1 | //////////////////////////////////////////////////////////////// |
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2 | version="version olga.lib 4.1.2.0 Feb_2019 "; // $Id$ |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: olga.lib Ore-localization in G-Algebras |
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6 | AUTHOR: Johannes Hoffmann, email: johannes.hoffmann at math.rwth-aachen.de |
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7 | |
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8 | OVERVIEW: |
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9 | Let A be a G-algebra. |
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10 | |
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11 | Current localization types: |
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12 | Type 0: monoidal |
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13 | - represented by a list of polys g_1,...,g_k that have to be contained in a |
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14 | commutative polynomial subring of A generated by a subset of the variables |
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15 | of A |
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16 | Type 1: geometric |
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17 | - only for algebras with an even number of variables where the first half |
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18 | induces a commutative polynomial subring B of A |
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19 | - represented by an ideal p, which has to be a prime ideal in B |
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20 | Type 2: rational |
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21 | - represented by an intvec v = [i_1,...,i_k] in the range 1..nvars(basering) |
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22 | |
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23 | Localization data is an int specifying the type and a def with the |
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24 | corresponding information. |
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25 | |
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26 | A fraction is represented as a vector with four entries: [s,r,p,t] |
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27 | Here, s^{-1}r is the left fraction representation, pt^{-1} is the right one. |
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28 | If s or t is zero, it means that the corresponding representation is not set. |
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29 | If both are zero, the fraction is not valid. |
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30 | |
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31 | A detailed description along with further examples can be found in our paper: |
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32 | Johannes Hoffmann, Viktor Levandovskyy: |
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33 | Constructive Arithmetics in Ore Localizations of Domains |
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34 | https://arxiv.org/abs/1712.01773 |
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35 | |
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36 | PROCEDURES: |
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37 | locStatus(int, def); |
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38 | report on the status/validity of the given localization data |
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39 | testLocData(int, def); |
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40 | check if the given data specifies a denominator set |
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41 | isInS(poly, int, def); |
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42 | determine if a polynomial is in a denominator set |
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43 | fracStatus(vector frac, int locType, def locData); |
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44 | report on the status/validity of the given fraction wrt. to the given |
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45 | localization data |
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46 | testFraction(vector, int, def); |
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47 | check if the given vector is a representation of a fraction in the specified |
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48 | localization |
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49 | leftOre(poly, poly, int, def) |
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50 | compute left Ore data |
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51 | rightOre(poly, poly, int, def) |
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52 | compute right Ore data |
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53 | convertRightToLeftFraction(vector, int, def); |
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54 | determine a left fraction representation of a given fraction |
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55 | convertLeftToRightFraction(vector, int, def); |
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56 | determine a right fraction representation of a given fraction |
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57 | addLeftFractions(vector, vector, int, def); |
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58 | add two left fractions in the specified localization |
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59 | multiplyLeftFractions(vector, vector, int, def); |
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60 | multiply two left fractions in the specified localization |
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61 | areEqualLeftFractions(vector, vector, int, def); |
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62 | check if two given fractions are equal |
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63 | isInvertibleLeftFraction(vector, int, def); |
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64 | check if a fraction is invertible in the specified localization |
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65 | (NOTE: check description for specific behaviour) |
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66 | invertLeftFraction(vector, int, def); |
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67 | invert a fraction in the specified localization |
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68 | (NOTE: check description for specific behaviour) |
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69 | isZeroFraction(vector); |
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70 | determine if the given fraction is equal to zero |
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71 | isOneFraction(vector); |
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72 | determine if the given fraction is equal to one |
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73 | normalizeMonoidal(list); |
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74 | determine a normal form for monoidal localization data |
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75 | normalizeRational(intvec); |
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76 | determine a normal form for rational localization data |
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77 | testOlga(); |
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78 | execute a series of internal testing procedures |
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79 | testOlgaExamples(); |
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80 | execute the examples of all procedures in this library |
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81 | "; |
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82 | |
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83 | LIB "dmodloc.lib"; // for polyVars |
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84 | LIB "ncpreim.lib"; |
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85 | LIB "elim.lib"; |
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86 | LIB "ncalg.lib"; |
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87 | ////////////////////////////////////////////////////////////////////// |
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88 | proc testOlgaExamples() |
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89 | "USAGE: testOlgaExamples() |
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90 | PURPOSE: execute the examples of all procedures in this library |
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91 | RETURN: nothing |
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92 | NOTE: |
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93 | EXAMPLE: " |
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94 | { |
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95 | example isInS; |
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96 | example leftOre; |
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97 | example rightOre; |
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98 | example convertRightToLeftFraction; |
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99 | example convertLeftToRightFraction; |
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100 | example addLeftFractions; |
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101 | example multiplyLeftFractions; |
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102 | example areEqualLeftFractions; |
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103 | example isInvertibleLeftFraction; |
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104 | example invertLeftFraction; |
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105 | } |
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106 | ////////////////////////////////////////////////////////////////////// |
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107 | proc testOlga() |
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108 | "USAGE: testOlga() |
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109 | PURPOSE: execute a series of internal testing procedures |
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110 | RETURN: nothing |
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111 | NOTE: |
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112 | EXAMPLE: " |
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113 | { |
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114 | print("testing olga.lib..."); |
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115 | testIsInS(); |
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116 | testLeftOre(); |
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117 | testRightOre(); |
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118 | testAddLeftFractions(); |
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119 | testMultiplyLeftFractions(); |
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120 | testAreEqualLeftFractions(); |
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121 | testConvertLeftToRightFraction(); |
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122 | testConvertRightToLeftFraction(); |
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123 | testIsInvertibleLeftFraction(); |
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124 | testInvertLeftFraction(); |
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125 | print("testing complete - olga.lib OK"); |
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126 | } |
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127 | ////////////////////////////////////////////////////////////////////// |
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128 | proc locStatus(int locType, def locData) |
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129 | "USAGE: locStatus(locType, locData), int locType, list/vector/intvec locData |
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130 | PURPOSE: determine the status of a set of localization data |
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131 | ASSUME: |
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132 | RETURN: list |
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133 | NOTE: - the first entry is 0 or 1, depending whether the input represents |
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134 | a valid localization |
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135 | - the second entry is a string with a status/error message |
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136 | EXAMPLE: example locStatus; shows example" |
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137 | { |
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138 | int i; |
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139 | if (locType < 0 || locType > 2) { |
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140 | string invalidTypeString = "invalid localization: type is " |
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141 | + string(locType) + ", valid types are:"; |
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142 | invalidTypeString = invalidTypeString + newline |
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143 | + "0 for a monoidal localization"; |
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144 | invalidTypeString = invalidTypeString + newline |
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145 | + "1 for a geometric localization"; |
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146 | invalidTypeString = invalidTypeString + newline |
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147 | + "2 for a rational localization"; |
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148 | return(list(0, invalidTypeString)); |
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149 | } |
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150 | string t = typeof(locData); |
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151 | if (t == "none") { |
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152 | return(list(0, "uninitialized or invalid localization:" |
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153 | + " locData has to be defined")); |
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154 | } |
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155 | if (locType == 0) |
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156 | { // monoidal localizations |
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157 | if (t != "list") |
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158 | { |
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159 | return(list(0, "for a monoidal localization, locData has to be of" |
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160 | + " type list, but is of type " + t)); |
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161 | } |
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162 | else |
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163 | { // locData is of type list |
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164 | if (size(locData) == 0) |
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165 | { |
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166 | return(list(0, "for a monoidal localization, locData has to be" |
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167 | + " a non-empty list")); |
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168 | } |
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169 | else |
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170 | { // locData is of type list and has at least one entry |
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171 | if (defined(basering)) {ideal listEntries;} |
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172 | for (i = 1; i <= size(locData); i++) |
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173 | { |
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174 | t = typeof(locData[i]); |
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175 | if (t != "poly" && t != "int" && t != "number") |
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176 | { |
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177 | return(list(0, "for a monoidal localization, locData" |
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178 | + " has to be a list of polys, ints or numbers, but" |
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179 | + " entry " + string(i) + " is " + string(locData[i]) |
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180 | + ", which is of type " + t)); |
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181 | } |
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182 | else |
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183 | { |
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184 | if (defined(basering)) |
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185 | { |
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186 | if (size(listEntries) == 0) |
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187 | { |
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188 | listEntries = locData[i]; |
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189 | } |
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190 | else |
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191 | { |
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192 | listEntries = listEntries, locData[i]; |
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193 | } |
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194 | } |
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195 | } |
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196 | } |
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197 | // locData is of type list, has at least one entry and all |
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198 | // entries are polys |
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199 | if (!defined(basering)) |
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200 | { |
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201 | return(list(0, "for a monoidal localization, the variables" |
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202 | + " occurring in the polys in locData have to induce a" |
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203 | + " commutative polynomial subring of basering")); |
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204 | } |
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205 | if (!inducesCommutativeSubring(listEntries)) |
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206 | { |
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207 | return(list(0, "for a monoidal localization, the variables" |
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208 | + " occurring in the polys in locData have to induce a" |
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209 | + " commutative polynomial subring of basering")); |
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210 | } |
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211 | } |
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212 | } |
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213 | } |
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214 | if (locType == 1) |
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215 | { // geometric localizations |
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216 | int n = nvars(basering) div 2; |
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217 | if (2*n != nvars(basering)) |
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218 | { |
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219 | return(list(0, "for a geometric localization, basering has to have" |
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220 | + " an even number of variables")); |
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221 | } |
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222 | else |
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223 | { |
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224 | int j; |
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225 | for (i = 1; i <= n; i++) |
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226 | { |
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227 | for (j = i + 1; j <= n; j++) |
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228 | { |
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229 | if (var(i)*var(j) != var(j)*var(i)) |
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230 | { |
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231 | return(list(0, "for a geometric localization, the" |
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232 | + " first half of the variables of basering has to" |
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233 | + " induce a commutative polynomial subring of" |
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234 | + " basering")); |
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235 | } |
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236 | } |
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237 | } |
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238 | } |
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239 | if (t != "ideal") |
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240 | { |
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241 | return(list(0, "for a geometric localization, locData has to be of" |
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242 | + " type ideal, but is of type " + t)); |
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243 | } |
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244 | for (i = 1; i <= size(locData); i++) |
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245 | { |
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246 | if (!polyVars(locData[i],1..n)) |
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247 | { |
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248 | return(list(0, "for a geometric localization, locData has to" |
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249 | + " be an ideal generated by polynomials containing only" |
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250 | + " variables from the first half of the variables")); |
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251 | } |
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252 | } |
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253 | } |
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254 | if (locType == 2) |
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255 | { // rational localizations |
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256 | if (t != "intvec") |
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257 | { |
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258 | return(list(0, "for a rational localization, locData has to be of" |
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259 | + " type intvec, but is of type " + t)); |
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260 | } |
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261 | else |
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262 | { // locData is of type intvec |
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263 | if(locData == 0) |
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264 | { |
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265 | return(list(0, "for a rational localization, locData has to be" |
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266 | + " a non-zero intvec")); |
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267 | } |
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268 | else |
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269 | { // locData is of type intvec and not zero |
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270 | if (!admissibleSub(locData)) |
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271 | { |
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272 | return(list(0, "for a rational localization, the variables" |
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273 | + " indexed by locData have to generate a sub-G-algebra" |
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274 | + " of the basering")); |
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275 | } |
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276 | } |
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277 | } |
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278 | } |
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279 | return(list(1, "valid localization")); |
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280 | } |
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281 | example |
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282 | { |
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283 | "EXAMPLE:"; echo = 2; |
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284 | locStatus(42, list(1)); |
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285 | def undef; |
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286 | locStatus(0, undef); |
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287 | string s; |
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288 | locStatus(0, s); |
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289 | list L; |
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290 | locStatus(0, L); |
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291 | L = s; |
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292 | print(L); |
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293 | locStatus(0, L); |
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294 | ring w = 0,(x,Dx,y,Dy),dp; |
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295 | def W = Weyl(1); |
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296 | setring W; |
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297 | W; |
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298 | locStatus(0, list(x, Dx)); |
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299 | ring R; |
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300 | setring R; |
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301 | R; |
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302 | locStatus(1, s); |
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303 | setring W; |
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304 | locStatus(1, s); |
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305 | ring t = 0,(x,y,Dx,Dy),dp; |
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306 | def T = Weyl(); |
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307 | setring T; |
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308 | T; |
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309 | locStatus(1, s); |
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310 | locStatus(1, ideal(Dx)); |
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311 | locStatus(2, s); |
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312 | intvec v; |
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313 | locStatus(2, v); |
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314 | locStatus(2, intvec(1,2)); |
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315 | } |
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316 | ////////////////////////////////////////////////////////////////////// |
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317 | proc testLocData(int locType, def locData) |
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318 | "USAGE: testLocData(locType, locData), int locType, |
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319 | list/vector/intvec locData |
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320 | PURPOSE: test if the given data specifies a denominator set wrt. the checks |
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321 | from locStatus |
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322 | ASSUME: |
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323 | RETURN: nothing |
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324 | NOTE: throws error if checks were not successful |
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325 | EXAMPLE: example testLocData; shows examples" |
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326 | { |
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327 | list stat = locStatus(locType, locData); |
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328 | if (!stat[1]) { |
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329 | ERROR(stat[2]); |
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330 | } else { |
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331 | return(); |
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332 | } |
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333 | } |
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334 | example |
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335 | { |
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336 | "EXAMPLE:"; echo = 2; |
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337 | ring R; setring R; |
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338 | testLocData(0, list(1)); // correct localization, no error |
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339 | testLocData(42, list(1)); // incorrect localization, results in error |
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340 | } |
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341 | ////////////////////////////////////////////////////////////////////// |
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342 | proc isInS(poly p, int locType, def locData, list #) |
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343 | "USAGE: isInS(p, locType, locData(, override)), poly p, int locType, |
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344 | list/vector/intvec locData(, int override) |
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345 | PURPOSE: determine if a polynomial is in a denominator set |
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346 | ASSUME: |
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347 | RETURN: int |
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348 | NOTE: - returns 0 or 1, depending whether p is in the denominator set |
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349 | specified by locType and locData |
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350 | - if override is set, will not normalize locData (use with care) |
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351 | EXAMPLE: example isInS; shows examples" |
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352 | { |
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353 | testLocData(locType, locData); |
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354 | if (p == 0) { // the zero polynomial is never a valid denominator |
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355 | return(0); |
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356 | } |
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357 | if (number(p) != 0) { |
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358 | // elements of the coefficient field are always valid denominators |
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359 | return(1); |
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360 | } |
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361 | int override; |
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362 | if (size(#) > 0) { |
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363 | if(typeof(#[1]) == "int") { |
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364 | override = #[1]; |
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365 | } |
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366 | } |
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367 | if (locType == 0) { |
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368 | ideal pFactors = commutativeFactorization(p, 1); |
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369 | list locFactors; |
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370 | if (override) { |
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371 | locFactors = locData; |
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372 | } else { |
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373 | locFactors = normalizeMonoidal(locData); |
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374 | } |
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375 | int i, j, foundFactor; |
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376 | for (i = 1; i <= size(pFactors); i++) { |
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377 | foundFactor = 0; |
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378 | for (j = 1; j <= size(locFactors); j++) { |
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379 | if (pFactors[i] == locFactors[j]) { |
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380 | foundFactor = 1; |
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381 | break; |
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382 | } |
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383 | } |
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384 | if (!foundFactor) { |
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385 | return(0); |
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386 | } |
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387 | } |
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388 | return(1); |
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389 | } |
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390 | if (locType == 1) { |
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391 | int n = nvars(basering) div 2; |
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392 | ideal I = p; |
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393 | ideal J = std(eliminateNC(I, (n+1)..(2*n))); |
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394 | ideal K = std(intersect(J, locData)); |
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395 | int i; |
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396 | for (i = 1; i <= size(J); i++) { |
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397 | if (NF(J[i], K) != 0) { |
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398 | return(1); |
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399 | } |
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400 | } |
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401 | } |
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402 | if (locType == 2) { |
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403 | if (!override) { |
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404 | locData = normalizeRational(locData); |
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405 | } |
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406 | int n = nvars(basering); |
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407 | if (size(locData) < n) { // there are variables to eliminate |
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408 | ideal I = p; |
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409 | intvec modLocData = intvecComplement(locData, 1..nvars(basering)); |
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410 | I = eliminateNC(I, modLocData); |
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411 | if (size(I)) { |
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412 | return(1); |
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413 | } |
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414 | } else { |
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415 | return(p != 0); |
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416 | } |
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417 | } |
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418 | return(0); |
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419 | } |
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420 | example |
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421 | { |
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422 | "EXAMPLE:"; echo = 2; |
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423 | ring R = 0,(x,y,Dx,Dy),dp; |
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424 | def S = Weyl(); |
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425 | setring S; S; |
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426 | // monoidal localization |
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427 | poly g1 = x^2*y+x+2; |
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428 | poly g2 = y^3+x*y; |
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429 | list L = g1,g2; |
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430 | poly g = g1^2*g2; |
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431 | poly f = g-1; |
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432 | isInS(g, 0, L); // g is in the denominator set |
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433 | isInS(f, 0, L); // f is NOT in the denominator set |
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434 | // geometric localization |
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435 | ideal p = x-1, y-3; |
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436 | g = x^2+y-3; |
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437 | f = (x-1)*g; |
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438 | isInS(g, 1, p); // g is in the denominator set |
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439 | isInS(f, 1, p); // f is NOT in the denominator set |
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440 | // rational localization |
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441 | intvec v = 2; |
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442 | g = y^5+17*y^2-4; |
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443 | f = x*y; |
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444 | isInS(g, 2, v); // g is in the denominator set |
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445 | isInS(f, 2, v); // f is NOT in the denominator set |
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446 | } |
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447 | ////////////////////////////////////////////////////////////////////// |
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448 | proc fracStatus(vector frac, int locType, def locData) |
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449 | "USAGE: fracStatus(frac, locType, locData), vector frac, int locType, |
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450 | list/intvec/vector locData |
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451 | PURPOSE: determine if the given vector is a representation of a fraction in the |
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452 | specified localization |
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453 | ASSUME: |
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454 | RETURN: list |
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455 | NOTE: - the first entry is 0 or 1, depending whether the input is valid |
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456 | - the second entry is a string with a status message |
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457 | EXAMPLE: example fracStatus; shows examples" |
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458 | { |
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459 | list locStat = locStatus(locType, locData); |
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460 | if (!locStat[1]) { // there is a problem with the localization data |
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461 | return(list(0, "invalid localization in fraction: "+ string(frac) |
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462 | + newline + " " + locStat[2])); |
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463 | } else { // the specified localization is valid |
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464 | if ((frac[1] == 0) && (frac[4] == 0)) { |
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465 | return(list(0, "vector is not a valid fraction: no denominator" |
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466 | + " specified in " + string(frac))); |
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467 | } |
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468 | if (frac[1] != 0) { // frac has a left representation |
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469 | if (!isInS(frac[1], locType, locData)) { |
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470 | return(list(0, "the left denominator " + string(frac[1]) |
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471 | + " of fraction " + string(frac) + " is not in the" |
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472 | + " denominator set of type " + string(locType) + " given by " |
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473 | + string(locData))); |
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474 | } |
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475 | } |
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476 | if (frac[4] != 0) { // frac has a right representation |
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477 | if (!isInS(frac[4], locType, locData)) { |
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478 | return(list(0, "the right denominator " + string(frac[4]) |
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479 | + " of fraction " + string(frac) + " is not in the" |
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480 | + " denominator set of type " + string(locType) + " given by " |
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481 | + string(locData))); |
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482 | } |
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483 | } |
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484 | if ((frac[1] != 0) && (frac[4] != 0)) { |
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485 | // frac has left and right representations |
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486 | if (frac[2]*frac[4] != frac[1]*frac[3]) { |
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487 | // the representations are not equal |
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488 | return(list(0, "left and right representation are not equal in:" |
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489 | + string(frac))); |
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490 | } |
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491 | } |
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492 | } |
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493 | return(list(1, "valid fraction")); |
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494 | } |
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495 | example |
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496 | { |
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497 | "EXAMPLE:"; echo = 2; |
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498 | ring r = QQ[x,y,Dx,Dy]; |
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499 | def R = Weyl(); |
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500 | setring R; |
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501 | fracStatus([1,0,0,0], 42, list(1)); |
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502 | list L = x; |
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503 | fracStatus([0,7,x,0], 0, L); |
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504 | fracStatus([Dx,Dy,0,0], 0, L); |
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505 | fracStatus([0,0,Dx,Dy], 0, L); |
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506 | fracStatus([x,Dx,Dy,x], 0, L); |
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507 | fracStatus([x,Dx,x*Dx+2,x^2], 0, L); |
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508 | } |
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509 | ////////////////////////////////////////////////////////////////////// |
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510 | proc testFraction(vector frac, int locType, def locData) |
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511 | "USAGE: testFraction(frac, locType, locData), vector frac, int locType, |
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512 | list/intvec/vector locData |
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513 | PURPOSE: test if the given vector is a representation of a fraction in the |
---|
514 | specified localization wrt. the checks from fracStatus |
---|
515 | ASSUME: |
---|
516 | RETURN: nothing |
---|
517 | NOTE: throws error if checks were not successful |
---|
518 | EXAMPLE: example testFraction; shows examples" |
---|
519 | { |
---|
520 | list stat = fracStatus(frac, locType, locData); |
---|
521 | if (!stat[1]) { |
---|
522 | ERROR(stat[2]); |
---|
523 | } else { |
---|
524 | return(); |
---|
525 | } |
---|
526 | } |
---|
527 | example |
---|
528 | { |
---|
529 | "EXAMPLE:"; echo = 2; |
---|
530 | ring r = QQ[x,y,Dx,Dy]; |
---|
531 | def R = Weyl(); |
---|
532 | setring R; |
---|
533 | list L = x; |
---|
534 | vector frac = [x,Dx,x*Dx+2,x^2]; |
---|
535 | testFraction(frac, 0, L); // correct localization, no error |
---|
536 | frac = [x,Dx,x*Dx,x^2]; |
---|
537 | testFraction(frac, 0, L); // incorrect localization, results in error |
---|
538 | } |
---|
539 | ////////////////////////////////////////////////////////////////////// |
---|
540 | proc leftOre(poly s, poly r, int locType, def locData) |
---|
541 | "USAGE: leftOre(s, r, locType, locData), poly s, r, int locType, |
---|
542 | list/vector/intvec locData |
---|
543 | PURPOSE: compute left Ore data for a given tuple (s,r) |
---|
544 | ASSUME: s is in the denominator set determined via locType and locData |
---|
545 | RETURN: list |
---|
546 | NOTE: - the first entry of the list is a vector [ts,tr] such that ts*r=tr*s |
---|
547 | - the second entry of the list is a description of all choices for ts |
---|
548 | EXAMPLE: example leftOre; shows examples" |
---|
549 | { |
---|
550 | testLocData(locType,locData); |
---|
551 | locData = normalizeLocalization(locType, locData); |
---|
552 | if(!isInS(s, locType, locData)) { |
---|
553 | ERROR("cannot find Ore-parameter since poly " + string(s) |
---|
554 | + " is not in the denominator set"); |
---|
555 | } |
---|
556 | return(ore(s, r, locType, locData, 0)); |
---|
557 | } |
---|
558 | example |
---|
559 | { |
---|
560 | "EXAMPLE:"; echo = 2; |
---|
561 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
562 | def S = Weyl(); |
---|
563 | setring S; S; |
---|
564 | // left Ore |
---|
565 | // monoidal localization |
---|
566 | poly g1 = x+3; |
---|
567 | poly g2 = x*y; |
---|
568 | list L = g1,g2; |
---|
569 | poly g = g1^2*g2; |
---|
570 | poly f = Dx; |
---|
571 | list rm = leftOre(g, f, 0, L); |
---|
572 | print(rm[1]); |
---|
573 | rm[2]; |
---|
574 | rm[1][2]*g-rm[1][1]*f; |
---|
575 | // geometric localization |
---|
576 | ideal p = x-1, y-3; |
---|
577 | f = Dx; |
---|
578 | g = x^2+y; |
---|
579 | list rg = leftOre(g, f, 1, p); |
---|
580 | print(rg[1]); |
---|
581 | rg[2]; |
---|
582 | rg[1][2]*g-rg[1][1]*f; |
---|
583 | // rational localization |
---|
584 | intvec rat = 1; |
---|
585 | f = Dx+Dy; |
---|
586 | g = x; |
---|
587 | list rr = leftOre(g, f, 2, rat); |
---|
588 | print(rr[1]); |
---|
589 | rr[2]; |
---|
590 | rr[1][2]*g-rr[1][1]*f; |
---|
591 | } |
---|
592 | ////////////////////////////////////////////////////////////////////// |
---|
593 | proc rightOre(poly s, poly r, int locType, def locData) |
---|
594 | "USAGE: rightOre(s, r, locType, locData), poly s, r, int locType, |
---|
595 | list/vector/intvec locData |
---|
596 | PURPOSE: compute right Ore data for a given tuple (s,r) |
---|
597 | ASSUME: s is in the denominator set determined via locType and locData |
---|
598 | RETURN: list |
---|
599 | NOTE: - the first entry of the list is a vector [ts,tr] such that r*ts=s*tr |
---|
600 | - the second entry of the list is a description of all choices for ts |
---|
601 | EXAMPLE: example rightOre; shows examples" |
---|
602 | { |
---|
603 | testLocData(locType,locData); |
---|
604 | locData = normalizeLocalization(locType, locData); |
---|
605 | if(!isInS(s, locType, locData)) { |
---|
606 | ERROR("cannot find Ore-parameter since poly " + string(s) |
---|
607 | + " is not in the denominator set"); |
---|
608 | } |
---|
609 | def bsRing = basering; |
---|
610 | if (locType == 0) { |
---|
611 | ideal modLocData = locData[1..size(locData)]; |
---|
612 | } |
---|
613 | def oppRing = opposite(bsRing); |
---|
614 | setring oppRing; |
---|
615 | if (locType == 0) { |
---|
616 | ideal oppModLocData = oppose(bsRing, modLocData); |
---|
617 | list oppLocData = oppModLocData[1..size(oppModLocData)]; |
---|
618 | } |
---|
619 | if (locType == 1) { |
---|
620 | ideal oppLocData = oppose(bsRing,locData); |
---|
621 | } |
---|
622 | if (locType == 2) { |
---|
623 | intvec oppLocData = locData; |
---|
624 | } |
---|
625 | poly oppS = oppose(bsRing, s); |
---|
626 | poly oppR = oppose(bsRing, r); |
---|
627 | list oppResult = ore(oppS, oppR, locType, oppLocData, 1); |
---|
628 | vector oppOreParas = oppResult[1]; |
---|
629 | ideal oppJ = oppResult[2]; |
---|
630 | setring bsRing; |
---|
631 | return(list(oppose(oppRing, oppOreParas), oppose(oppRing, oppJ))); |
---|
632 | } |
---|
633 | example |
---|
634 | { |
---|
635 | "EXAMPLE:"; echo = 2; |
---|
636 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
637 | def S = Weyl(); |
---|
638 | setring S; S; |
---|
639 | // monoidal localization |
---|
640 | poly g1 = x+3; |
---|
641 | poly g2 = x*y; |
---|
642 | list L = g1,g2; |
---|
643 | poly g = g1^2*g2; |
---|
644 | poly f = Dx; |
---|
645 | list rm = rightOre(g, f, 0, L); |
---|
646 | print(rm[1]); |
---|
647 | rm[2]; |
---|
648 | g*rm[1][2]-f*rm[1][1]; |
---|
649 | // geometric localization |
---|
650 | ideal p = x-1, y-3; |
---|
651 | f = Dx; |
---|
652 | g = x^2+y; |
---|
653 | list rg = rightOre(g, f, 1, p); |
---|
654 | print(rg[1]); |
---|
655 | rg[2]; |
---|
656 | g*rg[1][2]-f*rg[1][1]; |
---|
657 | // rational localization |
---|
658 | intvec rat = 1; |
---|
659 | f = Dx+Dy; |
---|
660 | g = x; |
---|
661 | list rr = rightOre(g, f, 2, rat); |
---|
662 | print(rr[1]); |
---|
663 | rr[2]; |
---|
664 | g*rr[1][2]-f*rr[1][1]; |
---|
665 | } |
---|
666 | ////////////////////////////////////////////////////////////////////// |
---|
667 | static proc ore(poly s, poly r, int locType, def locData, int rightOre) { |
---|
668 | // TODO: once ringlist-bug [#577] is PROPERLY fixed (probability: low), |
---|
669 | // use eliminateNC instead of eliminate in rightOre-rational and |
---|
670 | // rightOre-geometric |
---|
671 | // problem: the ordering on the opposite algebra is not recognized as |
---|
672 | // valid for eliminateNC, even though it is |
---|
673 | if (r == 0) { |
---|
674 | return([1,0], ideal(1)); |
---|
675 | } |
---|
676 | int i, j; |
---|
677 | // modification for monoidal localization |
---|
678 | if (locType == 0) { |
---|
679 | // computations will be carried out in the localization at the product |
---|
680 | // of all irreducible factors of s, thus all factors have to be |
---|
681 | // raised to the highest power occurring in the factorization of s |
---|
682 | list factorList = commutativeFactorization(s); |
---|
683 | ideal factors = factorList[1]; |
---|
684 | int maxPow = 0; |
---|
685 | intvec exponents = factorList[2]; |
---|
686 | for (i = 1; i <= size(exponents); i++) { |
---|
687 | if (exponents[i] > maxPow) { |
---|
688 | maxPow = exponents[i]; |
---|
689 | } |
---|
690 | } |
---|
691 | int expDiff; |
---|
692 | for (i = 1; i <= size(exponents); i++) { |
---|
693 | expDiff = maxPow - exponents[i]; |
---|
694 | s = s * factors[i]^expDiff; |
---|
695 | r = r * factors[i]^expDiff; |
---|
696 | } |
---|
697 | } |
---|
698 | // compute kernel of x maps to x*r+R*s |
---|
699 | ideal J = modulo(r, s); |
---|
700 | ideal I = std(J); |
---|
701 | // compute the poly in S |
---|
702 | poly ts; |
---|
703 | if (locType == 0) { // type 0: monoidal localization |
---|
704 | intvec sle = leadexp(s); |
---|
705 | intvec rle; |
---|
706 | int m = deg(r) + 1; // upper bound |
---|
707 | int dividesLeadingMonomial; |
---|
708 | // find the minimal possible exponent m such that lm(f_i)|lm(s)^m |
---|
709 | for (i = 1; i <= size(I); i++) { |
---|
710 | rle = leadexp(I[i]); |
---|
711 | dividesLeadingMonomial = 1; |
---|
712 | for (j = 1; j <= size(rle); j++) { |
---|
713 | if (sle[j] == 0 && rle[j] != 0) { |
---|
714 | // lm(f_i) divides no power of lm(s) |
---|
715 | dividesLeadingMonomial = 0; |
---|
716 | break; |
---|
717 | } |
---|
718 | } |
---|
719 | if (dividesLeadingMonomial) { |
---|
720 | int mf = 1; |
---|
721 | while (1) { |
---|
722 | if (mf * sle - rle >= 0) { |
---|
723 | break; // now lm(r_i) divides lm(s)^mf |
---|
724 | } else { |
---|
725 | mf++; |
---|
726 | } |
---|
727 | } |
---|
728 | if (mf < m) { |
---|
729 | m = mf; // mf is lower than the current minimum m |
---|
730 | } |
---|
731 | } |
---|
732 | } |
---|
733 | poly nf = NF(s^m, I); |
---|
734 | for (; m <= deg(r) + 1; m++) { // sic: m is initialized beforehand |
---|
735 | if (nf == 0) { // s^m is in I |
---|
736 | ts = s^m; |
---|
737 | ideal Js = s^m; |
---|
738 | break; |
---|
739 | } else { |
---|
740 | nf = NF(s*nf, I); |
---|
741 | } |
---|
742 | } |
---|
743 | } |
---|
744 | if (locType == 1) { // type 1: geometric localization |
---|
745 | int n = nvars(basering) div 2; |
---|
746 | if (rightOre == 1) { |
---|
747 | // in the right Ore setting, the order of variables is inverted |
---|
748 | poly elimVars = 1; |
---|
749 | for (i = 1; i <= n; i++) { |
---|
750 | elimVars = elimVars * var(i); |
---|
751 | } |
---|
752 | J = eliminate(I, elimVars); |
---|
753 | } else { |
---|
754 | J = eliminateNC(I, intvec((n+1)..(2*n))); |
---|
755 | } |
---|
756 | J = std(J); |
---|
757 | // compute ts, the generator of J with the smallest degree that does |
---|
758 | // not vanish at locData |
---|
759 | ideal K = intersect(J, locData); |
---|
760 | K = std(K); |
---|
761 | poly h, cand; |
---|
762 | ideal Js; |
---|
763 | ideal Q = std(locData); |
---|
764 | for (i = 1; i <= size(J); i++) { |
---|
765 | h = NF(J[i],Q); |
---|
766 | if (h != 0) { |
---|
767 | cand = NF(J[i],K); |
---|
768 | Js = Js + cand; |
---|
769 | if (ts == 0 || deg(cand) < deg(ts)) { |
---|
770 | ts = cand; |
---|
771 | } |
---|
772 | } |
---|
773 | } |
---|
774 | } |
---|
775 | if (locType == 2) { // type 2: rational localization |
---|
776 | int n = nvars(basering); |
---|
777 | if (size(locData) < n) { // there are variables to eliminate |
---|
778 | intvec modLocData; |
---|
779 | int check; |
---|
780 | modLocData = intvecComplement(locData, 1..n); |
---|
781 | // calculate modLocData = {1...n}\locData |
---|
782 | if (rightOre == 1) { |
---|
783 | // in the right Ore setting, the order of variables is inverted |
---|
784 | poly elimVars = 1; |
---|
785 | for (i = 1; i <= size(locData); i++) { |
---|
786 | elimVars = elimVars * var(locData[i]); |
---|
787 | } |
---|
788 | J = eliminate(I, elimVars); |
---|
789 | } else { |
---|
790 | J = eliminateNC(I, modLocData); |
---|
791 | } |
---|
792 | |
---|
793 | } else { // no variables to eliminate (total localization) |
---|
794 | J = I; |
---|
795 | } |
---|
796 | J = std(J); |
---|
797 | ts = J[1]; |
---|
798 | for (i = 2; i <= size(J); i++) { |
---|
799 | if (deg(J[i]) < deg(ts)) { |
---|
800 | ts = J[i]; // choose generator with lowest total degree |
---|
801 | } |
---|
802 | } |
---|
803 | ideal Js = J; |
---|
804 | } |
---|
805 | // calculate the other poly |
---|
806 | poly tr = division(ts*r, s)[1][1,1]; |
---|
807 | if (ts == 0 || tr*s-ts*r != 0) { |
---|
808 | string s; |
---|
809 | if (rightOre) { |
---|
810 | s = "right"; |
---|
811 | } else { |
---|
812 | s = "left"; |
---|
813 | } |
---|
814 | ERROR("no " + s + " Ore data could be found for s=" + string(s) |
---|
815 | + " and r=" + string(r)); |
---|
816 | } |
---|
817 | vector oreParas = [ts,tr]; |
---|
818 | list result = oreParas, Js; |
---|
819 | return (result); |
---|
820 | } |
---|
821 | ////////////////////////////////////////////////////////////////////// |
---|
822 | proc convertRightToLeftFraction(vector frac, int locType, def locData) |
---|
823 | "USAGE: convertRightToLeftFraction(frac, locType, locData), |
---|
824 | vector frac, int locType, list/vector/intvec locData |
---|
825 | PURPOSE: determine a left fraction representation of a given fraction |
---|
826 | ASSUME: |
---|
827 | RETURN: vector |
---|
828 | NOTE: - the returned vector contains a repr. of frac as a left fraction |
---|
829 | - if the left representation of frac is already specified, |
---|
830 | frac will be returned. |
---|
831 | EXAMPLE: example convertRightToLeftFraction; shows examples" |
---|
832 | { |
---|
833 | testLocData(locType, locData); |
---|
834 | testFraction(frac, locType, locData); |
---|
835 | if (frac[1] != 0) { // frac already has a left representation |
---|
836 | return (frac); |
---|
837 | } else { // frac has no left representation, but a right one |
---|
838 | vector oreParas = leftOre(frac[4], frac[3], locType, locData)[1]; |
---|
839 | vector result = [oreParas[1], oreParas[2], frac[3], frac[4]]; |
---|
840 | return (result); |
---|
841 | } |
---|
842 | } |
---|
843 | example |
---|
844 | { |
---|
845 | "EXAMPLE:"; echo = 2; |
---|
846 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
847 | def S = Weyl(); |
---|
848 | setring S; S; |
---|
849 | // monoidal localization |
---|
850 | poly g1 = x+3; |
---|
851 | poly g2 = x*y; |
---|
852 | list L = g1,g2; |
---|
853 | poly g = g1^2*g2; |
---|
854 | poly f = Dx; |
---|
855 | vector fracm = [0,0,f,g]; |
---|
856 | vector rm = convertRightToLeftFraction(fracm, 0, L); |
---|
857 | print(rm); |
---|
858 | rm[2]*g-rm[1]*f; |
---|
859 | // geometric localization |
---|
860 | ideal p = x-1, y-3; |
---|
861 | f = Dx; |
---|
862 | g = x^2+y; |
---|
863 | vector fracg = [0,0,f,g]; |
---|
864 | vector rg = convertRightToLeftFraction(fracg, 1, p); |
---|
865 | print(rg); |
---|
866 | rg[2]*g-rg[1]*f; |
---|
867 | // rational localization |
---|
868 | intvec rat = 1; |
---|
869 | f = Dx+Dy; |
---|
870 | g = x; |
---|
871 | vector fracr = [0,0,f,g]; |
---|
872 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
873 | print(rr); |
---|
874 | rr[2]*g-rr[1]*f; |
---|
875 | } |
---|
876 | ////////////////////////////////////////////////////////////////////// |
---|
877 | proc convertLeftToRightFraction(vector frac, int locType, def locData) |
---|
878 | "USAGE: convertLeftToRightFraction(frac, locType, locData), vector frac, |
---|
879 | int locType, list/vector/intvec locData |
---|
880 | PURPOSE: determine a right fraction representation of a given fraction |
---|
881 | ASSUME: |
---|
882 | RETURN: vector |
---|
883 | NOTE: - the returned vector contains a repr. of frac as a right fraction, |
---|
884 | - if the right representation of frac is already specified, |
---|
885 | frac will be returned. |
---|
886 | EXAMPLE: example convertLeftToRightFraction; shows examples" |
---|
887 | { |
---|
888 | testLocData(locType, locData); |
---|
889 | testFraction(frac, locType, locData); |
---|
890 | if (frac[4] != 0) { // frac already has a right representation |
---|
891 | return (frac); |
---|
892 | } else { // frac has no right representation, but a left one |
---|
893 | vector oreParas = rightOre(frac[1], frac[2], locType, locData)[1]; |
---|
894 | vector result = [frac[1], frac[2], oreParas[2], oreParas[1]]; |
---|
895 | return (result); |
---|
896 | } |
---|
897 | } |
---|
898 | example |
---|
899 | { |
---|
900 | "EXAMPLE:"; echo = 2; |
---|
901 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
902 | def S = Weyl(); |
---|
903 | setring S; S; |
---|
904 | // monoidal localization |
---|
905 | poly g = x; |
---|
906 | poly f = Dx; |
---|
907 | vector fracm = [g,f,0,0]; |
---|
908 | list L = g; |
---|
909 | vector rm = convertLeftToRightFraction(fracm, 0, L); |
---|
910 | print(rm); |
---|
911 | f*rm[4]-g*rm[3]; |
---|
912 | // geometric localization |
---|
913 | g = x+y; |
---|
914 | f = Dx+Dy; |
---|
915 | vector fracg = [g,f,0,0]; |
---|
916 | ideal p = x-1, y-3; |
---|
917 | vector rg = convertLeftToRightFraction(fracg, 1, p); |
---|
918 | print(rg); |
---|
919 | f*rg[4]-g*rg[3]; |
---|
920 | // rational localization |
---|
921 | intvec rat = 1; |
---|
922 | f = Dx+Dy; |
---|
923 | g = x; |
---|
924 | vector fracr = [g,f,0,0]; |
---|
925 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
926 | print(rr); |
---|
927 | f*rr[4]-g*rr[3]; |
---|
928 | } |
---|
929 | ////////////////////////////////////////////////////////////////////// |
---|
930 | proc isZeroFraction(vector frac) |
---|
931 | "USAGE: isZeroFraction(frac), vector frac |
---|
932 | PURPOSE: determine if the vector frac represents zero |
---|
933 | ASSUME: frac is a valid fraction |
---|
934 | RETURN: int |
---|
935 | NOTE: returns 1, if frac == 0; 0 otherwise |
---|
936 | EXAMPLE: example isZeroFraction; shows examples" |
---|
937 | { |
---|
938 | if (frac[1] == 0 && frac[4] == 0) { |
---|
939 | return(0); |
---|
940 | } |
---|
941 | if (frac[1] == 0) { // frac has no left representation |
---|
942 | if (frac[3] == 0) { // the right representation of frac is zero |
---|
943 | return(1); |
---|
944 | } |
---|
945 | } else { // frac has a left representation |
---|
946 | if (frac[2] == 0) { // the left representation of frac is zero |
---|
947 | return(1); |
---|
948 | } |
---|
949 | } |
---|
950 | return(0); |
---|
951 | } |
---|
952 | example |
---|
953 | { |
---|
954 | "EXAMPLE:"; echo = 2; |
---|
955 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
956 | def S = Weyl(); |
---|
957 | setring S; S; |
---|
958 | isZeroFraction([42,0,0,0]); |
---|
959 | isZeroFraction([0,0,Dx,3]); |
---|
960 | isZeroFraction([1,1,1,1]); |
---|
961 | } |
---|
962 | ////////////////////////////////////////////////////////////////////// |
---|
963 | proc isOneFraction(vector frac) |
---|
964 | "USAGE: isOneFraction(frac), vector frac |
---|
965 | PURPOSE: determine if the vector frac represents one |
---|
966 | ASSUME: frac is a valid fraction |
---|
967 | RETURN: int |
---|
968 | NOTE: 1, if frac == 1; 0 otherwise |
---|
969 | EXAMPLE: example isOneFraction; shows examples" |
---|
970 | { |
---|
971 | if (frac[1] == 0 && frac[4] == 0) { |
---|
972 | return(0); |
---|
973 | } |
---|
974 | if (frac[1] == 0) { // frac has no left representation |
---|
975 | if (frac[3] == frac[4]) { // the right representation of frac is zero |
---|
976 | return(1); |
---|
977 | } |
---|
978 | } else { // frac has a left representation |
---|
979 | if (frac[2] == frac[1]) { // the left representation of frac is zero |
---|
980 | return(1); |
---|
981 | } |
---|
982 | } |
---|
983 | return(0); |
---|
984 | } |
---|
985 | example |
---|
986 | { |
---|
987 | "EXAMPLE:"; echo = 2; |
---|
988 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
989 | def S = Weyl(); |
---|
990 | setring S; S; |
---|
991 | isOneFraction([42,42,0,0]); |
---|
992 | isOneFraction([0,0,Dx,3]); |
---|
993 | isOneFraction([1,0,0,1]); |
---|
994 | } |
---|
995 | ////////// arithmetic //////////////////////////////////////////////// |
---|
996 | proc addLeftFractions(vector a, vector b, int locType, def locData, list #) |
---|
997 | "USAGE: addLeftFractions(a, b, locType, locData(, override)), |
---|
998 | vector a, b, int locType, list/vector/intvec locData(, int override) |
---|
999 | PURPOSE: add two left fractions in the specified localization |
---|
1000 | ASSUME: |
---|
1001 | RETURN: vector |
---|
1002 | NOTE: the returned vector is the sum of a and b as fractions in the |
---|
1003 | localization specified by locType and locData. |
---|
1004 | EXAMPLE: example addLeftFractions; shows examples" |
---|
1005 | { |
---|
1006 | int override = 0; |
---|
1007 | if (size(#) > 1) { |
---|
1008 | if(typeof(#[1]) == "int") { |
---|
1009 | override = #[1]; |
---|
1010 | } |
---|
1011 | } |
---|
1012 | if (!override) { |
---|
1013 | testLocData(locType, locData); |
---|
1014 | testFraction(a, locType, locData); |
---|
1015 | testFraction(b, locType, locData); |
---|
1016 | } |
---|
1017 | // check for a shortcut |
---|
1018 | if (isZeroFraction(a)) { |
---|
1019 | return(b); |
---|
1020 | } |
---|
1021 | if (isZeroFraction(b)) { |
---|
1022 | return(a); |
---|
1023 | } |
---|
1024 | if (a[1] == 0) { // a has no left representation |
---|
1025 | a = convertRightToLeftFraction(a, locType, locData); |
---|
1026 | } |
---|
1027 | if (b[1] == 0) { // b has no left representation |
---|
1028 | b = convertRightToLeftFraction(b, locType, locData); |
---|
1029 | } |
---|
1030 | if (a[1] == b[1]) { // a and b have the same left denominator |
---|
1031 | return ([a[1],a[2] + b[2],0,0]); |
---|
1032 | } |
---|
1033 | // no shortcut found, use regular method |
---|
1034 | vector oreParas = leftOre(b[1], a[1], locType, locData)[1]; |
---|
1035 | vector result = [oreParas[1]*a[1],oreParas[1]*a[2]+oreParas[2]*b[2],0,0]; |
---|
1036 | return (result); |
---|
1037 | } |
---|
1038 | example |
---|
1039 | { |
---|
1040 | "EXAMPLE:"; echo = 2; |
---|
1041 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1042 | def S = Weyl(); |
---|
1043 | setring S; S; |
---|
1044 | // monoidal localization |
---|
1045 | poly g1 = x+3; |
---|
1046 | poly g2 = x*y+y; |
---|
1047 | list L = g1,g2; |
---|
1048 | poly s1 = g1; |
---|
1049 | poly s2 = g2; |
---|
1050 | poly r1 = Dx; |
---|
1051 | poly r2 = Dy; |
---|
1052 | vector frac1 = [s1,r1,0,0]; |
---|
1053 | vector frac2 = [s2,r2,0,0]; |
---|
1054 | vector rm = addLeftFractions(frac1, frac2, 0, L); |
---|
1055 | print(rm); |
---|
1056 | // geometric localization |
---|
1057 | ideal p = x-1, y-3; |
---|
1058 | vector rg = addLeftFractions(frac1, frac2, 1, p); |
---|
1059 | print(rg); |
---|
1060 | // rational localization |
---|
1061 | intvec v = 2; |
---|
1062 | s1 = y^2+y+1; |
---|
1063 | s2 = y-2; |
---|
1064 | r1 = Dx; |
---|
1065 | r2 = Dy; |
---|
1066 | frac1 = [s1,r1,0,0]; |
---|
1067 | frac2 = [s2,r2,0,0]; |
---|
1068 | vector rr = addLeftFractions(frac1, frac2, 2, v); |
---|
1069 | print(rr); |
---|
1070 | } |
---|
1071 | ////////////////////////////////////////////////////////////////////// |
---|
1072 | proc multiplyLeftFractions(vector a, vector b, int locType, def locData, list #) |
---|
1073 | "USAGE: multiplyLeftFractions(a, b, locType, locData(, override)), |
---|
1074 | vector a, b, int locType, list/vector/intvec locData, int override |
---|
1075 | PURPOSE: multiply two left fractions in the specified localization |
---|
1076 | ASSUME: |
---|
1077 | RETURN: vector |
---|
1078 | NOTE: the returned vector is the product of a and b as fractions in the |
---|
1079 | localization specified by locType and locData. |
---|
1080 | EXAMPLE: example multiplyLeftFractions; shows examples" |
---|
1081 | { |
---|
1082 | int override = 0; |
---|
1083 | if (size(#) > 1) { |
---|
1084 | if(typeof(#[1]) == "int") { |
---|
1085 | override = #[1]; |
---|
1086 | } |
---|
1087 | } |
---|
1088 | if (!override) { |
---|
1089 | testLocData(locType, locData); |
---|
1090 | testFraction(a, locType, locData); |
---|
1091 | testFraction(b, locType, locData); |
---|
1092 | } |
---|
1093 | // check for a shortcut |
---|
1094 | if (isZeroFraction(a) || isZeroFraction(b)) { |
---|
1095 | return([1,0,0,1]); |
---|
1096 | } |
---|
1097 | if (isOneFraction(a)) { |
---|
1098 | return(b); |
---|
1099 | } |
---|
1100 | if (isOneFraction(b)) { |
---|
1101 | return(a); |
---|
1102 | } |
---|
1103 | if(a[1] == 0) { |
---|
1104 | a = convertRightToLeftFraction(a, locType, locData); |
---|
1105 | } |
---|
1106 | if(b[1] == 0) { |
---|
1107 | b = convertRightToLeftFraction(b, locType, locData); |
---|
1108 | } |
---|
1109 | if( (a[2] == 0) || (b[2] == 0) ) { |
---|
1110 | return ([1,0,0,1]); |
---|
1111 | } |
---|
1112 | if (a[2]*b[1] == b[1]*a[2]) { // trivial solution of the Ore condition |
---|
1113 | return([b[1]*a[1],a[2]*b[2]]); |
---|
1114 | } |
---|
1115 | // no shortcut found, use regular method |
---|
1116 | vector oreParas = ore(b[1], a[2], locType, locData, 0)[1]; |
---|
1117 | vector result = [oreParas[1]*a[1],oreParas[2]*b[2],0,0]; |
---|
1118 | return (result); |
---|
1119 | } |
---|
1120 | example |
---|
1121 | { |
---|
1122 | "EXAMPLE:"; echo = 2; |
---|
1123 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1124 | def S = Weyl(); |
---|
1125 | setring S; S; |
---|
1126 | // monoidal localization |
---|
1127 | poly g1 = x+3; |
---|
1128 | poly g2 = x*y+y; |
---|
1129 | list L = g1,g2; |
---|
1130 | poly s1 = g1; |
---|
1131 | poly s2 = g2; |
---|
1132 | poly r1 = Dx; |
---|
1133 | poly r2 = Dy; |
---|
1134 | vector frac1 = [s1,r1,0,0]; |
---|
1135 | vector frac2 = [s2,r2,0,0]; |
---|
1136 | vector rm = multiplyLeftFractions(frac1, frac2, 0, L); |
---|
1137 | print(rm); |
---|
1138 | // geometric localization |
---|
1139 | ideal p = x-1, y-3; |
---|
1140 | vector rg = multiplyLeftFractions(frac1, frac2, 1, p); |
---|
1141 | print(rg); |
---|
1142 | // rational localization |
---|
1143 | intvec v = 2; |
---|
1144 | s1 = y^2+y+1; |
---|
1145 | s2 = y-2; |
---|
1146 | r1 = Dx; |
---|
1147 | r2 = Dy; |
---|
1148 | frac1 = [s1,r1,0,0]; |
---|
1149 | frac2 = [s2,r2,0,0]; |
---|
1150 | vector rr1 = multiplyLeftFractions(frac1, frac2, 2, v); |
---|
1151 | print(rr1); |
---|
1152 | vector rr2 = multiplyLeftFractions(frac2, frac1, 2, v); |
---|
1153 | print(rr2); |
---|
1154 | areEqualLeftFractions(rr1, rr2, 2, v); |
---|
1155 | } |
---|
1156 | ////////////////////////////////////////////////////////////////////// |
---|
1157 | proc areEqualLeftFractions(vector a, vector b, int locType, def locData) |
---|
1158 | "USAGE: areEqualLeftFractions(a, b, locType, locData), vector a, b, |
---|
1159 | int locType, list/vector/intvec locData |
---|
1160 | PURPOSE: check if two given fractions are equal |
---|
1161 | ASSUME: |
---|
1162 | RETURN: int |
---|
1163 | NOTE: returns 1 or 0, depending whether a=b as fractions in the |
---|
1164 | localization specified by locType and locData |
---|
1165 | EXAMPLE: example areEqualLeftFractions; shows examples" |
---|
1166 | { |
---|
1167 | testLocData(locType, locData); |
---|
1168 | testFraction(a, locType, locData); |
---|
1169 | testFraction(b, locType, locData); |
---|
1170 | if(a[1] == 0) { |
---|
1171 | a = convertRightToLeftFraction(a, locType, locData); |
---|
1172 | } |
---|
1173 | if(b[1] == 0) { |
---|
1174 | b = convertRightToLeftFraction(b, locType, locData); |
---|
1175 | } |
---|
1176 | vector negB = [b[1], -b[2], -b[3], b[4]]; |
---|
1177 | //testFraction(negB, locType, locData); // unnecessary check |
---|
1178 | vector result = addLeftFractions(a, negB, locType, locData); |
---|
1179 | return(isZeroFraction(result)); |
---|
1180 | } |
---|
1181 | example |
---|
1182 | { |
---|
1183 | "EXAMPLE:"; echo = 2; |
---|
1184 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1185 | def S = Weyl(); |
---|
1186 | setring S; S; |
---|
1187 | // monoidal |
---|
1188 | poly g1 = x*y+3; |
---|
1189 | poly g2 = y^3; |
---|
1190 | list L = g1,g2; |
---|
1191 | poly s1 = g1; |
---|
1192 | poly s2 = s1*g2; |
---|
1193 | poly s3 = s2; |
---|
1194 | poly r1 = Dx; |
---|
1195 | poly r2 = g2*r1; |
---|
1196 | poly r3 = s1*r1+3; |
---|
1197 | vector fracm1 = [s1,r1,0,0]; |
---|
1198 | vector fracm2 = [s2,r2,0,0]; |
---|
1199 | vector fracm3 = [s3,r3,0,0]; |
---|
1200 | areEqualLeftFractions(fracm1, fracm2, 0, L); |
---|
1201 | areEqualLeftFractions(fracm1, fracm3, 0, L); |
---|
1202 | areEqualLeftFractions(fracm2, fracm3, 0, L); |
---|
1203 | } |
---|
1204 | ////////////////////////////////////////////////////////////////////// |
---|
1205 | proc isInvertibleLeftFraction(vector frac, int locType, def locData) |
---|
1206 | "USAGE: isInvertibleLeftFraction(frac, locType, locData), vector frac, |
---|
1207 | int locType, list/vector/intvec locData |
---|
1208 | PURPOSE: check if a fraction is invertible in the specified localization |
---|
1209 | ASSUME: |
---|
1210 | RETURN: int |
---|
1211 | NOTE: - returns 1, if the numerator of frac is in the denominator set, |
---|
1212 | - returns 0, otherwise (NOTE: this does NOT mean that the fraction is |
---|
1213 | not invertible, it just means it could not be determined by the |
---|
1214 | method above). |
---|
1215 | EXAMPLE: example isInvertibleLeftFraction; shows examples" |
---|
1216 | { |
---|
1217 | testLocData(locType, locData); |
---|
1218 | testFraction(frac, locType, locData); |
---|
1219 | locData = normalizeLocalization(locType, locData); |
---|
1220 | if(frac[1] != 0) { // frac has a left representation |
---|
1221 | return(isInS(frac[2], locType, locData)); |
---|
1222 | } else { // frac has no left, but a right representation |
---|
1223 | return(isInS(frac[3], locType, locData)); |
---|
1224 | } |
---|
1225 | } |
---|
1226 | example |
---|
1227 | { |
---|
1228 | "EXAMPLE:"; echo = 2; |
---|
1229 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1230 | def S = Weyl(); |
---|
1231 | setring S; S; |
---|
1232 | poly g1 = x+3; |
---|
1233 | poly g2 = x*y; |
---|
1234 | list L = g1,g2; |
---|
1235 | vector frac = [g1*g2, 17, 0, 0]; |
---|
1236 | isInvertibleLeftFraction(frac, 0, L); |
---|
1237 | ideal p = x-1, y; |
---|
1238 | frac = [g1, x, 0, 0]; |
---|
1239 | isInvertibleLeftFraction(frac, 1, p); |
---|
1240 | intvec rat = 1,2; |
---|
1241 | frac = [g1*g2, Dx, 0, 0]; |
---|
1242 | isInvertibleLeftFraction(frac, 2, rat); |
---|
1243 | } |
---|
1244 | ////////////////////////////////////////////////////////////////////// |
---|
1245 | proc invertLeftFraction(vector frac, int locType, def locData) |
---|
1246 | "USAGE: invertLeftFraction(frac, locType, locData), vector frac, int locType, |
---|
1247 | list/vector/intvec locData |
---|
1248 | PURPOSE: invert a fraction in the specified localization |
---|
1249 | ASSUME: frac is invertible in the loc. specified by locType and locData |
---|
1250 | RETURN: vector |
---|
1251 | NOTE: - returns the multiplicative inverse of frac in the localization |
---|
1252 | specified by locType and locData, |
---|
1253 | - throws error if frac is not invertible (NOTE: this does NOT mean |
---|
1254 | that the fraction is not invertible, it just means it could not be |
---|
1255 | determined by the method listed above). |
---|
1256 | EXAMPLE: example invertLeftFraction; shows examples" |
---|
1257 | { |
---|
1258 | // standard tests are done by isInvertibleLeftFraction |
---|
1259 | if (isInvertibleLeftFraction(frac, locType, locData)) { |
---|
1260 | return([frac[2],frac[1],frac[4],frac[3]]); |
---|
1261 | } else { |
---|
1262 | return([1,0,0,1]); |
---|
1263 | } |
---|
1264 | } |
---|
1265 | example |
---|
1266 | { |
---|
1267 | "EXAMPLE:"; echo = 2; |
---|
1268 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1269 | def S = Weyl(); |
---|
1270 | setring S; S; |
---|
1271 | poly g1 = x+3; |
---|
1272 | poly g2 = x*y; |
---|
1273 | list L = g1,g2; |
---|
1274 | vector frac = [g1*g2, 17, 0, 0]; |
---|
1275 | print(invertLeftFraction(frac, 0, L)); |
---|
1276 | ideal p = x-1, y; |
---|
1277 | frac = [g1, x, 0, 0]; |
---|
1278 | print(invertLeftFraction(frac, 1, p)); |
---|
1279 | intvec rat = 1,2; |
---|
1280 | frac = [g1*g2, y, 0, 0]; |
---|
1281 | print(invertLeftFraction(frac, 2, rat)); |
---|
1282 | } |
---|
1283 | ////////////////////////////////////////////////////////////////////// |
---|
1284 | proc normalizeMonoidal(list L) |
---|
1285 | "USAGE: normalizeMonoidal(L), list L |
---|
1286 | PURPOSE: compute a normal form of monoidal localization data |
---|
1287 | RETURN: list |
---|
1288 | NOTE: given a list of polys, returns a list of all unique factors appearing |
---|
1289 | in the given polys |
---|
1290 | EXAMPLE: example normalizeMonoidal; shows examples" |
---|
1291 | { |
---|
1292 | ideal allFactors; |
---|
1293 | int i; |
---|
1294 | for (i = 1; i <= size(L); i++) { |
---|
1295 | allFactors = allFactors, commutativeFactorization(L[i],1); |
---|
1296 | } |
---|
1297 | allFactors = simplify(allFactors,1+2+4); |
---|
1298 | // simplify: divide by leading coefficients (1), |
---|
1299 | // purge zero generators (2), purge double entries (4) |
---|
1300 | ideal rev = sort(allFactors)[1]; // sort sorts ascendingly |
---|
1301 | return(list(rev[size(rev)..1])); // reverse order and cast to list |
---|
1302 | } |
---|
1303 | example |
---|
1304 | { |
---|
1305 | "EXAMPLE:"; echo = 2; |
---|
1306 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1307 | def S = Weyl(); setring S; |
---|
1308 | list L = x^2*y^3, (x+1)*(x*y-3*y^2+1); |
---|
1309 | L = normalizeMonoidal(L); |
---|
1310 | print(L); |
---|
1311 | } |
---|
1312 | ////////////////////////////////////////////////////////////////////// |
---|
1313 | proc normalizeRational(intvec v) |
---|
1314 | "USAGE: normalizeRational(v), intvec v |
---|
1315 | PURPOSE: compute a normal form of rational localization data |
---|
1316 | RETURN: intvec |
---|
1317 | NOTE: purges double entries and sorts ascendingly |
---|
1318 | EXAMPLE: example normalizeRational; shows examples" |
---|
1319 | { |
---|
1320 | int n = nvars(basering); |
---|
1321 | int i; |
---|
1322 | intvec result; |
---|
1323 | intvec occurring = 0:n; |
---|
1324 | for (i = 1; i <= size(v); i++) { |
---|
1325 | occurring[v[i]] = 1; |
---|
1326 | } |
---|
1327 | for (i = 1; i <= size(occurring); i++) { |
---|
1328 | if (occurring[i]) { |
---|
1329 | if (result == 0) { |
---|
1330 | result = i; |
---|
1331 | } else { |
---|
1332 | result = result, i; |
---|
1333 | } |
---|
1334 | } |
---|
1335 | } |
---|
1336 | return(result); |
---|
1337 | } |
---|
1338 | example |
---|
1339 | { |
---|
1340 | "EXAMPLE:"; echo = 2; |
---|
1341 | ring R; setring R; |
---|
1342 | intvec v = 9,5,9,3,1,5; |
---|
1343 | v = normalizeRational(v); |
---|
1344 | v; |
---|
1345 | } |
---|
1346 | ////////// internal functions //////////////////////////////////////// |
---|
1347 | static proc inducesCommutativeSubring(def input) |
---|
1348 | { |
---|
1349 | ideal vars = variables(input); |
---|
1350 | int i, j; |
---|
1351 | for (i = 1; i <= size(vars); i++) { |
---|
1352 | for (j = i + 1; j <= size(vars); j++) { |
---|
1353 | if (vars[i]*vars[j] != vars[j]*vars[i]) { |
---|
1354 | return(0); |
---|
1355 | } |
---|
1356 | } |
---|
1357 | } |
---|
1358 | return(1); |
---|
1359 | } |
---|
1360 | ////////////////////////////////////////////////////////////////////// |
---|
1361 | static proc commutativeFactorization(poly p, list #) |
---|
1362 | "USAGE: commutativeFactorization(p[, #]), poly p[, list #] |
---|
1363 | PURPOSE: compute a factorization of p ignoring non-commutative relations |
---|
1364 | RETURN: list or ideal |
---|
1365 | NOTE: the optional parameter is passed to factorize after changing to a |
---|
1366 | commutative ring, the result of factorize is transferred back to |
---|
1367 | basering |
---|
1368 | SEE ALSO: factorize |
---|
1369 | EXAMPLE: " |
---|
1370 | { |
---|
1371 | int factorType = 0; |
---|
1372 | if (size(#) > 0) { |
---|
1373 | if (typeof(#[1]) == "int") { |
---|
1374 | factorType = #[1]; |
---|
1375 | } |
---|
1376 | } |
---|
1377 | list RL = ringlist(basering); |
---|
1378 | if (size(RL) > 4) { |
---|
1379 | def bsRing = basering; |
---|
1380 | RL = RL[1..4]; |
---|
1381 | def commRing = ring(RL); |
---|
1382 | setring commRing; |
---|
1383 | poly commP = imap(bsRing, p); |
---|
1384 | if (factorType == 1) { |
---|
1385 | ideal commFactors = factorize(commP, 1); |
---|
1386 | setring bsRing; |
---|
1387 | return(imap(commRing, commFactors)); |
---|
1388 | } else { |
---|
1389 | list commFac = factorize(commP, factorType); |
---|
1390 | intvec exponents = commFac[2]; |
---|
1391 | ideal commFactors = commFac[1]; |
---|
1392 | setring bsRing; |
---|
1393 | list result; |
---|
1394 | result[1] = imap(commRing, commFactors); |
---|
1395 | result[2] = exponents; |
---|
1396 | return(result); |
---|
1397 | } |
---|
1398 | } else { |
---|
1399 | return(factorize(p, factorType)); |
---|
1400 | } |
---|
1401 | } |
---|
1402 | ////////////////////////////////////////////////////////////////////// |
---|
1403 | static proc normalizeLocalization(int locType, def locData) { |
---|
1404 | if (locType == 0) { |
---|
1405 | return(normalizeMonoidal(locData)); |
---|
1406 | } |
---|
1407 | if (locType == 1) { |
---|
1408 | return(std(locData)); |
---|
1409 | } |
---|
1410 | if (locType == 2) { |
---|
1411 | return(normalizeRational(locData)); |
---|
1412 | } |
---|
1413 | return(locData); |
---|
1414 | } |
---|
1415 | ////////////////////////////////////////////////////////////////////// |
---|
1416 | static proc intvecComplement(intvec v, intvec w) { // complement of v in w as sets |
---|
1417 | intvec result; |
---|
1418 | int i; |
---|
1419 | int j; |
---|
1420 | int foundMatch; |
---|
1421 | for (i = 1; i <= size(w); i++) { |
---|
1422 | foundMatch = 0; |
---|
1423 | for (j = 1; j <= size(v); j++) { |
---|
1424 | if (w[i] == v[j]) { |
---|
1425 | foundMatch = 1; |
---|
1426 | break; |
---|
1427 | } |
---|
1428 | } |
---|
1429 | if (!foundMatch) { // v[i] is not in w |
---|
1430 | if (result == 0) { |
---|
1431 | result = w[i]; |
---|
1432 | } else { |
---|
1433 | result = result, w[i]; |
---|
1434 | } |
---|
1435 | } |
---|
1436 | } |
---|
1437 | return(result); |
---|
1438 | } |
---|
1439 | ////////////////////////////////////////////////////////////////////// |
---|
1440 | ////////// internal testing procedures /////////////////////////////// |
---|
1441 | static proc testIsInS() |
---|
1442 | { |
---|
1443 | print(" testing isInS..."); |
---|
1444 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1445 | def S = Weyl(); |
---|
1446 | setring S; |
---|
1447 | // monoidal localization |
---|
1448 | poly g1 = x^2*y+x+2; |
---|
1449 | poly g2 = y^3+x*y; |
---|
1450 | list L = g1, g2; |
---|
1451 | poly g = g1^2*g2; |
---|
1452 | if (!isInS(g, 0, L)) { |
---|
1453 | ERROR("Weyl monoidal isInS direct positive failed"); |
---|
1454 | } |
---|
1455 | if (!isInS(y^2+x, 0, L)) { |
---|
1456 | ERROR("Weyl monoidal isInS indirect positive failed"); |
---|
1457 | } |
---|
1458 | if (isInS(g-1, 0, L)) { |
---|
1459 | ERROR("Weyl monoidal isInS negative failed"); |
---|
1460 | } |
---|
1461 | // geometric localization |
---|
1462 | ideal p = x-1, y-3; |
---|
1463 | g = x^2+y-3; |
---|
1464 | if (!isInS(g, 1, p)) { |
---|
1465 | ERROR("Weyl geometric isInS positive failed"); |
---|
1466 | } |
---|
1467 | if (isInS((x-1)*g, 1, p)) { |
---|
1468 | ERROR("Weyl geometric isInS negative failed"); |
---|
1469 | } |
---|
1470 | // rational localization |
---|
1471 | intvec v = 2; |
---|
1472 | if (!isInS(y^5+17*y^2-4, 2, v)) { |
---|
1473 | ERROR("Weyl rational isInS positive failed"); |
---|
1474 | } |
---|
1475 | if (isInS(x*y, 2, v)) { |
---|
1476 | ERROR("Weyl rational isInS negative failed"); |
---|
1477 | } |
---|
1478 | intvec w = 4,2,3,4,1; |
---|
1479 | if (!isInS(x*y*Dx*Dy,2,w)) { |
---|
1480 | ERROR("Weyl total rational isInS positive failed"); |
---|
1481 | } |
---|
1482 | if (isInS(0, 2, w)) { |
---|
1483 | ERROR("Weyl total rational isInS negative failed"); |
---|
1484 | } |
---|
1485 | print(" isInS OK"); |
---|
1486 | } |
---|
1487 | ////////////////////////////////////////////////////////////////////// |
---|
1488 | static proc testLeftOre() { |
---|
1489 | print(" testing leftOre..."); |
---|
1490 | // Weyl |
---|
1491 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
1492 | def ncW = Weyl(); |
---|
1493 | setring ncW; |
---|
1494 | //// monoidal localization |
---|
1495 | poly g1 = x+3; |
---|
1496 | poly g2 = x*y; |
---|
1497 | list L = g1,g2; |
---|
1498 | poly g = g1^2*g2; |
---|
1499 | poly f = Dx; |
---|
1500 | vector rm = leftOre(g, f, 0, L)[1]; |
---|
1501 | if (rm[1] == 0 || rm[2]*g-rm[1]*f != 0) { |
---|
1502 | ERROR("Weyl monoidal left Ore failed"); |
---|
1503 | } |
---|
1504 | //// geometric localization |
---|
1505 | vector p1,p2,p3,p4 = [1,3],[0,0],[-1,-2],[10,-180]; |
---|
1506 | vector rg; |
---|
1507 | ideal p; |
---|
1508 | list pVecs = p1,p2,p3,p4; |
---|
1509 | f = Dx; |
---|
1510 | g = x^2+y+3; |
---|
1511 | for(int i = 1; i <= 4; i = i + 1) { |
---|
1512 | p = x-pVecs[i][1], y-pVecs[i][2]; |
---|
1513 | rg = leftOre(g, f, 1, p)[1]; |
---|
1514 | if (rg[1] == 0 || rg[2]*g-rg[1]*f != 0) { |
---|
1515 | ERROR("Weyl geometric left Ore failed at maximal ideal" |
---|
1516 | + " induced by " + string(pVecs[i])); |
---|
1517 | } |
---|
1518 | } |
---|
1519 | //// rational localization |
---|
1520 | intvec rat = 1; |
---|
1521 | f = Dx+Dy; |
---|
1522 | g = x; |
---|
1523 | vector rr = leftOre(g, f, 2, rat)[1]; |
---|
1524 | if (rr[1] == 0 || rr[2]*g-rr[1]*f != 0) { |
---|
1525 | ERROR("Weyl rational left Ore failed"); |
---|
1526 | } |
---|
1527 | // shift rational localization |
---|
1528 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
1529 | matrix D[4][4]; |
---|
1530 | D[1,3] = Sx; |
---|
1531 | D[2,4] = Sy; |
---|
1532 | def ncS = nc_algebra(1, D); |
---|
1533 | setring ncS; |
---|
1534 | rat = 1; |
---|
1535 | poly f = Sx+Sy; |
---|
1536 | poly g = x; |
---|
1537 | vector rr = leftOre(g, f, 2, rat)[1]; |
---|
1538 | if (rr[1] == 0 || rr[2]*g-rr[1]*f != 0) { |
---|
1539 | ERROR("shift rational left Ore failed"); |
---|
1540 | } |
---|
1541 | // q-shift rational localization |
---|
1542 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
1543 | matrix C[4][4] = UpOneMatrix(4); |
---|
1544 | C[1,3] = q; |
---|
1545 | C[2,4] = q; |
---|
1546 | def ncQ = nc_algebra(C, 0); |
---|
1547 | setring ncQ; |
---|
1548 | rat = 1; |
---|
1549 | poly f = Qx+Qy; |
---|
1550 | poly g = x; |
---|
1551 | vector rr = leftOre(g, f, 2, rat)[1]; |
---|
1552 | if (rr[1] == 0 || rr[2]*g-rr[1]*f != 0) { |
---|
1553 | ERROR("q-shift rational left Ore failed"); |
---|
1554 | } |
---|
1555 | print(" leftOre OK"); |
---|
1556 | } |
---|
1557 | ////////////////////////////////////////////////////////////////////// |
---|
1558 | static proc testRightOre() { |
---|
1559 | print(" testing rightOre..."); |
---|
1560 | // Weyl |
---|
1561 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
1562 | def ncW = Weyl(); |
---|
1563 | setring ncW; |
---|
1564 | //// monoidal localization |
---|
1565 | poly g1 = x+3; |
---|
1566 | poly g2 = x*y; |
---|
1567 | list L = g1,g2; |
---|
1568 | poly g = x; |
---|
1569 | poly f = Dx; |
---|
1570 | vector rm = rightOre(g, f, 0, L)[1]; |
---|
1571 | if (rm[1] == 0 || f*rm[1]-g*rm[2] != 0) { |
---|
1572 | ERROR("Weyl monoidal right Ore failed"); |
---|
1573 | } |
---|
1574 | //// geometric localization |
---|
1575 | g = x+y; |
---|
1576 | f = Dx+Dy; |
---|
1577 | ideal p = x-1,y-3; |
---|
1578 | vector rg = rightOre(g, f, 1, p)[1]; |
---|
1579 | if (rg[1] == 0 || f*rg[1]-g*rg[2] != 0) { |
---|
1580 | ERROR("Weyl geometric right Ore failed"); |
---|
1581 | } |
---|
1582 | //// rational localization |
---|
1583 | intvec rat = 1; |
---|
1584 | f = Dx+Dy; |
---|
1585 | g = x; |
---|
1586 | vector rr = rightOre(g, f, 2, rat)[1]; |
---|
1587 | if (rr[1] == 0 || f*rr[1]-g*rr[2] != 0) { |
---|
1588 | ERROR("Weyl rational right Ore failed"); |
---|
1589 | } |
---|
1590 | // shift rational localization |
---|
1591 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
1592 | matrix D[4][4]; |
---|
1593 | D[1,3] = Sx; |
---|
1594 | D[2,4] = Sy; |
---|
1595 | def ncS = nc_algebra(1, D); |
---|
1596 | setring ncS; |
---|
1597 | rat = 1; |
---|
1598 | poly f = Sx+Sy; |
---|
1599 | poly g = x; |
---|
1600 | vector rr = rightOre(g, f, 2, rat)[1]; |
---|
1601 | if (rr[1] == 0 || f*rr[1]-g*rr[2] != 0) { |
---|
1602 | ERROR("shift rational right Ore failed"); |
---|
1603 | } |
---|
1604 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
1605 | matrix C[4][4] = UpOneMatrix(4); |
---|
1606 | C[1,3] = q; |
---|
1607 | C[2,4] = q; |
---|
1608 | def ncQ = nc_algebra(C, 0); |
---|
1609 | setring ncQ; |
---|
1610 | rat = 1; |
---|
1611 | poly f = Qx+Qy; |
---|
1612 | poly g = x; |
---|
1613 | vector rr = rightOre(g, f, 2, rat)[1]; |
---|
1614 | if (rr[1] == 0 || f*rr[1]-g*rr[2] != 0) { |
---|
1615 | ERROR("q-shift rational right Ore failed"); |
---|
1616 | } |
---|
1617 | print(" rightOre OK"); |
---|
1618 | } |
---|
1619 | ////////////////////////////////////////////////////////////////////// |
---|
1620 | static proc testAddLeftFractions() |
---|
1621 | { |
---|
1622 | print(" testing addLeftFractions..."); |
---|
1623 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1624 | def S = Weyl(); |
---|
1625 | setring S; |
---|
1626 | // monoidal localization |
---|
1627 | poly g1 = x+3; |
---|
1628 | poly g2 = x*y+y; |
---|
1629 | list L = g1,g2; |
---|
1630 | vector frac1 = [g1,Dx,0,0]; |
---|
1631 | vector frac2 = [g2,Dy,0,0]; |
---|
1632 | vector rm = addLeftFractions(frac1, frac2, 0, L); |
---|
1633 | if (rm[1] != x^2*y+4*x*y+3*y || rm[2] != x*y*Dx+y*Dx+x*Dy+3*Dy) { |
---|
1634 | ERROR("Weyl monoidal addition failed"); |
---|
1635 | } |
---|
1636 | // geometric localization |
---|
1637 | ideal p = x-1,y-3; |
---|
1638 | vector rg = addLeftFractions(frac1, frac2, 1, p); |
---|
1639 | if (rg[1] != x^2*y+4*x*y+3*y || rg[2] != x*y*Dx+y*Dx+x*Dy+3*Dy) { |
---|
1640 | ERROR("Weyl geometric addition failed"); |
---|
1641 | } |
---|
1642 | // rational localization |
---|
1643 | intvec v = 2; |
---|
1644 | frac1 = [y^2+y+1,Dx,0,0]; |
---|
1645 | frac2 = [y-2,Dy,0,0]; |
---|
1646 | vector rr = addLeftFractions(frac1, frac2, 2, v); |
---|
1647 | if (rr[1] != y^3-y^2-y-2 || rr[2] != y^2*Dy+y*Dx+y*Dy-2*Dx+Dy) { |
---|
1648 | ERROR("Weyl rational addition failed"); |
---|
1649 | } |
---|
1650 | print(" addLeftFractions OK"); |
---|
1651 | } |
---|
1652 | ////////////////////////////////////////////////////////////////////// |
---|
1653 | static proc testMultiplyLeftFractions() { |
---|
1654 | print(" testing multiplyLeftFractions..."); |
---|
1655 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1656 | def S = Weyl(); |
---|
1657 | setring S; |
---|
1658 | // monoidal localization |
---|
1659 | poly g1 = x+3; |
---|
1660 | poly g2 = x*y+y; |
---|
1661 | list L = g1,g2; |
---|
1662 | vector frac1 = [g1,Dx,0,0]; |
---|
1663 | vector frac2 = [g2,Dy,0,0]; |
---|
1664 | vector rm = multiplyLeftFractions(frac1, frac2, 0, L); |
---|
1665 | if (rm[1] != g1*g2^2 || rm[2] != x*y*Dx*Dy+y*Dx*Dy-y*Dy) { |
---|
1666 | ERROR("Weyl monoidal multiplication error"); |
---|
1667 | } |
---|
1668 | // geometric localization |
---|
1669 | ideal p = x-1,y-3; |
---|
1670 | vector rg = multiplyLeftFractions(frac1, frac2, 1, p); |
---|
1671 | if (rg[1] != g1*g2*(x+1) || rg[2] != x*Dx*Dy+Dx*Dy-Dy) { |
---|
1672 | ERROR("Weyl geometric multiplication error"); |
---|
1673 | } |
---|
1674 | // rational localization |
---|
1675 | intvec v = 2; |
---|
1676 | frac1 = [y^2+y+1,Dx,0,0]; |
---|
1677 | frac2 = [y-2,Dy,0,0]; |
---|
1678 | vector rr = multiplyLeftFractions(frac1, frac2, 2, v); |
---|
1679 | if (rr[1] != (y^2+y+1)*(y-2) || rr[2] != Dx*Dy) { |
---|
1680 | ERROR("Weyl rational multiplication (1*2) error"); |
---|
1681 | } |
---|
1682 | rr = multiplyLeftFractions(frac2, frac1, 2, v); |
---|
1683 | if (rr[1] != (y^2+y+1)^2*(y-2) || rr[2] != y^2*Dx*Dy+y*Dx*Dy-2*y*Dx+Dx*Dy-Dx) { |
---|
1684 | ERROR("Weyl rational multiplication (2*1) error"); |
---|
1685 | } |
---|
1686 | print(" multiplyLeftFractions OK"); |
---|
1687 | } |
---|
1688 | ////////////////////////////////////////////////////////////////////// |
---|
1689 | static proc testAreEqualLeftFractions() { |
---|
1690 | print(" testing areEqualLeftFractions..."); |
---|
1691 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1692 | def S = Weyl(); |
---|
1693 | setring S; |
---|
1694 | // monoidal |
---|
1695 | poly g1 = x*y+3; |
---|
1696 | poly g2 = y^3; |
---|
1697 | list L = g1,g2; |
---|
1698 | vector fracm1 = [g1,Dx,0,0]; |
---|
1699 | vector fracm2 = [g1*g2,g2*Dx,0,0]; |
---|
1700 | vector fracm3 = [g1*g2,g1*Dx+3,0,0]; |
---|
1701 | if (!areEqualLeftFractions(fracm1, fracm2, 0, L)) { |
---|
1702 | ERROR("Weyl monoidal positive basic comparison error"); |
---|
1703 | } |
---|
1704 | if (areEqualLeftFractions(fracm1, fracm3, 0, L)) { |
---|
1705 | ERROR("Weyl monoidal first negative basic comparison error"); |
---|
1706 | } |
---|
1707 | if (areEqualLeftFractions(fracm2, fracm3, 0, L)) { |
---|
1708 | ERROR("Weyl monoidal second negative basic comparison error"); |
---|
1709 | } |
---|
1710 | // geometric |
---|
1711 | ideal p = x+5, y-2; |
---|
1712 | vector fracg1 = [g1,Dx,0,0]; |
---|
1713 | vector fracg2 = [g1*g2,g2*Dx,0,0]; |
---|
1714 | vector fracg3 = [g1*g2,g1*Dx+3,0,0]; |
---|
1715 | if (!areEqualLeftFractions(fracg1, fracg2, 1, p)) { |
---|
1716 | ERROR("Weyl geometric positive basic comparison error"); |
---|
1717 | } |
---|
1718 | if (areEqualLeftFractions(fracg1, fracg3, 1, p)) { |
---|
1719 | ERROR("Weyl geometric first negative basic comparison error"); |
---|
1720 | } |
---|
1721 | if (areEqualLeftFractions(fracg2, fracg3, 1, p)) { |
---|
1722 | ERROR("Weyl geometric second negative basic comparison error"); |
---|
1723 | } |
---|
1724 | // rational |
---|
1725 | intvec rat = 1,4; |
---|
1726 | vector fracr1 = [x+Dy,Dx,0,0]; |
---|
1727 | vector fracr2 = [x*Dy*(x+Dy),x*Dx*Dy,0,0]; |
---|
1728 | vector fracr3 = [Dy*x*(x+Dy),x*Dx*Dy+1,0,0]; |
---|
1729 | if (!areEqualLeftFractions(fracr1, fracr2, 2, rat)) { |
---|
1730 | ERROR("Weyl rational positive basic comparison error"); |
---|
1731 | } |
---|
1732 | if (areEqualLeftFractions(fracr1, fracr3, 2, rat)) { |
---|
1733 | ERROR("Weyl rational first negative basic comparison error"); |
---|
1734 | } |
---|
1735 | if (areEqualLeftFractions(fracr2, fracr3, 2, rat)) { |
---|
1736 | ERROR("Weyl rational second negative basic comparison error"); |
---|
1737 | } |
---|
1738 | print(" areEqualLeftFractions OK"); |
---|
1739 | } |
---|
1740 | ////////////////////////////////////////////////////////////////////// |
---|
1741 | static proc testConvertLeftToRightFraction() { |
---|
1742 | print(" testing convertLeftToRightFraction..."); |
---|
1743 | // Weyl |
---|
1744 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
1745 | def ncW = Weyl(); |
---|
1746 | setring ncW; |
---|
1747 | //// monoidal localization |
---|
1748 | vector fracm = [x,Dx,0,0]; |
---|
1749 | list L = x; |
---|
1750 | vector rm = convertLeftToRightFraction(fracm, 0, L); |
---|
1751 | if (!fracStatus(rm, 0, L)[1]) { |
---|
1752 | ERROR("Weyl monoidal convertLeftToRightFraction failed"); |
---|
1753 | } |
---|
1754 | //// geometric localization |
---|
1755 | vector fracg = [x+y,Dx+Dy,0,0]; |
---|
1756 | ideal p = x-1,y-3; |
---|
1757 | vector rg = convertLeftToRightFraction(fracg, 1, p); |
---|
1758 | if (!fracStatus(rg, 1, p)[1]) { |
---|
1759 | ERROR("Weyl geometric convertLeftToRightFraction failed"); |
---|
1760 | } |
---|
1761 | //// rational localization |
---|
1762 | intvec rat = 1; |
---|
1763 | vector fracr = [x,Dx+Dy,0,0]; |
---|
1764 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
1765 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
1766 | ERROR("Weyl rational convertLeftToRightFraction failed"); |
---|
1767 | } |
---|
1768 | // shift rational localization |
---|
1769 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
1770 | matrix D[4][4]; |
---|
1771 | D[1,3] = Sx; |
---|
1772 | D[2,4] = Sy; |
---|
1773 | def ncS = nc_algebra(1, D); |
---|
1774 | setring ncS; |
---|
1775 | vector fracr = [x,Sx+Sy,0,0]; |
---|
1776 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
1777 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
1778 | ERROR("Shift rational convertLeftToRightFraction failed"); |
---|
1779 | } |
---|
1780 | // q-shift rational localization |
---|
1781 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
1782 | matrix C[4][4] = UpOneMatrix(4); |
---|
1783 | C[1,3] = q; |
---|
1784 | C[2,4] = q; |
---|
1785 | def ncQ = nc_algebra(C, 0); |
---|
1786 | setring ncQ; |
---|
1787 | vector fracr = [x,Qx+Qy,0,0]; |
---|
1788 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
1789 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
1790 | ERROR("q-shift rational convertLeftToRightFraction failed"); |
---|
1791 | } |
---|
1792 | print(" convertLeftToRightFraction OK"); |
---|
1793 | } |
---|
1794 | ////////////////////////////////////////////////////////////////////// |
---|
1795 | static proc testConvertRightToLeftFraction() { |
---|
1796 | print(" testing convertRightToLeftFraction..."); |
---|
1797 | // Weyl |
---|
1798 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
1799 | def ncW = Weyl(); |
---|
1800 | setring ncW; |
---|
1801 | //// monoidal localization |
---|
1802 | poly g1 = x+3; |
---|
1803 | poly g2 = x*y; |
---|
1804 | list L = g1,g2; |
---|
1805 | vector fracm = [0,0,Dx,g1^2*g2]; |
---|
1806 | vector rm = convertRightToLeftFraction(fracm, 0, L); |
---|
1807 | if (!fracStatus(rm, 0, L)[1]) { |
---|
1808 | ERROR("Weyl monoidal convertRightToLeftFraction failed"); |
---|
1809 | } |
---|
1810 | //// geometric localization |
---|
1811 | ideal p = x-1,y-3; |
---|
1812 | vector fracg = [0,0,Dx,x^2+y]; |
---|
1813 | vector rg = convertRightToLeftFraction(fracg, 1, p); |
---|
1814 | if (!fracStatus(rg, 1, p)[1]) { |
---|
1815 | ERROR("Weyl geometric convertRightToLeftFraction failed"); |
---|
1816 | } |
---|
1817 | //// rational localization |
---|
1818 | intvec rat = 1; |
---|
1819 | vector fracr = [0,0,Dx+Dy,x]; |
---|
1820 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
1821 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
1822 | ERROR("Weyl rational convertRightToLeftFraction failed"); |
---|
1823 | } |
---|
1824 | // shift rational localization |
---|
1825 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
1826 | matrix D[4][4]; |
---|
1827 | D[1,3] = Sx; |
---|
1828 | D[2,4] = Sy; |
---|
1829 | def ncS = nc_algebra(1, D); |
---|
1830 | setring ncS; |
---|
1831 | vector fracr = [0,0,Sx+Sy,x]; |
---|
1832 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
1833 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
1834 | ERROR("Shift rational convertRightToLeftFraction failed"); |
---|
1835 | } |
---|
1836 | // q-shift rational localization |
---|
1837 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
1838 | matrix C[4][4] = UpOneMatrix(4); |
---|
1839 | C[1,3] = q; |
---|
1840 | C[2,4] = q; |
---|
1841 | def ncQ = nc_algebra(C, 0); |
---|
1842 | setring ncQ; |
---|
1843 | vector fracr = [0,0,Qx+Qy,x]; |
---|
1844 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
1845 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
1846 | ERROR("q-shift rational convertRightToLeftFraction failed"); |
---|
1847 | } |
---|
1848 | print(" convertRightToLeftFraction OK"); |
---|
1849 | } |
---|
1850 | ////////////////////////////////////////////////////////////////////// |
---|
1851 | static proc testIsInvertibleLeftFraction() { |
---|
1852 | print(" testing isInvertibleLeftFraction..."); |
---|
1853 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1854 | def S = Weyl(); |
---|
1855 | setring S; |
---|
1856 | poly g1 = x+3; |
---|
1857 | poly g2 = x*y; |
---|
1858 | // monoidal |
---|
1859 | list L = g1, g2; |
---|
1860 | if (!isInvertibleLeftFraction([g1*g2,17,0,0], 0, L)) { |
---|
1861 | ERROR("Weyl monoidal positive invertibility test error"); |
---|
1862 | } |
---|
1863 | if (isInvertibleLeftFraction([g1*g2,Dx,0,0], 0, L)) { |
---|
1864 | ERROR("Weyl monoidal negative invertibility test error"); |
---|
1865 | } |
---|
1866 | if (!isInvertibleLeftFraction([1,1,1,1], 0, L)) { |
---|
1867 | ERROR("Weyl monoidal one invertibility test error"); |
---|
1868 | } |
---|
1869 | if (isInvertibleLeftFraction([1,0,0,1], 0, L)) { |
---|
1870 | ERROR("Weyl monoidal zero invertibility test error"); |
---|
1871 | } |
---|
1872 | // geometric |
---|
1873 | ideal p = x-1, y; |
---|
1874 | if (!isInvertibleLeftFraction([g1,3*x,0,0], 1, p)) { |
---|
1875 | ERROR("Weyl geometric positive invertibility test error"); |
---|
1876 | } |
---|
1877 | if (isInvertibleLeftFraction([g1,Dx,0,0], 0, L)) { |
---|
1878 | ERROR("Weyl geometric negative invertibility test error"); |
---|
1879 | } |
---|
1880 | if (!isInvertibleLeftFraction([1,1,1,1], 1, p)) { |
---|
1881 | ERROR("Weyl geometric one invertibility test error"); |
---|
1882 | } |
---|
1883 | if (isInvertibleLeftFraction([1,0,0,1], 1, p)) { |
---|
1884 | ERROR("Weyl geometric zero invertibility test error"); |
---|
1885 | } |
---|
1886 | // rational |
---|
1887 | intvec rat = 1,2; |
---|
1888 | if (!isInvertibleLeftFraction([g1*g2,y,0,0], 2, rat)) { |
---|
1889 | ERROR("Weyl rational positive invertibility test error"); |
---|
1890 | } |
---|
1891 | if (isInvertibleLeftFraction([g1*g2,Dx,0,0], 0, L)) { |
---|
1892 | ERROR("Weyl rational negative invertibility test error"); |
---|
1893 | } |
---|
1894 | if (!isInvertibleLeftFraction([1,1,1,1], 2, rat)) { |
---|
1895 | ERROR("Weyl rational one invertibility test error"); |
---|
1896 | } |
---|
1897 | if (isInvertibleLeftFraction([1,0,0,1], 2, rat)) { |
---|
1898 | ERROR("Weyl rational zero invertibility test error"); |
---|
1899 | } |
---|
1900 | print(" isInvertibleLeftFraction OK"); |
---|
1901 | } |
---|
1902 | ////////////////////////////////////////////////////////////////////// |
---|
1903 | static proc testInvertLeftFraction() { |
---|
1904 | print(" testing invertLeftFraction..."); |
---|
1905 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
1906 | def S = Weyl(); |
---|
1907 | setring S; |
---|
1908 | poly g1 = x+3; |
---|
1909 | poly g2 = x*y; |
---|
1910 | // monoidal |
---|
1911 | list L = g1, g2; |
---|
1912 | vector rm = [g1*g2, 17, 0, 0]; |
---|
1913 | vector rmInv = invertLeftFraction(rm, 0 , L); |
---|
1914 | if (!isOneFraction(multiplyLeftFractions(rm, rmInv, 0, L))) { |
---|
1915 | ERROR("Weyl monoidal inversion error"); |
---|
1916 | } |
---|
1917 | // geometric |
---|
1918 | ideal p = x-1, y; |
---|
1919 | vector rg = [g1, 3*x, 0, 0]; |
---|
1920 | vector rgInv = invertLeftFraction(rg, 1, p); |
---|
1921 | if (!isOneFraction(multiplyLeftFractions(rg, rgInv, 1, p))) { |
---|
1922 | ERROR("Weyl geometric inversion error"); |
---|
1923 | } |
---|
1924 | // rational |
---|
1925 | intvec rat = 1,2; |
---|
1926 | vector rr = [g1*g2, y, 0, 0]; |
---|
1927 | vector rrInv = invertLeftFraction(rr, 2, rat); |
---|
1928 | if (!isOneFraction(multiplyLeftFractions(rr, rrInv, 2, rat))) { |
---|
1929 | ERROR("Weyl rational inversion error"); |
---|
1930 | } |
---|
1931 | print(" invertLeftFraction OK"); |
---|
1932 | } |
---|
1933 | ////////////////////////////////////////////////////////////////////// |
---|