1 | /*****************************************************************************\ |
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2 | * Computer Algebra System SINGULAR |
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3 | \*****************************************************************************/ |
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4 | /** @file facFqSquarefree.cc |
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5 | * |
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6 | * This file provides functions for squarefrees factorizing over |
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7 | * \f$ F_{p} \f$ , \f$ F_{p}(\alpha ) \f$ or GF, as decribed in "Factoring |
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8 | * multivariate polynomials over a finite field" by L. Bernardin |
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9 | * |
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10 | * @author Martin Lee |
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11 | * |
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12 | **/ |
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13 | /*****************************************************************************/ |
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14 | |
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15 | #include "config.h" |
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16 | |
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17 | #include "canonicalform.h" |
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18 | |
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19 | #include "cf_gcd_smallp.h" |
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20 | #include "cf_iter.h" |
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21 | #include "cf_map.h" |
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22 | #include "cf_util.h" |
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23 | #include "cf_factory.h" |
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24 | #include "facFqSquarefree.h" |
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25 | |
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26 | static inline |
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27 | CanonicalForm |
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28 | pthRoot (const CanonicalForm & F, const int & q) |
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29 | { |
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30 | CanonicalForm A= F; |
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31 | int p= getCharacteristic (); |
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32 | if (A.inCoeffDomain()) |
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33 | { |
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34 | A= power (A, q/p); |
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35 | return A; |
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36 | } |
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37 | else |
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38 | { |
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39 | CanonicalForm buf= 0; |
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40 | for (CFIterator i= A; i.hasTerms(); i++) |
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41 | buf= buf + power(A.mvar(), i.exp()/p)*pthRoot (i.coeff(), q); |
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42 | return buf; |
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43 | } |
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44 | } |
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45 | |
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46 | CanonicalForm |
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47 | maxpthRoot (const CanonicalForm & F, const int & q, int& l) |
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48 | { |
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49 | CanonicalForm result= F; |
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50 | bool derivZero= true; |
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51 | l= 0; |
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52 | while (derivZero) |
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53 | { |
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54 | for (int i= 1; i <= result.level(); i++) |
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55 | { |
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56 | if (!deriv (result, Variable (i)).isZero()) |
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57 | { |
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58 | derivZero= false; |
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59 | break; |
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60 | } |
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61 | } |
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62 | if (!derivZero) |
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63 | break; |
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64 | result= pthRoot (result, q); |
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65 | l++; |
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66 | } |
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67 | return result; |
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68 | } |
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69 | |
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70 | static inline |
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71 | CFFList |
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72 | sqrfPosDer (const CanonicalForm & F, const Variable & x, |
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73 | CanonicalForm & c) |
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74 | { |
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75 | CanonicalForm b= deriv (F, x); |
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76 | c= gcd (F, b); |
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77 | CanonicalForm w= F/c; |
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78 | CanonicalForm v= b/c; |
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79 | CanonicalForm u= v - deriv (w, x); |
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80 | int j= 1; |
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81 | int p= getCharacteristic(); |
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82 | CanonicalForm g; |
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83 | CFFList result; |
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84 | while (j < p - 1 && degree(u) >= 0) |
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85 | { |
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86 | g= gcd (w, u); |
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87 | if (!g.inCoeffDomain()) |
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88 | result.append (CFFactor (g, j)); |
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89 | w= w/g; |
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90 | c= c/w; |
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91 | v= u/g; |
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92 | u= v - deriv (w, x); |
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93 | j++; |
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94 | } |
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95 | if (!w.inCoeffDomain()) |
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96 | result.append (CFFactor (w, j)); |
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97 | return result; |
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98 | } |
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99 | |
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100 | CFFList |
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101 | squarefreeFactorization (const CanonicalForm & F, const Variable & alpha) |
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102 | { |
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103 | int p= getCharacteristic(); |
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104 | CanonicalForm A= F; |
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105 | CFMap M; |
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106 | A= compress (A, M); |
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107 | Variable x= A.mvar(); |
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108 | int l= x.level(); |
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109 | int k; |
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110 | if (CFFactory::gettype() == GaloisFieldDomain) |
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111 | k= getGFDegree(); |
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112 | else if (alpha.level() != 1) |
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113 | k= degree (getMipo (alpha)); |
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114 | else |
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115 | k= 1; |
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116 | Variable buf, buf2; |
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117 | CanonicalForm tmp; |
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118 | |
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119 | CFFList tmp1, tmp2; |
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120 | bool found; |
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121 | for (int i= l; i > 0; i--) |
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122 | { |
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123 | buf= Variable (i); |
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124 | if (degree (deriv (A, buf)) >= 0) |
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125 | { |
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126 | tmp1= sqrfPosDer (A, buf, tmp); |
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127 | A= tmp; |
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128 | for (CFFListIterator j= tmp1; j.hasItem(); j++) |
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129 | { |
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130 | found= false; |
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131 | CFFListIterator k= tmp2; |
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132 | if (!k.hasItem() && !j.getItem().factor().inCoeffDomain()) tmp2.append (j.getItem()); |
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133 | else |
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134 | { |
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135 | for (; k.hasItem(); k++) |
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136 | { |
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137 | if (k.getItem().exp() == j.getItem().exp()) |
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138 | { |
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139 | k.getItem()= CFFactor (k.getItem().factor()*j.getItem().factor(), |
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140 | j.getItem().exp()); |
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141 | found= true; |
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142 | } |
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143 | } |
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144 | if (found == false && !j.getItem().factor().inCoeffDomain()) |
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145 | tmp2.append(j.getItem()); |
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146 | } |
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147 | } |
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148 | } |
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149 | } |
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150 | |
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151 | bool degcheck= false;; |
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152 | for (int i= l; i > 0; i--) |
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153 | if (degree (A, Variable(i)) >= p) |
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154 | degcheck= true; |
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155 | |
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156 | if (degcheck == false && tmp1.isEmpty() && tmp2.isEmpty()) |
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157 | return CFFList (CFFactor (F/Lc(F), 1)); |
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158 | |
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159 | CanonicalForm buffer= pthRoot (A, ipower (p, k)); |
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160 | |
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161 | tmp1= squarefreeFactorization (buffer, alpha); |
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162 | |
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163 | CFFList result; |
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164 | buf= alpha; |
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165 | for (CFFListIterator i= tmp2; i.hasItem(); i++) |
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166 | { |
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167 | for (CFFListIterator j= tmp1; j.hasItem(); j++) |
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168 | { |
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169 | tmp= gcd (i.getItem().factor(), j.getItem().factor()); |
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170 | i.getItem()= CFFactor (i.getItem().factor()/tmp, i.getItem().exp()); |
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171 | j.getItem()= CFFactor (j.getItem().factor()/tmp, j.getItem().exp()); |
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172 | if (!tmp.inCoeffDomain()) |
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173 | { |
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174 | tmp= M (tmp); |
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175 | result.append (CFFactor (tmp/Lc(tmp), |
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176 | j.getItem().exp()*p + i.getItem().exp())); |
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177 | } |
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178 | } |
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179 | } |
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180 | for (CFFListIterator i= tmp2; i.hasItem(); i++) |
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181 | { |
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182 | if (!i.getItem().factor().inCoeffDomain()) |
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183 | { |
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184 | tmp= M (i.getItem().factor()); |
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185 | result.append (CFFactor (tmp/Lc(tmp), i.getItem().exp())); |
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186 | } |
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187 | } |
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188 | for (CFFListIterator j= tmp1; j.hasItem(); j++) |
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189 | { |
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190 | if (!j.getItem().factor().inCoeffDomain()) |
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191 | { |
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192 | tmp= M (j.getItem().factor()); |
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193 | result.append (CFFactor (tmp/Lc(tmp), j.getItem().exp()*p)); |
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194 | } |
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195 | } |
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196 | return result; |
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197 | } |
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198 | |
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199 | CanonicalForm |
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200 | sqrfPart (const CanonicalForm& F, CanonicalForm& pthPower, |
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201 | const Variable& alpha) |
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202 | { |
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203 | if (F.inCoeffDomain()) |
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204 | { |
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205 | pthPower= 1; |
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206 | return F; |
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207 | } |
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208 | CFMap M; |
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209 | CanonicalForm A= compress (F, M); |
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210 | Variable vBuf= alpha; |
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211 | CanonicalForm w, v, b; |
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212 | pthPower= 1; |
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213 | CanonicalForm result; |
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214 | int i= 1; |
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215 | bool allZero= true; |
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216 | for (; i <= A.level(); i++) |
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217 | { |
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218 | if (!deriv (A, Variable (i)).isZero()) |
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219 | { |
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220 | allZero= false; |
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221 | break; |
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222 | } |
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223 | } |
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224 | if (allZero) |
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225 | { |
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226 | pthPower= F; |
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227 | return 1; |
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228 | } |
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229 | w= gcd (A, deriv (A, Variable (i))); |
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230 | |
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231 | b= A/w; |
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232 | result= b; |
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233 | if (degree (w) < 1) |
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234 | return M (result); |
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235 | i++; |
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236 | for (; i <= A.level(); i++) |
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237 | { |
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238 | if (!deriv (w, Variable (i)).isZero()) |
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239 | { |
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240 | b= w; |
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241 | w= gcd (w, deriv (w, Variable (i))); |
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242 | b /= w; |
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243 | if (degree (b) < 1) |
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244 | break; |
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245 | CanonicalForm g; |
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246 | g= gcd (b, result); |
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247 | if (degree (g) > 0) |
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248 | result *= b/g; |
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249 | if (degree (g) <= 0) |
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250 | result *= b; |
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251 | } |
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252 | } |
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253 | result= M (result); |
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254 | return result; |
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255 | } |
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256 | |
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