---|
1015 | { |
---|
1016 | int d1= degree (F, 1) + degree (G, 1) + 1; |
---|
1017 | d1 /= 2; |
---|
1018 | d1 += 1; |
---|
1019 | |
---|
1020 | zz_pEX F1, F2; |
---|
1021 | kronSubRecipro (F1, F2, F, d1, alpha); |
---|
1022 | zz_pEX G1, G2; |
---|
1023 | kronSubRecipro (G1, G2, G, d1, alpha); |
---|
1024 | |
---|
1025 | int k= d1*degree (M); |
---|
1026 | MulTrunc (F1, F1, G1, (long) k); |
---|
1027 | |
---|
1028 | int degtailF= degree (tailcoeff (F), 1); |
---|
1029 | int degtailG= degree (tailcoeff (G), 1); |
---|
1030 | int taildegF= taildegree (F); |
---|
1031 | int taildegG= taildegree (G); |
---|
1032 | int b= k + degtailF + degtailG - d1*(2+taildegF+taildegG); |
---|
1033 | |
---|
1034 | reverse (F2, F2); |
---|
1035 | reverse (G2, G2); |
---|
1036 | MulTrunc (F2, F2, G2, b + 1); |
---|
1037 | reverse (F2, F2, b); |
---|
1038 | |
---|
1039 | int d2= tmax (deg (F2)/d1, deg (F1)/d1); |
---|
1040 | return reverseSubst (F1, F2, d1, d2, alpha); |
---|
1041 | } |
---|
1042 | |
---|
1043 | CanonicalForm |
---|
1044 | mulMod2NTLFq (const CanonicalForm& F, const CanonicalForm& G, const |
---|
1045 | CanonicalForm& M) |
---|
1046 | { |
---|
1047 | Variable alpha; |
---|
1048 | CanonicalForm A= F; |
---|
1049 | CanonicalForm B= G; |
---|
1050 | |
---|
1051 | if (hasFirstAlgVar (A, alpha) || hasFirstAlgVar (B, alpha)) |
---|
1052 | { |
---|
1053 | int degAx= degree (A, 1); |
---|
1054 | int degAy= degree (A, 2); |
---|
1055 | int degBx= degree (B, 1); |
---|
1056 | int degBy= degree (B, 2); |
---|
1057 | int d1= degAx + degBx + 1; |
---|
1058 | int d2= tmax (degAy, degBy); |
---|
1059 | zz_p::init (getCharacteristic()); |
---|
1060 | zz_pX NTLMipo= convertFacCF2NTLzzpX (getMipo (alpha)); |
---|
1061 | zz_pE::init (NTLMipo); |
---|
1062 | |
---|
1063 | int degMipo= degree (getMipo (alpha)); |
---|
1064 | if ((d1 > 128/degMipo) && (d2 > 160/degMipo) && (degAy == degBy) && |
---|
1065 | (2*degAy > degree (M))) |
---|
1066 | return mulMod2NTLFqReci (A, B, M, alpha); |
---|
1067 | |
---|
1068 | zz_pEX NTLA= kronSub (A, d1, alpha); |
---|
1069 | zz_pEX NTLB= kronSub (B, d1, alpha); |
---|
1070 | |
---|
1071 | int k= d1*degree (M); |
---|
1072 | |
---|
1073 | MulTrunc (NTLA, NTLA, NTLB, (long) k); |
---|
1074 | |
---|
1075 | A= reverseSubst (NTLA, d1, alpha); |
---|
1076 | |
---|
1077 | return A; |
---|
1078 | } |
---|
1079 | else |
---|
1080 | return mulMod2NTLFp (A, B, M); |
---|
1081 | } |
---|
1082 | |
---|
1083 | CanonicalForm mulMod2 (const CanonicalForm& A, const CanonicalForm& B, |
---|
1084 | const CanonicalForm& M) |
---|
1085 | { |
---|
1086 | if (A.isZero() || B.isZero()) |
---|
1087 | return 0; |
---|
1088 | |
---|
1089 | ASSERT (M.isUnivariate(), "M must be univariate"); |
---|
1090 | |
---|
1091 | CanonicalForm F= mod (A, M); |
---|
1092 | CanonicalForm G= mod (B, M); |
---|
1093 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
---|
1094 | return F*G; |
---|
1095 | Variable y= M.mvar(); |
---|
1096 | int degF= degree (F, y); |
---|
1097 | int degG= degree (G, y); |
---|
1098 | |
---|
1099 | if ((degF < 1 && degG < 1) && (F.isUnivariate() && G.isUnivariate()) && |
---|
1100 | (F.level() == G.level())) |
---|
1101 | { |
---|
1102 | CanonicalForm result= mulNTL (F, G); |
---|
1103 | return mod (result, M); |
---|
1104 | } |
---|
1105 | else if (degF <= 1 && degG <= 1) |
---|
1106 | { |
---|
1107 | CanonicalForm result= F*G; |
---|
1108 | return mod (result, M); |
---|
1109 | } |
---|
1110 | |
---|
1111 | int sizeF= size (F); |
---|
1112 | int sizeG= size (G); |
---|
1113 | |
---|
1114 | int fallBackToNaive= 50; |
---|
1115 | if (sizeF < fallBackToNaive || sizeG < fallBackToNaive) |
---|
1116 | return mod (F*G, M); |
---|
1117 | |
---|
1118 | if (getCharacteristic() > 0 && CFFactory::gettype() != GaloisFieldDomain && |
---|
1119 | (((degF-degG) < 50 && degF > degG) || ((degG-degF) < 50 && degF <= degG))) |
---|
1120 | return mulMod2NTLFq (F, G, M); |
---|
1121 | |
---|
1122 | int m= (int) ceil (degree (M)/2.0); |
---|
1123 | if (degF >= m || degG >= m) |
---|
1124 | { |
---|
1125 | CanonicalForm MLo= power (y, m); |
---|
1126 | CanonicalForm MHi= power (y, degree (M) - m); |
---|
1127 | CanonicalForm F0= mod (F, MLo); |
---|
1128 | CanonicalForm F1= div (F, MLo); |
---|
1129 | CanonicalForm G0= mod (G, MLo); |
---|
1130 | CanonicalForm G1= div (G, MLo); |
---|
1131 | CanonicalForm F0G1= mulMod2 (F0, G1, MHi); |
---|
1132 | CanonicalForm F1G0= mulMod2 (F1, G0, MHi); |
---|
1133 | CanonicalForm F0G0= mulMod2 (F0, G0, M); |
---|
1134 | return F0G0 + MLo*(F0G1 + F1G0); |
---|
1135 | } |
---|
1136 | else |
---|
1137 | { |
---|
1138 | m= (int) ceil (tmax (degF, degG)/2.0); |
---|
1139 | CanonicalForm yToM= power (y, m); |
---|
1140 | CanonicalForm F0= mod (F, yToM); |
---|
1141 | CanonicalForm F1= div (F, yToM); |
---|
1142 | CanonicalForm G0= mod (G, yToM); |
---|
1143 | CanonicalForm G1= div (G, yToM); |
---|
1144 | CanonicalForm H00= mulMod2 (F0, G0, M); |
---|
1145 | CanonicalForm H11= mulMod2 (F1, G1, M); |
---|
1146 | CanonicalForm H01= mulMod2 (F0 + F1, G0 + G1, M); |
---|
1147 | return H11*yToM*yToM + (H01 - H11 - H00)*yToM + H00; |
---|
1148 | } |
---|
1149 | DEBOUTLN (cerr, "fatal end in mulMod2"); |
---|
1150 | } |
---|
1151 | |
---|
1152 | CanonicalForm mulMod (const CanonicalForm& A, const CanonicalForm& B, |
---|
1153 | const CFList& MOD) |
---|
1154 | { |
---|
1155 | if (A.isZero() || B.isZero()) |
---|
1156 | return 0; |
---|
1157 | |
---|
1158 | if (MOD.length() == 1) |
---|
1159 | return mulMod2 (A, B, MOD.getLast()); |
---|
1160 | |
---|
1161 | CanonicalForm M= MOD.getLast(); |
---|
1162 | CanonicalForm F= mod (A, M); |
---|
1163 | CanonicalForm G= mod (B, M); |
---|
1164 | if (F.inCoeffDomain() || G.inCoeffDomain()) |
---|
1165 | return F*G; |
---|
1166 | Variable y= M.mvar(); |
---|
1167 | int degF= degree (F, y); |
---|
1168 | int degG= degree (G, y); |
---|
1169 | |
---|
1170 | if ((degF <= 1 && F.level() <= M.level()) && |
---|
1171 | (degG <= 1 && G.level() <= M.level())) |
---|
1172 | { |
---|
1173 | CFList buf= MOD; |
---|
1174 | buf.removeLast(); |
---|
1175 | if (degF == 1 && degG == 1) |
---|
1176 | { |
---|
1177 | CanonicalForm F0= mod (F, y); |
---|
1178 | CanonicalForm F1= div (F, y); |
---|
1179 | CanonicalForm G0= mod (G, y); |
---|
1180 | CanonicalForm G1= div (G, y); |
---|
1181 | if (degree (M) > 2) |
---|
1182 | { |
---|
1183 | CanonicalForm H00= mulMod (F0, G0, buf); |
---|
1184 | CanonicalForm H11= mulMod (F1, G1, buf); |
---|
1185 | CanonicalForm H01= mulMod (F0 + F1, G0 + G1, buf); |
---|
1186 | return H11*y*y + (H01 - H00 - H11)*y + H00; |
---|
1187 | } |
---|
1188 | else //here degree (M) == 2 |
---|
1189 | { |
---|
1190 | buf.append (y); |
---|
1191 | CanonicalForm F0G1= mulMod (F0, G1, buf); |
---|
1192 | CanonicalForm F1G0= mulMod (F1, G0, buf); |
---|
1193 | CanonicalForm F0G0= mulMod (F0, G0, MOD); |
---|
1194 | CanonicalForm result= F0G0 + y*(F0G1 + F1G0); |
---|
1195 | return result; |
---|
1196 | } |
---|
1197 | } |
---|
1198 | else if (degF == 1 && degG == 0) |
---|
1199 | return mulMod (div (F, y), G, buf)*y + mulMod (mod (F, y), G, buf); |
---|
1200 | else if (degF == 0 && degG == 1) |
---|
1201 | return mulMod (div (G, y), F, buf)*y + mulMod (mod (G, y), F, buf); |
---|
1202 | else |
---|
1203 | return mulMod (F, G, buf); |
---|
1204 | } |
---|
1205 | int m= (int) ceil (degree (M)/2.0); |
---|
1206 | if (degF >= m || degG >= m) |
---|
1207 | { |
---|
1208 | CanonicalForm MLo= power (y, m); |
---|
1209 | CanonicalForm MHi= power (y, degree (M) - m); |
---|
1210 | CanonicalForm F0= mod (F, MLo); |
---|
1211 | CanonicalForm F1= div (F, MLo); |
---|
1212 | CanonicalForm G0= mod (G, MLo); |
---|
1213 | CanonicalForm G1= div (G, MLo); |
---|
1214 | CFList buf= MOD; |
---|
1215 | buf.removeLast(); |
---|
1216 | buf.append (MHi); |
---|
1217 | CanonicalForm F0G1= mulMod (F0, G1, buf); |
---|
1218 | CanonicalForm F1G0= mulMod (F1, G0, buf); |
---|
1219 | CanonicalForm F0G0= mulMod (F0, G0, MOD); |
---|
1220 | return F0G0 + MLo*(F0G1 + F1G0); |
---|
1221 | } |
---|
1222 | else |
---|
1223 | { |
---|
1224 | m= (int) ceil (tmax (degF, degG)/2.0); |
---|
1225 | CanonicalForm yToM= power (y, m); |
---|
1226 | CanonicalForm F0= mod (F, yToM); |
---|
1227 | CanonicalForm F1= div (F, yToM); |
---|
1228 | CanonicalForm G0= mod (G, yToM); |
---|
1229 | CanonicalForm G1= div (G, yToM); |
---|
1230 | CanonicalForm H00= mulMod (F0, G0, MOD); |
---|
1231 | CanonicalForm H11= mulMod (F1, G1, MOD); |
---|
1232 | CanonicalForm H01= mulMod (F0 + F1, G0 + G1, MOD); |
---|
1233 | return H11*yToM*yToM + (H01 - H11 - H00)*yToM + H00; |
---|
1234 | } |
---|
1235 | DEBOUTLN (cerr, "fatal end in mulMod"); |
---|
1236 | } |
---|
1237 | |
---|
1238 | CanonicalForm prodMod (const CFList& L, const CanonicalForm& M) |
---|
1239 | { |
---|
1240 | if (L.isEmpty()) |
---|
1241 | return 1; |
---|
1242 | int l= L.length(); |
---|
1243 | if (l == 1) |
---|
1244 | return mod (L.getFirst(), M); |
---|
1245 | else if (l == 2) { |
---|
1246 | CanonicalForm result= mulMod2 (L.getFirst(), L.getLast(), M); |
---|
1247 | return result; |
---|
1248 | } |
---|
1249 | else |
---|
1250 | { |
---|
1251 | l /= 2; |
---|
1252 | CFList tmp1, tmp2; |
---|
1253 | CFListIterator i= L; |
---|
1254 | CanonicalForm buf1, buf2; |
---|
1255 | for (int j= 1; j <= l; j++, i++) |
---|
1256 | tmp1.append (i.getItem()); |
---|
1257 | tmp2= Difference (L, tmp1); |
---|
1258 | buf1= prodMod (tmp1, M); |
---|
1259 | buf2= prodMod (tmp2, M); |
---|
1260 | CanonicalForm result= mulMod2 (buf1, buf2, M); |
---|
1261 | return result; |
---|
1262 | } |
---|
1263 | } |
---|
1264 | |
---|
1265 | CanonicalForm prodMod (const CFList& L, const CFList& M) |
---|
1266 | { |
---|
1267 | if (L.isEmpty()) |
---|
1268 | return 1; |
---|
1269 | else if (L.length() == 1) |
---|
1270 | return L.getFirst(); |
---|
1271 | else if (L.length() == 2) |
---|
1272 | return mulMod (L.getFirst(), L.getLast(), M); |
---|
1273 | else |
---|
1274 | { |
---|
1275 | int l= L.length()/2; |
---|
1276 | CFListIterator i= L; |
---|
1277 | CFList tmp1, tmp2; |
---|
1278 | CanonicalForm buf1, buf2; |
---|
1279 | for (int j= 1; j <= l; j++, i++) |
---|
1280 | tmp1.append (i.getItem()); |
---|
1281 | tmp2= Difference (L, tmp1); |
---|
1282 | buf1= prodMod (tmp1, M); |
---|
1283 | buf2= prodMod (tmp2, M); |
---|
1284 | return mulMod (buf1, buf2, M); |
---|
1285 | } |
---|
1286 | } |
---|
1287 | |
---|
1288 | |
---|
1289 | CanonicalForm reverse (const CanonicalForm& F, int d) |
---|
1290 | { |
---|
1291 | if (d == 0) |
---|
1292 | return F; |
---|
1293 | CanonicalForm A= F; |
---|
1294 | Variable y= Variable (2); |
---|
1295 | Variable x= Variable (1); |
---|
1296 | if (degree (A, x) > 0) |
---|
1297 | { |
---|
1298 | A= swapvar (A, x, y); |
---|
1299 | CanonicalForm result= 0; |
---|
1300 | CFIterator i= A; |
---|
1301 | while (d - i.exp() < 0) |
---|
1302 | i++; |
---|
1303 | |
---|
1304 | for (; i.hasTerms() && (d - i.exp() >= 0); i++) |
---|
1305 | result += swapvar (i.coeff(),x,y)*power (x, d - i.exp()); |
---|
1306 | return result; |
---|
1307 | } |
---|
1308 | else |
---|
1309 | return A*power (x, d); |
---|
1310 | } |
---|
1311 | |
---|
1312 | CanonicalForm |
---|
1313 | newtonInverse (const CanonicalForm& F, const int n, const CanonicalForm& M) |
---|
1314 | { |
---|
1315 | int l= ilog2(n); |
---|
1316 | |
---|
1317 | CanonicalForm g= mod (F, M)[0] [0]; |
---|
1318 | |
---|
1319 | ASSERT (!g.isZero(), "expected a unit"); |
---|
1320 | |
---|
1321 | Variable alpha; |
---|
1322 | |
---|
1323 | if (!g.isOne()) |
---|
1324 | g = 1/g; |
---|
1325 | Variable x= Variable (1); |
---|
1326 | CanonicalForm result; |
---|
1327 | int exp= 0; |
---|
1328 | if (n & 1) |
---|
1329 | { |
---|
1330 | result= g; |
---|
1331 | exp= 1; |
---|
1332 | } |
---|
1333 | CanonicalForm h; |
---|
1334 | |
---|
1335 | for (int i= 1; i <= l; i++) |
---|
1336 | { |
---|
1337 | h= mulMod2 (g, mod (F, power (x, (1 << i))), M); |
---|
1338 | h= mod (h, power (x, (1 << i)) - 1); |
---|
1339 | h= div (h, power (x, (1 << (i - 1)))); |
---|
1340 | h= mod (h, M); |
---|
1341 | g -= power (x, (1 << (i - 1)))* |
---|
1342 | mod (mulMod2 (g, h, M), power (x, (1 << (i - 1)))); |
---|
1343 | |
---|
1344 | if (n & (1 << i)) |
---|
1345 | { |
---|
1346 | if (exp) |
---|
1347 | { |
---|
1348 | h= mulMod2 (result, mod (F, power (x, exp + (1 << i))), M); |
---|
1349 | h= mod (h, power (x, exp + (1 << i)) - 1); |
---|
1350 | h= div (h, power (x, exp)); |
---|
1351 | h= mod (h, M); |
---|
1352 | result -= power(x, exp)*mod (mulMod2 (g, h, M), |
---|
1353 | power (x, (1 << i))); |
---|
1354 | exp += (1 << i); |
---|
1355 | } |
---|
1356 | else |
---|
1357 | { |
---|
1358 | exp= (1 << i); |
---|
1359 | result= g; |
---|
1360 | } |
---|
1361 | } |
---|
1362 | } |
---|
1363 | |
---|
1364 | return result; |
---|
1365 | } |
---|
1366 | |
---|
1367 | CanonicalForm |
---|
1368 | newtonDiv (const CanonicalForm& F, const CanonicalForm& G, const CanonicalForm& |
---|
1369 | M) |
---|
1370 | { |
---|
1371 | ASSERT (getCharacteristic() > 0, "positive characteristic expected"); |
---|
1372 | ASSERT (CFFactory::gettype() != GaloisFieldDomain, "no GF expected"); |
---|
1373 | |
---|
1374 | CanonicalForm A= mod (F, M); |
---|
1375 | CanonicalForm B= mod (G, M); |
---|
1376 | |
---|
1377 | Variable x= Variable (1); |
---|
1378 | int degA= degree (A, x); |
---|
1379 | int degB= degree (B, x); |
---|
1380 | int m= degA - degB; |
---|
1381 | if (m < 0) |
---|
1382 | return 0; |
---|
1383 | |
---|
1384 | Variable v; |
---|
1385 | CanonicalForm Q; |
---|
1386 | if (degB < 1 || CFFactory::gettype() == GaloisFieldDomain) |
---|
1387 | { |
---|
1388 | CanonicalForm R; |
---|
1389 | divrem2 (A, B, Q, R, M); |
---|
1390 | } |
---|
1391 | else |
---|
1392 | { |
---|
1393 | if (hasFirstAlgVar (A, v) || hasFirstAlgVar (B, v)) |
---|
1394 | { |
---|
1395 | CanonicalForm R= reverse (A, degA); |
---|
1396 | CanonicalForm revB= reverse (B, degB); |
---|
1397 | revB= newtonInverse (revB, m + 1, M); |
---|
1398 | Q= mulMod2 (R, revB, M); |
---|
1399 | Q= mod (Q, power (x, m + 1)); |
---|
1400 | Q= reverse (Q, m); |
---|
1401 | } |
---|
1402 | else |
---|
1403 | { |
---|
1404 | zz_pX mipo= convertFacCF2NTLzzpX (M); |
---|
1405 | Variable y= Variable (2); |
---|
1406 | zz_pEX NTLA, NTLB; |
---|
1407 | NTLA= convertFacCF2NTLzz_pEX (swapvar (A, x, y), mipo); |
---|
1408 | NTLB= convertFacCF2NTLzz_pEX (swapvar (B, x, y), mipo); |
---|
1409 | div (NTLA, NTLA, NTLB); |
---|
1410 | Q= convertNTLzz_pEX2CF (NTLA, x, y); |
---|
1411 | } |
---|
1412 | } |
---|
1413 | |
---|
1414 | return Q; |
---|
1415 | } |
---|
1416 | |
---|
1417 | void |
---|
1418 | newtonDivrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1419 | CanonicalForm& R, const CanonicalForm& M) |
---|
1420 | { |
---|
1421 | CanonicalForm A= mod (F, M); |
---|
1422 | CanonicalForm B= mod (G, M); |
---|
1423 | Variable x= Variable (1); |
---|
1424 | int degA= degree (A, x); |
---|
1425 | int degB= degree (B, x); |
---|
1426 | int m= degA - degB; |
---|
1427 | |
---|
1428 | if (m < 0) |
---|
1429 | { |
---|
1430 | R= A; |
---|
1431 | Q= 0; |
---|
1432 | return; |
---|
1433 | } |
---|
1434 | |
---|
1435 | Variable v; |
---|
1436 | if (degB <= 1 || CFFactory::gettype() == GaloisFieldDomain) |
---|
1437 | { |
---|
1438 | divrem2 (A, B, Q, R, M); |
---|
1439 | } |
---|
1440 | else |
---|
1441 | { |
---|
1442 | if (hasFirstAlgVar (A, v) || hasFirstAlgVar (B, v)) |
---|
1443 | { |
---|
1444 | R= reverse (A, degA); |
---|
1445 | |
---|
1446 | CanonicalForm revB= reverse (B, degB); |
---|
1447 | revB= newtonInverse (revB, m + 1, M); |
---|
1448 | Q= mulMod2 (R, revB, M); |
---|
1449 | |
---|
1450 | Q= mod (Q, power (x, m + 1)); |
---|
1451 | Q= reverse (Q, m); |
---|
1452 | |
---|
1453 | R= A - mulMod2 (Q, B, M); |
---|
1454 | } |
---|
1455 | else |
---|
1456 | { |
---|
1457 | zz_pX mipo= convertFacCF2NTLzzpX (M); |
---|
1458 | Variable y= Variable (2); |
---|
1459 | zz_pEX NTLA, NTLB; |
---|
1460 | NTLA= convertFacCF2NTLzz_pEX (swapvar (A, x, y), mipo); |
---|
1461 | NTLB= convertFacCF2NTLzz_pEX (swapvar (B, x, y), mipo); |
---|
1462 | zz_pEX NTLQ, NTLR; |
---|
1463 | DivRem (NTLQ, NTLR, NTLA, NTLB); |
---|
1464 | Q= convertNTLzz_pEX2CF (NTLQ, x, y); |
---|
1465 | R= convertNTLzz_pEX2CF (NTLR, x, y); |
---|
1466 | } |
---|
1467 | } |
---|
1468 | } |
---|
1469 | |
---|
1470 | static inline |
---|
1471 | CFList split (const CanonicalForm& F, const int m, const Variable& x) |
---|
1472 | { |
---|
1473 | CanonicalForm A= F; |
---|
1474 | CanonicalForm buf= 0; |
---|
1475 | bool swap= false; |
---|
1476 | if (degree (A, x) <= 0) |
---|
1477 | return CFList(A); |
---|
1478 | else if (x.level() != A.level()) |
---|
1479 | { |
---|
1480 | swap= true; |
---|
1481 | A= swapvar (A, x, A.mvar()); |
---|
1482 | } |
---|
1483 | |
---|
1484 | int j= (int) floor ((double) degree (A)/ m); |
---|
1485 | CFList result; |
---|
1486 | CFIterator i= A; |
---|
1487 | for (; j >= 0; j--) |
---|
1488 | { |
---|
1489 | while (i.hasTerms() && i.exp() - j*m >= 0) |
---|
1490 | { |
---|
1491 | if (swap) |
---|
1492 | buf += i.coeff()*power (A.mvar(), i.exp() - j*m); |
---|
1493 | else |
---|
1494 | buf += i.coeff()*power (x, i.exp() - j*m); |
---|
1495 | i++; |
---|
1496 | } |
---|
1497 | if (swap) |
---|
1498 | result.append (swapvar (buf, x, F.mvar())); |
---|
1499 | else |
---|
1500 | result.append (buf); |
---|
1501 | buf= 0; |
---|
1502 | } |
---|
1503 | return result; |
---|
1504 | } |
---|
1505 | |
---|
1506 | static inline |
---|
1507 | void divrem32 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1508 | CanonicalForm& R, const CFList& M); |
---|
1509 | |
---|
1510 | static inline |
---|
1511 | void divrem21 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1512 | CanonicalForm& R, const CFList& M) |
---|
1513 | { |
---|
1514 | CanonicalForm A= mod (F, M); |
---|
1515 | CanonicalForm B= mod (G, M); |
---|
1516 | Variable x= Variable (1); |
---|
1517 | int degB= degree (B, x); |
---|
1518 | int degA= degree (A, x); |
---|
1519 | if (degA < degB) |
---|
1520 | { |
---|
1521 | Q= 0; |
---|
1522 | R= A; |
---|
1523 | return; |
---|
1524 | } |
---|
1525 | ASSERT (2*degB > degA, "expected degree (F, 1) < 2*degree (G, 1)"); |
---|
1526 | if (degB < 1) |
---|
1527 | { |
---|
1528 | divrem (A, B, Q, R); |
---|
1529 | Q= mod (Q, M); |
---|
1530 | R= mod (R, M); |
---|
1531 | return; |
---|
1532 | } |
---|
1533 | |
---|
1534 | int m= (int) ceil ((double) (degB + 1)/2.0) + 1; |
---|
1535 | CFList splitA= split (A, m, x); |
---|
1536 | if (splitA.length() == 3) |
---|
1537 | splitA.insert (0); |
---|
1538 | if (splitA.length() == 2) |
---|
1539 | { |
---|
1540 | splitA.insert (0); |
---|
1541 | splitA.insert (0); |
---|
1542 | } |
---|
1543 | if (splitA.length() == 1) |
---|
1544 | { |
---|
1545 | splitA.insert (0); |
---|
1546 | splitA.insert (0); |
---|
1547 | splitA.insert (0); |
---|
1548 | } |
---|
1549 | |
---|
1550 | CanonicalForm xToM= power (x, m); |
---|
1551 | |
---|
1552 | CFListIterator i= splitA; |
---|
1553 | CanonicalForm H= i.getItem(); |
---|
1554 | i++; |
---|
1555 | H *= xToM; |
---|
1556 | H += i.getItem(); |
---|
1557 | i++; |
---|
1558 | H *= xToM; |
---|
1559 | H += i.getItem(); |
---|
1560 | i++; |
---|
1561 | |
---|
1562 | divrem32 (H, B, Q, R, M); |
---|
1563 | |
---|
1564 | CFList splitR= split (R, m, x); |
---|
1565 | if (splitR.length() == 1) |
---|
1566 | splitR.insert (0); |
---|
1567 | |
---|
1568 | H= splitR.getFirst(); |
---|
1569 | H *= xToM; |
---|
1570 | H += splitR.getLast(); |
---|
1571 | H *= xToM; |
---|
1572 | H += i.getItem(); |
---|
1573 | |
---|
1574 | CanonicalForm bufQ; |
---|
1575 | divrem32 (H, B, bufQ, R, M); |
---|
1576 | |
---|
1577 | Q *= xToM; |
---|
1578 | Q += bufQ; |
---|
1579 | return; |
---|
1580 | } |
---|
1581 | |
---|
1582 | static inline |
---|
1583 | void divrem32 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1584 | CanonicalForm& R, const CFList& M) |
---|
1585 | { |
---|
1586 | CanonicalForm A= mod (F, M); |
---|
1587 | CanonicalForm B= mod (G, M); |
---|
1588 | Variable x= Variable (1); |
---|
1589 | int degB= degree (B, x); |
---|
1590 | int degA= degree (A, x); |
---|
1591 | if (degA < degB) |
---|
1592 | { |
---|
1593 | Q= 0; |
---|
1594 | R= A; |
---|
1595 | return; |
---|
1596 | } |
---|
1597 | ASSERT (3*(degB/2) > degA, "expected degree (F, 1) < 3*(degree (G, 1)/2)"); |
---|
1598 | if (degB < 1) |
---|
1599 | { |
---|
1600 | divrem (A, B, Q, R); |
---|
1601 | Q= mod (Q, M); |
---|
1602 | R= mod (R, M); |
---|
1603 | return; |
---|
1604 | } |
---|
1605 | int m= (int) ceil ((double) (degB + 1)/ 2.0); |
---|
1606 | |
---|
1607 | CFList splitA= split (A, m, x); |
---|
1608 | CFList splitB= split (B, m, x); |
---|
1609 | |
---|
1610 | if (splitA.length() == 2) |
---|
1611 | { |
---|
1612 | splitA.insert (0); |
---|
1613 | } |
---|
1614 | if (splitA.length() == 1) |
---|
1615 | { |
---|
1616 | splitA.insert (0); |
---|
1617 | splitA.insert (0); |
---|
1618 | } |
---|
1619 | CanonicalForm xToM= power (x, m); |
---|
1620 | |
---|
1621 | CanonicalForm H; |
---|
1622 | CFListIterator i= splitA; |
---|
1623 | i++; |
---|
1624 | |
---|
1625 | if (degree (splitA.getFirst(), x) < degree (splitB.getFirst(), x)) |
---|
1626 | { |
---|
1627 | H= splitA.getFirst()*xToM + i.getItem(); |
---|
1628 | divrem21 (H, splitB.getFirst(), Q, R, M); |
---|
1629 | } |
---|
1630 | else |
---|
1631 | { |
---|
1632 | R= splitA.getFirst()*xToM + i.getItem() + splitB.getFirst() - |
---|
1633 | splitB.getFirst()*xToM; |
---|
1634 | Q= xToM - 1; |
---|
1635 | } |
---|
1636 | |
---|
1637 | H= mulMod (Q, splitB.getLast(), M); |
---|
1638 | |
---|
1639 | R= R*xToM + splitA.getLast() - H; |
---|
1640 | |
---|
1641 | while (degree (R, x) >= degB) |
---|
1642 | { |
---|
1643 | xToM= power (x, degree (R, x) - degB); |
---|
1644 | Q += LC (R, x)*xToM; |
---|
1645 | R -= mulMod (LC (R, x), B, M)*xToM; |
---|
1646 | Q= mod (Q, M); |
---|
1647 | R= mod (R, M); |
---|
1648 | } |
---|
1649 | |
---|
1650 | return; |
---|
1651 | } |
---|
1652 | |
---|
1653 | void divrem2 (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1654 | CanonicalForm& R, const CanonicalForm& M) |
---|
1655 | { |
---|
1656 | CanonicalForm A= mod (F, M); |
---|
1657 | CanonicalForm B= mod (G, M); |
---|
1658 | |
---|
1659 | if (B.inCoeffDomain()) |
---|
1660 | { |
---|
1661 | divrem (A, B, Q, R); |
---|
1662 | return; |
---|
1663 | } |
---|
1664 | if (A.inCoeffDomain() && !B.inCoeffDomain()) |
---|
1665 | { |
---|
1666 | Q= 0; |
---|
1667 | R= A; |
---|
1668 | return; |
---|
1669 | } |
---|
1670 | |
---|
1671 | if (B.level() < A.level()) |
---|
1672 | { |
---|
1673 | divrem (A, B, Q, R); |
---|
1674 | return; |
---|
1675 | } |
---|
1676 | if (A.level() > B.level()) |
---|
1677 | { |
---|
1678 | R= A; |
---|
1679 | Q= 0; |
---|
1680 | return; |
---|
1681 | } |
---|
1682 | if (B.level() == 1 && B.isUnivariate()) |
---|
1683 | { |
---|
1684 | divrem (A, B, Q, R); |
---|
1685 | return; |
---|
1686 | } |
---|
1687 | if (!(B.level() == 1 && B.isUnivariate()) && (A.level() == 1 && A.isUnivariate())) |
---|
1688 | { |
---|
1689 | Q= 0; |
---|
1690 | R= A; |
---|
1691 | return; |
---|
1692 | } |
---|
1693 | |
---|
1694 | Variable x= Variable (1); |
---|
1695 | int degB= degree (B, x); |
---|
1696 | if (degB > degree (A, x)) |
---|
1697 | { |
---|
1698 | Q= 0; |
---|
1699 | R= A; |
---|
1700 | return; |
---|
1701 | } |
---|
1702 | |
---|
1703 | CFList splitA= split (A, degB, x); |
---|
1704 | |
---|
1705 | CanonicalForm xToDegB= power (x, degB); |
---|
1706 | CanonicalForm H, bufQ; |
---|
1707 | Q= 0; |
---|
1708 | CFListIterator i= splitA; |
---|
1709 | H= i.getItem()*xToDegB; |
---|
1710 | i++; |
---|
1711 | H += i.getItem(); |
---|
1712 | CFList buf; |
---|
1713 | while (i.hasItem()) |
---|
1714 | { |
---|
1715 | buf= CFList (M); |
---|
1716 | divrem21 (H, B, bufQ, R, buf); |
---|
1717 | i++; |
---|
1718 | if (i.hasItem()) |
---|
1719 | H= R*xToDegB + i.getItem(); |
---|
1720 | Q *= xToDegB; |
---|
1721 | Q += bufQ; |
---|
1722 | } |
---|
1723 | return; |
---|
1724 | } |
---|
1725 | |
---|
1726 | void divrem (const CanonicalForm& F, const CanonicalForm& G, CanonicalForm& Q, |
---|
1727 | CanonicalForm& R, const CFList& MOD) |
---|
1728 | { |
---|
1729 | CanonicalForm A= mod (F, MOD); |
---|
1730 | CanonicalForm B= mod (G, MOD); |
---|
1731 | Variable x= Variable (1); |
---|
1732 | int degB= degree (B, x); |
---|
1733 | if (degB > degree (A, x)) |
---|
1734 | { |
---|
1735 | Q= 0; |
---|
1736 | R= A; |
---|
1737 | return; |
---|
1738 | } |
---|
1739 | |
---|
1740 | if (degB <= 0) |
---|
1741 | { |
---|
1742 | divrem (A, B, Q, R); |
---|
1743 | Q= mod (Q, MOD); |
---|
1744 | R= mod (R, MOD); |
---|
1745 | return; |
---|
1746 | } |
---|
1747 | CFList splitA= split (A, degB, x); |
---|
1748 | |
---|
1749 | CanonicalForm xToDegB= power (x, degB); |
---|
1750 | CanonicalForm H, bufQ; |
---|
1751 | Q= 0; |
---|
1752 | CFListIterator i= splitA; |
---|
1753 | H= i.getItem()*xToDegB; |
---|
1754 | i++; |
---|
1755 | H += i.getItem(); |
---|
1756 | while (i.hasItem()) |
---|
1757 | { |
---|
1758 | divrem21 (H, B, bufQ, R, MOD); |
---|
1759 | i++; |
---|
1760 | if (i.hasItem()) |
---|
1761 | H= R*xToDegB + i.getItem(); |
---|
1762 | Q *= xToDegB; |
---|
1763 | Q += bufQ; |
---|
1764 | } |
---|
1765 | return; |
---|
1766 | } |
---|
1767 | |
---|
1768 | void sortList (CFList& list, const Variable& x) |
---|
1769 | { |
---|
1770 | int l= 1; |
---|
1771 | int k= 1; |
---|
1772 | CanonicalForm buf; |
---|
1773 | CFListIterator m; |
---|
1774 | for (CFListIterator i= list; l <= list.length(); i++, l++) |
---|
1775 | { |
---|
1776 | for (CFListIterator j= list; k <= list.length() - l; k++) |
---|
1777 | { |
---|
1778 | m= j; |
---|
1779 | m++; |
---|
1780 | if (degree (j.getItem(), x) > degree (m.getItem(), x)) |
---|
1781 | { |
---|
1782 | buf= m.getItem(); |
---|
1783 | m.getItem()= j.getItem(); |
---|
1784 | j.getItem()= buf; |
---|
1785 | j++; |
---|
1786 | j.getItem()= m.getItem(); |
---|
1787 | } |
---|
1788 | else |
---|
1789 | j++; |
---|
1790 | } |
---|
1791 | k= 1; |
---|
1792 | } |
---|
1793 | } |
---|
1794 | |
---|
1795 | static inline |
---|
1796 | CFList diophantine (const CanonicalForm& F, const CFList& factors) |
---|
1797 | { |
---|
1798 | if (getCharacteristic() == 0) |
---|
1799 | { |
---|
1800 | Variable v; |
---|
1801 | bool hasAlgVar= hasFirstAlgVar (F, v); |
---|
1802 | for (CFListIterator i= factors; i.hasItem() && !hasAlgVar; i++) |
---|
1803 | hasAlgVar= hasFirstAlgVar (i.getItem(), v); |
---|
1804 | if (hasAlgVar) |
---|
1805 | { |
---|
1806 | CFList result= modularDiophant (F, factors, getMipo (v)); |
---|
1807 | return result; |
---|
1808 | } |
---|
1809 | } |
---|
1810 | |
---|
1811 | CanonicalForm buf1, buf2, buf3, S, T; |
---|
1812 | CFListIterator i= factors; |
---|
1813 | CFList result; |
---|
1814 | if (i.hasItem()) |
---|
1815 | i++; |
---|
1816 | buf1= F/factors.getFirst(); |
---|
1817 | buf2= divNTL (F, i.getItem()); |
---|
1818 | buf3= extgcd (buf1, buf2, S, T); |
---|
1819 | result.append (S); |
---|
1820 | result.append (T); |
---|
1821 | if (i.hasItem()) |
---|
1822 | i++; |
---|
1823 | for (; i.hasItem(); i++) |
---|
1824 | { |
---|
1825 | buf1= divNTL (F, i.getItem()); |
---|
1826 | buf3= extgcd (buf3, buf1, S, T); |
---|
1827 | CFListIterator k= factors; |
---|
1828 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
1829 | { |
---|
1830 | j.getItem()= mulNTL (j.getItem(), S); |
---|
1831 | j.getItem()= modNTL (j.getItem(), k.getItem()); |
---|
1832 | } |
---|
1833 | result.append (T); |
---|
1834 | } |
---|
1835 | return result; |
---|
1836 | } |
---|
1837 | |
---|
1838 | void |
---|
1839 | henselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
1840 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
1841 | CFArray& Pi, int j) |
---|
1842 | { |
---|
1843 | CanonicalForm E; |
---|
1844 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
1845 | Variable x= F.mvar(); |
---|
1846 | // compute the error |
---|
1847 | if (j == 1) |
---|
1848 | E= F[j]; |
---|
1849 | else |
---|
1850 | { |
---|
1851 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1852 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1853 | else |
---|
1854 | E= F[j]; |
---|
1855 | } |
---|
1856 | |
---|
1857 | CFArray buf= CFArray (diophant.length()); |
---|
1858 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
1859 | int k= 0; |
---|
1860 | CanonicalForm remainder; |
---|
1861 | // actual lifting |
---|
1862 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1863 | { |
---|
1864 | if (degree (bufFactors[k], x) > 0) |
---|
1865 | { |
---|
1866 | if (k > 0) |
---|
1867 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
1868 | else |
---|
1869 | remainder= E; |
---|
1870 | } |
---|
1871 | else |
---|
1872 | remainder= modNTL (E, bufFactors[k]); |
---|
1873 | |
---|
1874 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
1875 | if (degree (bufFactors[k], x) > 0) |
---|
1876 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
1877 | else |
---|
1878 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
1879 | } |
---|
1880 | for (k= 1; k < factors.length(); k++) |
---|
1881 | bufFactors[k] += xToJ*buf[k]; |
---|
1882 | |
---|
1883 | // update Pi [0] |
---|
1884 | int degBuf0= degree (bufFactors[0], x); |
---|
1885 | int degBuf1= degree (bufFactors[1], x); |
---|
1886 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1887 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
1888 | CanonicalForm uIZeroJ; |
---|
1889 | if (j == 1) |
---|
1890 | { |
---|
1891 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1892 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1893 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
1894 | else if (degBuf0 > 0) |
---|
1895 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
1896 | else if (degBuf1 > 0) |
---|
1897 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
1898 | else |
---|
1899 | uIZeroJ= 0; |
---|
1900 | Pi [0] += xToJ*uIZeroJ; |
---|
1901 | } |
---|
1902 | else |
---|
1903 | { |
---|
1904 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1905 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
1906 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
1907 | else if (degBuf0 > 0) |
---|
1908 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
1909 | else if (degBuf1 > 0) |
---|
1910 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
1911 | else |
---|
1912 | uIZeroJ= 0; |
---|
1913 | Pi [0] += xToJ*uIZeroJ; |
---|
1914 | } |
---|
1915 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1916 | for (k= 0; k < factors.length() - 1; k++) |
---|
1917 | tmp[k]= 0; |
---|
1918 | CFIterator one, two; |
---|
1919 | one= bufFactors [0]; |
---|
1920 | two= bufFactors [1]; |
---|
1921 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1922 | { |
---|
1923 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1924 | { |
---|
1925 | if (k != j - k + 1) |
---|
1926 | { |
---|
1927 | if ((one.hasTerms() && one.exp() == j - k + 1) && (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1928 | { |
---|
1929 | tmp[0] += mulNTL ((bufFactors[0] [k] + one.coeff()), (bufFactors[1] [k] + |
---|
1930 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
1931 | one++; |
---|
1932 | two++; |
---|
1933 | } |
---|
1934 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1935 | { |
---|
1936 | tmp[0] += mulNTL ((bufFactors[0] [k] + one.coeff()), bufFactors[1] [k]) - |
---|
1937 | M (k + 1, 1); |
---|
1938 | one++; |
---|
1939 | } |
---|
1940 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1941 | { |
---|
1942 | tmp[0] += mulNTL (bufFactors[0] [k], (bufFactors[1] [k] + two.coeff())) - |
---|
1943 | M (k + 1, 1); |
---|
1944 | two++; |
---|
1945 | } |
---|
1946 | } |
---|
1947 | else |
---|
1948 | { |
---|
1949 | tmp[0] += M (k + 1, 1); |
---|
1950 | } |
---|
1951 | } |
---|
1952 | } |
---|
1953 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1954 | |
---|
1955 | // update Pi [l] |
---|
1956 | int degPi, degBuf; |
---|
1957 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1958 | { |
---|
1959 | degPi= degree (Pi [l - 1], x); |
---|
1960 | degBuf= degree (bufFactors[l + 1], x); |
---|
1961 | if (degPi > 0 && degBuf > 0) |
---|
1962 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
1963 | if (j == 1) |
---|
1964 | { |
---|
1965 | if (degPi > 0 && degBuf > 0) |
---|
1966 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [0] + Pi [l - 1] [j], |
---|
1967 | bufFactors[l + 1] [0] + buf[l + 1]) - M (j + 1, l +1) - |
---|
1968 | M (1, l + 1)); |
---|
1969 | else if (degPi > 0) |
---|
1970 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [j], bufFactors[l + 1])); |
---|
1971 | else if (degBuf > 0) |
---|
1972 | Pi [l] += xToJ*(mulNTL (Pi [l - 1], buf[l + 1])); |
---|
1973 | } |
---|
1974 | else |
---|
1975 | { |
---|
1976 | if (degPi > 0 && degBuf > 0) |
---|
1977 | { |
---|
1978 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
1979 | uIZeroJ += mulNTL (Pi [l - 1] [0], buf [l + 1]); |
---|
1980 | } |
---|
1981 | else if (degPi > 0) |
---|
1982 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1]); |
---|
1983 | else if (degBuf > 0) |
---|
1984 | { |
---|
1985 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
1986 | uIZeroJ += mulNTL (Pi [l - 1], buf[l + 1]); |
---|
1987 | } |
---|
1988 | Pi[l] += xToJ*uIZeroJ; |
---|
1989 | } |
---|
1990 | one= bufFactors [l + 1]; |
---|
1991 | two= Pi [l - 1]; |
---|
1992 | if (two.hasTerms() && two.exp() == j + 1) |
---|
1993 | { |
---|
1994 | if (degBuf > 0 && degPi > 0) |
---|
1995 | { |
---|
1996 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1][0]); |
---|
1997 | two++; |
---|
1998 | } |
---|
1999 | else if (degPi > 0) |
---|
2000 | { |
---|
2001 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1]); |
---|
2002 | two++; |
---|
2003 | } |
---|
2004 | } |
---|
2005 | if (degBuf > 0 && degPi > 0) |
---|
2006 | { |
---|
2007 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2008 | { |
---|
2009 | if (k != j - k + 1) |
---|
2010 | { |
---|
2011 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2012 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2013 | { |
---|
2014 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), (Pi[l - 1] [k] + |
---|
2015 | two.coeff())) - M (k + 1, l + 1) - M (j - k + 2, l + 1); |
---|
2016 | one++; |
---|
2017 | two++; |
---|
2018 | } |
---|
2019 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2020 | { |
---|
2021 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), Pi[l - 1] [k]) - |
---|
2022 | M (k + 1, l + 1); |
---|
2023 | one++; |
---|
2024 | } |
---|
2025 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2026 | { |
---|
2027 | tmp[l] += mulNTL (bufFactors[l + 1] [k], (Pi[l - 1] [k] + two.coeff())) - |
---|
2028 | M (k + 1, l + 1); |
---|
2029 | two++; |
---|
2030 | } |
---|
2031 | } |
---|
2032 | else |
---|
2033 | tmp[l] += M (k + 1, l + 1); |
---|
2034 | } |
---|
2035 | } |
---|
2036 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2037 | } |
---|
2038 | return; |
---|
2039 | } |
---|
2040 | |
---|
2041 | void |
---|
2042 | henselLift12 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
2043 | CFList& diophant, CFMatrix& M, bool sort) |
---|
2044 | { |
---|
2045 | if (sort) |
---|
2046 | sortList (factors, Variable (1)); |
---|
2047 | Pi= CFArray (factors.length() - 1); |
---|
2048 | CFListIterator j= factors; |
---|
2049 | diophant= diophantine (F[0], factors); |
---|
2050 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
2051 | j++; |
---|
2052 | Pi [0]= mulNTL (j.getItem(), mod (factors.getFirst(), F.mvar())); |
---|
2053 | M (1, 1)= Pi [0]; |
---|
2054 | int i= 1; |
---|
2055 | if (j.hasItem()) |
---|
2056 | j++; |
---|
2057 | for (; j.hasItem(); j++, i++) |
---|
2058 | { |
---|
2059 | Pi [i]= mulNTL (Pi [i - 1], j.getItem()); |
---|
2060 | M (1, i + 1)= Pi [i]; |
---|
2061 | } |
---|
2062 | CFArray bufFactors= CFArray (factors.length()); |
---|
2063 | i= 0; |
---|
2064 | for (CFListIterator k= factors; k.hasItem(); i++, k++) |
---|
2065 | { |
---|
2066 | if (i == 0) |
---|
2067 | bufFactors[i]= mod (k.getItem(), F.mvar()); |
---|
2068 | else |
---|
2069 | bufFactors[i]= k.getItem(); |
---|
2070 | } |
---|
2071 | for (i= 1; i < l; i++) |
---|
2072 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
2073 | |
---|
2074 | CFListIterator k= factors; |
---|
2075 | for (i= 0; i < factors.length (); i++, k++) |
---|
2076 | k.getItem()= bufFactors[i]; |
---|
2077 | factors.removeFirst(); |
---|
2078 | return; |
---|
2079 | } |
---|
2080 | |
---|
2081 | void |
---|
2082 | henselLiftResume12 (const CanonicalForm& F, CFList& factors, int start, int |
---|
2083 | end, CFArray& Pi, const CFList& diophant, CFMatrix& M) |
---|
2084 | { |
---|
2085 | CFArray bufFactors= CFArray (factors.length()); |
---|
2086 | int i= 0; |
---|
2087 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
2088 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
2089 | { |
---|
2090 | if (i == 0) |
---|
2091 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
2092 | else |
---|
2093 | bufFactors[i]= k.getItem(); |
---|
2094 | } |
---|
2095 | for (i= start; i < end; i++) |
---|
2096 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
2097 | |
---|
2098 | CFListIterator k= factors; |
---|
2099 | for (i= 0; i < factors.length(); k++, i++) |
---|
2100 | k.getItem()= bufFactors [i]; |
---|
2101 | factors.removeFirst(); |
---|
2102 | return; |
---|
2103 | } |
---|
2104 | |
---|
2105 | static inline |
---|
2106 | CFList |
---|
2107 | biDiophantine (const CanonicalForm& F, const CFList& factors, const int d) |
---|
2108 | { |
---|
2109 | Variable y= F.mvar(); |
---|
2110 | CFList result; |
---|
2111 | if (y.level() == 1) |
---|
2112 | { |
---|
2113 | result= diophantine (F, factors); |
---|
2114 | return result; |
---|
2115 | } |
---|
2116 | else |
---|
2117 | { |
---|
2118 | CFList buf= factors; |
---|
2119 | for (CFListIterator i= buf; i.hasItem(); i++) |
---|
2120 | i.getItem()= mod (i.getItem(), y); |
---|
2121 | CanonicalForm A= mod (F, y); |
---|
2122 | int bufD= 1; |
---|
2123 | CFList recResult= biDiophantine (A, buf, bufD); |
---|
2124 | CanonicalForm e= 1; |
---|
2125 | CFList p; |
---|
2126 | CFArray bufFactors= CFArray (factors.length()); |
---|
2127 | CanonicalForm yToD= power (y, d); |
---|
2128 | int k= 0; |
---|
2129 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
2130 | { |
---|
2131 | bufFactors [k]= i.getItem(); |
---|
2132 | } |
---|
2133 | CanonicalForm b, quot; |
---|
2134 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
2135 | { |
---|
2136 | b= 1; |
---|
2137 | if (fdivides (bufFactors[k], F, quot)) |
---|
2138 | b= quot; |
---|
2139 | else |
---|
2140 | { |
---|
2141 | for (int l= 0; l < factors.length(); l++) |
---|
2142 | { |
---|
2143 | if (l == k) |
---|
2144 | continue; |
---|
2145 | else |
---|
2146 | { |
---|
2147 | b= mulMod2 (b, bufFactors[l], yToD); |
---|
2148 | } |
---|
2149 | } |
---|
2150 | } |
---|
2151 | p.append (b); |
---|
2152 | } |
---|
2153 | |
---|
2154 | CFListIterator j= p; |
---|
2155 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
2156 | e -= i.getItem()*j.getItem(); |
---|
2157 | |
---|
2158 | if (e.isZero()) |
---|
2159 | return recResult; |
---|
2160 | CanonicalForm coeffE; |
---|
2161 | CFList s; |
---|
2162 | result= recResult; |
---|
2163 | CanonicalForm g; |
---|
2164 | for (int i= 1; i < d; i++) |
---|
2165 | { |
---|
2166 | if (degree (e, y) > 0) |
---|
2167 | coeffE= e[i]; |
---|
2168 | else |
---|
2169 | coeffE= 0; |
---|
2170 | if (!coeffE.isZero()) |
---|
2171 | { |
---|
2172 | CFListIterator k= result; |
---|
2173 | CFListIterator l= p; |
---|
2174 | int ii= 0; |
---|
2175 | j= recResult; |
---|
2176 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
2177 | { |
---|
2178 | g= coeffE*j.getItem(); |
---|
2179 | if (degree (bufFactors[ii], y) <= 0) |
---|
2180 | g= mod (g, bufFactors[ii]); |
---|
2181 | else |
---|
2182 | g= mod (g, bufFactors[ii][0]); |
---|
2183 | k.getItem() += g*power (y, i); |
---|
2184 | e -= mulMod2 (g*power(y, i), l.getItem(), yToD); |
---|
2185 | DEBOUTLN (cerr, "mod (e, power (y, i + 1))= " << |
---|
2186 | mod (e, power (y, i + 1))); |
---|
2187 | } |
---|
2188 | } |
---|
2189 | if (e.isZero()) |
---|
2190 | break; |
---|
2191 | } |
---|
2192 | |
---|
2193 | DEBOUTLN (cerr, "mod (e, y)= " << mod (e, y)); |
---|
2194 | |
---|
2195 | #ifdef DEBUGOUTPUT |
---|
2196 | CanonicalForm test= 0; |
---|
2197 | j= p; |
---|
2198 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
2199 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
2200 | DEBOUTLN (cerr, "test= " << test); |
---|
2201 | #endif |
---|
2202 | return result; |
---|
2203 | } |
---|
2204 | } |
---|
2205 | |
---|
2206 | static inline |
---|
2207 | CFList |
---|
2208 | multiRecDiophantine (const CanonicalForm& F, const CFList& factors, |
---|
2209 | const CFList& recResult, const CFList& M, const int d) |
---|
2210 | { |
---|
2211 | Variable y= F.mvar(); |
---|
2212 | CFList result; |
---|
2213 | CFListIterator i; |
---|
2214 | CanonicalForm e= 1; |
---|
2215 | CFListIterator j= factors; |
---|
2216 | CFList p; |
---|
2217 | CFArray bufFactors= CFArray (factors.length()); |
---|
2218 | CanonicalForm yToD= power (y, d); |
---|
2219 | int k= 0; |
---|
2220 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
2221 | bufFactors [k]= i.getItem(); |
---|
2222 | CanonicalForm b, quot; |
---|
2223 | CFList buf= M; |
---|
2224 | buf.removeLast(); |
---|
2225 | buf.append (yToD); |
---|
2226 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
2227 | { |
---|
2228 | b= 1; |
---|
2229 | if (fdivides (bufFactors[k], F, quot)) |
---|
2230 | b= quot; |
---|
2231 | else |
---|
2232 | { |
---|
2233 | for (int l= 0; l < factors.length(); l++) |
---|
2234 | { |
---|
2235 | if (l == k) |
---|
2236 | continue; |
---|
2237 | else |
---|
2238 | { |
---|
2239 | b= mulMod (b, bufFactors[l], buf); |
---|
2240 | } |
---|
2241 | } |
---|
2242 | } |
---|
2243 | p.append (b); |
---|
2244 | } |
---|
2245 | j= p; |
---|
2246 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
2247 | e -= mulMod (i.getItem(), j.getItem(), M); |
---|
2248 | |
---|
2249 | if (e.isZero()) |
---|
2250 | return recResult; |
---|
2251 | CanonicalForm coeffE; |
---|
2252 | CFList s; |
---|
2253 | result= recResult; |
---|
2254 | CanonicalForm g; |
---|
2255 | for (int i= 1; i < d; i++) |
---|
2256 | { |
---|
2257 | if (degree (e, y) > 0) |
---|
2258 | coeffE= e[i]; |
---|
2259 | else |
---|
2260 | coeffE= 0; |
---|
2261 | if (!coeffE.isZero()) |
---|
2262 | { |
---|
2263 | CFListIterator k= result; |
---|
2264 | CFListIterator l= p; |
---|
2265 | j= recResult; |
---|
2266 | int ii= 0; |
---|
2267 | CanonicalForm dummy; |
---|
2268 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
2269 | { |
---|
2270 | g= mulMod (coeffE, j.getItem(), M); |
---|
2271 | if (degree (bufFactors[ii], y) <= 0) |
---|
2272 | divrem (g, mod (bufFactors[ii], Variable (y.level() - 1)), dummy, |
---|
2273 | g, M); |
---|
2274 | else |
---|
2275 | divrem (g, bufFactors[ii][0], dummy, g, M); |
---|
2276 | k.getItem() += g*power (y, i); |
---|
2277 | e -= mulMod (g*power (y, i), l.getItem(), M); |
---|
2278 | } |
---|
2279 | } |
---|
2280 | |
---|
2281 | if (e.isZero()) |
---|
2282 | break; |
---|
2283 | } |
---|
2284 | |
---|
2285 | #ifdef DEBUGOUTPUT |
---|
2286 | CanonicalForm test= 0; |
---|
2287 | j= p; |
---|
2288 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
2289 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
2290 | DEBOUTLN (cerr, "test= " << test); |
---|
2291 | #endif |
---|
2292 | return result; |
---|
2293 | } |
---|
2294 | |
---|
2295 | static inline |
---|
2296 | void |
---|
2297 | henselStep (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
2298 | const CFList& diophant, CFMatrix& M, CFArray& Pi, int j, |
---|
2299 | const CFList& MOD) |
---|
2300 | { |
---|
2301 | CanonicalForm E; |
---|
2302 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
2303 | Variable x= F.mvar(); |
---|
2304 | // compute the error |
---|
2305 | if (j == 1) |
---|
2306 | { |
---|
2307 | E= F[j]; |
---|
2308 | #ifdef DEBUGOUTPUT |
---|
2309 | CanonicalForm test= 1; |
---|
2310 | for (int i= 0; i < factors.length(); i++) |
---|
2311 | { |
---|
2312 | if (i == 0) |
---|
2313 | test= mulMod (test, mod (bufFactors [i], xToJ), MOD); |
---|
2314 | else |
---|
2315 | test= mulMod (test, bufFactors[i], MOD); |
---|
2316 | } |
---|
2317 | CanonicalForm test2= mod (F-test, xToJ); |
---|
2318 | |
---|
2319 | test2= mod (test2, MOD); |
---|
2320 | DEBOUTLN (cerr, "test= " << test2); |
---|
2321 | #endif |
---|
2322 | } |
---|
2323 | else |
---|
2324 | { |
---|
2325 | #ifdef DEBUGOUTPUT |
---|
2326 | CanonicalForm test= 1; |
---|
2327 | for (int i= 0; i < factors.length(); i++) |
---|
2328 | { |
---|
2329 | if (i == 0) |
---|
2330 | test *= mod (bufFactors [i], power (x, j)); |
---|
2331 | else |
---|
2332 | test *= bufFactors[i]; |
---|
2333 | } |
---|
2334 | test= mod (test, power (x, j)); |
---|
2335 | test= mod (test, MOD); |
---|
2336 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
2337 | DEBOUTLN (cerr, "test= " << test2); |
---|
2338 | #endif |
---|
2339 | |
---|
2340 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
2341 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
2342 | else |
---|
2343 | E= F[j]; |
---|
2344 | } |
---|
2345 | |
---|
2346 | CFArray buf= CFArray (diophant.length()); |
---|
2347 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
2348 | int k= 0; |
---|
2349 | // actual lifting |
---|
2350 | CanonicalForm dummy, rest1; |
---|
2351 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
2352 | { |
---|
2353 | if (degree (bufFactors[k], x) > 0) |
---|
2354 | { |
---|
2355 | if (k > 0) |
---|
2356 | divrem (E, bufFactors[k] [0], dummy, rest1, MOD); |
---|
2357 | else |
---|
2358 | rest1= E; |
---|
2359 | } |
---|
2360 | else |
---|
2361 | divrem (E, bufFactors[k], dummy, rest1, MOD); |
---|
2362 | |
---|
2363 | buf[k]= mulMod (i.getItem(), rest1, MOD); |
---|
2364 | |
---|
2365 | if (degree (bufFactors[k], x) > 0) |
---|
2366 | divrem (buf[k], bufFactors[k] [0], dummy, buf[k], MOD); |
---|
2367 | else |
---|
2368 | divrem (buf[k], bufFactors[k], dummy, buf[k], MOD); |
---|
2369 | } |
---|
2370 | for (k= 1; k < factors.length(); k++) |
---|
2371 | bufFactors[k] += xToJ*buf[k]; |
---|
2372 | |
---|
2373 | // update Pi [0] |
---|
2374 | int degBuf0= degree (bufFactors[0], x); |
---|
2375 | int degBuf1= degree (bufFactors[1], x); |
---|
2376 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2377 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
2378 | CanonicalForm uIZeroJ; |
---|
2379 | if (j == 1) |
---|
2380 | { |
---|
2381 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2382 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
2383 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
2384 | else if (degBuf0 > 0) |
---|
2385 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
2386 | else if (degBuf1 > 0) |
---|
2387 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
2388 | else |
---|
2389 | uIZeroJ= 0; |
---|
2390 | Pi [0] += xToJ*uIZeroJ; |
---|
2391 | } |
---|
2392 | else |
---|
2393 | { |
---|
2394 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2395 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
2396 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
2397 | else if (degBuf0 > 0) |
---|
2398 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
2399 | else if (degBuf1 > 0) |
---|
2400 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
2401 | else |
---|
2402 | uIZeroJ= 0; |
---|
2403 | Pi [0] += xToJ*uIZeroJ; |
---|
2404 | } |
---|
2405 | |
---|
2406 | CFArray tmp= CFArray (factors.length() - 1); |
---|
2407 | for (k= 0; k < factors.length() - 1; k++) |
---|
2408 | tmp[k]= 0; |
---|
2409 | CFIterator one, two; |
---|
2410 | one= bufFactors [0]; |
---|
2411 | two= bufFactors [1]; |
---|
2412 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2413 | { |
---|
2414 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2415 | { |
---|
2416 | if (k != j - k + 1) |
---|
2417 | { |
---|
2418 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2419 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2420 | { |
---|
2421 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2422 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
2423 | M (j - k + 2, 1); |
---|
2424 | one++; |
---|
2425 | two++; |
---|
2426 | } |
---|
2427 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2428 | { |
---|
2429 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
2430 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
2431 | one++; |
---|
2432 | } |
---|
2433 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2434 | { |
---|
2435 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
2436 | two.coeff()), MOD) - M (k + 1, 1); |
---|
2437 | two++; |
---|
2438 | } |
---|
2439 | } |
---|
2440 | else |
---|
2441 | { |
---|
2442 | tmp[0] += M (k + 1, 1); |
---|
2443 | } |
---|
2444 | } |
---|
2445 | } |
---|
2446 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
2447 | |
---|
2448 | // update Pi [l] |
---|
2449 | int degPi, degBuf; |
---|
2450 | for (int l= 1; l < factors.length() - 1; l++) |
---|
2451 | { |
---|
2452 | degPi= degree (Pi [l - 1], x); |
---|
2453 | degBuf= degree (bufFactors[l + 1], x); |
---|
2454 | if (degPi > 0 && degBuf > 0) |
---|
2455 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
2456 | if (j == 1) |
---|
2457 | { |
---|
2458 | if (degPi > 0 && degBuf > 0) |
---|
2459 | Pi [l] += xToJ*(mulMod ((Pi [l - 1] [0] + Pi [l - 1] [j]), |
---|
2460 | (bufFactors[l + 1] [0] + buf[l + 1]), MOD) - M (j + 1, l +1)- |
---|
2461 | M (1, l + 1)); |
---|
2462 | else if (degPi > 0) |
---|
2463 | Pi [l] += xToJ*(mulMod (Pi [l - 1] [j], bufFactors[l + 1], MOD)); |
---|
2464 | else if (degBuf > 0) |
---|
2465 | Pi [l] += xToJ*(mulMod (Pi [l - 1], buf[l + 1], MOD)); |
---|
2466 | } |
---|
2467 | else |
---|
2468 | { |
---|
2469 | if (degPi > 0 && degBuf > 0) |
---|
2470 | { |
---|
2471 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
2472 | uIZeroJ += mulMod (Pi [l - 1] [0], buf [l + 1], MOD); |
---|
2473 | } |
---|
2474 | else if (degPi > 0) |
---|
2475 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1], MOD); |
---|
2476 | else if (degBuf > 0) |
---|
2477 | { |
---|
2478 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
2479 | uIZeroJ += mulMod (Pi [l - 1], buf[l + 1], MOD); |
---|
2480 | } |
---|
2481 | Pi[l] += xToJ*uIZeroJ; |
---|
2482 | } |
---|
2483 | one= bufFactors [l + 1]; |
---|
2484 | two= Pi [l - 1]; |
---|
2485 | if (two.hasTerms() && two.exp() == j + 1) |
---|
2486 | { |
---|
2487 | if (degBuf > 0 && degPi > 0) |
---|
2488 | { |
---|
2489 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1][0], MOD); |
---|
2490 | two++; |
---|
2491 | } |
---|
2492 | else if (degPi > 0) |
---|
2493 | { |
---|
2494 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1], MOD); |
---|
2495 | two++; |
---|
2496 | } |
---|
2497 | } |
---|
2498 | if (degBuf > 0 && degPi > 0) |
---|
2499 | { |
---|
2500 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2501 | { |
---|
2502 | if (k != j - k + 1) |
---|
2503 | { |
---|
2504 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2505 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2506 | { |
---|
2507 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2508 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
2509 | M (j - k + 2, l + 1); |
---|
2510 | one++; |
---|
2511 | two++; |
---|
2512 | } |
---|
2513 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2514 | { |
---|
2515 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2516 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
2517 | one++; |
---|
2518 | } |
---|
2519 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2520 | { |
---|
2521 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
2522 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
2523 | two++; |
---|
2524 | } |
---|
2525 | } |
---|
2526 | else |
---|
2527 | tmp[l] += M (k + 1, l + 1); |
---|
2528 | } |
---|
2529 | } |
---|
2530 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2531 | } |
---|
2532 | |
---|
2533 | return; |
---|
2534 | } |
---|
2535 | |
---|
2536 | CFList |
---|
2537 | henselLift23 (const CFList& eval, const CFList& factors, const int* l, CFList& |
---|
2538 | diophant, CFArray& Pi, CFMatrix& M) |
---|
2539 | { |
---|
2540 | CFList buf= factors; |
---|
2541 | int k= 0; |
---|
2542 | int liftBoundBivar= l[k]; |
---|
2543 | diophant= biDiophantine (eval.getFirst(), buf, liftBoundBivar); |
---|
2544 | CFList MOD; |
---|
2545 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
2546 | CFArray bufFactors= CFArray (factors.length()); |
---|
2547 | k= 0; |
---|
2548 | CFListIterator j= eval; |
---|
2549 | j++; |
---|
2550 | buf.removeFirst(); |
---|
2551 | buf.insert (LC (j.getItem(), 1)); |
---|
2552 | for (CFListIterator i= buf; i.hasItem(); i++, k++) |
---|
2553 | bufFactors[k]= i.getItem(); |
---|
2554 | Pi= CFArray (factors.length() - 1); |
---|
2555 | CFListIterator i= buf; |
---|
2556 | i++; |
---|
2557 | Variable y= j.getItem().mvar(); |
---|
2558 | Pi [0]= mulMod (i.getItem(), mod (buf.getFirst(), y), MOD); |
---|
2559 | M (1, 1)= Pi [0]; |
---|
2560 | k= 1; |
---|
2561 | if (i.hasItem()) |
---|
2562 | i++; |
---|
2563 | for (; i.hasItem(); i++, k++) |
---|
2564 | { |
---|
2565 | Pi [k]= mulMod (Pi [k - 1], i.getItem(), MOD); |
---|
2566 | M (1, k + 1)= Pi [k]; |
---|
2567 | } |
---|
2568 | |
---|
2569 | for (int d= 1; d < l[1]; d++) |
---|
2570 | henselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
2571 | CFList result; |
---|
2572 | for (k= 1; k < factors.length(); k++) |
---|
2573 | result.append (bufFactors[k]); |
---|
2574 | return result; |
---|
2575 | } |
---|
2576 | |
---|
2577 | void |
---|
2578 | henselLiftResume (const CanonicalForm& F, CFList& factors, int start, int end, |
---|
2579 | CFArray& Pi, const CFList& diophant, CFMatrix& M, |
---|
2580 | const CFList& MOD) |
---|
2581 | { |
---|
2582 | CFArray bufFactors= CFArray (factors.length()); |
---|
2583 | int i= 0; |
---|
2584 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
2585 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
2586 | { |
---|
2587 | if (i == 0) |
---|
2588 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
2589 | else |
---|
2590 | bufFactors[i]= k.getItem(); |
---|
2591 | } |
---|
2592 | for (i= start; i < end; i++) |
---|
2593 | henselStep (F, factors, bufFactors, diophant, M, Pi, i, MOD); |
---|
2594 | |
---|
2595 | CFListIterator k= factors; |
---|
2596 | for (i= 0; i < factors.length(); k++, i++) |
---|
2597 | k.getItem()= bufFactors [i]; |
---|
2598 | factors.removeFirst(); |
---|
2599 | return; |
---|
2600 | } |
---|
2601 | |
---|
2602 | CFList |
---|
2603 | henselLift (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
2604 | diophant, CFArray& Pi, CFMatrix& M, const int lOld, const int |
---|
2605 | lNew) |
---|
2606 | { |
---|
2607 | diophant= multiRecDiophantine (F.getFirst(), factors, diophant, MOD, lOld); |
---|
2608 | int k= 0; |
---|
2609 | CFArray bufFactors= CFArray (factors.length()); |
---|
2610 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
2611 | { |
---|
2612 | if (k == 0) |
---|
2613 | bufFactors[k]= LC (F.getLast(), 1); |
---|
2614 | else |
---|
2615 | bufFactors[k]= i.getItem(); |
---|
2616 | } |
---|
2617 | CFList buf= factors; |
---|
2618 | buf.removeFirst(); |
---|
2619 | buf.insert (LC (F.getLast(), 1)); |
---|
2620 | CFListIterator i= buf; |
---|
2621 | i++; |
---|
2622 | Variable y= F.getLast().mvar(); |
---|
2623 | Variable x= F.getFirst().mvar(); |
---|
2624 | CanonicalForm xToLOld= power (x, lOld); |
---|
2625 | Pi [0]= mod (Pi[0], xToLOld); |
---|
2626 | M (1, 1)= Pi [0]; |
---|
2627 | k= 1; |
---|
2628 | if (i.hasItem()) |
---|
2629 | i++; |
---|
2630 | for (; i.hasItem(); i++, k++) |
---|
2631 | { |
---|
2632 | Pi [k]= mod (Pi [k], xToLOld); |
---|
2633 | M (1, k + 1)= Pi [k]; |
---|
2634 | } |
---|
2635 | |
---|
2636 | for (int d= 1; d < lNew; d++) |
---|
2637 | henselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
2638 | CFList result; |
---|
2639 | for (k= 1; k < factors.length(); k++) |
---|
2640 | result.append (bufFactors[k]); |
---|
2641 | return result; |
---|
2642 | } |
---|
2643 | |
---|
2644 | CFList |
---|
2645 | henselLift (const CFList& eval, const CFList& factors, const int* l, const int |
---|
2646 | lLength, bool sort) |
---|
2647 | { |
---|
2648 | CFList diophant; |
---|
2649 | CFList buf= factors; |
---|
2650 | buf.insert (LC (eval.getFirst(), 1)); |
---|
2651 | if (sort) |
---|
2652 | sortList (buf, Variable (1)); |
---|
2653 | CFArray Pi; |
---|
2654 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
2655 | CFList result= henselLift23 (eval, buf, l, diophant, Pi, M); |
---|
2656 | if (eval.length() == 2) |
---|
2657 | return result; |
---|
2658 | CFList MOD; |
---|
2659 | for (int i= 0; i < 2; i++) |
---|
2660 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2661 | CFListIterator j= eval; |
---|
2662 | j++; |
---|
2663 | CFList bufEval; |
---|
2664 | bufEval.append (j.getItem()); |
---|
2665 | j++; |
---|
2666 | |
---|
2667 | for (int i= 2; i < lLength && j.hasItem(); i++, j++) |
---|
2668 | { |
---|
2669 | result.insert (LC (bufEval.getFirst(), 1)); |
---|
2670 | bufEval.append (j.getItem()); |
---|
2671 | M= CFMatrix (l[i], factors.length()); |
---|
2672 | result= henselLift (bufEval, result, MOD, diophant, Pi, M, l[i - 1], l[i]); |
---|
2673 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2674 | bufEval.removeFirst(); |
---|
2675 | } |
---|
2676 | return result; |
---|
2677 | } |
---|
2678 | |
---|
2679 | void |
---|
2680 | henselStep122 (const CanonicalForm& F, const CFList& factors, |
---|
2681 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
2682 | CFArray& Pi, int j, const CFArray& /*LCs*/) |
---|
2683 | { |
---|
2684 | Variable x= F.mvar(); |
---|
2685 | CanonicalForm xToJ= power (x, j); |
---|
2686 | |
---|
2687 | CanonicalForm E; |
---|
2688 | // compute the error |
---|
2689 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
2690 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
2691 | else |
---|
2692 | E= F[j]; |
---|
2693 | |
---|
2694 | CFArray buf= CFArray (diophant.length()); |
---|
2695 | |
---|
2696 | int k= 0; |
---|
2697 | CanonicalForm remainder; |
---|
2698 | // actual lifting |
---|
2699 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
2700 | { |
---|
2701 | if (degree (bufFactors[k], x) > 0) |
---|
2702 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
2703 | else |
---|
2704 | remainder= modNTL (E, bufFactors[k]); |
---|
2705 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
2706 | if (degree (bufFactors[k], x) > 0) |
---|
2707 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
2708 | else |
---|
2709 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
2710 | } |
---|
2711 | |
---|
2712 | for (k= 0; k < factors.length(); k++) |
---|
2713 | bufFactors[k] += xToJ*buf[k]; |
---|
2714 | |
---|
2715 | // update Pi [0] |
---|
2716 | int degBuf0= degree (bufFactors[0], x); |
---|
2717 | int degBuf1= degree (bufFactors[1], x); |
---|
2718 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2719 | { |
---|
2720 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
2721 | if (j + 2 <= M.rows()) |
---|
2722 | M (j + 2, 1)= mulNTL (bufFactors[0] [j + 1], bufFactors[1] [j + 1]); |
---|
2723 | } |
---|
2724 | CanonicalForm uIZeroJ; |
---|
2725 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2726 | uIZeroJ= mulNTL(bufFactors[0][0],buf[1])+mulNTL (bufFactors[1][0], buf[0]); |
---|
2727 | else if (degBuf0 > 0) |
---|
2728 | uIZeroJ= mulNTL (buf[0], bufFactors[1]); |
---|
2729 | else if (degBuf1 > 0) |
---|
2730 | uIZeroJ= mulNTL (bufFactors[0], buf [1]); |
---|
2731 | else |
---|
2732 | uIZeroJ= 0; |
---|
2733 | Pi [0] += xToJ*uIZeroJ; |
---|
2734 | |
---|
2735 | CFArray tmp= CFArray (factors.length() - 1); |
---|
2736 | for (k= 0; k < factors.length() - 1; k++) |
---|
2737 | tmp[k]= 0; |
---|
2738 | CFIterator one, two; |
---|
2739 | one= bufFactors [0]; |
---|
2740 | two= bufFactors [1]; |
---|
2741 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
2742 | { |
---|
2743 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2744 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2745 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2746 | { |
---|
2747 | if (one.hasTerms() && two.hasTerms()) |
---|
2748 | { |
---|
2749 | if (k != j - k + 1) |
---|
2750 | { |
---|
2751 | if ((one.hasTerms() && one.exp() == j - k + 1) && + |
---|
2752 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2753 | { |
---|
2754 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()),(bufFactors[1][k] + |
---|
2755 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
2756 | one++; |
---|
2757 | two++; |
---|
2758 | } |
---|
2759 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2760 | { |
---|
2761 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), bufFactors[1] [k]) - |
---|
2762 | M (k + 1, 1); |
---|
2763 | one++; |
---|
2764 | } |
---|
2765 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2766 | { |
---|
2767 | tmp[0] += mulNTL (bufFactors[0][k],(bufFactors[1][k] + two.coeff())) - |
---|
2768 | M (k + 1, 1); |
---|
2769 | two++; |
---|
2770 | } |
---|
2771 | } |
---|
2772 | else |
---|
2773 | tmp[0] += M (k + 1, 1); |
---|
2774 | } |
---|
2775 | } |
---|
2776 | } |
---|
2777 | |
---|
2778 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
2779 | { |
---|
2780 | if (j + 2 <= M.rows()) |
---|
2781 | tmp [0] += mulNTL ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
2782 | (bufFactors [1] [j + 1] + bufFactors [1] [0])) |
---|
2783 | - M(1,1) - M (j + 2,1); |
---|
2784 | } |
---|
2785 | else if (degBuf0 >= j + 1) |
---|
2786 | { |
---|
2787 | if (degBuf1 > 0) |
---|
2788 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1] [0]); |
---|
2789 | else |
---|
2790 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1]); |
---|
2791 | } |
---|
2792 | else if (degBuf1 >= j + 1) |
---|
2793 | { |
---|
2794 | if (degBuf0 > 0) |
---|
2795 | tmp[0] += mulNTL (bufFactors [0] [0], bufFactors [1] [j + 1]); |
---|
2796 | else |
---|
2797 | tmp[0] += mulNTL (bufFactors [0], bufFactors [1] [j + 1]); |
---|
2798 | } |
---|
2799 | |
---|
2800 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
2801 | |
---|
2802 | int degPi, degBuf; |
---|
2803 | for (int l= 1; l < factors.length() - 1; l++) |
---|
2804 | { |
---|
2805 | degPi= degree (Pi [l - 1], x); |
---|
2806 | degBuf= degree (bufFactors[l + 1], x); |
---|
2807 | if (degPi > 0 && degBuf > 0) |
---|
2808 | { |
---|
2809 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
2810 | if (j + 2 <= M.rows()) |
---|
2811 | M (j + 2, l + 1)= mulNTL (Pi [l - 1][j + 1], bufFactors[l + 1] [j + 1]); |
---|
2812 | } |
---|
2813 | |
---|
2814 | if (degPi > 0 && degBuf > 0) |
---|
2815 | uIZeroJ= mulNTL (Pi[l -1] [0], buf[l + 1]) + |
---|
2816 | mulNTL (uIZeroJ, bufFactors[l+1] [0]); |
---|
2817 | else if (degPi > 0) |
---|
2818 | uIZeroJ= mulNTL (uIZeroJ, bufFactors[l + 1]); |
---|
2819 | else if (degBuf > 0) |
---|
2820 | uIZeroJ= mulNTL (Pi[l - 1], buf[1]); |
---|
2821 | else |
---|
2822 | uIZeroJ= 0; |
---|
2823 | |
---|
2824 | Pi [l] += xToJ*uIZeroJ; |
---|
2825 | |
---|
2826 | one= bufFactors [l + 1]; |
---|
2827 | two= Pi [l - 1]; |
---|
2828 | if (degBuf > 0 && degPi > 0) |
---|
2829 | { |
---|
2830 | while (one.hasTerms() && one.exp() > j) one++; |
---|
2831 | while (two.hasTerms() && two.exp() > j) two++; |
---|
2832 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
2833 | { |
---|
2834 | if (k != j - k + 1) |
---|
2835 | { |
---|
2836 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
2837 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
2838 | { |
---|
2839 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2840 | (Pi[l - 1] [k] + two.coeff())) - M (k + 1, l + 1) - |
---|
2841 | M (j - k + 2, l + 1); |
---|
2842 | one++; |
---|
2843 | two++; |
---|
2844 | } |
---|
2845 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
2846 | { |
---|
2847 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), |
---|
2848 | Pi[l - 1] [k]) - M (k + 1, l + 1); |
---|
2849 | one++; |
---|
2850 | } |
---|
2851 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
2852 | { |
---|
2853 | tmp[l] += mulNTL (bufFactors[l + 1] [k], |
---|
2854 | (Pi[l - 1] [k] + two.coeff())) - M (k + 1, l + 1); |
---|
2855 | two++; |
---|
2856 | } |
---|
2857 | } |
---|
2858 | else |
---|
2859 | tmp[l] += M (k + 1, l + 1); |
---|
2860 | } |
---|
2861 | } |
---|
2862 | |
---|
2863 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
2864 | { |
---|
2865 | if (j + 2 <= M.rows()) |
---|
2866 | tmp [l] += mulNTL ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
2867 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
2868 | ) - M(1,l+1) - M (j + 2,l+1); |
---|
2869 | } |
---|
2870 | else if (degPi >= j + 1) |
---|
2871 | { |
---|
2872 | if (degBuf > 0) |
---|
2873 | tmp[l] += mulNTL (Pi [l - 1] [j+1], bufFactors [l + 1] [0]); |
---|
2874 | else |
---|
2875 | tmp[l] += mulNTL (Pi [l - 1] [j+1], bufFactors [l + 1]); |
---|
2876 | } |
---|
2877 | else if (degBuf >= j + 1) |
---|
2878 | { |
---|
2879 | if (degPi > 0) |
---|
2880 | tmp[l] += mulNTL (Pi [l - 1] [0], bufFactors [l + 1] [j + 1]); |
---|
2881 | else |
---|
2882 | tmp[l] += mulNTL (Pi [l - 1], bufFactors [l + 1] [j + 1]); |
---|
2883 | } |
---|
2884 | |
---|
2885 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
2886 | } |
---|
2887 | return; |
---|
2888 | } |
---|
2889 | |
---|
2890 | void |
---|
2891 | henselLift122 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
2892 | CFList& diophant, CFMatrix& M, const CFArray& LCs, bool sort) |
---|
2893 | { |
---|
2894 | if (sort) |
---|
2895 | sortList (factors, Variable (1)); |
---|
2896 | Pi= CFArray (factors.length() - 2); |
---|
2897 | CFList bufFactors2= factors; |
---|
2898 | bufFactors2.removeFirst(); |
---|
2899 | diophant= diophantine (F[0], bufFactors2); |
---|
2900 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
2901 | |
---|
2902 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
2903 | int i= 0; |
---|
2904 | for (CFListIterator k= bufFactors2; k.hasItem(); i++, k++) |
---|
2905 | bufFactors[i]= replaceLc (k.getItem(), LCs [i]); |
---|
2906 | |
---|
2907 | Variable x= F.mvar(); |
---|
2908 | if (degree (bufFactors[0], x) > 0 && degree (bufFactors [1], x) > 0) |
---|
2909 | { |
---|
2910 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1] [0]); |
---|
2911 | Pi [0]= M (1, 1) + (mulNTL (bufFactors [0] [1], bufFactors[1] [0]) + |
---|
2912 | mulNTL (bufFactors [0] [0], bufFactors [1] [1]))*x; |
---|
2913 | } |
---|
2914 | else if (degree (bufFactors[0], x) > 0) |
---|
2915 | { |
---|
2916 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1]); |
---|
2917 | Pi [0]= M (1, 1) + |
---|
2918 | mulNTL (bufFactors [0] [1], bufFactors[1])*x; |
---|
2919 | } |
---|
2920 | else if (degree (bufFactors[1], x) > 0) |
---|
2921 | { |
---|
2922 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1] [0]); |
---|
2923 | Pi [0]= M (1, 1) + |
---|
2924 | mulNTL (bufFactors [0], bufFactors[1] [1])*x; |
---|
2925 | } |
---|
2926 | else |
---|
2927 | { |
---|
2928 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1]); |
---|
2929 | Pi [0]= M (1, 1); |
---|
2930 | } |
---|
2931 | |
---|
2932 | for (i= 1; i < Pi.size(); i++) |
---|
2933 | { |
---|
2934 | if (degree (Pi[i-1], x) > 0 && degree (bufFactors [i+1], x) > 0) |
---|
2935 | { |
---|
2936 | M (1,i+1)= mulNTL (Pi[i-1] [0], bufFactors[i+1] [0]); |
---|
2937 | Pi [i]= M (1,i+1) + (mulNTL (Pi[i-1] [1], bufFactors[i+1] [0]) + |
---|
2938 | mulNTL (Pi[i-1] [0], bufFactors [i+1] [1]))*x; |
---|
2939 | } |
---|
2940 | else if (degree (Pi[i-1], x) > 0) |
---|
2941 | { |
---|
2942 | M (1,i+1)= mulNTL (Pi[i-1] [0], bufFactors [i+1]); |
---|
2943 | Pi [i]= M(1,i+1) + mulNTL (Pi[i-1] [1], bufFactors[i+1])*x; |
---|
2944 | } |
---|
2945 | else if (degree (bufFactors[i+1], x) > 0) |
---|
2946 | { |
---|
2947 | M (1,i+1)= mulNTL (Pi[i-1], bufFactors [i+1] [0]); |
---|
2948 | Pi [i]= M (1,i+1) + mulNTL (Pi[i-1], bufFactors[i+1] [1])*x; |
---|
2949 | } |
---|
2950 | else |
---|
2951 | { |
---|
2952 | M (1,i+1)= mulNTL (Pi [i-1], bufFactors [i+1]); |
---|
2953 | Pi [i]= M (1,i+1); |
---|
2954 | } |
---|
2955 | } |
---|
2956 | |
---|
2957 | for (i= 1; i < l; i++) |
---|
2958 | henselStep122 (F, bufFactors2, bufFactors, diophant, M, Pi, i, LCs); |
---|
2959 | |
---|
2960 | factors= CFList(); |
---|
2961 | for (i= 0; i < bufFactors.size(); i++) |
---|
2962 | factors.append (bufFactors[i]); |
---|
2963 | return; |
---|
2964 | } |
---|
2965 | |
---|
2966 | |
---|
2967 | /// solve \f$ E=sum_{i= 1}^{r}{\sigma_{i}prod_{j=1, j\neq i}^{r}{f_{i}}}\f$ |
---|
2968 | /// mod M, products contains \f$ prod_{j=1, j\neq i}^{r}{f_{i}}} \f$ |
---|
2969 | static inline |
---|
2970 | CFList |
---|
2971 | diophantine (const CFList& recResult, const CFList& factors, |
---|
2972 | const CFList& products, const CFList& M, const CanonicalForm& E, |
---|
2973 | bool& bad) |
---|
2974 | { |
---|
2975 | if (M.isEmpty()) |
---|
2976 | { |
---|
2977 | CFList result; |
---|
2978 | CFListIterator j= factors; |
---|
2979 | CanonicalForm buf; |
---|
2980 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
2981 | { |
---|
2982 | ASSERT (E.isUnivariate() || E.inCoeffDomain(), |
---|
2983 | "constant or univariate poly expected"); |
---|
2984 | ASSERT (i.getItem().isUnivariate() || i.getItem().inCoeffDomain(), |
---|
2985 | "constant or univariate poly expected"); |
---|
2986 | ASSERT (j.getItem().isUnivariate() || j.getItem().inCoeffDomain(), |
---|
2987 | "constant or univariate poly expected"); |
---|
2988 | buf= mulNTL (E, i.getItem()); |
---|
2989 | result.append (modNTL (buf, j.getItem())); |
---|
2990 | } |
---|
2991 | return result; |
---|
2992 | } |
---|
2993 | Variable y= M.getLast().mvar(); |
---|
2994 | CFList bufFactors= factors; |
---|
2995 | for (CFListIterator i= bufFactors; i.hasItem(); i++) |
---|
2996 | i.getItem()= mod (i.getItem(), y); |
---|
2997 | CFList bufProducts= products; |
---|
2998 | for (CFListIterator i= bufProducts; i.hasItem(); i++) |
---|
2999 | i.getItem()= mod (i.getItem(), y); |
---|
3000 | CFList buf= M; |
---|
3001 | buf.removeLast(); |
---|
3002 | CanonicalForm bufE= mod (E, y); |
---|
3003 | CFList recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
3004 | bufE, bad); |
---|
3005 | |
---|
3006 | if (bad) |
---|
3007 | return CFList(); |
---|
3008 | |
---|
3009 | CanonicalForm e= E; |
---|
3010 | CFListIterator j= products; |
---|
3011 | for (CFListIterator i= recDiophantine; i.hasItem(); i++, j++) |
---|
3012 | e -= i.getItem()*j.getItem(); |
---|
3013 | |
---|
3014 | CFList result= recDiophantine; |
---|
3015 | int d= degree (M.getLast()); |
---|
3016 | CanonicalForm coeffE; |
---|
3017 | for (int i= 1; i < d; i++) |
---|
3018 | { |
---|
3019 | if (degree (e, y) > 0) |
---|
3020 | coeffE= e[i]; |
---|
3021 | else |
---|
3022 | coeffE= 0; |
---|
3023 | if (!coeffE.isZero()) |
---|
3024 | { |
---|
3025 | CFListIterator k= result; |
---|
3026 | recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
3027 | coeffE, bad); |
---|
3028 | if (bad) |
---|
3029 | return CFList(); |
---|
3030 | CFListIterator l= products; |
---|
3031 | for (j= recDiophantine; j.hasItem(); j++, k++, l++) |
---|
3032 | { |
---|
3033 | k.getItem() += j.getItem()*power (y, i); |
---|
3034 | e -= j.getItem()*power (y, i)*l.getItem(); |
---|
3035 | } |
---|
3036 | } |
---|
3037 | if (e.isZero()) |
---|
3038 | break; |
---|
3039 | } |
---|
3040 | if (!e.isZero()) |
---|
3041 | { |
---|
3042 | bad= true; |
---|
3043 | return CFList(); |
---|
3044 | } |
---|
3045 | return result; |
---|
3046 | } |
---|
3047 | |
---|
3048 | static inline |
---|
3049 | void |
---|
3050 | henselStep2 (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
3051 | const CFList& diophant, CFMatrix& M, CFArray& Pi, |
---|
3052 | const CFList& products, int j, const CFList& MOD, bool& noOneToOne) |
---|
3053 | { |
---|
3054 | CanonicalForm E; |
---|
3055 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
3056 | Variable x= F.mvar(); |
---|
3057 | |
---|
3058 | // compute the error |
---|
3059 | #ifdef DEBUGOUTPUT |
---|
3060 | CanonicalForm test= 1; |
---|
3061 | for (int i= 0; i < factors.length(); i++) |
---|
3062 | { |
---|
3063 | if (i == 0) |
---|
3064 | test *= mod (bufFactors [i], power (x, j)); |
---|
3065 | else |
---|
3066 | test *= bufFactors[i]; |
---|
3067 | } |
---|
3068 | test= mod (test, power (x, j)); |
---|
3069 | test= mod (test, MOD); |
---|
3070 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
3071 | DEBOUTLN (cerr, "test= " << test2); |
---|
3072 | #endif |
---|
3073 | |
---|
3074 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
3075 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
3076 | else |
---|
3077 | E= F[j]; |
---|
3078 | |
---|
3079 | CFArray buf= CFArray (diophant.length()); |
---|
3080 | |
---|
3081 | // actual lifting |
---|
3082 | CFList diophantine2= diophantine (diophant, factors, products, MOD, E, |
---|
3083 | noOneToOne); |
---|
3084 | |
---|
3085 | if (noOneToOne) |
---|
3086 | return; |
---|
3087 | |
---|
3088 | int k= 0; |
---|
3089 | for (CFListIterator i= diophantine2; k < factors.length(); k++, i++) |
---|
3090 | { |
---|
3091 | buf[k]= i.getItem(); |
---|
3092 | bufFactors[k] += xToJ*i.getItem(); |
---|
3093 | } |
---|
3094 | |
---|
3095 | // update Pi [0] |
---|
3096 | int degBuf0= degree (bufFactors[0], x); |
---|
3097 | int degBuf1= degree (bufFactors[1], x); |
---|
3098 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
3099 | { |
---|
3100 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
3101 | if (j + 2 <= M.rows()) |
---|
3102 | M (j + 2, 1)= mulMod (bufFactors[0] [j + 1], bufFactors[1] [j + 1], MOD); |
---|
3103 | } |
---|
3104 | CanonicalForm uIZeroJ; |
---|
3105 | |
---|
3106 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
3107 | uIZeroJ= mulMod (bufFactors[0] [0], buf[1], MOD) + |
---|
3108 | mulMod (bufFactors[1] [0], buf[0], MOD); |
---|
3109 | else if (degBuf0 > 0) |
---|
3110 | uIZeroJ= mulMod (buf[0], bufFactors[1], MOD); |
---|
3111 | else if (degBuf1 > 0) |
---|
3112 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
3113 | else |
---|
3114 | uIZeroJ= 0; |
---|
3115 | Pi [0] += xToJ*uIZeroJ; |
---|
3116 | |
---|
3117 | CFArray tmp= CFArray (factors.length() - 1); |
---|
3118 | for (k= 0; k < factors.length() - 1; k++) |
---|
3119 | tmp[k]= 0; |
---|
3120 | CFIterator one, two; |
---|
3121 | one= bufFactors [0]; |
---|
3122 | two= bufFactors [1]; |
---|
3123 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
3124 | { |
---|
3125 | while (one.hasTerms() && one.exp() > j) one++; |
---|
3126 | while (two.hasTerms() && two.exp() > j) two++; |
---|
3127 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
3128 | { |
---|
3129 | if (k != j - k + 1) |
---|
3130 | { |
---|
3131 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
3132 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
3133 | { |
---|
3134 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
3135 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
3136 | M (j - k + 2, 1); |
---|
3137 | one++; |
---|
3138 | two++; |
---|
3139 | } |
---|
3140 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
3141 | { |
---|
3142 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
3143 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
3144 | one++; |
---|
3145 | } |
---|
3146 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
3147 | { |
---|
3148 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
3149 | two.coeff()), MOD) - M (k + 1, 1); |
---|
3150 | two++; |
---|
3151 | } |
---|
3152 | } |
---|
3153 | else |
---|
3154 | { |
---|
3155 | tmp[0] += M (k + 1, 1); |
---|
3156 | } |
---|
3157 | } |
---|
3158 | } |
---|
3159 | |
---|
3160 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
3161 | { |
---|
3162 | if (j + 2 <= M.rows()) |
---|
3163 | tmp [0] += mulMod ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
3164 | (bufFactors [1] [j + 1] + bufFactors [1] [0]), MOD) |
---|
3165 | - M(1,1) - M (j + 2,1); |
---|
3166 | } |
---|
3167 | else if (degBuf0 >= j + 1) |
---|
3168 | { |
---|
3169 | if (degBuf1 > 0) |
---|
3170 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1] [0], MOD); |
---|
3171 | else |
---|
3172 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1], MOD); |
---|
3173 | } |
---|
3174 | else if (degBuf1 >= j + 1) |
---|
3175 | { |
---|
3176 | if (degBuf0 > 0) |
---|
3177 | tmp[0] += mulMod (bufFactors [0] [0], bufFactors [1] [j + 1], MOD); |
---|
3178 | else |
---|
3179 | tmp[0] += mulMod (bufFactors [0], bufFactors [1] [j + 1], MOD); |
---|
3180 | } |
---|
3181 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
3182 | |
---|
3183 | // update Pi [l] |
---|
3184 | int degPi, degBuf; |
---|
3185 | for (int l= 1; l < factors.length() - 1; l++) |
---|
3186 | { |
---|
3187 | degPi= degree (Pi [l - 1], x); |
---|
3188 | degBuf= degree (bufFactors[l + 1], x); |
---|
3189 | if (degPi > 0 && degBuf > 0) |
---|
3190 | { |
---|
3191 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
3192 | if (j + 2 <= M.rows()) |
---|
3193 | M (j + 2, l + 1)= mulMod (Pi [l - 1] [j + 1], bufFactors[l + 1] [j + 1], |
---|
3194 | MOD); |
---|
3195 | } |
---|
3196 | |
---|
3197 | if (degPi > 0 && degBuf > 0) |
---|
3198 | uIZeroJ= mulMod (Pi[l -1] [0], buf[l + 1], MOD) + |
---|
3199 | mulMod (uIZeroJ, bufFactors[l+1] [0], MOD); |
---|
3200 | else if (degPi > 0) |
---|
3201 | uIZeroJ= mulMod (uIZeroJ, bufFactors[l + 1], MOD); |
---|
3202 | else if (degBuf > 0) |
---|
3203 | uIZeroJ= mulMod (Pi[l - 1], buf[1], MOD); |
---|
3204 | else |
---|
3205 | uIZeroJ= 0; |
---|
3206 | |
---|
3207 | Pi [l] += xToJ*uIZeroJ; |
---|
3208 | |
---|
3209 | one= bufFactors [l + 1]; |
---|
3210 | two= Pi [l - 1]; |
---|
3211 | if (degBuf > 0 && degPi > 0) |
---|
3212 | { |
---|
3213 | while (one.hasTerms() && one.exp() > j) one++; |
---|
3214 | while (two.hasTerms() && two.exp() > j) two++; |
---|
3215 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
3216 | { |
---|
3217 | if (k != j - k + 1) |
---|
3218 | { |
---|
3219 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
3220 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
3221 | { |
---|
3222 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
3223 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
3224 | M (j - k + 2, l + 1); |
---|
3225 | one++; |
---|
3226 | two++; |
---|
3227 | } |
---|
3228 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
3229 | { |
---|
3230 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
3231 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
3232 | one++; |
---|
3233 | } |
---|
3234 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
3235 | { |
---|
3236 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
3237 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
3238 | two++; |
---|
3239 | } |
---|
3240 | } |
---|
3241 | else |
---|
3242 | tmp[l] += M (k + 1, l + 1); |
---|
3243 | } |
---|
3244 | } |
---|
3245 | |
---|
3246 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
3247 | { |
---|
3248 | if (j + 2 <= M.rows()) |
---|
3249 | tmp [l] += mulMod ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
3250 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
3251 | , MOD) - M(1,l+1) - M (j + 2,l+1); |
---|
3252 | } |
---|
3253 | else if (degPi >= j + 1) |
---|
3254 | { |
---|
3255 | if (degBuf > 0) |
---|
3256 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1] [0], MOD); |
---|
3257 | else |
---|
3258 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1], MOD); |
---|
3259 | } |
---|
3260 | else if (degBuf >= j + 1) |
---|
3261 | { |
---|
3262 | if (degPi > 0) |
---|
3263 | tmp[l] += mulMod (Pi [l - 1] [0], bufFactors [l + 1] [j + 1], MOD); |
---|
3264 | else |
---|
3265 | tmp[l] += mulMod (Pi [l - 1], bufFactors [l + 1] [j + 1], MOD); |
---|
3266 | } |
---|
3267 | |
---|
3268 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
3269 | } |
---|
3270 | return; |
---|
3271 | } |
---|
3272 | |
---|
3273 | // wrt. Variable (1) |
---|
3274 | CanonicalForm replaceLC (const CanonicalForm& F, const CanonicalForm& c) |
---|
3275 | { |
---|
3276 | if (degree (F, 1) <= 0) |
---|
3277 | return c; |
---|
3278 | else |
---|
3279 | { |
---|
3280 | CanonicalForm result= swapvar (F, Variable (F.level() + 1), Variable (1)); |
---|
3281 | result += (swapvar (c, Variable (F.level() + 1), Variable (1)) |
---|
3282 | - LC (result))*power (result.mvar(), degree (result)); |
---|
3283 | return swapvar (result, Variable (F.level() + 1), Variable (1)); |
---|
3284 | } |
---|
3285 | } |
---|
3286 | |
---|
3287 | CFList |
---|
3288 | henselLift232 (const CFList& eval, const CFList& factors, int* l, CFList& |
---|
3289 | diophant, CFArray& Pi, CFMatrix& M, const CFList& LCs1, |
---|
3290 | const CFList& LCs2, bool& bad) |
---|
3291 | { |
---|
3292 | CFList buf= factors; |
---|
3293 | int k= 0; |
---|
3294 | int liftBoundBivar= l[k]; |
---|
3295 | CFList bufbuf= factors; |
---|
3296 | Variable v= Variable (2); |
---|
3297 | |
---|
3298 | CFList MOD; |
---|
3299 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
3300 | CFArray bufFactors= CFArray (factors.length()); |
---|
3301 | k= 0; |
---|
3302 | CFListIterator j= eval; |
---|
3303 | j++; |
---|
3304 | CFListIterator iter1= LCs1; |
---|
3305 | CFListIterator iter2= LCs2; |
---|
3306 | iter1++; |
---|
3307 | iter2++; |
---|
3308 | bufFactors[0]= replaceLC (buf.getFirst(), iter1.getItem()); |
---|
3309 | bufFactors[1]= replaceLC (buf.getLast(), iter2.getItem()); |
---|
3310 | |
---|
3311 | CFListIterator i= buf; |
---|
3312 | i++; |
---|
3313 | Variable y= j.getItem().mvar(); |
---|
3314 | if (y.level() != 3) |
---|
3315 | y= Variable (3); |
---|
3316 | |
---|
3317 | Pi[0]= mod (Pi[0], power (v, liftBoundBivar)); |
---|
3318 | M (1, 1)= Pi[0]; |
---|
3319 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
3320 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
3321 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
3322 | else if (degree (bufFactors[0], y) > 0) |
---|
3323 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
3324 | else if (degree (bufFactors[1], y) > 0) |
---|
3325 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
3326 | |
---|
3327 | CFList products; |
---|
3328 | for (int i= 0; i < bufFactors.size(); i++) |
---|
3329 | { |
---|
3330 | if (degree (bufFactors[i], y) > 0) |
---|
3331 | products.append (eval.getFirst()/bufFactors[i] [0]); |
---|
3332 | else |
---|
3333 | products.append (eval.getFirst()/bufFactors[i]); |
---|
3334 | } |
---|
3335 | |
---|
3336 | for (int d= 1; d < l[1]; d++) |
---|
3337 | { |
---|
3338 | henselStep2 (j.getItem(), buf, bufFactors, diophant, M, Pi, products, d, MOD, bad); |
---|
3339 | if (bad) |
---|
3340 | return CFList(); |
---|
3341 | } |
---|
3342 | CFList result; |
---|
3343 | for (k= 0; k < factors.length(); k++) |
---|
3344 | result.append (bufFactors[k]); |
---|
3345 | return result; |
---|
3346 | } |
---|
3347 | |
---|
3348 | |
---|
3349 | CFList |
---|
3350 | henselLift2 (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
3351 | diophant, CFArray& Pi, CFMatrix& M, const int lOld, int& |
---|
3352 | lNew, const CFList& LCs1, const CFList& LCs2, bool& bad) |
---|
3353 | { |
---|
3354 | int k= 0; |
---|
3355 | CFArray bufFactors= CFArray (factors.length()); |
---|
3356 | bufFactors[0]= replaceLC (factors.getFirst(), LCs1.getLast()); |
---|
3357 | bufFactors[1]= replaceLC (factors.getLast(), LCs2.getLast()); |
---|
3358 | CFList buf= factors; |
---|
3359 | Variable y= F.getLast().mvar(); |
---|
3360 | Variable x= F.getFirst().mvar(); |
---|
3361 | CanonicalForm xToLOld= power (x, lOld); |
---|
3362 | Pi [0]= mod (Pi[0], xToLOld); |
---|
3363 | M (1, 1)= Pi [0]; |
---|
3364 | |
---|
3365 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
3366 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
3367 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
3368 | else if (degree (bufFactors[0], y) > 0) |
---|
3369 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
3370 | else if (degree (bufFactors[1], y) > 0) |
---|
3371 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
3372 | |
---|
3373 | CFList products; |
---|
3374 | CanonicalForm quot; |
---|
3375 | for (int i= 0; i < bufFactors.size(); i++) |
---|
3376 | { |
---|
3377 | if (degree (bufFactors[i], y) > 0) |
---|
3378 | { |
---|
3379 | if (!fdivides (bufFactors[i] [0], F.getFirst(), quot)) |
---|
3380 | { |
---|
3381 | bad= true; |
---|
3382 | return CFList(); |
---|
3383 | } |
---|
3384 | products.append (quot); |
---|
3385 | } |
---|
3386 | else |
---|
3387 | { |
---|
3388 | if (!fdivides (bufFactors[i], F.getFirst(), quot)) |
---|
3389 | { |
---|
3390 | bad= true; |
---|
3391 | return CFList(); |
---|
3392 | } |
---|
3393 | products.append (quot); |
---|
3394 | } |
---|
3395 | } |
---|
3396 | |
---|
3397 | for (int d= 1; d < lNew; d++) |
---|
3398 | { |
---|
3399 | henselStep2 (F.getLast(), buf, bufFactors, diophant, M, Pi, products, d, MOD, bad); |
---|
3400 | if (bad) |
---|
3401 | return CFList(); |
---|
3402 | } |
---|
3403 | |
---|
3404 | CFList result; |
---|
3405 | for (k= 0; k < factors.length(); k++) |
---|
3406 | result.append (bufFactors[k]); |
---|
3407 | return result; |
---|
3408 | } |
---|
3409 | |
---|
3410 | CFList |
---|
3411 | henselLift2 (const CFList& eval, const CFList& factors, int* l, const int |
---|
3412 | lLength, bool sort, const CFList& LCs1, const CFList& LCs2, |
---|
3413 | const CFArray& Pi, const CFList& diophant, bool& bad) |
---|
3414 | { |
---|
3415 | CFList bufDiophant= diophant; |
---|
3416 | CFList buf= factors; |
---|
3417 | if (sort) |
---|
3418 | sortList (buf, Variable (1)); |
---|
3419 | CFArray bufPi= Pi; |
---|
3420 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
3421 | CFList result= henselLift232(eval, buf, l, bufDiophant, bufPi, M, LCs1, LCs2, |
---|
3422 | bad); |
---|
3423 | if (bad) |
---|
3424 | return CFList(); |
---|
3425 | |
---|
3426 | if (eval.length() == 2) |
---|
3427 | return result; |
---|
3428 | CFList MOD; |
---|
3429 | for (int i= 0; i < 2; i++) |
---|
3430 | MOD.append (power (Variable (i + 2), l[i])); |
---|
3431 | CFListIterator j= eval; |
---|
3432 | j++; |
---|
3433 | CFList bufEval; |
---|
3434 | bufEval.append (j.getItem()); |
---|
3435 | j++; |
---|
3436 | CFListIterator jj= LCs1; |
---|
3437 | CFListIterator jjj= LCs2; |
---|
3438 | CFList bufLCs1, bufLCs2; |
---|
3439 | jj++, jjj++; |
---|
3440 | bufLCs1.append (jj.getItem()); |
---|
3441 | bufLCs2.append (jjj.getItem()); |
---|
3442 | jj++, jjj++; |
---|
3443 | |
---|
3444 | for (int i= 2; i < lLength && j.hasItem(); i++, j++, jj++, jjj++) |
---|
3445 | { |
---|
3446 | bufEval.append (j.getItem()); |
---|
3447 | bufLCs1.append (jj.getItem()); |
---|
3448 | bufLCs2.append (jjj.getItem()); |
---|
3449 | M= CFMatrix (l[i], factors.length()); |
---|
3450 | result= henselLift2 (bufEval, result, MOD, bufDiophant, bufPi, M, l[i - 1], |
---|
3451 | l[i], bufLCs1, bufLCs2, bad); |
---|
3452 | if (bad) |
---|
3453 | return CFList(); |
---|
3454 | MOD.append (power (Variable (i + 2), l[i])); |
---|
3455 | bufEval.removeFirst(); |
---|
3456 | bufLCs1.removeFirst(); |
---|
3457 | bufLCs2.removeFirst(); |
---|
3458 | } |
---|
3459 | return result; |
---|
3460 | } |
---|
3461 | |
---|
3462 | CFList |
---|
3463 | nonMonicHenselLift23 (const CanonicalForm& F, const CFList& factors, const |
---|
3464 | CFList& LCs, CFList& diophant, CFArray& Pi, int liftBound, |
---|
3465 | int bivarLiftBound, bool& bad) |
---|
3466 | { |
---|
3467 | CFList bufFactors2= factors; |
---|
3468 | |
---|
3469 | Variable y= Variable (2); |
---|
3470 | for (CFListIterator i= bufFactors2; i.hasItem(); i++) |
---|
3471 | i.getItem()= mod (i.getItem(), y); |
---|
3472 | |
---|
3473 | CanonicalForm bufF= F; |
---|
3474 | bufF= mod (bufF, y); |
---|
3475 | bufF= mod (bufF, Variable (3)); |
---|
3476 | |
---|
3477 | diophant= diophantine (bufF, bufFactors2); |
---|
3478 | |
---|
3479 | CFMatrix M= CFMatrix (liftBound, bufFactors2.length() - 1); |
---|
3480 | |
---|
3481 | Pi= CFArray (bufFactors2.length() - 1); |
---|
3482 | |
---|
3483 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
3484 | CFListIterator j= LCs; |
---|
3485 | int i= 0; |
---|
3486 | for (CFListIterator k= factors; k.hasItem(); j++, k++, i++) |
---|
3487 | bufFactors[i]= replaceLC (k.getItem(), j.getItem()); |
---|
3488 | |
---|
3489 | //initialise Pi |
---|
3490 | Variable v= Variable (3); |
---|
3491 | CanonicalForm yToL= power (y, bivarLiftBound); |
---|
3492 | if (degree (bufFactors[0], v) > 0 && degree (bufFactors [1], v) > 0) |
---|
3493 | { |
---|
3494 | M (1, 1)= mulMod2 (bufFactors [0] [0], bufFactors[1] [0], yToL); |
---|
3495 | Pi [0]= M (1,1) + (mulMod2 (bufFactors [0] [1], bufFactors[1] [0], yToL) + |
---|
3496 | mulMod2 (bufFactors [0] [0], bufFactors [1] [1], yToL))*v; |
---|
3497 | } |
---|
3498 | else if (degree (bufFactors[0], v) > 0) |
---|
3499 | { |
---|
3500 | M (1,1)= mulMod2 (bufFactors [0] [0], bufFactors [1], yToL); |
---|
3501 | Pi [0]= M(1,1) + mulMod2 (bufFactors [0] [1], bufFactors[1], yToL)*v; |
---|
3502 | } |
---|
3503 | else if (degree (bufFactors[1], v) > 0) |
---|
3504 | { |
---|
3505 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1] [0], yToL); |
---|
3506 | Pi [0]= M (1,1) + mulMod2 (bufFactors [0], bufFactors[1] [1], yToL)*v; |
---|
3507 | } |
---|
3508 | else |
---|
3509 | { |
---|
3510 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1], yToL); |
---|
3511 | Pi [0]= M (1,1); |
---|
3512 | } |
---|
3513 | |
---|
3514 | for (i= 1; i < Pi.size(); i++) |
---|
3515 | { |
---|
3516 | if (degree (Pi[i-1], v) > 0 && degree (bufFactors [i+1], v) > 0) |
---|
3517 | { |
---|
3518 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors[i+1] [0], yToL); |
---|
3519 | Pi [i]= M (1,i+1) + (mulMod2 (Pi[i-1] [1], bufFactors[i+1] [0], yToL) + |
---|
3520 | mulMod2 (Pi[i-1] [0], bufFactors [i+1] [1], yToL))*v; |
---|
3521 | } |
---|
3522 | else if (degree (Pi[i-1], v) > 0) |
---|
3523 | { |
---|
3524 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors [i+1], yToL); |
---|
3525 | Pi [i]= M(1,i+1) + mulMod2 (Pi[i-1] [1], bufFactors[i+1], yToL)*v; |
---|
3526 | } |
---|
3527 | else if (degree (bufFactors[i+1], v) > 0) |
---|
3528 | { |
---|
3529 | M (1,i+1)= mulMod2 (Pi[i-1], bufFactors [i+1] [0], yToL); |
---|
3530 | Pi [i]= M (1,i+1) + mulMod2 (Pi[i-1], bufFactors[i+1] [1], yToL)*v; |
---|
3531 | } |
---|
3532 | else |
---|
3533 | { |
---|
3534 | M (1,i+1)= mulMod2 (Pi [i-1], bufFactors [i+1], yToL); |
---|
3535 | Pi [i]= M (1,i+1); |
---|
3536 | } |
---|
3537 | } |
---|
3538 | |
---|
3539 | CFList products; |
---|
3540 | bufF= mod (F, Variable (3)); |
---|
3541 | for (CFListIterator k= factors; k.hasItem(); k++) |
---|
3542 | products.append (bufF/k.getItem()); |
---|
3543 | |
---|
3544 | CFList MOD= CFList (power (v, liftBound)); |
---|
3545 | MOD.insert (yToL); |
---|
3546 | for (int d= 1; d < liftBound; d++) |
---|
3547 | { |
---|
3548 | henselStep2 (F, factors, bufFactors, diophant, M, Pi, products, d, MOD, bad); |
---|
3549 | if (bad) |
---|
3550 | return CFList(); |
---|
3551 | } |
---|
3552 | |
---|
3553 | CFList result; |
---|
3554 | for (i= 0; i < factors.length(); i++) |
---|
3555 | result.append (bufFactors[i]); |
---|
3556 | return result; |
---|
3557 | } |
---|
3558 | |
---|
3559 | CFList |
---|
3560 | nonMonicHenselLift (const CFList& F, const CFList& factors, const CFList& LCs, |
---|
3561 | CFList& diophant, CFArray& Pi, CFMatrix& M, const int lOld, |
---|
3562 | int& lNew, const CFList& MOD, bool& noOneToOne |
---|
3563 | ) |
---|
3564 | { |
---|
3565 | |
---|
3566 | int k= 0; |
---|
3567 | CFArray bufFactors= CFArray (factors.length()); |
---|
3568 | CFListIterator j= LCs; |
---|
3569 | for (CFListIterator i= factors; i.hasItem(); i++, j++, k++) |
---|
3570 | bufFactors [k]= replaceLC (i.getItem(), j.getItem()); |
---|
3571 | |
---|
3572 | Variable y= F.getLast().mvar(); |
---|
3573 | Variable x= F.getFirst().mvar(); |
---|
3574 | CanonicalForm xToLOld= power (x, lOld); |
---|
3575 | |
---|
3576 | Pi [0]= mod (Pi[0], xToLOld); |
---|
3577 | M (1, 1)= Pi [0]; |
---|
3578 | |
---|
3579 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
3580 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
3581 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
3582 | else if (degree (bufFactors[0], y) > 0) |
---|
3583 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
3584 | else if (degree (bufFactors[1], y) > 0) |
---|
3585 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
3586 | |
---|
3587 | for (int i= 1; i < Pi.size(); i++) |
---|
3588 | { |
---|
3589 | Pi [i]= mod (Pi [i], xToLOld); |
---|
3590 | M (1, i + 1)= Pi [i]; |
---|
3591 | |
---|
3592 | if (degree (Pi[i-1], y) > 0 && degree (bufFactors [i+1], y) > 0) |
---|
3593 | Pi [i] += (mulMod (Pi[i-1] [1], bufFactors[i+1] [0], MOD) + |
---|
3594 | mulMod (Pi[i-1] [0], bufFactors [i+1] [1], MOD))*y; |
---|
3595 | else if (degree (Pi[i-1], y) > 0) |
---|
3596 | Pi [i] += mulMod (Pi[i-1] [1], bufFactors[i+1], MOD)*y; |
---|
3597 | else if (degree (bufFactors[i+1], y) > 0) |
---|
3598 | Pi [i] += mulMod (Pi[i-1], bufFactors[i+1] [1], MOD)*y; |
---|
3599 | } |
---|
3600 | |
---|
3601 | CFList products; |
---|
3602 | CanonicalForm quot, bufF= F.getFirst(); |
---|
3603 | |
---|
3604 | for (int i= 0; i < bufFactors.size(); i++) |
---|
3605 | { |
---|
3606 | if (degree (bufFactors[i], y) > 0) |
---|
3607 | { |
---|
3608 | if (!fdivides (bufFactors[i] [0], bufF, quot)) |
---|
3609 | { |
---|
3610 | noOneToOne= true; |
---|
3611 | return factors; |
---|
3612 | } |
---|
3613 | products.append (quot); |
---|
3614 | } |
---|
3615 | else |
---|
3616 | { |
---|
3617 | if (!fdivides (bufFactors[i], bufF, quot)) |
---|
3618 | { |
---|
3619 | noOneToOne= true; |
---|
3620 | return factors; |
---|
3621 | } |
---|
3622 | products.append (quot); |
---|
3623 | } |
---|
3624 | } |
---|
3625 | |
---|
3626 | for (int d= 1; d < lNew; d++) |
---|
3627 | { |
---|
3628 | henselStep2 (F.getLast(), factors, bufFactors, diophant, M, Pi, products, d, |
---|
3629 | MOD, noOneToOne); |
---|
3630 | if (noOneToOne) |
---|
3631 | return CFList(); |
---|
3632 | } |
---|
3633 | |
---|
3634 | CFList result; |
---|
3635 | for (k= 0; k < factors.length(); k++) |
---|
3636 | result.append (bufFactors[k]); |
---|
3637 | return result; |
---|
3638 | } |
---|
3639 | |
---|
3640 | CFList |
---|
3641 | nonMonicHenselLift (const CFList& eval, const CFList& factors, |
---|
3642 | CFList* const& LCs, CFList& diophant, CFArray& Pi, |
---|
3643 | int* liftBound, int length, bool& noOneToOne |
---|
3644 | ) |
---|
3645 | { |
---|
3646 | CFList bufDiophant= diophant; |
---|
3647 | CFList buf= factors; |
---|
3648 | CFArray bufPi= Pi; |
---|
3649 | CFMatrix M= CFMatrix (liftBound[1], factors.length() - 1); |
---|
3650 | int k= 0; |
---|
3651 | |
---|
3652 | CFList result= |
---|
3653 | nonMonicHenselLift23 (eval.getFirst(), factors, LCs [0], diophant, bufPi, |
---|
3654 | liftBound[1], liftBound[0], noOneToOne); |
---|
3655 | |
---|
3656 | if (noOneToOne) |
---|
3657 | return CFList(); |
---|
3658 | |
---|
3659 | if (eval.length() == 1) |
---|
3660 | return result; |
---|
3661 | |
---|
3662 | k++; |
---|
3663 | CFList MOD; |
---|
3664 | for (int i= 0; i < 2; i++) |
---|
3665 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
3666 | |
---|
3667 | CFListIterator j= eval; |
---|
3668 | CFList bufEval; |
---|
3669 | bufEval.append (j.getItem()); |
---|
3670 | j++; |
---|
3671 | |
---|
3672 | for (int i= 2; i <= length && j.hasItem(); i++, j++, k++) |
---|
3673 | { |
---|
3674 | bufEval.append (j.getItem()); |
---|
3675 | M= CFMatrix (liftBound[i], factors.length() - 1); |
---|
3676 | result= nonMonicHenselLift (bufEval, result, LCs [i-1], diophant, bufPi, M, |
---|
3677 | liftBound[i-1], liftBound[i], MOD, noOneToOne); |
---|
3678 | if (noOneToOne) |
---|
3679 | return result; |
---|
3680 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
3681 | bufEval.removeFirst(); |
---|
3682 | } |
---|
3683 | |
---|
3684 | return result; |
---|
3685 | } |
---|
3686 | |
---|
3687 | #endif |
---|
3688 | /* HAVE_NTL */ |
---|
3689 | |
---|