1 | #ifndef GFANLIB_TROPICALSTRATEGY_H |
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2 | #define GFANLIB_TROPICALSTRATEGY_H |
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3 | |
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4 | #include <gfanlib/gfanlib_vector.h> |
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5 | #include <gfanlib/gfanlib_zcone.h> |
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6 | #include <polys/simpleideals.h> |
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7 | #include <kernel/ideals.h> // for idSize |
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8 | #include <set> |
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9 | #include <callgfanlib_conversion.h> |
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10 | #include <containsMonomial.h> |
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11 | #include <flip.h> |
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12 | #include <initial.h> |
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13 | #include <witness.h> |
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14 | |
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15 | #ifndef NDEBUG |
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16 | |
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17 | #include <Singular/ipshell.h> // for isPrime(int i) |
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18 | #include <adjustWeights.h> |
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19 | #include <ppinitialReduction.h> |
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20 | |
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21 | #endif |
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22 | |
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23 | /** \file |
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24 | * implementation of the class tropicalStrategy |
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25 | * |
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26 | * tropicalStrategy is a class that contains information relevant for |
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27 | * computing tropical varieties that is dependent on the valuation of the coefficient field. |
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28 | * It represents the mutable part of an overall framework that is capable of |
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29 | * computing tropical varieties both over coefficient fields without valuation and |
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30 | * with valuation (currently: only p-adic valuation over rational numbers) |
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31 | */ |
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32 | |
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33 | typedef gfan::ZVector (*wAdjAlg1)(gfan::ZVector); |
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34 | typedef gfan::ZVector (*wAdjAlg2)(gfan::ZVector,gfan::ZVector); |
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35 | typedef bool (*redAlg)(ideal,ring,number); |
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36 | |
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37 | class tropicalStrategy |
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38 | { |
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39 | private: |
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40 | /** |
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41 | * polynomial ring over a field with valuation |
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42 | */ |
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43 | ring originalRing; |
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44 | /** |
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45 | * input ideal, assumed to be a homogeneous prime ideal |
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46 | */ |
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47 | ideal originalIdeal; |
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48 | /** |
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49 | * the expected Dimension of the polyhedral output, |
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50 | * i.e. the dimension of the ideal if valuation trivial |
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51 | * or the dimension of the ideal plus one if valuation non-trivial |
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52 | * (as the output is supposed to be intersected with a hyperplane) |
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53 | */ |
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54 | int expectedDimension; |
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55 | /** |
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56 | * the homogeneity space of the Grobner fan |
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57 | */ |
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58 | gfan::ZCone linealitySpace; |
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59 | /** |
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60 | * polynomial ring over the valuation ring extended by one extra variable t |
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61 | */ |
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62 | ring startingRing; |
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63 | /** |
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64 | * preimage of the input ideal under the map that sends t to the uniformizing parameter |
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65 | */ |
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66 | ideal startingIdeal; |
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67 | /** |
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68 | * uniformizing parameter in the valuation ring |
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69 | */ |
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70 | number uniformizingParameter; |
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71 | /** |
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72 | * polynomial ring over the residue field |
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73 | */ |
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74 | ring shortcutRing; |
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75 | |
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76 | /** |
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77 | * true if valuation non-trivial, false otherwise |
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78 | */ |
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79 | bool onlyLowerHalfSpace; |
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80 | |
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81 | /** |
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82 | * A function such that: |
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83 | * Given weight w, returns a strictly positive weight u such that an ideal satisfying |
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84 | * the valuation-sepcific homogeneity conditions is weighted homogeneous with respect to w |
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85 | * if and only if it is homogeneous with respect to u |
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86 | */ |
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87 | gfan::ZVector (*weightAdjustingAlgorithm1) (const gfan::ZVector &w); |
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88 | /** |
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89 | * A function such that: |
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90 | * Given strictly positive weight w and weight v, |
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91 | * returns a strictly positive weight u such that on an ideal that is weighted homogeneous |
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92 | * with respect to w the weights u and v coincide |
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93 | */ |
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94 | gfan::ZVector (*weightAdjustingAlgorithm2) (const gfan::ZVector &v, const gfan::ZVector &w); |
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95 | /** |
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96 | * A function that reduces the generators of an ideal I so that |
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97 | * the inequalities and equations of the Groebner cone can be read off. |
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98 | */ |
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99 | bool (*extraReductionAlgorithm) (ideal I, ring r, number p); |
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100 | |
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101 | ring copyAndChangeCoefficientRing(const ring r) const; |
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102 | ring copyAndChangeOrderingWP(const ring r, const gfan::ZVector &w, const gfan::ZVector &v) const; |
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103 | ring copyAndChangeOrderingLS(const ring r, const gfan::ZVector &w, const gfan::ZVector &v) const; |
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104 | |
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105 | /** |
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106 | * if valuation non-trivial, checks whether the generating system contains p-t |
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107 | * otherwise returns true |
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108 | */ |
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109 | bool checkForUniformizingBinomial(const ideal I, const ring r) const; |
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110 | |
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111 | /** |
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112 | * if valuation non-trivial, checks whether the genearting system contains p |
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113 | * otherwise returns true |
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114 | */ |
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115 | bool checkForUniformizingParameter(const ideal inI, const ring r) const; |
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116 | int findPositionOfUniformizingBinomial(const ideal I, const ring r) const; |
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117 | void putUniformizingBinomialInFront(ideal I, const ring r, const number q) const; |
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118 | |
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119 | public: |
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120 | |
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121 | /** |
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122 | * Constructor for the trivial valuation case |
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123 | */ |
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124 | tropicalStrategy(const ideal I, const ring r, const bool completelyHomogeneous=true, const bool completeSpace=true); |
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125 | /** |
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126 | * Constructor for the non-trivial valuation case |
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127 | * p is the uniformizing parameter of the valuation |
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128 | */ |
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129 | tropicalStrategy(const ideal J, const number p, const ring s); |
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130 | /** |
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131 | * copy constructor |
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132 | */ |
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133 | tropicalStrategy(const tropicalStrategy& currentStrategy); |
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134 | |
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135 | |
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136 | #ifndef NDEBUG |
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137 | tropicalStrategy(); |
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138 | static tropicalStrategy debugStrategy(const ideal startIdeal, number unifParameter, ring startRing); |
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139 | #endif |
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140 | |
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141 | /** |
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142 | * destructor |
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143 | */ |
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144 | ~tropicalStrategy(); |
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145 | /** |
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146 | * assignment operator |
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147 | **/ |
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148 | tropicalStrategy& operator=(const tropicalStrategy& currentStrategy); |
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149 | |
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150 | bool isConstantCoefficientCase() const |
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151 | { |
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152 | bool b = (uniformizingParameter==NULL); |
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153 | return b; |
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154 | } |
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155 | bool isValuationTrivial() const |
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156 | { |
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157 | bool b = (uniformizingParameter==NULL); |
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158 | return b; |
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159 | } |
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160 | bool isValuationNonTrivial() const |
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161 | { |
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162 | bool b = (uniformizingParameter!=NULL); |
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163 | return b; |
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164 | } |
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165 | |
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166 | /** |
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167 | * returns the polynomial ring over the field with valuation |
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168 | */ |
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169 | ring getOriginalRing() const |
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170 | { |
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171 | rTest(originalRing); |
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172 | return originalRing; |
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173 | } |
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174 | |
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175 | /** |
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176 | * returns the input ideal over the field with valuation |
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177 | */ |
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178 | ideal getOriginalIdeal() const |
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179 | { |
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180 | if (originalIdeal) id_Test(originalIdeal,originalRing); |
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181 | return originalIdeal; |
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182 | } |
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183 | |
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184 | /** |
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185 | * returns the polynomial ring over the valuation ring |
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186 | */ |
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187 | ring getStartingRing() const |
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188 | { |
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189 | if (startingRing) rTest(startingRing); |
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190 | return startingRing; |
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191 | } |
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192 | |
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193 | /** |
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194 | * returns the input ideal |
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195 | */ |
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196 | ideal getStartingIdeal() const |
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197 | { |
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198 | if (startingIdeal) id_Test(startingIdeal,startingRing); |
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199 | return startingIdeal; |
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200 | } |
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201 | |
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202 | /** |
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203 | * returns the expected Dimension of the polyhedral output |
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204 | */ |
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205 | int getExpectedDimension() const |
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206 | { |
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207 | return expectedDimension; |
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208 | } |
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209 | |
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210 | /** |
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211 | * returns the uniformizing parameter in the valuation ring |
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212 | */ |
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213 | number getUniformizingParameter() const |
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214 | { |
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215 | if (uniformizingParameter) n_Test(uniformizingParameter,startingRing->cf); |
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216 | return uniformizingParameter; |
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217 | } |
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218 | |
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219 | ring getShortcutRing() const |
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220 | { |
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221 | if (shortcutRing) rTest(shortcutRing); |
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222 | return shortcutRing; |
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223 | } |
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224 | |
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225 | /** |
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226 | * returns the homogeneity space of the preimage ideal |
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227 | */ |
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228 | gfan::ZCone getHomogeneitySpace() const |
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229 | { |
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230 | return linealitySpace; |
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231 | } |
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232 | |
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233 | /** |
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234 | * returns true, if v is contained in the homogeneity space; false otherwise |
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235 | */ |
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236 | bool homogeneitySpaceContains(const gfan::ZVector &v) const |
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237 | { |
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238 | return linealitySpace.contains(v); |
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239 | } |
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240 | |
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241 | /** |
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242 | * returns true, if valuation non-trivial, false otherwise |
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243 | */ |
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244 | bool restrictToLowerHalfSpace() const |
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245 | { |
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246 | return onlyLowerHalfSpace; |
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247 | } |
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248 | |
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249 | /** |
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250 | * Given weight w, returns a strictly positive weight u such that an ideal satisfying |
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251 | * the valuation-sepcific homogeneity conditions is weighted homogeneous with respect to w |
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252 | * if and only if it is homogeneous with respect to u |
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253 | */ |
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254 | gfan::ZVector adjustWeightForHomogeneity(gfan::ZVector w) const |
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255 | { |
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256 | return this->weightAdjustingAlgorithm1(w); |
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257 | } |
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258 | |
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259 | /** |
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260 | * Given strictly positive weight w and weight v, |
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261 | * returns a strictly positive weight u such that on an ideal that is weighted homogeneous |
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262 | * with respect to w the weights u and v coincide |
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263 | */ |
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264 | gfan::ZVector adjustWeightUnderHomogeneity(gfan::ZVector v, gfan::ZVector w) const |
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265 | { |
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266 | return this->weightAdjustingAlgorithm2(v,w); |
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267 | } |
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268 | |
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269 | gfan::ZVector negateWeight(const gfan::ZVector &w) const |
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270 | { |
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271 | gfan::ZVector wNeg(w.size()); |
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272 | |
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273 | if (this->isValuationNonTrivial()) |
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274 | { |
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275 | wNeg[0]=w[0]; |
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276 | for (unsigned i=1; i<w.size(); i++) |
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277 | wNeg[i]=w[i]; |
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278 | } |
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279 | else |
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280 | wNeg = -w; |
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281 | |
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282 | return wNeg; |
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283 | } |
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284 | |
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285 | /** |
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286 | * If valuation trivial, returns a copy of r with a positive weight prepended, |
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287 | * such that any ideal homogeneous with respect to w is homogeneous with respect to that weight. |
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288 | * If valuation non-trivial, changes the coefficient ring to the residue field. |
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289 | */ |
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290 | ring getShortcutRingPrependingWeight(const ring r, const gfan::ZVector &w) const; |
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291 | |
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292 | /** |
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293 | * reduces the generators of an ideal I so that |
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294 | * the inequalities and equations of the Groebner cone can be read off. |
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295 | */ |
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296 | bool reduce(ideal I, const ring r) const; |
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297 | |
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298 | void pReduce(ideal I, const ring r) const; |
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299 | |
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300 | /** |
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301 | * If given w, assuming w is in the Groebner cone of the ordering on r |
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302 | * and I is a standard basis with respect to that ordering, |
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303 | * checks whether the initial ideal of I with respect to w contains a monomial. |
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304 | * If no w is given, assuming that I is already an initial form of some ideal, |
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305 | * checks whether I contains a monomial. |
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306 | * In both cases returns a monomial, if it contains one, returns NULL otherwise. |
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307 | **/ |
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308 | std::pair<poly,int> checkInitialIdealForMonomial(const ideal I, const ring r, const gfan::ZVector &w=0) const; |
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309 | |
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310 | /** |
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311 | * given generators of the initial ideal, computes its standard basis |
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312 | */ |
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313 | ideal computeStdOfInitialIdeal(const ideal inI, const ring r) const; |
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314 | |
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315 | /** |
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316 | * suppose w a weight in maximal groebner cone of > |
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317 | * suppose I (initially) reduced standard basis w.r.t. > and inI initial forms of its elements w.r.t. w |
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318 | * suppose inJ elements of initial ideal that are homogeneous w.r.t w |
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319 | * returns J elements of ideal whose initial form w.r.t. w are inI |
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320 | * in particular, if w lies also inthe maximal groebner cone of another ordering >' |
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321 | * and inJ is a standard basis of the initial ideal w.r.t. >' |
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322 | * then the returned J will be a standard baiss of the ideal w.r.t. >' |
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323 | */ |
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324 | ideal computeWitness(const ideal inJ, const ideal inI, const ideal I, const ring r) const; |
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325 | |
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326 | ideal computeLift(const ideal inJs, const ring s, const ideal inIr, const ideal Ir, const ring r) const; |
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327 | |
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328 | /** |
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329 | * given an interior point of a groebner cone |
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330 | * computes the groebner cone adjacent to it |
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331 | */ |
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332 | std::pair<ideal,ring> computeFlip(const ideal Ir, const ring r, const gfan::ZVector &interiorPoint, const gfan::ZVector &facetNormal) const; |
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333 | }; |
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334 | |
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335 | #ifndef NDEBUG |
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336 | BOOLEAN computeWitnessDebug(leftv res, leftv args); |
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337 | BOOLEAN computeFlipDebug(leftv res, leftv args); |
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338 | #endif |
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339 | |
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340 | #endif |
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