1 | // |
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2 | version="$Id$"; |
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3 | category="Factorization"; |
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4 | info=" |
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5 | LIBRARY: absfact.lib Absolute factorization for characteristic 0 |
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6 | AUTHORS: Wolfram Decker, decker at math.uni-sb.de |
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7 | Gregoire Lecerf, lecerf at math.uvsq.fr |
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8 | Gerhard Pfister, pfister at mathematik.uni-kl.de |
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9 | |
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10 | OVERVIEW: |
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11 | A library for computing the absolute factorization of multivariate |
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12 | polynomials f with coefficients in a field K of characteristic zero. |
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13 | Using Trager's idea, the implemented algorithm computes an absolutely |
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14 | irreducible factor by factorizing over some finite extension field L |
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15 | (which is chosen such that V(f) has a smooth point with coordinates in L). |
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16 | Then a minimal extension field is determined making use of the |
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17 | Rothstein-Trager partial fraction decomposition algorithm. |
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18 | |
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19 | REFERENCES: |
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20 | G. Cheze, G. Lecerf: Lifting and recombination techniques for absolute |
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21 | factorization. Journal of Complexity, 23(3):380-420, 2007. |
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22 | |
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23 | KEYWORDS: factorization; absolute factorization. |
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24 | SEE ALSO: factorize |
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25 | |
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26 | PROCEDURES: |
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27 | absFactorize(); absolute factorization of poly |
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28 | absBiFactorize(); absolute factorization of bivariate poly |
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29 | "; |
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30 | |
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31 | //////////////////////////////////////////////////////////////////// |
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32 | static proc partialDegree(poly p, int i) |
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33 | "USAGE: partialDegree(p,i); p poly, i int |
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34 | RETURN: int, the degree of p in the i-th variable |
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35 | " |
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36 | { |
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37 | int n = nvars(basering); |
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38 | intvec tmp; |
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39 | tmp[n] = 0; |
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40 | tmp[i] = 1; |
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41 | return(deg(p,tmp)); |
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42 | } |
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43 | //////////////////////////////////////////////////////////////////// |
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44 | static proc belongTo(string s, list l) |
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45 | "USAGE: belongTo(s,l); s string, l list |
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46 | RETURN: 1 if s belongs to l, 0 otherwise |
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47 | " |
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48 | { |
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49 | string tmp; |
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50 | for(int i = 1; i <= size(l); i++) { |
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51 | tmp = l[i]; |
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52 | if (tmp == s) { |
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53 | return(1); |
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54 | } |
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55 | } |
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56 | return(0); |
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57 | } |
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58 | //////////////////////////////////////////////////////////////////// |
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59 | static proc variableWithSmallestPositiveDegree(poly p) |
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60 | "USAGE: variableWithSmallestPositiveDegree(p); p poly |
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61 | RETURN: int; 0 if p is constant. Otherwise, the index of the |
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62 | variable which has the smallest positive degree in p. |
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63 | " |
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64 | { |
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65 | int n = nvars(basering); |
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66 | int v = 0; |
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67 | int d = deg(p); |
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68 | int d_loc; |
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69 | for(int i = 1; i <= n; i++) { |
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70 | d_loc = partialDegree(p, i); |
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71 | if (d_loc >= 1 and d_loc <= d) { |
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72 | v = i; |
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73 | d = d_loc; |
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74 | } |
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75 | } |
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76 | return(v); |
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77 | } |
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78 | //////////////////////////////////////////////////////////////////// |
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79 | static proc smallestProperSimpleFactor(poly p) |
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80 | "USAGE: smallestProperSimpleFactor(p); p poly |
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81 | RETURN: poly: a proper irreducible simple factor of p of smallest |
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82 | degree. If no such factor exists, 0 is returned. |
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83 | " |
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84 | { |
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85 | list p_facts = factorize(p); |
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86 | int s = size(p_facts[1]); |
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87 | int d = deg(p)+1; |
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88 | poly q = 0; |
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89 | poly f; |
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90 | int e; |
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91 | for(int i = 1; i <= s; i++) |
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92 | { |
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93 | f = p_facts[1][i]; |
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94 | e = deg(f); |
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95 | if (e >= 1 and e < d and p_facts[2][i] == 1) |
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96 | { |
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97 | q = f / leadcoef(f); |
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98 | d = e; |
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99 | } |
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100 | } |
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101 | return(q); |
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102 | } |
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103 | //////////////////////////////////////////////////////////////////// |
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104 | static proc smallestProperFactor(poly p) |
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105 | "USAGE: smallestProperFactor(p); p poly |
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106 | RETURN: poly: a proper irreducible factor of p of smallest degree. |
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107 | If p is constant, 0 is returned. |
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108 | " |
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109 | { |
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110 | list p_facts = factorize(p); |
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111 | int s = size(p_facts[1]); |
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112 | int d = deg(p)+1; |
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113 | poly q = 0; |
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114 | poly f; |
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115 | int e; |
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116 | for(int i = 1; i <= s; i++) |
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117 | { |
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118 | f = p_facts[1][i]; |
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119 | e = deg(f); |
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120 | if (e >= 1 and e < d) |
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121 | { |
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122 | q = f / leadcoef(f); |
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123 | d = e; |
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124 | } |
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125 | } |
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126 | return(q); |
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127 | } |
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128 | //////////////////////////////////////////////////////////////////// |
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129 | static proc extensionContainingSmoothPoint(poly p, int m) |
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130 | "USAGE: extensionContainingSmoothPoint(p,m); p poly, m int |
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131 | RETURN: poly: an irreducible univariate polynomial that defines an |
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132 | algebraic extension of the current ground field that contains |
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133 | a smooth point of the hypersurface defined by p=0. |
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134 | " |
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135 | { |
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136 | int n = nvars(basering) - 1; |
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137 | poly q = 0; |
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138 | int i; |
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139 | list a; |
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140 | for(i=1;i<=n+1;i++){a[i] = 0;} |
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141 | a[m] = var(n+1); |
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142 | // The list a is to be taken with random entries in [-e, e]. |
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143 | // Every 10 * n trial, e is incremented by 1. |
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144 | int e = 1; |
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145 | int nbtrial = 0; |
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146 | map h; |
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147 | while (q == 0) |
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148 | { |
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149 | h = basering, a[1..n+1]; |
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150 | q = smallestProperSimpleFactor(h(p)); |
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151 | for(i = 1; i <= n ; i = i + 1) |
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152 | { |
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153 | if (i != m) |
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154 | { |
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155 | a[i] = random(-e, e); |
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156 | } |
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157 | } |
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158 | nbtrial++; |
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159 | if (nbtrial >= 10 * n) |
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160 | { |
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161 | e = e + 1; |
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162 | nbtrial = 0; |
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163 | } |
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164 | } |
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165 | return(q); |
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166 | } |
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167 | //////////////////////////////////////////////////////////////////// |
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168 | static proc RothsteinTragerResultant(poly g, poly f, int m) |
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169 | "USAGE: RothsteinTragerResultant(g,f,m); g,f poly, m int |
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170 | RETURN: poly |
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171 | NOTE: To be called by the RothsteinTrager procedure only. |
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172 | " |
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173 | { |
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174 | def MPz = basering; |
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175 | int n = nvars(MPz) - 1; |
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176 | int d = partialDegree(f, m); |
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177 | poly df = diff(f, var(m)); |
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178 | list a; |
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179 | int i; |
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180 | for(i=1;i<=n+1;i++){ a[i] = 0; } |
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181 | a[m] = var(m); |
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182 | poly q = 0; |
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183 | int e = 1; |
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184 | int nbtrial = 0; |
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185 | map h; |
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186 | while (q == 0) |
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187 | { |
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188 | h = MPz, a[1..n+1]; |
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189 | q = resultant(h(f), h(df) * var(n+1) - h(g), var(m)); |
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190 | if (deg(q) == d) |
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191 | { |
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192 | return(q/leadcoef(q)); |
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193 | } |
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194 | q = 0; |
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195 | for(i = 1; i <= n ; i++) |
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196 | { |
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197 | if (i != m) |
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198 | { |
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199 | a[i] = random(-e, e); |
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200 | } |
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201 | } |
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202 | nbtrial++; |
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203 | if (nbtrial >= 10 * n) |
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204 | { |
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205 | e++; |
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206 | nbtrial = 0; |
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207 | } |
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208 | } |
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209 | } |
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210 | //////////////////////////////////////////////////////////////////// |
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211 | static proc RothsteinTrager(list g, poly p, int m, int expectedDegQ) |
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212 | "USAGE: RothsteinTrager(g,p,m,d); g list, p poly, m,d int |
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213 | RETURN: list L consisting of two entries of type poly |
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214 | NOTE: the return value is the Rothstein-Trager partial fraction |
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215 | decomposition of the quotient s/p, where s is a generic linear |
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216 | combination of the elements of g. The genericity via d |
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217 | (the expected degree of L[1]). |
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218 | " |
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219 | { |
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220 | def MPz = basering; |
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221 | int n = nvars(MPz) - 1; |
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222 | poly dp = diff(p, var(m)); |
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223 | int r = size(g); |
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224 | list a; |
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225 | int i; |
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226 | for(i=1;i<=r;i++){a[i] = 0;} |
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227 | a[r] = 1; |
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228 | int nbtrial = 0; |
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229 | int e = 1; |
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230 | poly s; |
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231 | poly q; |
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232 | while (1) |
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233 | { |
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234 | s = 0; |
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235 | for(i = 1; i <= r; i++){s = s + a[i] * g[i];} |
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236 | q = RothsteinTragerResultant(s, p, m); |
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237 | q = smallestProperFactor(q); |
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238 | if (deg(q) == expectedDegQ) |
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239 | { |
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240 | // Go into the quotient by q(z)=0 |
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241 | ring MP_z = (0,var(n+1)), (x(1..n)), dp; |
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242 | list lMP_z = ringlist(MP_z); |
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243 | lMP_z[1][4] = ideal(imap(MPz,q)); |
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244 | list tmp = ringlist(MPz)[2]; |
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245 | lMP_z[2] = list(tmp[1..n]); |
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246 | def MPq = ring(lMP_z); |
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247 | setring(MPq); |
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248 | poly f = gcd(imap(MPz, p), par(1) * imap(MPz, dp) - imap(MPz, s)); |
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249 | f = f / leadcoef(f); |
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250 | setring(MPz); |
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251 | return(list(q, imap(MPq, f))); |
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252 | } |
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253 | for(i = 1; i <= r ; i++) |
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254 | { |
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255 | a[i] = random(-e, e); |
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256 | } |
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257 | nbtrial++; |
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258 | if (nbtrial >= 10 * r) |
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259 | { |
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260 | e++; |
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261 | nbtrial = 0; |
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262 | } |
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263 | } |
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264 | } |
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265 | //////////////////////////////////////////////////////////////////// |
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266 | static proc absFactorizeIrreducible(poly p) |
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267 | "USAGE: absFactorizeIrreducible(p); p poly |
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268 | ASSUME: p is an irreducible polynomial that does not depend on the last |
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269 | variable @z of the basering. |
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270 | RETURN: list L of two polynomials: q=L[1] is an irreducible polynomial of |
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271 | minimal degree in @z such that p has an absolute factor |
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272 | over K[@z]/<q>, and f represents such an absolute factor. |
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273 | " |
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274 | { |
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275 | int dblevel = printlevel - voice + 2; |
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276 | dbprint(dblevel,"Entering absfact.lib::absFactorizeIrreducible with ",p); |
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277 | def MPz = basering; |
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278 | int d = deg(p); |
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279 | int n = nvars(MPz) - 1; |
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280 | |
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281 | if (d < 1) |
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282 | { |
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283 | return(list(var(n+1), p)); |
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284 | } |
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285 | int m = variableWithSmallestPositiveDegree(p); |
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286 | // var(m) is now considered as the main variable. |
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287 | |
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288 | poly q = extensionContainingSmoothPoint(p, m); |
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289 | int r = deg(q); |
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290 | if (r == 1) |
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291 | { |
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292 | return(list(var(n+1), p)); |
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293 | } |
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294 | |
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295 | list tmp = ringlist(MPz)[2]; |
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296 | // Go into the quotient by q(z)=0 |
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297 | ring MP_z = (0,var(n+1)), (x(1..n)), dp; |
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298 | list lMP_z = ringlist(MP_z); |
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299 | lMP_z[1][4] = ideal(imap(MPz,q)); |
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300 | lMP_z[2] = list(tmp[1..n]); |
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301 | def MPq = ring(lMP_z); |
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302 | setring(MPq); |
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303 | dbprint(dblevel-1,"Factoring in algebraic extension"); |
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304 | // "Factoring p in the algebraic extension..."; |
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305 | poly p_loc = imap(MPz, p); |
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306 | poly f = smallestProperSimpleFactor(p_loc); |
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307 | int degf = deg(f); |
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308 | |
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309 | if (degf == d) |
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310 | { |
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311 | setring(MPz); |
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312 | return(list(var(n+1), p)); |
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313 | } |
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314 | |
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315 | if (degf * r == d) |
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316 | { |
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317 | setring(MPz); |
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318 | return(list(q, imap(MPq, f))); |
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319 | } |
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320 | dbprint(dblevel-1,"Absolutely irreducible factor found"); |
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321 | dbprint(dblevel,"Minimizing field extension"); |
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322 | // "Need to find a minimal extension"; |
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323 | poly co_f = p_loc / f; |
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324 | poly e = diff(f, var(m)) * co_f; |
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325 | setring(MPz); |
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326 | poly e = imap(MPq, e); |
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327 | list g; |
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328 | int i; |
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329 | for(i = 1; i <= r; i++) |
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330 | { |
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331 | g[i] = subst(e, var(n+1), 0); |
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332 | e = diff(e, var(n+1)); |
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333 | } |
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334 | return(RothsteinTrager(g, p, m, d div degf)); |
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335 | } |
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336 | |
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337 | //////////////////////////////////////////////////////////////////// |
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338 | proc absFactorize(poly p, list #) |
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339 | "USAGE: absFactorize(p [,s]); p poly, s string |
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340 | ASSUME: coefficient field is the field of rational numbers or a |
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341 | transcendental extension thereof |
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342 | RETURN: ring @code{R} which is obtained from the current basering |
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343 | by adding a new parameter (if a string @code{s} is given as a |
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344 | second input, the new parameter gets this string as name). The ring |
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345 | @code{R} comes with a list @code{absolute_factors} with the |
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346 | following entries: |
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347 | @format |
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348 | absolute_factors[1]: ideal (the absolute factors) |
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349 | absolute_factors[2]: intvec (the multiplicities) |
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350 | absolute_factors[3]: ideal (the minimal polynomials) |
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351 | absolute_factors[4]: int (total number of nontriv. absolute factors) |
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352 | @end format |
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353 | The entry @code{absolute_factors[1][1]} is a constant, the |
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354 | entry @code{absolute_factors[3][1]} is the parameter added to the |
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355 | current ring.@* |
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356 | Each of the remaining entries @code{absolute_factors[1][j]} stands for |
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357 | a class of conjugated absolute factors. The corresponding entry |
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358 | @code{absolute_factors[3][j]} is the minimal polynomial of the |
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359 | field extension over which the factor is minimally defined (its degree |
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360 | is the number of conjugates in the class). If the entry |
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361 | @code{absolute_factors[3][j]} coincides with @code{absolute_factors[3][1]}, |
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362 | no field extension was necessary for the @code{j}th (class of) |
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363 | absolute factor(s). |
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364 | NOTE: All factors are presented denominator- and content-free. The constant |
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365 | factor (first entry) is chosen such that the product of all (!) the |
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366 | (denominator- and content-free) absolute factors of @code{p} equals |
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367 | @code{p / absolute_factors[1][1]}. |
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368 | SEE ALSO: factorize, absPrimdecGTZ |
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369 | EXAMPLE: example absFactorize; shows an example |
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370 | " |
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371 | { |
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372 | int dblevel = printlevel - voice + 2; |
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373 | dbprint(dblevel,"Entering absfact.lib::absFactorize with ",p); |
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374 | def MP = basering; |
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375 | int i; |
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376 | if (char(MP) != 0) |
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377 | { |
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378 | ERROR("// absfact.lib::absFactorize is only implemented for "+ |
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379 | "characteristic 0"); |
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380 | } |
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381 | if(minpoly!=0) |
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382 | { |
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383 | ERROR("// absfact.lib::absFactorize is not implemented for algebraic " |
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384 | +"extensions"); |
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385 | } |
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386 | |
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387 | int n = nvars(MP); |
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388 | int pa=npars(MP); |
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389 | list lMP= ringlist(MP); |
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390 | list buflMP= lMP; |
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391 | intvec vv,vk; |
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392 | for(i=1;i<=n;i++){vv[i]=1;} |
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393 | vk=vv,1; |
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394 | |
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395 | //if the basering has parameters, add the parameters to the variables |
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396 | //takes care about coefficients and possible denominators |
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397 | if(pa>0) |
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398 | { |
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399 | poly qh=cleardenom(p); |
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400 | if (p==0) |
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401 | { |
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402 | number cok=0; |
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403 | } |
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404 | else |
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405 | { |
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406 | number cok=leadcoef(p)/leadcoef(qh); |
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407 | } |
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408 | p=qh; |
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409 | string sp; |
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410 | for(i=1;i<=npars(basering);i++) |
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411 | { |
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412 | sp=string(par(i)); |
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413 | sp=sp[2..size(sp)-1]; |
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414 | lMP[2][n+i]=sp; |
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415 | vv=vv,1; |
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416 | } |
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417 | lMP[1]=0; |
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418 | n=n+npars(MP); |
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419 | } |
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420 | |
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421 | // MPz is obtained by adding the new variable @z to MP |
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422 | // ordering is wp(1...1) |
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423 | // All the above subroutines work in MPz |
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424 | string newvar; |
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425 | if(size(#)>0) |
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426 | { |
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427 | if(typeof(#[1])=="string") |
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428 | { |
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429 | newvar=#[1]; |
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430 | } |
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431 | else |
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432 | { |
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433 | newvar = "a"; |
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434 | } |
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435 | } |
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436 | else |
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437 | { |
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438 | newvar = "a"; |
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439 | } |
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440 | if (newvar=="a") |
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441 | { |
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442 | if(belongTo(newvar, lMP[2])||defined(a)){newvar = "b";} |
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443 | if(belongTo(newvar, lMP[2])||defined(b)){newvar = "c";} |
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444 | if(belongTo(newvar, lMP[2])||defined(c)){newvar = "@c";} |
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445 | while(belongTo(newvar, lMP[2])) |
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446 | { |
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447 | newvar = "@" + newvar; |
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448 | } |
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449 | } |
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450 | lMP[2][n+1] = newvar; |
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451 | |
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452 | // new ordering |
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453 | vv=vv,1; |
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454 | list orst; |
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455 | orst[1]=list("wp",vv); |
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456 | orst[2]=list("C",0); |
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457 | lMP[3]=orst; |
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458 | |
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459 | def MPz = ring(lMP); |
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460 | setring(MPz); |
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461 | poly p=imap(MP,p); |
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462 | |
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463 | // special treatment in the homogeneous case, dropping one variable |
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464 | // by substituting the first variable by 1 |
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465 | int ho=homog(p); |
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466 | if(ho) |
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467 | { |
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468 | int dh=deg(p); |
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469 | p=subst(p,var(1),1); |
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470 | int di=deg(p); |
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471 | } |
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472 | list rat_facts = factorize(p); |
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473 | int s = size(rat_facts[1]); |
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474 | list tmpf; // absolute factors |
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475 | intvec tmpm; // respective multiplicities |
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476 | tmpf[1] = list(var(n+1), leadcoef(imap(MP,p))); |
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477 | tmpm[1] = 1; |
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478 | poly tmp; |
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479 | for(i = 2; i <= s; i++) |
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480 | { |
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481 | tmp = rat_facts[1][i]; |
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482 | tmp = tmp / leadcoef(tmp); |
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483 | tmpf[i] = absFactorizeIrreducible(tmp); |
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484 | tmpm[i] = rat_facts[2][i]; |
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485 | } |
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486 | // the homogeneous case, homogenizing the result |
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487 | // the new variable has to have degree 0 |
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488 | // need to change the ring |
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489 | if(ho) |
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490 | { |
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491 | list ll=ringlist(MPz); |
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492 | vv[size(vv)]=0; |
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493 | ll[3][1][2]=vv; |
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494 | def MPhelp=ring(ll); |
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495 | setring(MPhelp); |
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496 | list tmpf=imap(MPz,tmpf); |
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497 | for(i=2;i<=size(tmpf);i++) |
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498 | { |
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499 | tmpf[i][2]=homog(tmpf[i][2],var(1)); |
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500 | } |
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501 | if(dh>di) |
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502 | { |
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503 | tmpf[size(tmpf)+1]=list(var(n+1),var(1)); |
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504 | tmpm[size(tmpm)+1]=dh-di; |
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505 | } |
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506 | setring(MPz); |
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507 | tmpf=imap(MPhelp,tmpf); |
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508 | } |
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509 | // in case of parameters we have to go back to the old ring |
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510 | // taking care about constant factors |
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511 | if(pa) |
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512 | { |
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513 | setring(MP); |
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514 | n=nvars(MP); |
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515 | list lM=ringlist(MP); |
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516 | orst[1]=list("wp",vk); |
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517 | orst[2]=list("C",0); |
---|
518 | lM[2][n+1] = newvar; |
---|
519 | lM[3]=orst; |
---|
520 | def MPout=ring(lM); |
---|
521 | setring(MPout); |
---|
522 | list tmpf=imap(MPz,tmpf); |
---|
523 | number cok=imap(MP,cok); |
---|
524 | tmpf[1][2]=cok*tmpf[1][2]; |
---|
525 | } |
---|
526 | else |
---|
527 | { |
---|
528 | def MPout=MPz; |
---|
529 | } |
---|
530 | // if we want the output as string |
---|
531 | if(size(#)>0) |
---|
532 | { |
---|
533 | if(typeof(#[1])=="int") |
---|
534 | { |
---|
535 | if(#[1]==77) |
---|
536 | { // undocumented feature for Gerhard's absPrimdecGTZ |
---|
537 | if (size(tmpf)<2){ list abs_fac = list(var(n+1),poly(1)); } |
---|
538 | else { list abs_fac=tmpf[2..size(tmpf)]; } |
---|
539 | abs_fac=abs_fac,newvar; |
---|
540 | return(string(abs_fac)); |
---|
541 | } |
---|
542 | } |
---|
543 | } |
---|
544 | // preparing the output for SINGULAR standard |
---|
545 | // a list: factors(ideal),multiplicities(intvec),minpolys(ideal), |
---|
546 | // number of factors in the absolute factorization |
---|
547 | // the output(except the coefficient) should have no denominators |
---|
548 | // and no content |
---|
549 | ideal facts,minpols; |
---|
550 | intvec mults; |
---|
551 | int nfacts; |
---|
552 | number co=1; |
---|
553 | minpols[1]=tmpf[1][1]; |
---|
554 | facts[1]=tmpf[1][2]; //the coefficient |
---|
555 | for(i=2;i<=size(tmpf);i++) |
---|
556 | { |
---|
557 | minpols[i]=cleardenom(tmpf[i][1]); |
---|
558 | facts[i]=cleardenom(tmpf[i][2]); |
---|
559 | co=co*(leadcoef(tmpf[i][2])/leadcoef(facts[i]))^(deg(minpols[i])*tmpm[i]); |
---|
560 | } |
---|
561 | facts[1]=facts[1]*co; |
---|
562 | for(i=1;i<=size(tmpm);i++) |
---|
563 | { |
---|
564 | mults[i]=tmpm[i]; |
---|
565 | } |
---|
566 | for(i=2;i<=size(mults);i++) |
---|
567 | { |
---|
568 | nfacts=nfacts+mults[i]*deg(minpols[i]); |
---|
569 | } |
---|
570 | list absolute_factors=facts,mults,minpols,nfacts; |
---|
571 | |
---|
572 | // create ring with extra parameter `newvar` for output: |
---|
573 | setring(MP); |
---|
574 | list Lout=ringlist(MP); |
---|
575 | if(!pa) |
---|
576 | { |
---|
577 | list Lpar=list(char(MP),list(newvar),list(list("lp",intvec(1))),ideal(0)); |
---|
578 | } |
---|
579 | else |
---|
580 | { |
---|
581 | list Lpar=Lout[1]; |
---|
582 | Lpar[2][size(Lpar[2])+1]=newvar; |
---|
583 | vv=Lpar[3][1][2]; |
---|
584 | vv=vv,1; |
---|
585 | Lpar[3][1][2]=vv; |
---|
586 | } |
---|
587 | Lout[1]=Lpar; |
---|
588 | def MPo=ring(Lout); |
---|
589 | setring(MPo); |
---|
590 | list absolute_factors=imap(MPout,absolute_factors); |
---|
591 | export absolute_factors; |
---|
592 | setring(MP); |
---|
593 | |
---|
594 | dbprint( printlevel-voice+3," |
---|
595 | // 'absFactorize' created a ring, in which a list absolute_factors (the |
---|
596 | // absolute factors) is stored. |
---|
597 | // To access the list of absolute factors, type (if the name S was assigned |
---|
598 | // to the return value): |
---|
599 | setring(S); absolute_factors; "); |
---|
600 | return(MPo); |
---|
601 | } |
---|
602 | example |
---|
603 | { |
---|
604 | "EXAMPLE:"; echo = 2; |
---|
605 | ring R = (0), (x,y), lp; |
---|
606 | poly p = (-7*x^2 + 2*x*y^2 + 6*x + y^4 + 14*y^2 + 47)*(5x2+y2)^3*(x-y)^4; |
---|
607 | def S = absFactorize(p) ; |
---|
608 | setring(S); |
---|
609 | absolute_factors; |
---|
610 | } |
---|
611 | |
---|
612 | //////////////////////////////////////////////////////////////////////////////// |
---|
613 | proc absBiFactorize(poly p, list #) |
---|
614 | "USAGE: absBiFactorize(p [,s]); p poly, s string |
---|
615 | ASSUME: coefficient field is the field of rational numbers or a |
---|
616 | transcendental extension thereof |
---|
617 | RETURN: ring @code{R} which is obtained from the current basering |
---|
618 | by adding a new parameter (if a string @code{s} is given as a |
---|
619 | second input, the new parameter gets this string as name). The ring |
---|
620 | @code{R} comes with a list @code{absolute_factors} with the |
---|
621 | following entries: |
---|
622 | @format |
---|
623 | absolute_factors[1]: ideal (the absolute factors) |
---|
624 | absolute_factors[2]: intvec (the multiplicities) |
---|
625 | absolute_factors[3]: ideal (the minimal polynomials) |
---|
626 | absolute_factors[4]: int (total number of nontriv. absolute factors) |
---|
627 | @end format |
---|
628 | The entry @code{absolute_factors[1][1]} is a constant, the |
---|
629 | entry @code{absolute_factors[3][1]} is the parameter added to the |
---|
630 | current ring.@* |
---|
631 | Each of the remaining entries @code{absolute_factors[1][j]} stands for |
---|
632 | a class of conjugated absolute factors. The corresponding entry |
---|
633 | @code{absolute_factors[3][j]} is the minimal polynomial of the |
---|
634 | field extension over which the factor is minimally defined (its degree |
---|
635 | is the number of conjugates in the class). If the entry |
---|
636 | @code{absolute_factors[3][j]} coincides with @code{absolute_factors[3][1]}, |
---|
637 | no field extension was necessary for the @code{j}th (class of) |
---|
638 | absolute factor(s). |
---|
639 | NOTE: The input polynomial p is assumed to be bivariate or to contain one para- |
---|
640 | meter and one variable. All factors are presented denominator- and |
---|
641 | content-free. The constant factor (first entry) is chosen such that the |
---|
642 | product of all (!) the (denominator- and content-free) absolute factors |
---|
643 | of @code{p} equals @code{p / absolute_factors[1][1]}. |
---|
644 | SEE ALSO: factorize, absPrimdecGTZ, absFactorize |
---|
645 | EXAMPLE: example absBiFactorize; shows an example |
---|
646 | " |
---|
647 | { |
---|
648 | int dblevel = printlevel - voice + 2; |
---|
649 | dbprint(dblevel,"Entering absfact.lib::absBiFactorize with ",p); |
---|
650 | def MP = basering; |
---|
651 | int i; |
---|
652 | if (char(MP) != 0) |
---|
653 | { |
---|
654 | ERROR("// absfact.lib::absBiFactorize is only implemented for "+ |
---|
655 | "characteristic 0"); |
---|
656 | } |
---|
657 | if(minpoly!=0) |
---|
658 | { |
---|
659 | ERROR("// absfact.lib::absBiFactorize is not implemented for algebraic " |
---|
660 | +"extensions"); |
---|
661 | } |
---|
662 | |
---|
663 | int n = nvars(MP); |
---|
664 | int pa=npars(MP); |
---|
665 | list lMP= ringlist(MP); |
---|
666 | intvec vv,vk; |
---|
667 | for(i=1;i<=n;i++){vv[i]=1;} |
---|
668 | vk=vv,1; |
---|
669 | |
---|
670 | //if the basering has parameters, add the parameters to the variables |
---|
671 | //takes care about coefficients and possible denominators |
---|
672 | if(pa>0) |
---|
673 | { |
---|
674 | poly qh=cleardenom(p); |
---|
675 | if (p==0) |
---|
676 | { |
---|
677 | number cok=0; |
---|
678 | } |
---|
679 | else |
---|
680 | { |
---|
681 | number cok=leadcoef(p)/leadcoef(qh); |
---|
682 | } |
---|
683 | p=qh; |
---|
684 | string sp; |
---|
685 | for(i=1;i<=npars(basering);i++) |
---|
686 | { |
---|
687 | sp=string(par(i)); |
---|
688 | sp=sp[2..size(sp)-1]; |
---|
689 | lMP[2][n+i]=sp; |
---|
690 | vv=vv,1; |
---|
691 | } |
---|
692 | lMP[1]=0; |
---|
693 | n=n+npars(MP); |
---|
694 | } |
---|
695 | |
---|
696 | // MPz is obtained by adding the new variable @z to MP |
---|
697 | // ordering is wp(1...1) |
---|
698 | // All the above subroutines work in MPz |
---|
699 | string newvar; |
---|
700 | if(size(#)>0) |
---|
701 | { |
---|
702 | if(typeof(#[1])=="string") |
---|
703 | { |
---|
704 | newvar=#[1]; |
---|
705 | } |
---|
706 | else |
---|
707 | { |
---|
708 | newvar = "a"; |
---|
709 | } |
---|
710 | } |
---|
711 | else |
---|
712 | { |
---|
713 | newvar = "a"; |
---|
714 | } |
---|
715 | if (newvar=="a") |
---|
716 | { |
---|
717 | if(belongTo(newvar, lMP[2])||defined(a)){newvar = "b";} |
---|
718 | if(belongTo(newvar, lMP[2])||defined(b)){newvar = "c";} |
---|
719 | if(belongTo(newvar, lMP[2])||defined(c)){newvar = "@c";} |
---|
720 | while(belongTo(newvar, lMP[2])) |
---|
721 | { |
---|
722 | newvar = "@" + newvar; |
---|
723 | } |
---|
724 | } |
---|
725 | |
---|
726 | // create ring with extra parameter `newvar` for output: |
---|
727 | setring(MP); |
---|
728 | list Lout=ringlist(MP); |
---|
729 | if(!pa) |
---|
730 | { |
---|
731 | list Lpar=list(char(MP),list(newvar),list(list("lp",intvec(1))),ideal(0)); |
---|
732 | } |
---|
733 | else |
---|
734 | { |
---|
735 | list Lpar=Lout[1]; |
---|
736 | Lpar[2][size(Lpar[2])+1]=newvar; |
---|
737 | vv=Lpar[3][1][2]; |
---|
738 | vv=vv,1; |
---|
739 | Lpar[3][1][2]=vv; |
---|
740 | } |
---|
741 | Lout[1]=Lpar; |
---|
742 | def MPo=ring(Lout); |
---|
743 | setring(MPo); |
---|
744 | |
---|
745 | poly p=imap(MP,p); |
---|
746 | |
---|
747 | if (size (variables (p)) > 2) |
---|
748 | { |
---|
749 | ERROR("// absfact.lib::absBiFactorize is not implemented for polynomials " |
---|
750 | +"with more than 2 variables"); |
---|
751 | } |
---|
752 | |
---|
753 | // special treatment in the homogeneous case, dropping one variable |
---|
754 | // by substituting the first variable by 1 |
---|
755 | int ho=homog(p); |
---|
756 | if(ho) |
---|
757 | { |
---|
758 | int dh=deg(p); |
---|
759 | p=subst(p,var(1),1); |
---|
760 | int di=deg(p); |
---|
761 | } |
---|
762 | |
---|
763 | list tmpf=absBiFact (p); |
---|
764 | |
---|
765 | // the homogeneous case, homogenizing the result |
---|
766 | // the new variable has to have degree 0 |
---|
767 | // need to change the ring |
---|
768 | if(ho) |
---|
769 | { |
---|
770 | list ll=ringlist(MPo); |
---|
771 | vv[size(vv)]=0; |
---|
772 | ll[3][1][2]=vv; |
---|
773 | def MPhelp=ring(ll); |
---|
774 | setring(MPhelp); |
---|
775 | list tmpf=imap(MPo,tmpf); |
---|
776 | for(i=2;i<=size(tmpf[1]);i++) |
---|
777 | { |
---|
778 | tmpf[1][i]=homog(tmpf[1][i],var(1)); |
---|
779 | } |
---|
780 | if(dh>di) |
---|
781 | { |
---|
782 | tmpf[1][size(tmpf[1])+1]=var(1); |
---|
783 | tmpf[2][size(tmpf[2])+1]=dh-di; |
---|
784 | tmpf[3][size(tmpf[3])+1]=par(npars(MPo)); |
---|
785 | tmpf[4]= tmpf[4]+dh-di; |
---|
786 | } |
---|
787 | setring(MPo); |
---|
788 | tmpf=imap(MPhelp,tmpf); |
---|
789 | } |
---|
790 | |
---|
791 | if (pa) |
---|
792 | { |
---|
793 | number cok=imap(MP,cok); |
---|
794 | tmpf[1][1]=cok*tmpf[1][1]; |
---|
795 | } |
---|
796 | |
---|
797 | // if we want the output as string |
---|
798 | if(size(#)>0) |
---|
799 | { |
---|
800 | if(typeof(#[1])=="int") |
---|
801 | { |
---|
802 | if(#[1]==77) |
---|
803 | { // undocumented feature for Gerhard's absPrimdecGTZ |
---|
804 | if (size(tmpf[1])<2){ list abs_fac = list(var(n+1),poly(1)); } |
---|
805 | else |
---|
806 | { |
---|
807 | list abs_fac= tmpf[3][2]; |
---|
808 | abs_fac= abs_fac, tmpf[1][2]; |
---|
809 | for (i= 3; i <= size(tmpf[1]); i++) |
---|
810 | { |
---|
811 | abs_fac=abs_fac,tmpf[3][i]; |
---|
812 | abs_fac=abs_fac,tmpf[1][i]; |
---|
813 | } |
---|
814 | } |
---|
815 | abs_fac=abs_fac,newvar; |
---|
816 | return(string(abs_fac)); //TODO muss ich Parameter zu Variable machen |
---|
817 | } |
---|
818 | } |
---|
819 | } |
---|
820 | |
---|
821 | list absolute_factors= tmpf; |
---|
822 | export absolute_factors; |
---|
823 | setring(MP); |
---|
824 | |
---|
825 | dbprint( printlevel-voice+3," |
---|
826 | // 'absBiFactorize' created a ring, in which a list absolute_factors (the |
---|
827 | // absolute factors) is stored. |
---|
828 | // To access the list of absolute factors, type (if the name S was assigned |
---|
829 | // to the return value): |
---|
830 | setring(S); absolute_factors; "); |
---|
831 | return(MPo); |
---|
832 | } |
---|
833 | example |
---|
834 | { |
---|
835 | "EXAMPLE:"; echo = 2; |
---|
836 | ring R = (0), (x,y), lp; |
---|
837 | poly p = (-7*x^2 + 2*x*y^2 + 6*x + y^4 + 14*y^2 + 47)*(5x2+y2)^3*(x-y)^4; |
---|
838 | def S = absBiFactorize(p) ; |
---|
839 | setring(S); |
---|
840 | absolute_factors; |
---|
841 | } |
---|
842 | |
---|
843 | |
---|
844 | /* |
---|
845 | ring r=0,(x,t),dp; |
---|
846 | poly p=x^4+(t^3-2t^2-2t)*x^3-(t^5-2t^4-t^2-2t-1)*x^2 |
---|
847 | -(t^6-4t^5+t^4+6t^3+2t^2)*x+(t^6-4t^5+2t^4+4t^3+t^2); |
---|
848 | def S = absFactorize(p,"s"); |
---|
849 | setring(S); |
---|
850 | absolute_factors; |
---|
851 | |
---|
852 | ring r1=(0,a,b),(x,y),dp; |
---|
853 | poly p=(a3-a2b+27ab3-27b4)/(a+b5)*x2+(a2+27b3)*y; |
---|
854 | def S = absFactorize(p); |
---|
855 | setring(S); |
---|
856 | absolute_factors; |
---|
857 | |
---|
858 | ring r2=0,(x,y,z,w),dp; |
---|
859 | poly f=(x2+y2+z2)^2+w4; |
---|
860 | def S =absFactorize(f); |
---|
861 | setring(S); |
---|
862 | absolute_factors; |
---|
863 | |
---|
864 | ring r=0,(x),dp; |
---|
865 | poly p=0; |
---|
866 | def S = absFactorize(p); |
---|
867 | setring(S); |
---|
868 | absolute_factors; |
---|
869 | |
---|
870 | ring r=0,(x),dp; |
---|
871 | poly p=7/11; |
---|
872 | def S = absFactorize(p); |
---|
873 | setring(S); |
---|
874 | absolute_factors; |
---|
875 | |
---|
876 | ring r=(0,a,b),(x,y),dp; |
---|
877 | poly p=0; |
---|
878 | def S = absFactorize(p); |
---|
879 | setring(S); |
---|
880 | absolute_factors; |
---|
881 | |
---|
882 | ring r=(0,a,b),(x,y),dp; |
---|
883 | poly p=(a+1)/b; |
---|
884 | def S = absFactorize(p); |
---|
885 | setring(S); |
---|
886 | absolute_factors; |
---|
887 | |
---|
888 | ring r=(0,a,b),(x,y),dp; |
---|
889 | poly p=(a+1)/b*x; |
---|
890 | def S = absFactorize(p,"s"); |
---|
891 | setring(S); |
---|
892 | absolute_factors; |
---|
893 | |
---|
894 | ring r=(0,a,b),(x,y),dp; |
---|
895 | poly p=(a+1)/b*x + 1; |
---|
896 | def S = absFactorize(p,"s"); |
---|
897 | setring(S); |
---|
898 | absolute_factors; |
---|
899 | |
---|
900 | ring r=(0,a,b),(x,y),dp; |
---|
901 | poly p=(a+1)/b*x + y; |
---|
902 | def S = absFactorize(p,"s"); |
---|
903 | setring(S); |
---|
904 | absolute_factors; |
---|
905 | |
---|
906 | ring r=0,(x,t),dp; |
---|
907 | poly p=x^4+(t^3-2t^2-2t)*x^3-(t^5-2t^4-t^2-2t-1)*x^2 |
---|
908 | -(t^6-4t^5+t^4+6t^3+2t^2)*x+(t^6-4t^5+2t^4+4t^3+t^2); |
---|
909 | def S = absFactorize(p,"s"); |
---|
910 | setring(S); |
---|
911 | absolute_factors; |
---|
912 | |
---|
913 | ring r1=(0,a,b),(x,y),dp; |
---|
914 | poly p=(a3-a2b+27ab3-27b4)/(a+b5)*x2+(a2+27b3)*y; |
---|
915 | def S = absFactorize(p); |
---|
916 | setring(S); |
---|
917 | absolute_factors; |
---|
918 | |
---|
919 | ring r2=0,(x,y,z,w),dp; |
---|
920 | poly f=(x2+y2+z2)^2+w4; |
---|
921 | def S =absFactorize(f); |
---|
922 | setring(S); |
---|
923 | absolute_factors; |
---|
924 | |
---|
925 | ring r3=0,(x,y,z,w),dp; |
---|
926 | poly f=(x2+y2+z2)^4+w8; |
---|
927 | def S =absFactorize(f); |
---|
928 | setring(S); |
---|
929 | absolute_factors; |
---|
930 | |
---|
931 | ring r4=0,(x,y),dp; |
---|
932 | poly f=y6-(2x2-2x-14)*y4-(4x3+35x2-6x-47)*y2+14x4-12x3-94x2; |
---|
933 | def S=absFactorize(f); |
---|
934 | setring(S); |
---|
935 | absolute_factors; |
---|
936 | |
---|
937 | ring R1 = 0, x, dp; |
---|
938 | def S1 = absFactorize(x4-2); |
---|
939 | setring(S1); |
---|
940 | absolute_factors; |
---|
941 | |
---|
942 | ring R3 = 0, (x,y), dp; |
---|
943 | poly f = x2y4+y6+2x3y2+2xy4-7x4+7x2y2+14y4+6x3+6xy2+47x2+47y2; |
---|
944 | def S3 = absFactorize(f); |
---|
945 | setring(S3); |
---|
946 | absolute_factors; |
---|
947 | |
---|
948 | ring R4 = 0, (x,y), dp; |
---|
949 | poly f = y4+2*xy2-7*x2+14*y2+6*x+47; |
---|
950 | def S4 = absFactorize(f); |
---|
951 | setring(S4); |
---|
952 | absolute_factors; |
---|
953 | |
---|
954 | */ |
---|