[44168a] | 1 | //////////////////////////////////////////////////////////////// |
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[6391eb] | 2 | version="version olga.lib 4.1.2.0 Feb_2019 "; // $Id$ |
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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 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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[44168a] | 174 | t = typeof(locData[i]); |
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[2ceb9f] | 175 | if (t != "poly" && t != "int" && t != "number") |
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| 176 | { |
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[44168a] | 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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[2ceb9f] | 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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[44168a] | 188 | listEntries = locData[i]; |
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[2ceb9f] | 189 | } |
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| 190 | else |
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| 191 | { |
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[44168a] | 192 | listEntries = listEntries, locData[i]; |
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| 193 | } |
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[2ceb9f] | 194 | } |
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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 214 | if (locType == 1) |
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| 215 | { // geometric localizations |
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[44168a] | 216 | int n = nvars(basering) div 2; |
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[2ceb9f] | 217 | if (2*n != nvars(basering)) |
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| 218 | { |
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[44168a] | 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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[2ceb9f] | 221 | } |
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| 222 | else |
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| 223 | { |
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[44168a] | 224 | int j; |
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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 239 | if (t != "ideal") |
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| 240 | { |
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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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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[2ceb9f] | 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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[44168a] | 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 | ideal I = p; |
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| 407 | intvec modLocData = intvecComplement(locData, 1..nvars(basering)); |
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| 408 | I = eliminateNC(I, modLocData); |
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| 409 | if (size(I)) { |
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| 410 | return(1); |
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| 411 | } |
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| 412 | } |
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| 413 | return(0); |
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| 414 | } |
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| 415 | example |
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| 416 | { |
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| 417 | "EXAMPLE:"; echo = 2; |
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| 418 | ring R = 0,(x,y,Dx,Dy),dp; |
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| 419 | def S = Weyl(); |
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| 420 | setring S; S; |
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| 421 | // monoidal localization |
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| 422 | poly g1 = x^2*y+x+2; |
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| 423 | poly g2 = y^3+x*y; |
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| 424 | list L = g1,g2; |
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| 425 | poly g = g1^2*g2; |
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| 426 | poly f = g-1; |
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| 427 | isInS(g, 0, L); // g is in the denominator set |
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| 428 | isInS(f, 0, L); // f is NOT in the denominator set |
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| 429 | // geometric localization |
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| 430 | ideal p = x-1, y-3; |
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| 431 | g = x^2+y-3; |
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| 432 | f = (x-1)*g; |
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| 433 | isInS(g, 1, p); // g is in the denominator set |
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| 434 | isInS(f, 1, p); // f is NOT in the denominator set |
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| 435 | // rational localization |
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| 436 | intvec v = 2; |
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| 437 | g = y^5+17*y^2-4; |
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| 438 | f = x*y; |
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| 439 | isInS(g, 2, v); // g is in the denominator set |
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| 440 | isInS(f, 2, v); // f is NOT in the denominator set |
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| 441 | } |
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| 442 | ////////////////////////////////////////////////////////////////////// |
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| 443 | proc fracStatus(vector frac, int locType, def locData) |
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| 444 | "USAGE: fracStatus(frac, locType, locData), vector frac, int locType, |
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| 445 | list/intvec/vector locData |
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| 446 | PURPOSE: determine if the given vector is a representation of a fraction in the |
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| 447 | specified localization |
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| 448 | ASSUME: |
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| 449 | RETURN: list |
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| 450 | NOTE: - the first entry is 0 or 1, depending whether the input is valid |
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| 451 | - the second entry is a string with a status message |
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| 452 | EXAMPLE: example fracStatus; shows examples" |
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| 453 | { |
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| 454 | list locStat = locStatus(locType, locData); |
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| 455 | if (!locStat[1]) { // there is a problem with the localization data |
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| 456 | return(list(0, "invalid localization in fraction: "+ string(frac) |
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| 457 | + newline + " " + locStat[2])); |
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| 458 | } else { // the specified localization is valid |
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| 459 | if ((frac[1] == 0) && (frac[4] == 0)) { |
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| 460 | return(list(0, "vector is not a valid fraction: no denominator" |
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| 461 | + " specified in " + string(frac))); |
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| 462 | } |
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| 463 | if (frac[1] != 0) { // frac has a left representation |
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| 464 | if (!isInS(frac[1], locType, locData)) { |
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| 465 | return(list(0, "the left denominator " + string(frac[1]) |
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| 466 | + " of fraction " + string(frac) + " is not in the" |
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| 467 | + " denominator set of type " + string(locType) + " given by " |
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| 468 | + string(locData))); |
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| 469 | } |
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| 470 | } |
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| 471 | if (frac[4] != 0) { // frac has a right representation |
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| 472 | if (!isInS(frac[4], locType, locData)) { |
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| 473 | return(list(0, "the right denominator " + string(frac[4]) |
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| 474 | + " of fraction " + string(frac) + " is not in the" |
---|
| 475 | + " denominator set of type " + string(locType) + " given by " |
---|
| 476 | + string(locData))); |
---|
| 477 | } |
---|
| 478 | } |
---|
| 479 | if ((frac[1] != 0) && (frac[4] != 0)) { |
---|
| 480 | // frac has left and right representations |
---|
| 481 | if (frac[2]*frac[4] != frac[1]*frac[3]) { |
---|
| 482 | // the representations are not equal |
---|
| 483 | return(list(0, "left and right representation are not equal in:" |
---|
| 484 | + string(frac))); |
---|
| 485 | } |
---|
| 486 | } |
---|
| 487 | } |
---|
| 488 | return(list(1, "valid fraction")); |
---|
| 489 | } |
---|
| 490 | example |
---|
| 491 | { |
---|
| 492 | "EXAMPLE:"; echo = 2; |
---|
| 493 | ring r = QQ[x,y,Dx,Dy]; |
---|
| 494 | def R = Weyl(); |
---|
| 495 | setring R; |
---|
| 496 | fracStatus([1,0,0,0], 42, list(1)); |
---|
| 497 | list L = x; |
---|
| 498 | fracStatus([0,7,x,0], 0, L); |
---|
| 499 | fracStatus([Dx,Dy,0,0], 0, L); |
---|
| 500 | fracStatus([0,0,Dx,Dy], 0, L); |
---|
| 501 | fracStatus([x,Dx,Dy,x], 0, L); |
---|
| 502 | fracStatus([x,Dx,x*Dx+2,x^2], 0, L); |
---|
| 503 | } |
---|
| 504 | ////////////////////////////////////////////////////////////////////// |
---|
| 505 | proc testFraction(vector frac, int locType, def locData) |
---|
| 506 | "USAGE: testFraction(frac, locType, locData), vector frac, int locType, |
---|
| 507 | list/intvec/vector locData |
---|
| 508 | PURPOSE: test if the given vector is a representation of a fraction in the |
---|
| 509 | specified localization wrt. the checks from fracStatus |
---|
| 510 | ASSUME: |
---|
| 511 | RETURN: nothing |
---|
| 512 | NOTE: throws error if checks were not successful |
---|
| 513 | EXAMPLE: example testFraction; shows examples" |
---|
| 514 | { |
---|
| 515 | list stat = fracStatus(frac, locType, locData); |
---|
| 516 | if (!stat[1]) { |
---|
| 517 | ERROR(stat[2]); |
---|
| 518 | } else { |
---|
| 519 | return(); |
---|
| 520 | } |
---|
| 521 | } |
---|
| 522 | example |
---|
| 523 | { |
---|
| 524 | "EXAMPLE:"; echo = 2; |
---|
| 525 | ring r = QQ[x,y,Dx,Dy]; |
---|
| 526 | def R = Weyl(); |
---|
| 527 | setring R; |
---|
| 528 | list L = x; |
---|
| 529 | vector frac = [x,Dx,x*Dx+2,x^2]; |
---|
| 530 | testFraction(frac, 0, L); // correct localization, no error |
---|
| 531 | frac = [x,Dx,x*Dx,x^2]; |
---|
| 532 | testFraction(frac, 0, L); // incorrect localization, results in error |
---|
| 533 | } |
---|
| 534 | ////////////////////////////////////////////////////////////////////// |
---|
| 535 | proc leftOre(poly s, poly r, int locType, def locData) |
---|
| 536 | "USAGE: leftOre(s, r, locType, locData), poly s, r, int locType, |
---|
| 537 | list/vector/intvec locData |
---|
| 538 | PURPOSE: compute left Ore data for a given tuple (s,r) |
---|
| 539 | ASSUME: s is in the denominator set determined via locType and locData |
---|
| 540 | RETURN: list |
---|
| 541 | NOTE: - the first entry of the list is a vector [ts,tr] such that ts*r=tr*s |
---|
| 542 | - the second entry of the list is a description of all choices for ts |
---|
| 543 | EXAMPLE: example leftOre; shows examples" |
---|
| 544 | { |
---|
| 545 | testLocData(locType,locData); |
---|
| 546 | locData = normalizeLocalization(locType, locData); |
---|
| 547 | if(!isInS(s, locType, locData)) { |
---|
| 548 | ERROR("cannot find Ore-parameter since poly " + string(s) |
---|
| 549 | + " is not in the denominator set"); |
---|
| 550 | } |
---|
| 551 | return(ore(s, r, locType, locData, 0)); |
---|
| 552 | } |
---|
| 553 | example |
---|
| 554 | { |
---|
| 555 | "EXAMPLE:"; echo = 2; |
---|
| 556 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 557 | def S = Weyl(); |
---|
| 558 | setring S; S; |
---|
| 559 | // left Ore |
---|
| 560 | // monoidal localization |
---|
| 561 | poly g1 = x+3; |
---|
| 562 | poly g2 = x*y; |
---|
| 563 | list L = g1,g2; |
---|
| 564 | poly g = g1^2*g2; |
---|
| 565 | poly f = Dx; |
---|
| 566 | list rm = leftOre(g, f, 0, L); |
---|
| 567 | print(rm[1]); |
---|
| 568 | rm[2]; |
---|
| 569 | rm[1][2]*g-rm[1][1]*f; |
---|
| 570 | // geometric localization |
---|
| 571 | ideal p = x-1, y-3; |
---|
| 572 | f = Dx; |
---|
| 573 | g = x^2+y; |
---|
| 574 | list rg = leftOre(g, f, 1, p); |
---|
| 575 | print(rg[1]); |
---|
| 576 | rg[2]; |
---|
| 577 | rg[1][2]*g-rg[1][1]*f; |
---|
| 578 | // rational localization |
---|
| 579 | intvec rat = 1; |
---|
| 580 | f = Dx+Dy; |
---|
| 581 | g = x; |
---|
| 582 | list rr = leftOre(g, f, 2, rat); |
---|
| 583 | print(rr[1]); |
---|
| 584 | rr[2]; |
---|
| 585 | rr[1][2]*g-rr[1][1]*f; |
---|
| 586 | } |
---|
| 587 | ////////////////////////////////////////////////////////////////////// |
---|
| 588 | proc rightOre(poly s, poly r, int locType, def locData) |
---|
| 589 | "USAGE: rightOre(s, r, locType, locData), poly s, r, int locType, |
---|
| 590 | list/vector/intvec locData |
---|
| 591 | PURPOSE: compute right Ore data for a given tuple (s,r) |
---|
| 592 | ASSUME: s is in the denominator set determined via locType and locData |
---|
| 593 | RETURN: list |
---|
| 594 | NOTE: - the first entry of the list is a vector [ts,tr] such that r*ts=s*tr |
---|
| 595 | - the second entry of the list is a description of all choices for ts |
---|
| 596 | EXAMPLE: example rightOre; shows examples" |
---|
| 597 | { |
---|
| 598 | testLocData(locType,locData); |
---|
| 599 | locData = normalizeLocalization(locType, locData); |
---|
| 600 | if(!isInS(s, locType, locData)) { |
---|
| 601 | ERROR("cannot find Ore-parameter since poly " + string(s) |
---|
| 602 | + " is not in the denominator set"); |
---|
| 603 | } |
---|
| 604 | def bsRing = basering; |
---|
| 605 | if (locType == 0) { |
---|
| 606 | ideal modLocData = locData[1..size(locData)]; |
---|
| 607 | } |
---|
| 608 | def oppRing = opposite(bsRing); |
---|
| 609 | setring oppRing; |
---|
| 610 | if (locType == 0) { |
---|
| 611 | ideal oppModLocData = oppose(bsRing, modLocData); |
---|
| 612 | list oppLocData = oppModLocData[1..size(oppModLocData)]; |
---|
| 613 | } |
---|
| 614 | if (locType == 1) { |
---|
| 615 | ideal oppLocData = oppose(bsRing,locData); |
---|
| 616 | } |
---|
| 617 | if (locType == 2) { |
---|
| 618 | intvec oppLocData = locData; |
---|
| 619 | } |
---|
| 620 | poly oppS = oppose(bsRing, s); |
---|
| 621 | poly oppR = oppose(bsRing, r); |
---|
| 622 | list oppResult = ore(oppS, oppR, locType, oppLocData, 1); |
---|
| 623 | vector oppOreParas = oppResult[1]; |
---|
| 624 | ideal oppJ = oppResult[2]; |
---|
| 625 | setring bsRing; |
---|
| 626 | return(list(oppose(oppRing, oppOreParas), oppose(oppRing, oppJ))); |
---|
| 627 | } |
---|
| 628 | example |
---|
| 629 | { |
---|
| 630 | "EXAMPLE:"; echo = 2; |
---|
| 631 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 632 | def S = Weyl(); |
---|
| 633 | setring S; S; |
---|
| 634 | // monoidal localization |
---|
| 635 | poly g1 = x+3; |
---|
| 636 | poly g2 = x*y; |
---|
| 637 | list L = g1,g2; |
---|
| 638 | poly g = g1^2*g2; |
---|
| 639 | poly f = Dx; |
---|
| 640 | list rm = rightOre(g, f, 0, L); |
---|
| 641 | print(rm[1]); |
---|
| 642 | rm[2]; |
---|
| 643 | g*rm[1][2]-f*rm[1][1]; |
---|
| 644 | // geometric localization |
---|
| 645 | ideal p = x-1, y-3; |
---|
| 646 | f = Dx; |
---|
| 647 | g = x^2+y; |
---|
| 648 | list rg = rightOre(g, f, 1, p); |
---|
| 649 | print(rg[1]); |
---|
| 650 | rg[2]; |
---|
| 651 | g*rg[1][2]-f*rg[1][1]; |
---|
| 652 | // rational localization |
---|
| 653 | intvec rat = 1; |
---|
| 654 | f = Dx+Dy; |
---|
| 655 | g = x; |
---|
| 656 | list rr = rightOre(g, f, 2, rat); |
---|
| 657 | print(rr[1]); |
---|
| 658 | rr[2]; |
---|
| 659 | g*rr[1][2]-f*rr[1][1]; |
---|
| 660 | } |
---|
| 661 | ////////////////////////////////////////////////////////////////////// |
---|
| 662 | static proc ore(poly s, poly r, int locType, def locData, int rightOre) { |
---|
| 663 | // TODO: once ringlist-bug [#577] is PROPERLY fixed (probability: low), |
---|
| 664 | // use eliminateNC instead of eliminate in rightOre-rational and |
---|
| 665 | // rightOre-geometric |
---|
| 666 | // problem: the ordering on the opposite algebra is not recognized as |
---|
| 667 | // valid for eliminateNC, even though it is |
---|
| 668 | if (r == 0) { |
---|
| 669 | return([1,0], ideal(1)); |
---|
| 670 | } |
---|
| 671 | int i, j; |
---|
| 672 | // modification for monoidal localization |
---|
| 673 | if (locType == 0) { |
---|
| 674 | // computations will be carried out in the localization at the product |
---|
| 675 | // of all irreducible factors of s, thus all factors have to be |
---|
| 676 | // raised to the highest power occurring in the factorization of s |
---|
| 677 | list factorList = commutativeFactorization(s); |
---|
| 678 | ideal factors = factorList[1]; |
---|
| 679 | int maxPow = 0; |
---|
| 680 | intvec exponents = factorList[2]; |
---|
| 681 | for (i = 1; i <= size(exponents); i++) { |
---|
| 682 | if (exponents[i] > maxPow) { |
---|
| 683 | maxPow = exponents[i]; |
---|
| 684 | } |
---|
| 685 | } |
---|
| 686 | int expDiff; |
---|
| 687 | for (i = 1; i <= size(exponents); i++) { |
---|
| 688 | expDiff = maxPow - exponents[i]; |
---|
| 689 | s = s * factors[i]^expDiff; |
---|
| 690 | r = r * factors[i]^expDiff; |
---|
| 691 | } |
---|
| 692 | } |
---|
| 693 | // compute kernel of x maps to x*r+R*s |
---|
| 694 | ideal J = modulo(r, s); |
---|
| 695 | ideal I = std(J); |
---|
| 696 | // compute the poly in S |
---|
| 697 | poly ts; |
---|
| 698 | if (locType == 0) { // type 0: monoidal localization |
---|
| 699 | intvec sle = leadexp(s); |
---|
| 700 | intvec rle; |
---|
| 701 | int m = deg(r) + 1; // upper bound |
---|
| 702 | int dividesLeadingMonomial; |
---|
| 703 | // find the minimal possible exponent m such that lm(f_i)|lm(s)^m |
---|
| 704 | for (i = 1; i <= size(I); i++) { |
---|
| 705 | rle = leadexp(I[i]); |
---|
| 706 | dividesLeadingMonomial = 1; |
---|
| 707 | for (j = 1; j <= size(rle); j++) { |
---|
| 708 | if (sle[j] == 0 && rle[j] != 0) { |
---|
| 709 | // lm(f_i) divides no power of lm(s) |
---|
| 710 | dividesLeadingMonomial = 0; |
---|
| 711 | break; |
---|
| 712 | } |
---|
| 713 | } |
---|
| 714 | if (dividesLeadingMonomial) { |
---|
| 715 | int mf = 1; |
---|
| 716 | while (1) { |
---|
| 717 | if (mf * sle - rle >= 0) { |
---|
| 718 | break; // now lm(r_i) divides lm(s)^mf |
---|
| 719 | } else { |
---|
| 720 | mf++; |
---|
| 721 | } |
---|
| 722 | } |
---|
| 723 | if (mf < m) { |
---|
| 724 | m = mf; // mf is lower than the current minimum m |
---|
| 725 | } |
---|
| 726 | } |
---|
| 727 | } |
---|
| 728 | poly nf = NF(s^m, I); |
---|
| 729 | for (; m <= deg(r) + 1; m++) { // sic: m is initialized beforehand |
---|
| 730 | if (nf == 0) { // s^m is in I |
---|
| 731 | ts = s^m; |
---|
| 732 | ideal Js = s^m; |
---|
| 733 | break; |
---|
| 734 | } else { |
---|
| 735 | nf = NF(s*nf, I); |
---|
| 736 | } |
---|
| 737 | } |
---|
| 738 | } |
---|
| 739 | if (locType == 1) { // type 1: geometric localization |
---|
| 740 | int n = nvars(basering) div 2; |
---|
| 741 | if (rightOre == 1) { |
---|
| 742 | // in the right Ore setting, the order of variables is inverted |
---|
| 743 | poly elimVars = 1; |
---|
| 744 | for (i = 1; i <= n; i++) { |
---|
| 745 | elimVars = elimVars * var(i); |
---|
| 746 | } |
---|
| 747 | J = eliminate(I, elimVars); |
---|
| 748 | } else { |
---|
| 749 | J = eliminateNC(I, intvec((n+1)..(2*n))); |
---|
| 750 | } |
---|
| 751 | J = std(J); |
---|
| 752 | // compute ts, the generator of J with the smallest degree that does |
---|
| 753 | // not vanish at locData |
---|
| 754 | ideal K = intersect(J, locData); |
---|
| 755 | K = std(K); |
---|
| 756 | poly h, cand; |
---|
| 757 | ideal Js; |
---|
| 758 | ideal Q = std(locData); |
---|
| 759 | for (i = 1; i <= size(J); i++) { |
---|
| 760 | h = NF(J[i],Q); |
---|
| 761 | if (h != 0) { |
---|
| 762 | cand = NF(J[i],K); |
---|
| 763 | Js = Js + cand; |
---|
| 764 | if (ts == 0 || deg(cand) < deg(ts)) { |
---|
| 765 | ts = cand; |
---|
| 766 | } |
---|
| 767 | } |
---|
| 768 | } |
---|
| 769 | } |
---|
| 770 | if (locType == 2) { // type 2: rational localization |
---|
| 771 | int n = nvars(basering); |
---|
| 772 | if (size(locData) < n) { // there are variables to eliminate |
---|
| 773 | intvec modLocData; |
---|
| 774 | int check; |
---|
| 775 | modLocData = intvecComplement(locData, 1..n); |
---|
| 776 | // calculate modLocData = {1...n}\locData |
---|
| 777 | if (rightOre == 1) { |
---|
| 778 | // in the right Ore setting, the order of variables is inverted |
---|
| 779 | poly elimVars = 1; |
---|
| 780 | for (i = 1; i <= size(locData); i++) { |
---|
| 781 | elimVars = elimVars * var(locData[i]); |
---|
| 782 | } |
---|
| 783 | J = eliminate(I, elimVars); |
---|
| 784 | } else { |
---|
| 785 | J = eliminateNC(I, modLocData); |
---|
| 786 | } |
---|
| 787 | |
---|
| 788 | } else { // no variables to eliminate (total localization) |
---|
| 789 | J = I; |
---|
| 790 | } |
---|
| 791 | J = std(J); |
---|
| 792 | ts = J[1]; |
---|
| 793 | for (i = 2; i <= size(J); i++) { |
---|
| 794 | if (deg(J[i]) < deg(ts)) { |
---|
| 795 | ts = J[i]; // choose generator with lowest total degree |
---|
| 796 | } |
---|
| 797 | } |
---|
| 798 | ideal Js = J; |
---|
| 799 | } |
---|
| 800 | // calculate the other poly |
---|
| 801 | poly tr = division(ts*r, s)[1][1,1]; |
---|
| 802 | if (ts == 0 || tr*s-ts*r != 0) { |
---|
| 803 | string s; |
---|
| 804 | if (rightOre) { |
---|
| 805 | s = "right"; |
---|
| 806 | } else { |
---|
| 807 | s = "left"; |
---|
| 808 | } |
---|
| 809 | ERROR("no " + s + " Ore data could be found for s=" + string(s) |
---|
| 810 | + " and r=" + string(r)); |
---|
| 811 | } |
---|
| 812 | vector oreParas = [ts,tr]; |
---|
| 813 | list result = oreParas, Js; |
---|
| 814 | return (result); |
---|
| 815 | } |
---|
| 816 | ////////////////////////////////////////////////////////////////////// |
---|
| 817 | proc convertRightToLeftFraction(vector frac, int locType, def locData) |
---|
| 818 | "USAGE: convertRightToLeftFraction(frac, locType, locData), |
---|
| 819 | vector frac, int locType, list/vector/intvec locData |
---|
| 820 | PURPOSE: determine a left fraction representation of a given fraction |
---|
| 821 | ASSUME: |
---|
| 822 | RETURN: vector |
---|
| 823 | NOTE: - the returned vector contains a repr. of frac as a left fraction |
---|
| 824 | - if the left representation of frac is already specified, |
---|
| 825 | frac will be returned. |
---|
| 826 | EXAMPLE: example convertRightToLeftFraction; shows examples" |
---|
| 827 | { |
---|
| 828 | testLocData(locType, locData); |
---|
| 829 | testFraction(frac, locType, locData); |
---|
| 830 | if (frac[1] != 0) { // frac already has a left representation |
---|
| 831 | return (frac); |
---|
| 832 | } else { // frac has no left representation, but a right one |
---|
| 833 | vector oreParas = leftOre(frac[4], frac[3], locType, locData)[1]; |
---|
| 834 | vector result = [oreParas[1], oreParas[2], frac[3], frac[4]]; |
---|
| 835 | return (result); |
---|
| 836 | } |
---|
| 837 | } |
---|
| 838 | example |
---|
| 839 | { |
---|
| 840 | "EXAMPLE:"; echo = 2; |
---|
| 841 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 842 | def S = Weyl(); |
---|
| 843 | setring S; S; |
---|
| 844 | // monoidal localization |
---|
| 845 | poly g1 = x+3; |
---|
| 846 | poly g2 = x*y; |
---|
| 847 | list L = g1,g2; |
---|
| 848 | poly g = g1^2*g2; |
---|
| 849 | poly f = Dx; |
---|
| 850 | vector fracm = [0,0,f,g]; |
---|
| 851 | vector rm = convertRightToLeftFraction(fracm, 0, L); |
---|
| 852 | print(rm); |
---|
| 853 | rm[2]*g-rm[1]*f; |
---|
| 854 | // geometric localization |
---|
| 855 | ideal p = x-1, y-3; |
---|
| 856 | f = Dx; |
---|
| 857 | g = x^2+y; |
---|
| 858 | vector fracg = [0,0,f,g]; |
---|
| 859 | vector rg = convertRightToLeftFraction(fracg, 1, p); |
---|
| 860 | print(rg); |
---|
| 861 | rg[2]*g-rg[1]*f; |
---|
| 862 | // rational localization |
---|
| 863 | intvec rat = 1; |
---|
| 864 | f = Dx+Dy; |
---|
| 865 | g = x; |
---|
| 866 | vector fracr = [0,0,f,g]; |
---|
| 867 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
| 868 | print(rr); |
---|
| 869 | rr[2]*g-rr[1]*f; |
---|
| 870 | } |
---|
| 871 | ////////////////////////////////////////////////////////////////////// |
---|
| 872 | proc convertLeftToRightFraction(vector frac, int locType, def locData) |
---|
| 873 | "USAGE: convertLeftToRightFraction(frac, locType, locData), vector frac, |
---|
| 874 | int locType, list/vector/intvec locData |
---|
| 875 | PURPOSE: determine a right fraction representation of a given fraction |
---|
| 876 | ASSUME: |
---|
| 877 | RETURN: vector |
---|
| 878 | NOTE: - the returned vector contains a repr. of frac as a right fraction, |
---|
| 879 | - if the right representation of frac is already specified, |
---|
| 880 | frac will be returned. |
---|
| 881 | EXAMPLE: example convertLeftToRightFraction; shows examples" |
---|
| 882 | { |
---|
| 883 | testLocData(locType, locData); |
---|
| 884 | testFraction(frac, locType, locData); |
---|
| 885 | if (frac[4] != 0) { // frac already has a right representation |
---|
| 886 | return (frac); |
---|
| 887 | } else { // frac has no right representation, but a left one |
---|
| 888 | vector oreParas = rightOre(frac[1], frac[2], locType, locData)[1]; |
---|
| 889 | vector result = [frac[1], frac[2], oreParas[2], oreParas[1]]; |
---|
| 890 | return (result); |
---|
| 891 | } |
---|
| 892 | } |
---|
| 893 | example |
---|
| 894 | { |
---|
| 895 | "EXAMPLE:"; echo = 2; |
---|
| 896 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 897 | def S = Weyl(); |
---|
| 898 | setring S; S; |
---|
| 899 | // monoidal localization |
---|
| 900 | poly g = x; |
---|
| 901 | poly f = Dx; |
---|
| 902 | vector fracm = [g,f,0,0]; |
---|
| 903 | list L = g; |
---|
| 904 | vector rm = convertLeftToRightFraction(fracm, 0, L); |
---|
| 905 | print(rm); |
---|
| 906 | f*rm[4]-g*rm[3]; |
---|
| 907 | // geometric localization |
---|
| 908 | g = x+y; |
---|
| 909 | f = Dx+Dy; |
---|
| 910 | vector fracg = [g,f,0,0]; |
---|
| 911 | ideal p = x-1, y-3; |
---|
| 912 | vector rg = convertLeftToRightFraction(fracg, 1, p); |
---|
| 913 | print(rg); |
---|
| 914 | f*rg[4]-g*rg[3]; |
---|
| 915 | // rational localization |
---|
| 916 | intvec rat = 1; |
---|
| 917 | f = Dx+Dy; |
---|
| 918 | g = x; |
---|
| 919 | vector fracr = [g,f,0,0]; |
---|
| 920 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
| 921 | print(rr); |
---|
| 922 | f*rr[4]-g*rr[3]; |
---|
| 923 | } |
---|
| 924 | ////////////////////////////////////////////////////////////////////// |
---|
| 925 | proc isZeroFraction(vector frac) |
---|
| 926 | "USAGE: isZeroFraction(frac), vector frac |
---|
| 927 | PURPOSE: determine if the vector frac represents zero |
---|
| 928 | ASSUME: frac is a valid fraction |
---|
| 929 | RETURN: int |
---|
| 930 | NOTE: returns 1, if frac == 0; 0 otherwise |
---|
| 931 | EXAMPLE: example isZeroFraction; shows examples" |
---|
| 932 | { |
---|
| 933 | if (frac[1] == 0 && frac[4] == 0) { |
---|
| 934 | return(0); |
---|
| 935 | } |
---|
| 936 | if (frac[1] == 0) { // frac has no left representation |
---|
| 937 | if (frac[3] == 0) { // the right representation of frac is zero |
---|
| 938 | return(1); |
---|
| 939 | } |
---|
| 940 | } else { // frac has a left representation |
---|
| 941 | if (frac[2] == 0) { // the left representation of frac is zero |
---|
| 942 | return(1); |
---|
| 943 | } |
---|
| 944 | } |
---|
| 945 | return(0); |
---|
| 946 | } |
---|
| 947 | example |
---|
| 948 | { |
---|
| 949 | "EXAMPLE:"; echo = 2; |
---|
| 950 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 951 | def S = Weyl(); |
---|
| 952 | setring S; S; |
---|
| 953 | isZeroFraction([42,0,0,0]); |
---|
| 954 | isZeroFraction([0,0,Dx,3]); |
---|
| 955 | isZeroFraction([1,1,1,1]); |
---|
| 956 | } |
---|
| 957 | ////////////////////////////////////////////////////////////////////// |
---|
| 958 | proc isOneFraction(vector frac) |
---|
| 959 | "USAGE: isOneFraction(frac), vector frac |
---|
| 960 | PURPOSE: determine if the vector frac represents one |
---|
| 961 | ASSUME: frac is a valid fraction |
---|
| 962 | RETURN: int |
---|
| 963 | NOTE: 1, if frac == 1; 0 otherwise |
---|
| 964 | EXAMPLE: example isOneFraction; shows examples" |
---|
| 965 | { |
---|
| 966 | if (frac[1] == 0 && frac[4] == 0) { |
---|
| 967 | return(0); |
---|
| 968 | } |
---|
| 969 | if (frac[1] == 0) { // frac has no left representation |
---|
| 970 | if (frac[3] == frac[4]) { // the right representation of frac is zero |
---|
| 971 | return(1); |
---|
| 972 | } |
---|
| 973 | } else { // frac has a left representation |
---|
| 974 | if (frac[2] == frac[1]) { // the left representation of frac is zero |
---|
| 975 | return(1); |
---|
| 976 | } |
---|
| 977 | } |
---|
| 978 | return(0); |
---|
| 979 | } |
---|
| 980 | example |
---|
| 981 | { |
---|
| 982 | "EXAMPLE:"; echo = 2; |
---|
| 983 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 984 | def S = Weyl(); |
---|
| 985 | setring S; S; |
---|
| 986 | isOneFraction([42,42,0,0]); |
---|
| 987 | isOneFraction([0,0,Dx,3]); |
---|
| 988 | isOneFraction([1,0,0,1]); |
---|
| 989 | } |
---|
| 990 | ////////// arithmetic //////////////////////////////////////////////// |
---|
| 991 | proc addLeftFractions(vector a, vector b, int locType, def locData, list #) |
---|
| 992 | "USAGE: addLeftFractions(a, b, locType, locData(, override)), |
---|
| 993 | vector a, b, int locType, list/vector/intvec locData(, int override) |
---|
| 994 | PURPOSE: add two left fractions in the specified localization |
---|
| 995 | ASSUME: |
---|
| 996 | RETURN: vector |
---|
| 997 | NOTE: the returned vector is the sum of a and b as fractions in the |
---|
| 998 | localization specified by locType and locData. |
---|
| 999 | EXAMPLE: example addLeftFractions; shows examples" |
---|
| 1000 | { |
---|
| 1001 | int override = 0; |
---|
| 1002 | if (size(#) > 1) { |
---|
| 1003 | if(typeof(#[1]) == "int") { |
---|
| 1004 | override = #[1]; |
---|
| 1005 | } |
---|
| 1006 | } |
---|
| 1007 | if (!override) { |
---|
| 1008 | testLocData(locType, locData); |
---|
| 1009 | testFraction(a, locType, locData); |
---|
| 1010 | testFraction(b, locType, locData); |
---|
| 1011 | } |
---|
| 1012 | // check for a shortcut |
---|
| 1013 | if (isZeroFraction(a)) { |
---|
| 1014 | return(b); |
---|
| 1015 | } |
---|
| 1016 | if (isZeroFraction(b)) { |
---|
| 1017 | return(a); |
---|
| 1018 | } |
---|
| 1019 | if (a[1] == 0) { // a has no left representation |
---|
| 1020 | a = convertRightToLeftFraction(a, locType, locData); |
---|
| 1021 | } |
---|
| 1022 | if (b[1] == 0) { // b has no left representation |
---|
| 1023 | b = convertRightToLeftFraction(b, locType, locData); |
---|
| 1024 | } |
---|
| 1025 | if (a[1] == b[1]) { // a and b have the same left denominator |
---|
| 1026 | return ([a[1],a[2] + b[2],0,0]); |
---|
| 1027 | } |
---|
| 1028 | // no shortcut found, use regular method |
---|
| 1029 | vector oreParas = leftOre(b[1], a[1], locType, locData)[1]; |
---|
| 1030 | vector result = [oreParas[1]*a[1],oreParas[1]*a[2]+oreParas[2]*b[2],0,0]; |
---|
| 1031 | return (result); |
---|
| 1032 | } |
---|
| 1033 | example |
---|
| 1034 | { |
---|
| 1035 | "EXAMPLE:"; echo = 2; |
---|
| 1036 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1037 | def S = Weyl(); |
---|
| 1038 | setring S; S; |
---|
| 1039 | // monoidal localization |
---|
| 1040 | poly g1 = x+3; |
---|
| 1041 | poly g2 = x*y+y; |
---|
| 1042 | list L = g1,g2; |
---|
| 1043 | poly s1 = g1; |
---|
| 1044 | poly s2 = g2; |
---|
| 1045 | poly r1 = Dx; |
---|
| 1046 | poly r2 = Dy; |
---|
| 1047 | vector frac1 = [s1,r1,0,0]; |
---|
| 1048 | vector frac2 = [s2,r2,0,0]; |
---|
| 1049 | vector rm = addLeftFractions(frac1, frac2, 0, L); |
---|
| 1050 | print(rm); |
---|
| 1051 | // geometric localization |
---|
| 1052 | ideal p = x-1, y-3; |
---|
| 1053 | vector rg = addLeftFractions(frac1, frac2, 1, p); |
---|
| 1054 | print(rg); |
---|
| 1055 | // rational localization |
---|
| 1056 | intvec v = 2; |
---|
| 1057 | s1 = y^2+y+1; |
---|
| 1058 | s2 = y-2; |
---|
| 1059 | r1 = Dx; |
---|
| 1060 | r2 = Dy; |
---|
| 1061 | frac1 = [s1,r1,0,0]; |
---|
| 1062 | frac2 = [s2,r2,0,0]; |
---|
| 1063 | vector rr = addLeftFractions(frac1, frac2, 2, v); |
---|
| 1064 | print(rr); |
---|
| 1065 | } |
---|
| 1066 | ////////////////////////////////////////////////////////////////////// |
---|
| 1067 | proc multiplyLeftFractions(vector a, vector b, int locType, def locData, list #) |
---|
| 1068 | "USAGE: multiplyLeftFractions(a, b, locType, locData(, override)), |
---|
| 1069 | vector a, b, int locType, list/vector/intvec locData, int override |
---|
| 1070 | PURPOSE: multiply two left fractions in the specified localization |
---|
| 1071 | ASSUME: |
---|
| 1072 | RETURN: vector |
---|
| 1073 | NOTE: the returned vector is the product of a and b as fractions in the |
---|
| 1074 | localization specified by locType and locData. |
---|
| 1075 | EXAMPLE: example multiplyLeftFractions; shows examples" |
---|
| 1076 | { |
---|
| 1077 | int override = 0; |
---|
| 1078 | if (size(#) > 1) { |
---|
| 1079 | if(typeof(#[1]) == "int") { |
---|
| 1080 | override = #[1]; |
---|
| 1081 | } |
---|
| 1082 | } |
---|
| 1083 | if (!override) { |
---|
| 1084 | testLocData(locType, locData); |
---|
| 1085 | testFraction(a, locType, locData); |
---|
| 1086 | testFraction(b, locType, locData); |
---|
| 1087 | } |
---|
| 1088 | // check for a shortcut |
---|
| 1089 | if (isZeroFraction(a) || isZeroFraction(b)) { |
---|
| 1090 | return([1,0,0,1]); |
---|
| 1091 | } |
---|
| 1092 | if (isOneFraction(a)) { |
---|
| 1093 | return(b); |
---|
| 1094 | } |
---|
| 1095 | if (isOneFraction(b)) { |
---|
| 1096 | return(a); |
---|
| 1097 | } |
---|
| 1098 | if(a[1] == 0) { |
---|
| 1099 | a = convertRightToLeftFraction(a, locType, locData); |
---|
| 1100 | } |
---|
| 1101 | if(b[1] == 0) { |
---|
| 1102 | b = convertRightToLeftFraction(b, locType, locData); |
---|
| 1103 | } |
---|
| 1104 | if( (a[2] == 0) || (b[2] == 0) ) { |
---|
| 1105 | return ([1,0,0,1]); |
---|
| 1106 | } |
---|
| 1107 | if (a[2]*b[1] == b[1]*a[2]) { // trivial solution of the Ore condition |
---|
| 1108 | return([b[1]*a[1],a[2]*b[2]]); |
---|
| 1109 | } |
---|
| 1110 | // no shortcut found, use regular method |
---|
| 1111 | vector oreParas = ore(b[1], a[2], locType, locData, 0)[1]; |
---|
| 1112 | vector result = [oreParas[1]*a[1],oreParas[2]*b[2],0,0]; |
---|
| 1113 | return (result); |
---|
| 1114 | } |
---|
| 1115 | example |
---|
| 1116 | { |
---|
| 1117 | "EXAMPLE:"; echo = 2; |
---|
| 1118 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1119 | def S = Weyl(); |
---|
| 1120 | setring S; S; |
---|
| 1121 | // monoidal localization |
---|
| 1122 | poly g1 = x+3; |
---|
| 1123 | poly g2 = x*y+y; |
---|
| 1124 | list L = g1,g2; |
---|
| 1125 | poly s1 = g1; |
---|
| 1126 | poly s2 = g2; |
---|
| 1127 | poly r1 = Dx; |
---|
| 1128 | poly r2 = Dy; |
---|
| 1129 | vector frac1 = [s1,r1,0,0]; |
---|
| 1130 | vector frac2 = [s2,r2,0,0]; |
---|
| 1131 | vector rm = multiplyLeftFractions(frac1, frac2, 0, L); |
---|
| 1132 | print(rm); |
---|
| 1133 | // geometric localization |
---|
| 1134 | ideal p = x-1, y-3; |
---|
| 1135 | vector rg = multiplyLeftFractions(frac1, frac2, 1, p); |
---|
| 1136 | print(rg); |
---|
| 1137 | // rational localization |
---|
| 1138 | intvec v = 2; |
---|
| 1139 | s1 = y^2+y+1; |
---|
| 1140 | s2 = y-2; |
---|
| 1141 | r1 = Dx; |
---|
| 1142 | r2 = Dy; |
---|
| 1143 | frac1 = [s1,r1,0,0]; |
---|
| 1144 | frac2 = [s2,r2,0,0]; |
---|
| 1145 | vector rr1 = multiplyLeftFractions(frac1, frac2, 2, v); |
---|
| 1146 | print(rr1); |
---|
| 1147 | vector rr2 = multiplyLeftFractions(frac2, frac1, 2, v); |
---|
| 1148 | print(rr2); |
---|
| 1149 | areEqualLeftFractions(rr1, rr2, 2, v); |
---|
| 1150 | } |
---|
| 1151 | ////////////////////////////////////////////////////////////////////// |
---|
| 1152 | proc areEqualLeftFractions(vector a, vector b, int locType, def locData) |
---|
| 1153 | "USAGE: areEqualLeftFractions(a, b, locType, locData), vector a, b, |
---|
| 1154 | int locType, list/vector/intvec locData |
---|
| 1155 | PURPOSE: check if two given fractions are equal |
---|
| 1156 | ASSUME: |
---|
| 1157 | RETURN: int |
---|
| 1158 | NOTE: returns 1 or 0, depending whether a=b as fractions in the |
---|
| 1159 | localization specified by locType and locData |
---|
| 1160 | EXAMPLE: example areEqualLeftFractions; shows examples" |
---|
| 1161 | { |
---|
| 1162 | testLocData(locType, locData); |
---|
| 1163 | testFraction(a, locType, locData); |
---|
| 1164 | testFraction(b, locType, locData); |
---|
| 1165 | if(a[1] == 0) { |
---|
| 1166 | a = convertRightToLeftFraction(a, locType, locData); |
---|
| 1167 | } |
---|
| 1168 | if(b[1] == 0) { |
---|
| 1169 | b = convertRightToLeftFraction(b, locType, locData); |
---|
| 1170 | } |
---|
| 1171 | vector negB = [b[1], -b[2], -b[3], b[4]]; |
---|
| 1172 | //testFraction(negB, locType, locData); // unnecessary check |
---|
| 1173 | vector result = addLeftFractions(a, negB, locType, locData); |
---|
| 1174 | return(isZeroFraction(result)); |
---|
| 1175 | } |
---|
| 1176 | example |
---|
| 1177 | { |
---|
| 1178 | "EXAMPLE:"; echo = 2; |
---|
| 1179 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1180 | def S = Weyl(); |
---|
| 1181 | setring S; S; |
---|
| 1182 | // monoidal |
---|
| 1183 | poly g1 = x*y+3; |
---|
| 1184 | poly g2 = y^3; |
---|
| 1185 | list L = g1,g2; |
---|
| 1186 | poly s1 = g1; |
---|
| 1187 | poly s2 = s1*g2; |
---|
| 1188 | poly s3 = s2; |
---|
| 1189 | poly r1 = Dx; |
---|
| 1190 | poly r2 = g2*r1; |
---|
| 1191 | poly r3 = s1*r1+3; |
---|
| 1192 | vector fracm1 = [s1,r1,0,0]; |
---|
| 1193 | vector fracm2 = [s2,r2,0,0]; |
---|
| 1194 | vector fracm3 = [s3,r3,0,0]; |
---|
| 1195 | areEqualLeftFractions(fracm1, fracm2, 0, L); |
---|
| 1196 | areEqualLeftFractions(fracm1, fracm3, 0, L); |
---|
| 1197 | areEqualLeftFractions(fracm2, fracm3, 0, L); |
---|
| 1198 | } |
---|
| 1199 | ////////////////////////////////////////////////////////////////////// |
---|
| 1200 | proc isInvertibleLeftFraction(vector frac, int locType, def locData) |
---|
| 1201 | "USAGE: isInvertibleLeftFraction(frac, locType, locData), vector frac, |
---|
| 1202 | int locType, list/vector/intvec locData |
---|
| 1203 | PURPOSE: check if a fraction is invertible in the specified localization |
---|
| 1204 | ASSUME: |
---|
| 1205 | RETURN: int |
---|
| 1206 | NOTE: - returns 1, if the numerator of frac is in the denominator set, |
---|
| 1207 | - returns 0, otherwise (NOTE: this does NOT mean that the fraction is |
---|
| 1208 | not invertible, it just means it could not be determined by the |
---|
| 1209 | method above). |
---|
| 1210 | EXAMPLE: example isInvertibleLeftFraction; shows examples" |
---|
| 1211 | { |
---|
| 1212 | testLocData(locType, locData); |
---|
| 1213 | testFraction(frac, locType, locData); |
---|
| 1214 | locData = normalizeLocalization(locType, locData); |
---|
| 1215 | if(frac[1] != 0) { // frac has a left representation |
---|
| 1216 | return(isInS(frac[2], locType, locData)); |
---|
| 1217 | } else { // frac has no left, but a right representation |
---|
| 1218 | return(isInS(frac[3], locType, locData)); |
---|
| 1219 | } |
---|
| 1220 | } |
---|
| 1221 | example |
---|
| 1222 | { |
---|
| 1223 | "EXAMPLE:"; echo = 2; |
---|
| 1224 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1225 | def S = Weyl(); |
---|
| 1226 | setring S; S; |
---|
| 1227 | poly g1 = x+3; |
---|
| 1228 | poly g2 = x*y; |
---|
| 1229 | list L = g1,g2; |
---|
| 1230 | vector frac = [g1*g2, 17, 0, 0]; |
---|
| 1231 | isInvertibleLeftFraction(frac, 0, L); |
---|
| 1232 | ideal p = x-1, y; |
---|
| 1233 | frac = [g1, x, 0, 0]; |
---|
| 1234 | isInvertibleLeftFraction(frac, 1, p); |
---|
| 1235 | intvec rat = 1,2; |
---|
| 1236 | frac = [g1*g2, Dx, 0, 0]; |
---|
| 1237 | isInvertibleLeftFraction(frac, 2, rat); |
---|
| 1238 | } |
---|
| 1239 | ////////////////////////////////////////////////////////////////////// |
---|
| 1240 | proc invertLeftFraction(vector frac, int locType, def locData) |
---|
| 1241 | "USAGE: invertLeftFraction(frac, locType, locData), vector frac, int locType, |
---|
| 1242 | list/vector/intvec locData |
---|
| 1243 | PURPOSE: invert a fraction in the specified localization |
---|
| 1244 | ASSUME: frac is invertible in the loc. specified by locType and locData |
---|
| 1245 | RETURN: vector |
---|
| 1246 | NOTE: - returns the multiplicative inverse of frac in the localization |
---|
| 1247 | specified by locType and locData, |
---|
| 1248 | - throws error if frac is not invertible (NOTE: this does NOT mean |
---|
| 1249 | that the fraction is not invertible, it just means it could not be |
---|
| 1250 | determined by the method listed above). |
---|
| 1251 | EXAMPLE: example invertLeftFraction; shows examples" |
---|
| 1252 | { |
---|
| 1253 | // standard tests are done by isInvertibleLeftFraction |
---|
| 1254 | if (isInvertibleLeftFraction(frac, locType, locData)) { |
---|
| 1255 | return([frac[2],frac[1],frac[4],frac[3]]); |
---|
| 1256 | } else { |
---|
| 1257 | return([1,0,0,1]); |
---|
| 1258 | } |
---|
| 1259 | } |
---|
| 1260 | example |
---|
| 1261 | { |
---|
| 1262 | "EXAMPLE:"; echo = 2; |
---|
| 1263 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1264 | def S = Weyl(); |
---|
| 1265 | setring S; S; |
---|
| 1266 | poly g1 = x+3; |
---|
| 1267 | poly g2 = x*y; |
---|
| 1268 | list L = g1,g2; |
---|
| 1269 | vector frac = [g1*g2, 17, 0, 0]; |
---|
| 1270 | print(invertLeftFraction(frac, 0, L)); |
---|
| 1271 | ideal p = x-1, y; |
---|
| 1272 | frac = [g1, x, 0, 0]; |
---|
| 1273 | print(invertLeftFraction(frac, 1, p)); |
---|
| 1274 | intvec rat = 1,2; |
---|
| 1275 | frac = [g1*g2, y, 0, 0]; |
---|
| 1276 | print(invertLeftFraction(frac, 2, rat)); |
---|
| 1277 | } |
---|
| 1278 | ////////////////////////////////////////////////////////////////////// |
---|
| 1279 | proc normalizeMonoidal(list L) |
---|
| 1280 | "USAGE: normalizeMonoidal(L), list L |
---|
| 1281 | PURPOSE: compute a normal form of monoidal localization data |
---|
| 1282 | RETURN: list |
---|
| 1283 | NOTE: given a list of polys, returns a list of all unique factors appearing |
---|
| 1284 | in the given polys |
---|
| 1285 | EXAMPLE: example normalizeMonoidal; shows examples" |
---|
| 1286 | { |
---|
| 1287 | ideal allFactors; |
---|
| 1288 | int i; |
---|
| 1289 | for (i = 1; i <= size(L); i++) { |
---|
| 1290 | allFactors = allFactors, commutativeFactorization(L[i],1); |
---|
| 1291 | } |
---|
| 1292 | allFactors = simplify(allFactors,1+2+4); |
---|
| 1293 | // simplify: divide by leading coefficients (1), |
---|
| 1294 | // purge zero generators (2), purge double entries (4) |
---|
| 1295 | ideal rev = sort(allFactors)[1]; // sort sorts ascendingly |
---|
| 1296 | return(list(rev[size(rev)..1])); // reverse order and cast to list |
---|
| 1297 | } |
---|
| 1298 | example |
---|
| 1299 | { |
---|
| 1300 | "EXAMPLE:"; echo = 2; |
---|
| 1301 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1302 | def S = Weyl(); setring S; |
---|
| 1303 | list L = x^2*y^3, (x+1)*(x*y-3*y^2+1); |
---|
| 1304 | L = normalizeMonoidal(L); |
---|
| 1305 | print(L); |
---|
| 1306 | } |
---|
| 1307 | ////////////////////////////////////////////////////////////////////// |
---|
| 1308 | proc normalizeRational(intvec v) |
---|
| 1309 | "USAGE: normalizeRational(v), intvec v |
---|
| 1310 | PURPOSE: compute a normal form of rational localization data |
---|
| 1311 | RETURN: intvec |
---|
| 1312 | NOTE: purges double entries and sorts ascendingly |
---|
| 1313 | EXAMPLE: example normalizeRational; shows examples" |
---|
| 1314 | { |
---|
| 1315 | int n = nvars(basering); |
---|
| 1316 | int i; |
---|
| 1317 | intvec result; |
---|
| 1318 | intvec occurring = 0:n; |
---|
| 1319 | for (i = 1; i <= size(v); i++) { |
---|
| 1320 | occurring[v[i]] = 1; |
---|
| 1321 | } |
---|
| 1322 | for (i = 1; i <= size(occurring); i++) { |
---|
| 1323 | if (occurring[i]) { |
---|
| 1324 | if (result == 0) { |
---|
| 1325 | result = i; |
---|
| 1326 | } else { |
---|
| 1327 | result = result, i; |
---|
| 1328 | } |
---|
| 1329 | } |
---|
| 1330 | } |
---|
| 1331 | return(result); |
---|
| 1332 | } |
---|
| 1333 | example |
---|
| 1334 | { |
---|
| 1335 | "EXAMPLE:"; echo = 2; |
---|
| 1336 | ring R; setring R; |
---|
| 1337 | intvec v = 9,5,9,3,1,5; |
---|
| 1338 | v = normalizeRational(v); |
---|
| 1339 | v; |
---|
| 1340 | } |
---|
| 1341 | ////////// internal functions //////////////////////////////////////// |
---|
| 1342 | static proc inducesCommutativeSubring(def input) |
---|
| 1343 | { |
---|
| 1344 | ideal vars = variables(input); |
---|
| 1345 | int i, j; |
---|
| 1346 | for (i = 1; i <= size(vars); i++) { |
---|
| 1347 | for (j = i + 1; j <= size(vars); j++) { |
---|
| 1348 | if (vars[i]*vars[j] != vars[j]*vars[i]) { |
---|
| 1349 | return(0); |
---|
| 1350 | } |
---|
| 1351 | } |
---|
| 1352 | } |
---|
| 1353 | return(1); |
---|
| 1354 | } |
---|
| 1355 | ////////////////////////////////////////////////////////////////////// |
---|
| 1356 | static proc commutativeFactorization(poly p, list #) |
---|
| 1357 | "USAGE: commutativeFactorization(p[, #]), poly p[, list #] |
---|
| 1358 | PURPOSE: compute a factorization of p ignoring non-commutative relations |
---|
| 1359 | RETURN: list or ideal |
---|
| 1360 | NOTE: the optional parameter is passed to factorize after changing to a |
---|
[a1b40a] | 1361 | commutative ring, the result of factorize is transferred back to |
---|
[44168a] | 1362 | basering |
---|
| 1363 | SEE ALSO: factorize |
---|
| 1364 | EXAMPLE: " |
---|
| 1365 | { |
---|
| 1366 | int factorType = 0; |
---|
| 1367 | if (size(#) > 0) { |
---|
| 1368 | if (typeof(#[1]) == "int") { |
---|
| 1369 | factorType = #[1]; |
---|
| 1370 | } |
---|
| 1371 | } |
---|
| 1372 | list RL = ringlist(basering); |
---|
| 1373 | if (size(RL) > 4) { |
---|
| 1374 | def bsRing = basering; |
---|
| 1375 | RL = RL[1..4]; |
---|
| 1376 | def commRing = ring(RL); |
---|
| 1377 | setring commRing; |
---|
| 1378 | poly commP = imap(bsRing, p); |
---|
| 1379 | if (factorType == 1) { |
---|
| 1380 | ideal commFactors = factorize(commP, 1); |
---|
| 1381 | setring bsRing; |
---|
| 1382 | return(imap(commRing, commFactors)); |
---|
| 1383 | } else { |
---|
| 1384 | list commFac = factorize(commP, factorType); |
---|
| 1385 | intvec exponents = commFac[2]; |
---|
| 1386 | ideal commFactors = commFac[1]; |
---|
| 1387 | setring bsRing; |
---|
| 1388 | list result; |
---|
| 1389 | result[1] = imap(commRing, commFactors); |
---|
| 1390 | result[2] = exponents; |
---|
| 1391 | return(result); |
---|
| 1392 | } |
---|
| 1393 | } else { |
---|
| 1394 | return(factorize(p, factorType)); |
---|
| 1395 | } |
---|
| 1396 | } |
---|
| 1397 | ////////////////////////////////////////////////////////////////////// |
---|
| 1398 | static proc normalizeLocalization(int locType, def locData) { |
---|
| 1399 | if (locType == 0) { |
---|
| 1400 | return(normalizeMonoidal(locData)); |
---|
| 1401 | } |
---|
| 1402 | if (locType == 1) { |
---|
| 1403 | return(std(locData)); |
---|
| 1404 | } |
---|
| 1405 | if (locType == 2) { |
---|
| 1406 | return(normalizeRational(locData)); |
---|
| 1407 | } |
---|
| 1408 | return(locData); |
---|
| 1409 | } |
---|
| 1410 | ////////////////////////////////////////////////////////////////////// |
---|
| 1411 | static proc intvecComplement(intvec v, intvec w) { // complement of v in w as sets |
---|
| 1412 | intvec result; |
---|
| 1413 | int i; |
---|
| 1414 | int j; |
---|
| 1415 | int foundMatch; |
---|
| 1416 | for (i = 1; i <= size(w); i++) { |
---|
| 1417 | foundMatch = 0; |
---|
| 1418 | for (j = 1; j <= size(v); j++) { |
---|
| 1419 | if (w[i] == v[j]) { |
---|
| 1420 | foundMatch = 1; |
---|
| 1421 | break; |
---|
| 1422 | } |
---|
| 1423 | } |
---|
| 1424 | if (!foundMatch) { // v[i] is not in w |
---|
| 1425 | if (result == 0) { |
---|
| 1426 | result = w[i]; |
---|
| 1427 | } else { |
---|
| 1428 | result = result, w[i]; |
---|
| 1429 | } |
---|
| 1430 | } |
---|
| 1431 | } |
---|
| 1432 | return(result); |
---|
| 1433 | } |
---|
| 1434 | ////////////////////////////////////////////////////////////////////// |
---|
| 1435 | ////////// internal testing procedures /////////////////////////////// |
---|
| 1436 | static proc testIsInS() |
---|
| 1437 | { |
---|
| 1438 | print(" testing isInS..."); |
---|
| 1439 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1440 | def S = Weyl(); |
---|
| 1441 | setring S; |
---|
| 1442 | // monoidal localization |
---|
| 1443 | poly g1 = x^2*y+x+2; |
---|
| 1444 | poly g2 = y^3+x*y; |
---|
| 1445 | list L = g1, g2; |
---|
| 1446 | poly g = g1^2*g2; |
---|
| 1447 | if (!isInS(g, 0, L)) { |
---|
| 1448 | ERROR("Weyl monoidal isInS direct positive failed"); |
---|
| 1449 | } |
---|
| 1450 | if (!isInS(y^2+x, 0, L)) { |
---|
| 1451 | ERROR("Weyl monoidal isInS indirect positive failed"); |
---|
| 1452 | } |
---|
| 1453 | if (isInS(g-1, 0, L)) { |
---|
| 1454 | ERROR("Weyl monoidal isInS negative failed"); |
---|
| 1455 | } |
---|
| 1456 | // geometric localization |
---|
| 1457 | ideal p = x-1, y-3; |
---|
| 1458 | g = x^2+y-3; |
---|
| 1459 | if (!isInS(g, 1, p)) { |
---|
| 1460 | ERROR("Weyl geometric isInS positive failed"); |
---|
| 1461 | } |
---|
| 1462 | if (isInS((x-1)*g, 1, p)) { |
---|
| 1463 | ERROR("Weyl geometric isInS negative failed"); |
---|
| 1464 | } |
---|
| 1465 | // rational localization |
---|
| 1466 | intvec v = 2; |
---|
| 1467 | if (!isInS(y^5+17*y^2-4, 2, v)) { |
---|
| 1468 | ERROR("Weyl rational isInS positive failed"); |
---|
| 1469 | } |
---|
| 1470 | if (isInS(x*y, 2, v)) { |
---|
| 1471 | ERROR("Weyl rational isInS negative failed"); |
---|
| 1472 | } |
---|
| 1473 | print(" isInS OK"); |
---|
| 1474 | } |
---|
| 1475 | ////////////////////////////////////////////////////////////////////// |
---|
| 1476 | static proc testLeftOre() { |
---|
| 1477 | print(" testing leftOre..."); |
---|
| 1478 | // Weyl |
---|
| 1479 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
| 1480 | def ncW = Weyl(); |
---|
| 1481 | setring ncW; |
---|
| 1482 | //// monoidal localization |
---|
| 1483 | poly g1 = x+3; |
---|
| 1484 | poly g2 = x*y; |
---|
| 1485 | list L = g1,g2; |
---|
| 1486 | poly g = g1^2*g2; |
---|
| 1487 | poly f = Dx; |
---|
| 1488 | vector rm = leftOre(g, f, 0, L)[1]; |
---|
| 1489 | if (rm[1] == 0 || rm[2]*g-rm[1]*f != 0) { |
---|
| 1490 | ERROR("Weyl monoidal left Ore failed"); |
---|
| 1491 | } |
---|
| 1492 | //// geometric localization |
---|
| 1493 | vector p1,p2,p3,p4 = [1,3],[0,0],[-1,-2],[10,-180]; |
---|
| 1494 | vector rg; |
---|
| 1495 | ideal p; |
---|
| 1496 | list pVecs = p1,p2,p3,p4; |
---|
| 1497 | f = Dx; |
---|
| 1498 | g = x^2+y+3; |
---|
| 1499 | for(int i = 1; i <= 4; i = i + 1) { |
---|
| 1500 | p = x-pVecs[i][1], y-pVecs[i][2]; |
---|
| 1501 | rg = leftOre(g, f, 1, p)[1]; |
---|
| 1502 | if (rg[1] == 0 || rg[2]*g-rg[1]*f != 0) { |
---|
| 1503 | ERROR("Weyl geometric left Ore failed at maximal ideal" |
---|
| 1504 | + " induced by " + string(pVecs[i])); |
---|
| 1505 | } |
---|
| 1506 | } |
---|
| 1507 | //// rational localization |
---|
| 1508 | intvec rat = 1; |
---|
| 1509 | f = Dx+Dy; |
---|
| 1510 | g = x; |
---|
| 1511 | vector rr = leftOre(g, f, 2, rat)[1]; |
---|
| 1512 | if (rr[1] == 0 || rr[2]*g-rr[1]*f != 0) { |
---|
| 1513 | ERROR("Weyl rational left Ore failed"); |
---|
| 1514 | } |
---|
| 1515 | // shift rational localization |
---|
| 1516 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
| 1517 | matrix D[4][4]; |
---|
| 1518 | D[1,3] = Sx; |
---|
| 1519 | D[2,4] = Sy; |
---|
| 1520 | def ncS = nc_algebra(1, D); |
---|
| 1521 | setring ncS; |
---|
| 1522 | rat = 1; |
---|
| 1523 | poly f = Sx+Sy; |
---|
| 1524 | poly g = x; |
---|
| 1525 | vector rr = leftOre(g, f, 2, rat)[1]; |
---|
| 1526 | if (rr[1] == 0 || rr[2]*g-rr[1]*f != 0) { |
---|
| 1527 | ERROR("shift rational left Ore failed"); |
---|
| 1528 | } |
---|
| 1529 | // q-shift rational localization |
---|
| 1530 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
| 1531 | matrix C[4][4] = UpOneMatrix(4); |
---|
| 1532 | C[1,3] = q; |
---|
| 1533 | C[2,4] = q; |
---|
| 1534 | def ncQ = nc_algebra(C, 0); |
---|
| 1535 | setring ncQ; |
---|
| 1536 | rat = 1; |
---|
| 1537 | poly f = Qx+Qy; |
---|
| 1538 | poly g = x; |
---|
| 1539 | vector rr = leftOre(g, f, 2, rat)[1]; |
---|
| 1540 | if (rr[1] == 0 || rr[2]*g-rr[1]*f != 0) { |
---|
| 1541 | ERROR("q-shift rational left Ore failed"); |
---|
| 1542 | } |
---|
| 1543 | print(" leftOre OK"); |
---|
| 1544 | } |
---|
| 1545 | ////////////////////////////////////////////////////////////////////// |
---|
| 1546 | static proc testRightOre() { |
---|
| 1547 | print(" testing rightOre..."); |
---|
| 1548 | // Weyl |
---|
| 1549 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
| 1550 | def ncW = Weyl(); |
---|
| 1551 | setring ncW; |
---|
| 1552 | //// monoidal localization |
---|
| 1553 | poly g1 = x+3; |
---|
| 1554 | poly g2 = x*y; |
---|
| 1555 | list L = g1,g2; |
---|
| 1556 | poly g = x; |
---|
| 1557 | poly f = Dx; |
---|
| 1558 | vector rm = rightOre(g, f, 0, L)[1]; |
---|
| 1559 | if (rm[1] == 0 || f*rm[1]-g*rm[2] != 0) { |
---|
| 1560 | ERROR("Weyl monoidal right Ore failed"); |
---|
| 1561 | } |
---|
| 1562 | //// geometric localization |
---|
| 1563 | g = x+y; |
---|
| 1564 | f = Dx+Dy; |
---|
| 1565 | ideal p = x-1,y-3; |
---|
| 1566 | vector rg = rightOre(g, f, 1, p)[1]; |
---|
| 1567 | if (rg[1] == 0 || f*rg[1]-g*rg[2] != 0) { |
---|
| 1568 | ERROR("Weyl geometric right Ore failed"); |
---|
| 1569 | } |
---|
| 1570 | //// rational localization |
---|
| 1571 | intvec rat = 1; |
---|
| 1572 | f = Dx+Dy; |
---|
| 1573 | g = x; |
---|
| 1574 | vector rr = rightOre(g, f, 2, rat)[1]; |
---|
| 1575 | if (rr[1] == 0 || f*rr[1]-g*rr[2] != 0) { |
---|
| 1576 | ERROR("Weyl rational right Ore failed"); |
---|
| 1577 | } |
---|
| 1578 | // shift rational localization |
---|
| 1579 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
| 1580 | matrix D[4][4]; |
---|
| 1581 | D[1,3] = Sx; |
---|
| 1582 | D[2,4] = Sy; |
---|
| 1583 | def ncS = nc_algebra(1, D); |
---|
| 1584 | setring ncS; |
---|
| 1585 | rat = 1; |
---|
| 1586 | poly f = Sx+Sy; |
---|
| 1587 | poly g = x; |
---|
| 1588 | vector rr = rightOre(g, f, 2, rat)[1]; |
---|
| 1589 | if (rr[1] == 0 || f*rr[1]-g*rr[2] != 0) { |
---|
| 1590 | ERROR("shift rational right Ore failed"); |
---|
| 1591 | } |
---|
| 1592 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
| 1593 | matrix C[4][4] = UpOneMatrix(4); |
---|
| 1594 | C[1,3] = q; |
---|
| 1595 | C[2,4] = q; |
---|
| 1596 | def ncQ = nc_algebra(C, 0); |
---|
| 1597 | setring ncQ; |
---|
| 1598 | rat = 1; |
---|
| 1599 | poly f = Qx+Qy; |
---|
| 1600 | poly g = x; |
---|
| 1601 | vector rr = rightOre(g, f, 2, rat)[1]; |
---|
| 1602 | if (rr[1] == 0 || f*rr[1]-g*rr[2] != 0) { |
---|
| 1603 | ERROR("q-shift rational right Ore failed"); |
---|
| 1604 | } |
---|
| 1605 | print(" rightOre OK"); |
---|
| 1606 | } |
---|
| 1607 | ////////////////////////////////////////////////////////////////////// |
---|
| 1608 | static proc testAddLeftFractions() |
---|
| 1609 | { |
---|
| 1610 | print(" testing addLeftFractions..."); |
---|
| 1611 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1612 | def S = Weyl(); |
---|
| 1613 | setring S; |
---|
| 1614 | // monoidal localization |
---|
| 1615 | poly g1 = x+3; |
---|
| 1616 | poly g2 = x*y+y; |
---|
| 1617 | list L = g1,g2; |
---|
| 1618 | vector frac1 = [g1,Dx,0,0]; |
---|
| 1619 | vector frac2 = [g2,Dy,0,0]; |
---|
| 1620 | vector rm = addLeftFractions(frac1, frac2, 0, L); |
---|
| 1621 | if (rm[1] != x^2*y+4*x*y+3*y || rm[2] != x*y*Dx+y*Dx+x*Dy+3*Dy) { |
---|
| 1622 | ERROR("Weyl monoidal addition failed"); |
---|
| 1623 | } |
---|
| 1624 | // geometric localization |
---|
| 1625 | ideal p = x-1,y-3; |
---|
| 1626 | vector rg = addLeftFractions(frac1, frac2, 1, p); |
---|
| 1627 | if (rg[1] != x^2*y+4*x*y+3*y || rg[2] != x*y*Dx+y*Dx+x*Dy+3*Dy) { |
---|
| 1628 | ERROR("Weyl geometric addition failed"); |
---|
| 1629 | } |
---|
| 1630 | // rational localization |
---|
| 1631 | intvec v = 2; |
---|
| 1632 | frac1 = [y^2+y+1,Dx,0,0]; |
---|
| 1633 | frac2 = [y-2,Dy,0,0]; |
---|
| 1634 | vector rr = addLeftFractions(frac1, frac2, 2, v); |
---|
| 1635 | if (rr[1] != y^3-y^2-y-2 || rr[2] != y^2*Dy+y*Dx+y*Dy-2*Dx+Dy) { |
---|
| 1636 | ERROR("Weyl rational addition failed"); |
---|
| 1637 | } |
---|
| 1638 | print(" addLeftFractions OK"); |
---|
| 1639 | } |
---|
| 1640 | ////////////////////////////////////////////////////////////////////// |
---|
| 1641 | static proc testMultiplyLeftFractions() { |
---|
| 1642 | print(" testing multiplyLeftFractions..."); |
---|
| 1643 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1644 | def S = Weyl(); |
---|
| 1645 | setring S; |
---|
| 1646 | // monoidal localization |
---|
| 1647 | poly g1 = x+3; |
---|
| 1648 | poly g2 = x*y+y; |
---|
| 1649 | list L = g1,g2; |
---|
| 1650 | vector frac1 = [g1,Dx,0,0]; |
---|
| 1651 | vector frac2 = [g2,Dy,0,0]; |
---|
| 1652 | vector rm = multiplyLeftFractions(frac1, frac2, 0, L); |
---|
| 1653 | if (rm[1] != g1*g2^2 || rm[2] != x*y*Dx*Dy+y*Dx*Dy-y*Dy) { |
---|
| 1654 | ERROR("Weyl monoidal multiplication error"); |
---|
| 1655 | } |
---|
| 1656 | // geometric localization |
---|
| 1657 | ideal p = x-1,y-3; |
---|
| 1658 | vector rg = multiplyLeftFractions(frac1, frac2, 1, p); |
---|
| 1659 | if (rg[1] != g1*g2*(x+1) || rg[2] != x*Dx*Dy+Dx*Dy-Dy) { |
---|
| 1660 | ERROR("Weyl geometric multiplication error"); |
---|
| 1661 | } |
---|
| 1662 | // rational localization |
---|
| 1663 | intvec v = 2; |
---|
| 1664 | frac1 = [y^2+y+1,Dx,0,0]; |
---|
| 1665 | frac2 = [y-2,Dy,0,0]; |
---|
| 1666 | vector rr = multiplyLeftFractions(frac1, frac2, 2, v); |
---|
| 1667 | if (rr[1] != (y^2+y+1)*(y-2) || rr[2] != Dx*Dy) { |
---|
| 1668 | ERROR("Weyl rational multiplication (1*2) error"); |
---|
| 1669 | } |
---|
| 1670 | rr = multiplyLeftFractions(frac2, frac1, 2, v); |
---|
| 1671 | 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) { |
---|
| 1672 | ERROR("Weyl rational multiplication (2*1) error"); |
---|
| 1673 | } |
---|
| 1674 | print(" multiplyLeftFractions OK"); |
---|
| 1675 | } |
---|
| 1676 | ////////////////////////////////////////////////////////////////////// |
---|
| 1677 | static proc testAreEqualLeftFractions() { |
---|
| 1678 | print(" testing areEqualLeftFractions..."); |
---|
| 1679 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1680 | def S = Weyl(); |
---|
| 1681 | setring S; |
---|
| 1682 | // monoidal |
---|
| 1683 | poly g1 = x*y+3; |
---|
| 1684 | poly g2 = y^3; |
---|
| 1685 | list L = g1,g2; |
---|
| 1686 | vector fracm1 = [g1,Dx,0,0]; |
---|
| 1687 | vector fracm2 = [g1*g2,g2*Dx,0,0]; |
---|
| 1688 | vector fracm3 = [g1*g2,g1*Dx+3,0,0]; |
---|
| 1689 | if (!areEqualLeftFractions(fracm1, fracm2, 0, L)) { |
---|
| 1690 | ERROR("Weyl monoidal positive basic comparison error"); |
---|
| 1691 | } |
---|
| 1692 | if (areEqualLeftFractions(fracm1, fracm3, 0, L)) { |
---|
| 1693 | ERROR("Weyl monoidal first negative basic comparison error"); |
---|
| 1694 | } |
---|
| 1695 | if (areEqualLeftFractions(fracm2, fracm3, 0, L)) { |
---|
| 1696 | ERROR("Weyl monoidal second negative basic comparison error"); |
---|
| 1697 | } |
---|
| 1698 | // geometric |
---|
| 1699 | ideal p = x+5, y-2; |
---|
| 1700 | vector fracg1 = [g1,Dx,0,0]; |
---|
| 1701 | vector fracg2 = [g1*g2,g2*Dx,0,0]; |
---|
| 1702 | vector fracg3 = [g1*g2,g1*Dx+3,0,0]; |
---|
| 1703 | if (!areEqualLeftFractions(fracg1, fracg2, 1, p)) { |
---|
| 1704 | ERROR("Weyl geometric positive basic comparison error"); |
---|
| 1705 | } |
---|
| 1706 | if (areEqualLeftFractions(fracg1, fracg3, 1, p)) { |
---|
| 1707 | ERROR("Weyl geometric first negative basic comparison error"); |
---|
| 1708 | } |
---|
| 1709 | if (areEqualLeftFractions(fracg2, fracg3, 1, p)) { |
---|
| 1710 | ERROR("Weyl geometric second negative basic comparison error"); |
---|
| 1711 | } |
---|
| 1712 | // rational |
---|
| 1713 | intvec rat = 1,4; |
---|
| 1714 | vector fracr1 = [x+Dy,Dx,0,0]; |
---|
| 1715 | vector fracr2 = [x*Dy*(x+Dy),x*Dx*Dy,0,0]; |
---|
| 1716 | vector fracr3 = [Dy*x*(x+Dy),x*Dx*Dy+1,0,0]; |
---|
| 1717 | if (!areEqualLeftFractions(fracr1, fracr2, 2, rat)) { |
---|
| 1718 | ERROR("Weyl rational positive basic comparison error"); |
---|
| 1719 | } |
---|
| 1720 | if (areEqualLeftFractions(fracr1, fracr3, 2, rat)) { |
---|
| 1721 | ERROR("Weyl rational first negative basic comparison error"); |
---|
| 1722 | } |
---|
| 1723 | if (areEqualLeftFractions(fracr2, fracr3, 2, rat)) { |
---|
| 1724 | ERROR("Weyl rational second negative basic comparison error"); |
---|
| 1725 | } |
---|
| 1726 | print(" areEqualLeftFractions OK"); |
---|
| 1727 | } |
---|
| 1728 | ////////////////////////////////////////////////////////////////////// |
---|
| 1729 | static proc testConvertLeftToRightFraction() { |
---|
| 1730 | print(" testing convertLeftToRightFraction..."); |
---|
| 1731 | // Weyl |
---|
| 1732 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
| 1733 | def ncW = Weyl(); |
---|
| 1734 | setring ncW; |
---|
| 1735 | //// monoidal localization |
---|
| 1736 | vector fracm = [x,Dx,0,0]; |
---|
| 1737 | list L = x; |
---|
| 1738 | vector rm = convertLeftToRightFraction(fracm, 0, L); |
---|
| 1739 | if (!fracStatus(rm, 0, L)[1]) { |
---|
| 1740 | ERROR("Weyl monoidal convertLeftToRightFraction failed"); |
---|
| 1741 | } |
---|
| 1742 | //// geometric localization |
---|
| 1743 | vector fracg = [x+y,Dx+Dy,0,0]; |
---|
| 1744 | ideal p = x-1,y-3; |
---|
| 1745 | vector rg = convertLeftToRightFraction(fracg, 1, p); |
---|
| 1746 | if (!fracStatus(rg, 1, p)[1]) { |
---|
| 1747 | ERROR("Weyl geometric convertLeftToRightFraction failed"); |
---|
| 1748 | } |
---|
| 1749 | //// rational localization |
---|
| 1750 | intvec rat = 1; |
---|
| 1751 | vector fracr = [x,Dx+Dy,0,0]; |
---|
| 1752 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
| 1753 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
| 1754 | ERROR("Weyl rational convertLeftToRightFraction failed"); |
---|
| 1755 | } |
---|
| 1756 | // shift rational localization |
---|
| 1757 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
| 1758 | matrix D[4][4]; |
---|
| 1759 | D[1,3] = Sx; |
---|
| 1760 | D[2,4] = Sy; |
---|
| 1761 | def ncS = nc_algebra(1, D); |
---|
| 1762 | setring ncS; |
---|
| 1763 | vector fracr = [x,Sx+Sy,0,0]; |
---|
| 1764 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
| 1765 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
| 1766 | ERROR("Shift rational convertLeftToRightFraction failed"); |
---|
| 1767 | } |
---|
| 1768 | // q-shift rational localization |
---|
| 1769 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
| 1770 | matrix C[4][4] = UpOneMatrix(4); |
---|
| 1771 | C[1,3] = q; |
---|
| 1772 | C[2,4] = q; |
---|
| 1773 | def ncQ = nc_algebra(C, 0); |
---|
| 1774 | setring ncQ; |
---|
| 1775 | vector fracr = [x,Qx+Qy,0,0]; |
---|
| 1776 | vector rr = convertLeftToRightFraction(fracr, 2, rat); |
---|
| 1777 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
| 1778 | ERROR("q-shift rational convertLeftToRightFraction failed"); |
---|
| 1779 | } |
---|
| 1780 | print(" convertLeftToRightFraction OK"); |
---|
| 1781 | } |
---|
| 1782 | ////////////////////////////////////////////////////////////////////// |
---|
| 1783 | static proc testConvertRightToLeftFraction() { |
---|
| 1784 | print(" testing convertRightToLeftFraction..."); |
---|
| 1785 | // Weyl |
---|
| 1786 | ring W = 0,(x,y,Dx,Dy),dp; |
---|
| 1787 | def ncW = Weyl(); |
---|
| 1788 | setring ncW; |
---|
| 1789 | //// monoidal localization |
---|
| 1790 | poly g1 = x+3; |
---|
| 1791 | poly g2 = x*y; |
---|
| 1792 | list L = g1,g2; |
---|
| 1793 | vector fracm = [0,0,Dx,g1^2*g2]; |
---|
| 1794 | vector rm = convertRightToLeftFraction(fracm, 0, L); |
---|
| 1795 | if (!fracStatus(rm, 0, L)[1]) { |
---|
| 1796 | ERROR("Weyl monoidal convertRightToLeftFraction failed"); |
---|
| 1797 | } |
---|
| 1798 | //// geometric localization |
---|
| 1799 | ideal p = x-1,y-3; |
---|
| 1800 | vector fracg = [0,0,Dx,x^2+y]; |
---|
| 1801 | vector rg = convertRightToLeftFraction(fracg, 1, p); |
---|
| 1802 | if (!fracStatus(rg, 1, p)[1]) { |
---|
| 1803 | ERROR("Weyl geometric convertRightToLeftFraction failed"); |
---|
| 1804 | } |
---|
| 1805 | //// rational localization |
---|
| 1806 | intvec rat = 1; |
---|
| 1807 | vector fracr = [0,0,Dx+Dy,x]; |
---|
| 1808 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
| 1809 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
| 1810 | ERROR("Weyl rational convertRightToLeftFraction failed"); |
---|
| 1811 | } |
---|
| 1812 | // shift rational localization |
---|
| 1813 | ring S = 0,(x,y,Sx,Sy),dp; |
---|
| 1814 | matrix D[4][4]; |
---|
| 1815 | D[1,3] = Sx; |
---|
| 1816 | D[2,4] = Sy; |
---|
| 1817 | def ncS = nc_algebra(1, D); |
---|
| 1818 | setring ncS; |
---|
| 1819 | vector fracr = [0,0,Sx+Sy,x]; |
---|
| 1820 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
| 1821 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
| 1822 | ERROR("Shift rational convertRightToLeftFraction failed"); |
---|
| 1823 | } |
---|
| 1824 | // q-shift rational localization |
---|
| 1825 | ring Q = (0,q),(x,y,Qx,Qy),dp; |
---|
| 1826 | matrix C[4][4] = UpOneMatrix(4); |
---|
| 1827 | C[1,3] = q; |
---|
| 1828 | C[2,4] = q; |
---|
| 1829 | def ncQ = nc_algebra(C, 0); |
---|
| 1830 | setring ncQ; |
---|
| 1831 | vector fracr = [0,0,Qx+Qy,x]; |
---|
| 1832 | vector rr = convertRightToLeftFraction(fracr, 2, rat); |
---|
| 1833 | if (!fracStatus(rr, 2, rat)[1]) { |
---|
| 1834 | ERROR("q-shift rational convertRightToLeftFraction failed"); |
---|
| 1835 | } |
---|
| 1836 | print(" convertRightToLeftFraction OK"); |
---|
| 1837 | } |
---|
| 1838 | ////////////////////////////////////////////////////////////////////// |
---|
| 1839 | static proc testIsInvertibleLeftFraction() { |
---|
| 1840 | print(" testing isInvertibleLeftFraction..."); |
---|
| 1841 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1842 | def S = Weyl(); |
---|
| 1843 | setring S; |
---|
| 1844 | poly g1 = x+3; |
---|
| 1845 | poly g2 = x*y; |
---|
| 1846 | // monoidal |
---|
| 1847 | list L = g1, g2; |
---|
| 1848 | if (!isInvertibleLeftFraction([g1*g2,17,0,0], 0, L)) { |
---|
| 1849 | ERROR("Weyl monoidal positive invertibility test error"); |
---|
| 1850 | } |
---|
| 1851 | if (isInvertibleLeftFraction([g1*g2,Dx,0,0], 0, L)) { |
---|
| 1852 | ERROR("Weyl monoidal negative invertibility test error"); |
---|
| 1853 | } |
---|
| 1854 | if (!isInvertibleLeftFraction([1,1,1,1], 0, L)) { |
---|
| 1855 | ERROR("Weyl monoidal one invertibility test error"); |
---|
| 1856 | } |
---|
| 1857 | if (isInvertibleLeftFraction([1,0,0,1], 0, L)) { |
---|
| 1858 | ERROR("Weyl monoidal zero invertibility test error"); |
---|
| 1859 | } |
---|
| 1860 | // geometric |
---|
| 1861 | ideal p = x-1, y; |
---|
| 1862 | if (!isInvertibleLeftFraction([g1,3*x,0,0], 1, p)) { |
---|
| 1863 | ERROR("Weyl geometric positive invertibility test error"); |
---|
| 1864 | } |
---|
| 1865 | if (isInvertibleLeftFraction([g1,Dx,0,0], 0, L)) { |
---|
| 1866 | ERROR("Weyl geometric negative invertibility test error"); |
---|
| 1867 | } |
---|
| 1868 | if (!isInvertibleLeftFraction([1,1,1,1], 1, p)) { |
---|
| 1869 | ERROR("Weyl geometric one invertibility test error"); |
---|
| 1870 | } |
---|
| 1871 | if (isInvertibleLeftFraction([1,0,0,1], 1, p)) { |
---|
| 1872 | ERROR("Weyl geometric zero invertibility test error"); |
---|
| 1873 | } |
---|
| 1874 | // rational |
---|
| 1875 | intvec rat = 1,2; |
---|
| 1876 | if (!isInvertibleLeftFraction([g1*g2,y,0,0], 2, rat)) { |
---|
| 1877 | ERROR("Weyl rational positive invertibility test error"); |
---|
| 1878 | } |
---|
| 1879 | if (isInvertibleLeftFraction([g1*g2,Dx,0,0], 0, L)) { |
---|
| 1880 | ERROR("Weyl rational negative invertibility test error"); |
---|
| 1881 | } |
---|
| 1882 | if (!isInvertibleLeftFraction([1,1,1,1], 2, rat)) { |
---|
| 1883 | ERROR("Weyl rational one invertibility test error"); |
---|
| 1884 | } |
---|
| 1885 | if (isInvertibleLeftFraction([1,0,0,1], 2, rat)) { |
---|
| 1886 | ERROR("Weyl rational zero invertibility test error"); |
---|
| 1887 | } |
---|
| 1888 | print(" isInvertibleLeftFraction OK"); |
---|
| 1889 | } |
---|
| 1890 | ////////////////////////////////////////////////////////////////////// |
---|
| 1891 | static proc testInvertLeftFraction() { |
---|
| 1892 | print(" testing invertLeftFraction..."); |
---|
| 1893 | ring R = 0,(x,y,Dx,Dy),dp; |
---|
| 1894 | def S = Weyl(); |
---|
| 1895 | setring S; |
---|
| 1896 | poly g1 = x+3; |
---|
| 1897 | poly g2 = x*y; |
---|
| 1898 | // monoidal |
---|
| 1899 | list L = g1, g2; |
---|
| 1900 | vector rm = [g1*g2, 17, 0, 0]; |
---|
| 1901 | vector rmInv = invertLeftFraction(rm, 0 , L); |
---|
| 1902 | if (!isOneFraction(multiplyLeftFractions(rm, rmInv, 0, L))) { |
---|
| 1903 | ERROR("Weyl monoidal inversion error"); |
---|
| 1904 | } |
---|
| 1905 | // geometric |
---|
| 1906 | ideal p = x-1, y; |
---|
| 1907 | vector rg = [g1, 3*x, 0, 0]; |
---|
| 1908 | vector rgInv = invertLeftFraction(rg, 1, p); |
---|
| 1909 | if (!isOneFraction(multiplyLeftFractions(rg, rgInv, 1, p))) { |
---|
| 1910 | ERROR("Weyl geometric inversion error"); |
---|
| 1911 | } |
---|
| 1912 | // rational |
---|
| 1913 | intvec rat = 1,2; |
---|
| 1914 | vector rr = [g1*g2, y, 0, 0]; |
---|
| 1915 | vector rrInv = invertLeftFraction(rr, 2, rat); |
---|
| 1916 | if (!isOneFraction(multiplyLeftFractions(rr, rrInv, 2, rat))) { |
---|
| 1917 | ERROR("Weyl rational inversion error"); |
---|
| 1918 | } |
---|
| 1919 | print(" invertLeftFraction OK"); |
---|
| 1920 | } |
---|
| 1921 | ////////////////////////////////////////////////////////////////////// |
---|