1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file facFqFactorize.h |
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5 | * |
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6 | * This file provides functions for factorizing a multivariate polynomial over |
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7 | * \f$ F_{p} \f$ , \f$ F_{p}(\alpha ) \f$ or GF. |
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8 | * |
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9 | * ABSTRACT: So far factor recombination is done naive! |
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10 | * |
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11 | * @author Martin Lee |
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12 | * |
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13 | * @internal @version \$Id$ |
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14 | * |
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15 | **/ |
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16 | /*****************************************************************************/ |
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17 | |
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18 | #ifndef FAC_FQ_FACTORIZE_H |
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19 | #define FAC_FQ_FACTORIZE_H |
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20 | |
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21 | #include <config.h> |
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22 | |
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23 | #include "facFqBivar.h" |
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24 | #include "DegreePattern.h" |
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25 | #include "ExtensionInfo.h" |
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26 | #include "cf_util.h" |
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27 | #include "facFqSquarefree.h" |
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28 | #include "facFqBivarUtil.h" |
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29 | |
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30 | /// Factorization over a finite field |
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31 | /// |
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32 | /// @return @a multiFactorize returns a factorization of F |
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33 | /// @sa biFactorize(), extFactorize() |
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34 | CFList |
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35 | multiFactorize (const CanonicalForm& F, ///< [in] poly to be factored |
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36 | const ExtensionInfo& info ///< [in] info about extension |
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37 | ); |
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38 | |
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39 | /// factorize a squarefree multivariate polynomial over \f$ F_{p} \f$ |
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40 | /// |
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41 | /// @return @a FpSqrfFactorize returns a list of monic factors, the first |
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42 | /// element is the leading coefficient. |
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43 | /// @sa FqSqrfFactorize(), GFSqrfFactorize() |
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44 | inline |
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45 | CFList FpSqrfFactorize (const CanonicalForm & F ///< [in] a multivariate poly |
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46 | ) |
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47 | { |
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48 | if (getNumVars (F) == 2) |
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49 | return FpBiSqrfFactorize (F); |
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50 | ExtensionInfo info= ExtensionInfo (false); |
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51 | CFList result= multiFactorize (F, info); |
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52 | result.insert (Lc(F)); |
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53 | return result; |
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54 | } |
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55 | |
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56 | /// factorize a squarefree multivariate polynomial over \f$ F_{p} (\alpha ) \f$ |
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57 | /// |
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58 | /// @return @a FqSqrfFactorize returns a list of monic factors, the first |
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59 | /// element is the leading coefficient. |
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60 | /// @sa FpSqrfFactorize(), GFSqrfFactorize() |
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61 | inline |
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62 | CFList FqSqrfFactorize (const CanonicalForm & F, ///< [in] a multivariate poly |
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63 | const Variable& alpha ///< [in] algebraic variable |
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64 | ) |
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65 | { |
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66 | if (getNumVars (F) == 2) |
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67 | return FqBiSqrfFactorize (F, alpha); |
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68 | ExtensionInfo info= ExtensionInfo (alpha, false); |
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69 | CFList result= multiFactorize (F, info); |
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70 | result.insert (Lc(F)); |
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71 | return result; |
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72 | } |
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73 | |
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74 | /// factorize a squarefree multivariate polynomial over GF |
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75 | /// |
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76 | /// @return @a GFSqrfFactorize returns a list of monic factors, the first |
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77 | /// element is the leading coefficient. |
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78 | /// @sa FpSqrfFactorize(), FqSqrfFactorize() |
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79 | inline |
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80 | CFList GFSqrfFactorize (const CanonicalForm & F ///< [in] a multivariate poly |
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81 | ) |
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82 | { |
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83 | ASSERT (CFFactory::gettype() == GaloisFieldDomain, |
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84 | "GF as base field expected"); |
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85 | if (getNumVars (F) == 2) |
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86 | return GFBiSqrfFactorize (F); |
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87 | ExtensionInfo info= ExtensionInfo (getGFDegree(), gf_name, false); |
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88 | CFList result= multiFactorize (F, info); |
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89 | result.insert (Lc(F)); |
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90 | return result; |
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91 | } |
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92 | |
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93 | /// factorize a multivariate polynomial over \f$ F_{p} \f$ |
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94 | /// |
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95 | /// @return @a FpFactorize returns a list of monic factors with |
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96 | /// multiplicity, the first element is the leading coefficient. |
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97 | /// @sa FqFactorize(), GFFactorize() |
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98 | inline |
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99 | CFFList FpFactorize (const CanonicalForm& F ///< [in] a multivariate poly |
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100 | ) |
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101 | { |
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102 | if (getNumVars (F) == 2) |
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103 | return FpBiFactorize (F); |
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104 | ExtensionInfo info= ExtensionInfo (false); |
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105 | Variable a= Variable (1); |
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106 | CanonicalForm LcF= Lc (F); |
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107 | CanonicalForm pthRoot, A; |
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108 | CanonicalForm sqrfP= sqrfPart (F/Lc(F), pthRoot, a); |
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109 | CFList buf, bufRoot; |
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110 | CFFList result, resultRoot; |
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111 | int p= getCharacteristic(); |
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112 | int l; |
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113 | if (degree (pthRoot) > 0) |
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114 | { |
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115 | pthRoot= maxpthRoot (pthRoot, p, l); |
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116 | result= FpFactorize (pthRoot); |
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117 | result.removeFirst(); |
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118 | for (CFFListIterator i= result; i.hasItem(); i++) |
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119 | i.getItem()= CFFactor(i.getItem().factor(),i.getItem().exp()*ipower(p,l)); |
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120 | result.insert (CFFactor (LcF, 1)); |
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121 | return result; |
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122 | } |
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123 | else |
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124 | { |
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125 | buf= multiFactorize (sqrfP, info); |
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126 | A= F/LcF; |
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127 | result= multiplicity (A, buf); |
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128 | } |
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129 | if (degree (A) > 0) |
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130 | { |
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131 | resultRoot= FpFactorize (A); |
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132 | resultRoot.removeFirst(); |
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133 | result= Union (result, resultRoot); |
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134 | } |
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135 | result.insert (CFFactor (LcF, 1)); |
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136 | return result; |
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137 | } |
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138 | |
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139 | /// factorize a multivariate polynomial over \f$ F_{p} (\alpha ) \f$ |
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140 | /// |
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141 | /// @return @a FqFactorize returns a list of monic factors with |
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142 | /// multiplicity, the first element is the leading coefficient. |
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143 | /// @sa FpFactorize(), GFFactorize() |
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144 | inline |
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145 | CFFList FqFactorize (const CanonicalForm& F, ///< [in] a multivariate poly |
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146 | const Variable& alpha ///< [in] algebraic variable |
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147 | ) |
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148 | { |
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149 | if (getNumVars (F) == 2) |
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150 | return FqBiFactorize (F, alpha); |
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151 | ExtensionInfo info= ExtensionInfo (alpha, false); |
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152 | CanonicalForm LcF= Lc (F); |
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153 | CanonicalForm pthRoot, A; |
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154 | CanonicalForm sqrfP= sqrfPart (F/Lc(F), pthRoot, alpha); |
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155 | CFList buf, bufRoot; |
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156 | CFFList result, resultRoot; |
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157 | int p= getCharacteristic(); |
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158 | int q= ipower (p, degree (getMipo (alpha))); |
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159 | int l; |
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160 | if (degree (pthRoot) > 0) |
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161 | { |
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162 | pthRoot= maxpthRoot (pthRoot, q, l); |
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163 | result= FqFactorize (pthRoot, alpha); |
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164 | result.removeFirst(); |
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165 | for (CFFListIterator i= result; i.hasItem(); i++) |
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166 | i.getItem()= CFFactor(i.getItem().factor(),i.getItem().exp()*ipower(p,l)); |
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167 | result.insert (CFFactor (LcF, 1)); |
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168 | return result; |
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169 | } |
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170 | else |
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171 | { |
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172 | buf= multiFactorize (sqrfP, info); |
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173 | A= F/LcF; |
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174 | result= multiplicity (A, buf); |
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175 | } |
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176 | if (degree (A) > 0) |
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177 | { |
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178 | resultRoot= FqFactorize (A, alpha); |
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179 | resultRoot.removeFirst(); |
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180 | result= Union (result, resultRoot); |
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181 | } |
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182 | result.insert (CFFactor (LcF, 1)); |
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183 | return result; |
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184 | } |
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185 | |
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186 | /// factorize a multivariate polynomial over GF |
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187 | /// |
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188 | /// @return @a GFFactorize returns a list of monic factors with |
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189 | /// multiplicity, the first element is the leading coefficient. |
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190 | /// @sa FpFactorize(), FqFactorize() |
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191 | inline |
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192 | CFFList GFFactorize (const CanonicalForm& F ///< [in] a multivariate poly |
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193 | ) |
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194 | { |
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195 | ASSERT (CFFactory::gettype() == GaloisFieldDomain, |
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196 | "GF as base field expected"); |
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197 | if (getNumVars (F) == 2) |
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198 | return GFBiFactorize (F); |
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199 | Variable a= Variable (1); |
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200 | ExtensionInfo info= ExtensionInfo (getGFDegree(), gf_name, false); |
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201 | CanonicalForm LcF= Lc (F); |
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202 | CanonicalForm pthRoot, A; |
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203 | CanonicalForm sqrfP= sqrfPart (F/LcF, pthRoot, a); |
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204 | CFList buf; |
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205 | CFFList result, resultRoot; |
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206 | int p= getCharacteristic(); |
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207 | int q= ipower (p, getGFDegree()); |
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208 | int l; |
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209 | if (degree (pthRoot) > 0) |
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210 | { |
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211 | pthRoot= maxpthRoot (pthRoot, q, l); |
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212 | result= GFFactorize (pthRoot); |
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213 | result.removeFirst(); |
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214 | for (CFFListIterator i= result; i.hasItem(); i++) |
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215 | i.getItem()= CFFactor(i.getItem().factor(),i.getItem().exp()*ipower(p,l)); |
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216 | result.insert (CFFactor (LcF, 1)); |
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217 | return result; |
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218 | } |
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219 | else |
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220 | { |
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221 | buf= multiFactorize (sqrfP, info); |
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222 | A= F/LcF; |
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223 | result= multiplicity (A, buf); |
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224 | } |
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225 | if (degree (A) > 0) |
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226 | { |
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227 | resultRoot= GFFactorize (A); |
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228 | resultRoot.removeFirst(); |
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229 | result= Union (result, resultRoot); |
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230 | } |
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231 | result.insert (CFFactor (LcF, 1)); |
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232 | return result; |
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233 | } |
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234 | |
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235 | /// naive factor recombination for bivariate factorization. |
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236 | /// Uses precomputed data to exclude combinations that are not possible. |
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237 | /// |
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238 | /// @return @a monicFactorRecombi returns a list of factors of F, that are |
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239 | /// monic wrt Variable (1). |
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240 | /// @sa extFactorRecombination(), factorRecombination(), biFactorizer() |
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241 | CFList |
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242 | monicFactorRecombi ( |
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243 | const CFList& factors, ///< [in] list of lifted factors |
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244 | ///< that are monic wrt Variable (1) |
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245 | const CanonicalForm& F, ///< [in] bivariate poly |
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246 | const CanonicalForm& M, ///< [in] Variable (2)^liftBound |
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247 | const DegreePattern& degs ///< [in] degree pattern |
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248 | ); |
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249 | |
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250 | /// detects factors of @a F at stage @a deg of Hensel lifting. |
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251 | /// No combinations of more than one factor are tested. Lift bound and possible |
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252 | /// degree pattern are updated. |
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253 | /// |
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254 | /// @return @a earlyMonicFactorDetect returns a list of factors of F (possibly |
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255 | /// incomplete) that are monic wrt Variable (1) |
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256 | /// @sa monicFactorRecombi(), earlyFactorDetection(), monicFactorDetect(), |
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257 | /// biFactorizer() |
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258 | CFList |
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259 | earlyMonicFactorDetect ( |
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260 | CanonicalForm& F, ///< [in,out] poly to be factored, returns |
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261 | ///< poly divided by detected factors in case |
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262 | ///< of success |
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263 | CFList& factors, ///< [in,out] list of factors lifted up to |
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264 | ///< @a deg, returns a list of factors |
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265 | ///< without detected factors |
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266 | int& adaptedLiftBound, ///< [in,out] adapted lift bound |
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267 | DegreePattern& degs, ///< [in,out] degree pattern, is updated |
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268 | ///< whenever we find a factor |
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269 | bool& success, ///< [in,out] indicating success |
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270 | int deg, ///< [in] stage of Hensel lifting |
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271 | const int bound ///< [in] degree (A, 2) + 1 + |
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272 | ///< degree (LC (A, 1), 2), where A is the |
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273 | ///< multivariate polynomial corresponding to |
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274 | ///< F. |
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275 | ); |
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276 | |
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277 | /// Bivariate factorization. In contrast to biFactorize() the factors returned |
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278 | /// are monic wrt Variable (1), if @a F is not irreducible. And no factorization |
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279 | /// wrt Variable (2) are performed. However, |
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280 | /// |
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281 | /// @return @a biFactorizer returns a list of factors that are monic wrt |
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282 | /// Variable (1), if @a F is irreducible @a F is returned |
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283 | /// @sa multiFactorize(), biFactorize() |
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284 | CFList biFactorizer (const CanonicalForm& F, ///< [in] a bivariate poly |
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285 | const Variable& alpha, ///< [in] algebraic variable |
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286 | CanonicalForm& bivarEval, ///< [in,out] a valid evaluation |
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287 | ///< point |
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288 | int& liftBound ///< [in,out] lift bound, may be |
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289 | ///< adapted during Hensel |
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290 | ///< lifting |
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291 | ); |
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292 | |
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293 | /// Naive factor recombination for multivariate factorization over an extension |
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294 | /// of the initial field. No precomputed is used to exclude combinations. |
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295 | /// |
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296 | /// @return @a extFactorRecombination returns a list of factors of @a F, whose |
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297 | /// shift to zero is reversed. |
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298 | /// @sa factorRecombination() |
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299 | CFList |
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300 | extFactorRecombination ( |
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301 | const CFList& factors, ///< [in] list of lifted factors |
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302 | ///< that are monic wrt Variable (1) |
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303 | const CanonicalForm& F, ///< [in] poly to be factored |
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304 | const CFList& M, ///< [in] a list of powers of |
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305 | ///< Variables |
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306 | const ExtensionInfo& info, ///< [in] info about extension |
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307 | const CFList& evaluation ///< [in] evaluation point |
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308 | ); |
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309 | |
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310 | /// Naive factor recombination for multivariate factorization. |
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311 | /// No precomputed is used to exclude combinations. |
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312 | /// |
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313 | /// @return @a factorRecombination returns a list of factors of @a F |
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314 | /// @sa extFactorRecombination() |
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315 | CFList |
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316 | factorRecombination (const CanonicalForm& F,///< [in] poly to be factored |
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317 | const CFList& factors, ///< [in] list of lifted factors |
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318 | ///< that are monic wrt Variable (1) |
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319 | const CFList& M ///< [in] a list of powers of |
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320 | ///< Variables |
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321 | ); |
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322 | |
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323 | /// Lift bound adaption. Essentially an early factor detection but only the lift |
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324 | /// bound is adapted. |
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325 | /// |
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326 | /// @return @a liftBoundAdaption returns an adapted lift bound. |
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327 | /// @sa earlyFactorDetect(), earlyFactorDetection() |
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328 | int |
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329 | liftBoundAdaption (const CanonicalForm& F, ///< [in] a poly |
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330 | const CFList& factors, ///< [in] list of list of lifted |
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331 | ///< factors that are monic wrt |
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332 | ///< Variable (1) |
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333 | bool& success, ///< [in,out] indicates that no |
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334 | ///< further lifting is necessary |
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335 | const int deg, ///< [in] stage of Hensel lifting |
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336 | const CFList& MOD, ///< [in] a list of powers of |
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337 | ///< Variables |
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338 | const int bound ///< [in] initial lift bound |
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339 | ); |
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340 | |
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341 | /// Lift bound adaption over an extension of the initial field. Essentially an |
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342 | ///early factor detection but only the lift bound is adapted. |
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343 | /// |
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344 | /// @return @a liftBoundAdaption returns an adapted lift bound. |
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345 | /// @sa earlyFactorDetect(), earlyFactorDetection() |
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346 | int |
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347 | extLiftBoundAdaption ( |
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348 | const CanonicalForm& F, ///< [in] a poly |
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349 | const CFList& factors, ///< [in] list of list of lifted |
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350 | ///< factors that are monic wrt |
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351 | bool& success, ///< [in,out] indicates that no further |
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352 | ///< lifting is necessary |
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353 | const ExtensionInfo& info, ///< [in] info about extension |
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354 | const CFList& eval, ///< [in] evaluation point |
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355 | const int deg, ///< [in] stage of Hensel lifting |
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356 | const CFList& MOD, ///< [in] a list of powers of |
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357 | ///< Variables |
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358 | const int bound ///< [in] initial lift bound |
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359 | ); |
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360 | |
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361 | /// detects factors of @a F at stage @a deg of Hensel lifting. |
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362 | /// No combinations of more than one factor are tested. Lift bound is adapted. |
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363 | /// |
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364 | /// @return @a earlyFactorDetect returns a list of factors of F (possibly |
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365 | /// incomplete), in case of success. Otherwise an empty list. |
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366 | /// @sa factorRecombination(), extEarlyFactorDetect() |
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367 | CFList |
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368 | earlyFactorDetect ( |
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369 | CanonicalForm& F, ///< [in,out] poly to be factored, |
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370 | ///< returns poly divided by detected |
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371 | ///< factors in case of success |
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372 | CFList& factors, ///< [in,out] list of factors lifted up |
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373 | ///< to @a deg, returns a list of factors |
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374 | ///< without detected factors |
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375 | int& adaptedLiftBound, ///< [in,out] adapted lift bound |
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376 | bool& success, ///< [in,out] indicating success |
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377 | const int deg, ///< [in] stage of Hensel lifting |
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378 | const CFList& MOD, ///< [in] a list of powers of |
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379 | ///< Variables |
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380 | const int bound ///< [in] initial lift bound |
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381 | ); |
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382 | |
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383 | /// detects factors of @a F at stage @a deg of Hensel lifting. |
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384 | /// No combinations of more than one factor are tested. Lift bound is adapted. |
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385 | /// |
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386 | /// @return @a extEarlyFactorDetect returns a list of factors of F (possibly |
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387 | /// incomplete), whose shift to zero is reversed, in case of success. |
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388 | /// Otherwise an empty list. |
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389 | /// @sa factorRecombination(), earlyFactorDetection() |
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390 | CFList |
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391 | extEarlyFactorDetect ( |
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392 | CanonicalForm& F, ///< [in,out] poly to be factored, |
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393 | ///< returns poly divided by detected |
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394 | ///< factors in case of success |
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395 | CFList& factors, ///< [in,out] list of factors lifted up |
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396 | ///< to @a deg, returns a list of factors |
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397 | ///< without detected factors |
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398 | int& adaptedLiftBound, ///< [in,out] adapted lift bound |
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399 | bool& success, ///< [in,out] indicating succes |
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400 | const ExtensionInfo& info, ///< [in] info about extension |
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401 | const CFList& eval, ///< [in] evaluation point |
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402 | const int deg, ///< [in] stage of Hensel lifting |
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403 | const CFList& MOD, ///< [in] a list of powers of Variables |
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404 | const int bound ///< [in] initial lift bound |
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405 | ); |
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406 | |
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407 | /// evaluation point search for multivariate factorization, |
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408 | /// looks for a (F.level() - 1)-tuple such that the resulting univariate |
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409 | /// polynomial has main variable F.mvar(), is squarefree and its degree |
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410 | /// coincides with degree(F) and the bivariate one is primitive wrt. |
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411 | /// Variable(1), fails if there are no valid evaluation points, eval contains |
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412 | /// the intermediate evaluated polynomials. |
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413 | /// |
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414 | /// @return @a evalPoints returns an evaluation point, which is valid if and |
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415 | /// only if fail == false. |
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416 | CFList |
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417 | evalPoints (const CanonicalForm& F, ///< [in] a compressed poly |
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418 | CFList & eval, ///< [in,out] an empty list, returns @a F |
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419 | ///< successive evaluated |
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420 | const Variable& alpha, ///< [in] algebraic variable |
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421 | CFList& list, ///< [in,out] a list of points already |
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422 | ///< considered, a point is encoded as a |
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423 | ///< poly of degree F.level()-1 in |
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424 | ///< Variable(1) |
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425 | const bool& GF, ///< [in] GF? |
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426 | bool& fail ///< [in,out] indicates failure |
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427 | ); |
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428 | |
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429 | /// hensel Lifting and early factor detection |
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430 | /// |
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431 | /// @return @a henselLiftAndEarly returns monic (wrt Variable (1)) lifted |
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432 | /// factors without factors which have been detected at an early stage |
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433 | /// of Hensel lifting |
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434 | /// @sa earlyFactorDetectn(), extEarlyFactorDetect() |
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435 | CFList |
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436 | henselLiftAndEarly ( |
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437 | CanonicalForm& A, ///< [in,out] poly to be factored, |
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438 | ///< returns poly divided by detected |
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439 | ///< factors, in case of success |
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440 | CFList& MOD, ///< [in,out] a list of powers of |
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441 | ///< Variables |
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442 | int*& liftBounds, ///< [in,out] initial lift bounds, returns |
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443 | ///< adapted lift bounds |
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444 | bool& earlySuccess, ///< [in,out] indicating success |
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445 | CFList& earlyFactors, ///< [in,out] early factors |
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446 | const CFList& Aeval, ///< [in] @a A successively evaluated at |
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447 | ///< elements of @a evaluation |
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448 | const CFList& biFactors, ///< [in] bivariate factors |
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449 | const CFList& evaluation, ///< [in] evaluation point |
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450 | const ExtensionInfo& info ///< [in] info about extension |
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451 | ); |
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452 | |
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453 | /// Factorization over an extension of initial field |
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454 | /// |
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455 | /// @return @a extFactorize returns factorization of F over initial field |
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456 | /// @sa extBiFactorize(), multiFactorize() |
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457 | CFList |
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458 | extFactorize (const CanonicalForm& F, ///< [in] poly to be factored |
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459 | const ExtensionInfo& info ///< [in] info about extension |
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460 | ); |
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461 | |
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462 | #endif |
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463 | /* FAC_FQ_FACTORIZE_H */ |
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464 | |
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