1 | // Last change 12.02.2001 (Eric Westenberger) |
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2 | /////////////////////////////////////////////////////////////////////////////// |
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3 | version="$Id: zeroset.lib,v 1.11 2005-04-25 10:13:07 Singular Exp $"; |
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4 | category="Symbolic-numerical solving"; |
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5 | info=" |
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6 | LIBRARY: zeroset.lib Procedures For Roots and Factorization |
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7 | AUTHOR: Thomas Bayer, email: tbayer@mathematik.uni-kl.de |
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8 | http://wwwmayr.informatik.tu-muenchen.de/personen/bayert/ |
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9 | Current Adress: Institut fuer Informatik, TU Muenchen |
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10 | |
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11 | OVERVIEW: |
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12 | Algorithms for finding the zero-set of a zero-dim. ideal in Q(a)[x_1,..,x_n], |
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13 | Roots and Factorization of univariate polynomials over Q(a)[t] |
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14 | where a is an algebraic number. Written in the frame of the |
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15 | diploma thesis (advisor: Prof. Gert-Martin Greuel) 'Computations of moduli |
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16 | spaces of semiquasihomogeneous singularities and an implementation in Singular'. |
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17 | This library is meant as a preliminary extension of the functionality |
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18 | of Singular for univariate factorization of polynomials over simple algebraic |
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19 | extensions in characteristic 0. |
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20 | Subprocedures with postfix 'Main' require that the ring contains a variable |
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21 | 'a' and no parameters, and the ideal 'mpoly', where 'minpoly' from the |
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22 | basering is stored. |
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23 | |
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24 | PROCEDURES: |
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25 | EGCD(f, g) gcd over an algebraic extension field of Q |
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26 | Factor(f) factorization of f over an algebraic extension field |
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27 | Quotient(f, g) quotient q of f w.r.t. g (in f = q*g + remainder) |
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28 | Remainder(f,g) remainder of the division of f by g |
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29 | Roots(f) computes all roots of f in an extension field of Q |
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30 | SQFRNorm(f) norm of f (f must be squarefree) |
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31 | ZeroSet(I) zero-set of the 0-dim. ideal I |
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32 | |
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33 | AUXILIARY PROCEDURES: |
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34 | EGCDMain(f, g) gcd over an algebraic extension field of Q |
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35 | FactorMain(f) factorization of f over an algebraic extension field |
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36 | InvertNumberMain(c) inverts an element of an algebraic extension field |
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37 | QuotientMain(f, g) quotient of f w.r.t. g |
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38 | RemainderMain(f,g) remainder of the division of f by g |
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39 | RootsMain(f) computes all roots of f, might extend the ground field |
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40 | SQFRNormMain(f) norm of f (f must be squarefree) |
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41 | ContainedQ(data, f) f in data ? |
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42 | SameQ(a, b) a == b (list a,b) |
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43 | "; |
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44 | |
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45 | LIB "primitiv.lib"; |
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46 | LIB "primdec.lib"; |
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47 | |
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48 | // note : return a ring : ring need not be exported !!! |
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49 | |
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50 | // Artihmetic in Q(a)[x] without built-in procedures |
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51 | // assume basering = Q[x,a] and minpoly is represented by mpoly(a). |
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52 | // the algorithms are taken from "Polynomial Algorithms in Computer Algebra", |
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53 | // F. Winkler, Springer Verlag Wien, 1996. |
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54 | |
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55 | |
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56 | // To do : |
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57 | // squarefree factorization |
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58 | // multiplicities |
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59 | |
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60 | // Improvement : |
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61 | // a main problem is the growth of the coefficients. Try Roots(x7 - 1) |
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62 | // return ideal mpoly ! |
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63 | // mpoly is not monic, comes from primitive_extra |
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64 | |
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65 | // IMPLEMENTATION |
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66 | // |
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67 | // In procedures with name 'proc-name'Main a polynomial ring over a simple |
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68 | // extension field is represented as Q[x...,a] together with the ideal |
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69 | // 'mpoly' (attribute "isSB"). The arithmetic in the extension field is |
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70 | // implemented in the procedures in the procedures 'MultPolys' (multiplication) |
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71 | // and 'InvertNumber' (inversion). After addition and substraction one should |
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72 | // apply 'SimplifyPoly' to the result to reduce the result w.r.t. 'mpoly'. |
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73 | // This is done by reducing each coefficient seperately, which is more |
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74 | // efficient for polynomials with many terms. |
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75 | |
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76 | |
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77 | /////////////////////////////////////////////////////////////////////////////// |
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78 | |
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79 | proc Roots(poly f) |
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80 | "USAGE: Roots(f); where f is a polynomial |
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81 | PURPOSE: compute all roots of f in a finite extension of the ground field |
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82 | without multiplicities. |
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83 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
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84 | containing a list 'roots' and polynomials 'newA' and 'f': |
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85 | @format |
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86 | - 'roots' is the list of roots of the polynomial f (no multiplicities) |
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87 | - if the ground field is Q(a') and the extension field is Q(a), then |
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88 | 'newA' is the representation of a' in Q(a). |
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89 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
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90 | then 'newA' = 'a'. |
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91 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
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92 | - 'f' is the polynomial f in Q(a) (a' being substituted by 'newA') |
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93 | @end format |
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94 | ASSUME: ground field to be Q or a simple extension of Q given by a minpoly |
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95 | EXAMPLE: example Roots; shows an example |
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96 | " |
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97 | { |
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98 | int dbPrt = printlevel-voice+3; |
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99 | |
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100 | // create a new ring where par(1) is replaced by the variable |
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101 | // with the same name or, if basering does not contain a parameter, |
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102 | // with a new variable 'a'. |
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103 | |
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104 | def ROB = basering; |
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105 | def ROR = TransferRing(basering); |
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106 | setring ROR; |
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107 | export(ROR); |
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108 | |
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109 | // get the polynomial f and find the roots |
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110 | |
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111 | poly f = imap(ROB, f); |
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112 | list result = RootsMain(f); // find roots of f |
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113 | |
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114 | // store the roots and the the new representation of 'a' and transform |
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115 | // the coefficients of f. |
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116 | |
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117 | list roots = result[1]; |
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118 | poly newA = result[2]; |
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119 | map F = basering, maxideal(1); |
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120 | F[nvars(basering)] = newA; |
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121 | poly fn = SimplifyPoly(F(f)); |
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122 | |
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123 | // create a new ring with minploy = mpoly[1] (from ROR) |
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124 | |
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125 | def RON = NewBaseRing(); |
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126 | setring(RON); |
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127 | list roots = imap(ROR, roots); |
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128 | poly newA = imap(ROR, newA); |
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129 | poly f = imap(ROR, fn); |
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130 | kill(ROR); |
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131 | export(roots); |
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132 | export(newA); |
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133 | export(f); dbprint(dbPrt," |
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134 | // 'Roots' created a new ring which contains the list 'roots' and |
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135 | // the polynomials 'f' and 'newA' |
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136 | // To access the roots, newA and the new representation of f, type |
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137 | def R = Roots(f); setring R; roots; newA; f; |
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138 | "); |
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139 | return(RON); |
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140 | } |
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141 | example |
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142 | {"EXAMPLE:"; echo = 2; |
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143 | ring R = (0,a), x, lp; |
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144 | minpoly = a2+1; |
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145 | poly f = x3 - a; |
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146 | def R1 = Roots(f); |
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147 | setring R1; |
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148 | minpoly; |
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149 | newA; |
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150 | f; |
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151 | roots; |
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152 | map F; |
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153 | F[1] = roots[1]; |
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154 | F(f); |
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155 | } |
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156 | |
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157 | /////////////////////////////////////////////////////////////////////////////// |
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158 | |
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159 | proc RootsMain(poly f) |
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160 | "USAGE: RootsMain(f); where f is a polynomial |
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161 | PURPOSE: compute all roots of f in a finite extension of the ground field |
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162 | without multiplicities. |
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163 | RETURN: list, all entries are polynomials |
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164 | @format |
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165 | _[1] = roots of f, each entry is a polynomial |
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166 | _[2] = 'newA' - if the ground field is Q(a') and the extension field |
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167 | is Q(a), then 'newA' is the representation of a' in Q(a) |
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168 | _[3] = minpoly of the algebraic extension of the ground field |
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169 | @end format |
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170 | ASSUME: basering = Q[x,a] ideal mpoly must be defined, it might be 0! |
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171 | NOTE: might change the ideal mpoly !! |
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172 | EXAMPLE: example Roots; shows an example |
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173 | " |
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174 | { |
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175 | int i, linFactors, nlinFactors, dbPrt; |
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176 | intvec wt = 1,0; // deg(a) = 0 |
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177 | list factorList, nlFactors, nlMult, roots, result; |
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178 | poly fa, lc; |
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179 | |
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180 | dbPrt = printlevel-voice+3; |
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181 | |
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182 | // factor f in Q(a)[t] to obtain the roots lying in Q(a) |
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183 | // firstly, find roots of the linear factors, |
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184 | // nonlinear factors are processed later |
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185 | |
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186 | dbprint(dbPrt, "Roots of " + string(f) + ", minimal polynomial = " + string(mpoly[1])); |
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187 | factorList = FactorMain(f); // Factorize f |
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188 | dbprint(dbPrt, (" prime factors of f are : " + string(factorList[1]))); |
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189 | |
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190 | linFactors = 0; |
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191 | nlinFactors = 0; |
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192 | for(i = 2; i <= size(factorList[1]); i = i + 1) { // find linear and nonlinear factors |
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193 | fa = factorList[1][i]; |
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194 | if(deg(fa, wt) == 1) { |
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195 | linFactors++; // get the root from the linear factor |
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196 | lc = LeadTerm(fa, 1)[3]; |
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197 | fa = MultPolys(InvertNumberMain(lc), fa); // make factor monic |
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198 | roots[linFactors] = var(1) - fa; // fa is monic !! |
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199 | } |
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200 | else { // ignore nonlinear factors |
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201 | nlinFactors++; |
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202 | nlFactors[nlinFactors] = factorList[1][i]; |
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203 | nlMult[nlinFactors] = factorList[2][i]; |
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204 | } |
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205 | } |
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206 | if(linFactors == size(factorList[1]) - 1) { // all roots of f are contained in the ground field |
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207 | result[1] = roots; |
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208 | result[2] = var(2); |
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209 | result[3] = mpoly[1]; |
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210 | return(result); |
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211 | } |
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212 | |
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213 | // process the nonlinear factors, i.e., extend the ground field |
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214 | // where a nonlinear factor (irreducible) is a minimal polynomial |
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215 | // compute the primitive element of this extension |
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216 | |
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217 | ideal primElem, minPolys, Fid; |
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218 | list partSol; |
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219 | map F, Xchange; |
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220 | poly f1, newA, mp, oldMinPoly; |
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221 | |
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222 | Fid = mpoly; |
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223 | F[1] = var(1); |
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224 | Xchange[1] = var(2); // the variables have to be exchanged |
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225 | Xchange[2] = var(1); // for the use of 'primitive' |
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226 | |
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227 | if(nlinFactors == 1) { // one nl factor |
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228 | |
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229 | // compute the roots of the nonlinear (irreducible, monic) factor f1 of f |
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230 | // by extending the basefield by a' with minimal polynomial f1 |
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231 | // Then call Roots(f1) to find the roots of f1 over the new base field |
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232 | |
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233 | f1 = nlFactors[1]; |
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234 | if(mpoly[1] != 0) { |
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235 | mp = mpoly[1]; |
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236 | minPolys = Xchange(mp), Xchange(f1); |
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237 | primElem = primitive_extra(minPolys); // no random coord. change |
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238 | mpoly = std(primElem[1]); |
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239 | F = basering, maxideal(1); |
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240 | F[2] = primElem[2]; // transfer all to the new representation |
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241 | newA = primElem[2]; // new representation of a |
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242 | f1 = SimplifyPoly(F(f1)); //reduce(F(f1), mpoly); |
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243 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
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244 | } |
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245 | else { |
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246 | mpoly = std(Xchange(f1)); |
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247 | newA = var(2); |
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248 | } |
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249 | result[3] = mpoly[1]; |
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250 | oldMinPoly = mpoly[1]; |
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251 | partSol = RootsMain(f1); // find roots of f1 over extended field |
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252 | |
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253 | if(oldMinPoly != partSol[3]) { // minpoly has changed ? |
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254 | // all previously computed roots must be transformed |
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255 | // because the minpoly has changed |
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256 | result[3] = partSol[3]; // new minpoly |
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257 | F[2] = partSol[2]; // new representation of algebraic number |
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258 | if(size(roots) > 0) {roots = SimplifyData(F(roots)); } |
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259 | newA = SimplifyPoly(F(newA)); // F(newA); |
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260 | } |
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261 | roots = roots + partSol[1]; // add roots |
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262 | result[2] = newA; |
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263 | result[1] = roots; |
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264 | } |
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265 | else { // more than one nonlinear (irreducible) factor (f_1,...,f_r) |
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266 | // solve each of them by RootsMain(f_i), append their roots |
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267 | // change the minpoly and transform all previously computed |
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268 | // roots if necessary. |
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269 | // Note that the for-loop is more or less book-keeping |
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270 | |
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271 | newA = var(2); |
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272 | result[2] = newA; |
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273 | for(i = 1; i <= size(nlFactors); i = i + 1) { |
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274 | oldMinPoly = mpoly[1]; |
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275 | partSol = RootsMain(nlFactors[i]); // main work |
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276 | nlFactors[i] = 0; // delete factor |
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277 | result[3] = partSol[3]; // store minpoly |
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278 | |
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279 | // book-keeping starts here as in the case 1 nonlinear factor |
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280 | |
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281 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
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282 | F = basering, maxideal(1); |
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283 | F[2] = partSol[2]; // transfer all to the new representation |
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284 | newA = SimplifyPoly(F(newA)); // F(newA); new representation of a |
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285 | result[2] = newA; |
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286 | if(i < size(nlFactors)) { |
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287 | nlFactors = SimplifyData(F(nlFactors)); |
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288 | } // transform remaining factors |
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289 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
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290 | } |
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291 | roots = roots + partSol[1]; // transform roots |
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292 | result[1] = roots; |
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293 | } // end more than one nl factor |
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294 | |
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295 | } |
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296 | return(result); |
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297 | } |
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298 | |
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299 | /////////////////////////////////////////////////////////////////////////////// |
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300 | |
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301 | proc ZeroSet(ideal I, list #) |
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302 | "USAGE: ZeroSet(I [,opt] ); I=ideal, opt=integer |
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303 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
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304 | of the ground field. |
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305 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
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306 | containing a list 'zeroset', a polynomial 'newA', and an |
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307 | ideal 'id': |
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308 | @format |
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309 | - 'zeroset' is the list of the zeros of the ideal I, each zero is an ideal. |
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310 | - if the ground field is Q(a') and the extension field is Q(a), then |
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311 | 'newA' is the representation of a' in Q(a). |
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312 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
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313 | then 'newA' = 'a'. |
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314 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
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315 | - 'id' is the ideal I in Q(a)[x_1,...] (a' substituted by 'newA') |
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316 | @end format |
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317 | ASSUME: dim(I) = 0, and ground field to be Q or a simple extension of Q given |
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318 | by a minpoly. |
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319 | OPTIONS: opt = 0 no primary decomposition (default) |
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320 | opt > 0 primary decomposition |
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321 | NOTE: If I contains an algebraic number (parameter) then 'I' must be |
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322 | transformed w.r.t. 'newA' in the new ring. |
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323 | EXAMPLE: example ZeroSet; shows an example |
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324 | " |
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325 | { |
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326 | int primaryDecQ, dbPrt; |
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327 | list rp; |
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328 | |
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329 | dbPrt = printlevel-voice+2; |
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330 | |
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331 | if(size(#) > 0) { primaryDecQ = #[1]; } |
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332 | else { primaryDecQ = 0; } |
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333 | |
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334 | // create a new ring 'ZSR' with one additional variable instead of the |
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335 | // parameter |
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336 | // if the basering does not contain a parameter then 'a' is used as the |
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337 | // additional variable. |
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338 | |
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339 | def RZSB = basering; |
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340 | def ZSR = TransferRing(RZSB); |
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341 | setring ZSR; |
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342 | |
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343 | // get ideal I and find the zero-set |
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344 | |
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345 | ideal id = std(imap(RZSB, I)); |
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346 | // print(dim(id)); |
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347 | if(dim(id) > 1) { // new variable adjoined to ZSR |
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348 | ERROR(" ideal not zerodimensional "); |
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349 | } |
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350 | |
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351 | list result = ZeroSetMain(id, primaryDecQ); |
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352 | |
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353 | // store the zero-set, minimal polynomial and the new representative of 'a' |
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354 | |
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355 | list zeroset = result[1]; |
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356 | poly newA = result[2]; |
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357 | poly minPoly = result[3][1]; |
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358 | |
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359 | // transform the generators of the ideal I w.r.t. the new representation |
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360 | // of 'a' |
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361 | |
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362 | map F = basering, maxideal(1); |
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363 | F[nvars(basering)] = newA; |
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364 | id = SimplifyData(F(id)); |
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365 | |
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366 | // create a new ring with minpoly = minPoly |
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367 | |
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368 | def RZBN = NewBaseRing(); |
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369 | setring RZBN; |
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370 | |
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371 | list zeroset = imap(ZSR, zeroset); |
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372 | poly newA = imap(ZSR, newA); |
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373 | ideal id = imap(ZSR, id); |
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374 | kill(ZSR); |
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375 | |
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376 | export(id); |
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377 | export(zeroset); |
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378 | export(newA); |
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379 | dbprint(dbPrt," |
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380 | // 'ZeroSet' created a new ring which contains the list 'zeroset', the ideal |
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381 | // 'id' and the polynomial 'newA'. 'id' is the ideal of the input transformed |
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382 | // w.r.t. 'newA'. |
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383 | // To access the zero-set, 'newA' and the new representation of the ideal, type |
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384 | def R = ZeroSet(I); setring R; zeroset; newA; id; |
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385 | "); |
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386 | setring RZSB; |
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387 | return(RZBN); |
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388 | } |
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389 | example |
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390 | {"EXAMPLE:"; echo = 2; |
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391 | ring R = (0,a), (x,y,z), lp; |
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392 | minpoly = a2 + 1; |
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393 | ideal I = x2 - 1/2, a*z - 1, y - 2; |
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394 | def T = ZeroSet(I); |
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395 | setring T; |
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396 | minpoly; |
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397 | newA; |
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398 | id; |
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399 | zeroset; |
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400 | map F1 = basering, zeroset[1]; |
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401 | map F2 = basering, zeroset[2]; |
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402 | F1(id); |
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403 | F2(id); |
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404 | } |
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405 | |
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406 | /////////////////////////////////////////////////////////////////////////////// |
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407 | |
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408 | proc InvertNumberMain(poly f) |
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409 | "USAGE: InvertNumberMain(f); where f is a polynomial |
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410 | PURPOSE: compute 1/f if f is a number in Q(a) i.e., f is represented by a |
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411 | polynomial in Q[a]. |
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412 | RETURN: poly 1/f |
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413 | ASSUME: basering = Q[x_1,...,x_n,a], ideal mpoly must be defined and != 0 ! |
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414 | " |
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415 | { |
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416 | if(diff(f, var(1)) != 0) { ERROR("number must not contain variable !");} |
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417 | |
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418 | int n = nvars(basering); |
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419 | def RINB = basering; |
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420 | string ringSTR = "ring RINR = 0, " + string(var(n)) + ", dp;"; |
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421 | execute(ringSTR); // new ring = Q[a] |
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422 | |
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423 | list gcdList; |
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424 | poly f, g, inv; |
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425 | |
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426 | f = imap(RINB, f); |
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427 | g = imap(RINB, mpoly)[1]; |
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428 | |
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429 | if(diff(f, var(1)) != 0) { inv = extgcd(f, g)[2]; } // f contains var(1) |
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430 | else { inv = 1/f;} // f element in Q |
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431 | |
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432 | setring(RINB); |
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433 | return(imap(RINR, inv)); |
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434 | } |
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435 | |
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436 | /////////////////////////////////////////////////////////////////////////////// |
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437 | |
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438 | proc MultPolys(poly f, poly g) |
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439 | "USAGE: MultPolys(f, g); poly f,g |
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440 | PURPOSE: multiply the polynomials f and g and reduce them w.r.t. mpoly |
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441 | RETURN: poly f*g |
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442 | ASSUME: basering = Q[x,a], ideal mpoly must be defined, it might be 0 ! |
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443 | " |
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444 | { |
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445 | return(SimplifyPoly(f * g)); |
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446 | } |
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447 | |
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448 | /////////////////////////////////////////////////////////////////////////////// |
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449 | |
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450 | proc LeadTerm(poly f, int i) |
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451 | "USAGE: LeadTerm(f); poly f, int i |
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452 | PURPOSE: compute the leading coef and term of f w.r.t var(i), where the last |
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453 | ring variable is treated as a parameter. |
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454 | RETURN: list of polynomials |
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455 | _[1] = leading term |
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456 | _[2] = leading monomial |
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457 | _[3] = leading coefficient |
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458 | ASSUME: basering = Q[x_1,...,x_n,a] |
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459 | " |
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460 | { |
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461 | list result; |
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462 | matrix co = coef(f, var(i)); |
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463 | result[1] = co[1, 1]*co[2, 1]; |
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464 | result[2] = co[1, 1]; |
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465 | result[3] = co[2, 1]; |
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466 | return(result); |
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467 | } |
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468 | |
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469 | /////////////////////////////////////////////////////////////////////////////// |
---|
470 | |
---|
471 | proc Quotient(poly f, poly g) |
---|
472 | "USAGE: Quotient(f, g); where f,g are polynomials; |
---|
473 | PURPOSE: compute the quotient q and remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
474 | RETURN: list of polynomials |
---|
475 | @format |
---|
476 | _[1] = quotient q |
---|
477 | _[2] = remainder r |
---|
478 | @end format |
---|
479 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
480 | EXAMPLE: example Quotient; shows an example |
---|
481 | " |
---|
482 | { |
---|
483 | def QUOB = basering; |
---|
484 | def QUOR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
485 | setring QUOR; |
---|
486 | export(QUOR); |
---|
487 | poly f = imap(QUOB, f); |
---|
488 | poly g = imap(QUOB, g); |
---|
489 | list result = QuotientMain(f, g); |
---|
490 | |
---|
491 | setring(QUOB); |
---|
492 | list result = imap(QUOR, result); |
---|
493 | kill(QUOR); |
---|
494 | return(result); |
---|
495 | } |
---|
496 | example |
---|
497 | {"EXAMPLE:"; echo = 2; |
---|
498 | ring R = (0,a), x, lp; |
---|
499 | minpoly = a2+1; |
---|
500 | poly f = x4 - 2; |
---|
501 | poly g = x - a; |
---|
502 | list qr = Quotient(f, g); |
---|
503 | qr; |
---|
504 | qr[1]*g + qr[2] - f; |
---|
505 | } |
---|
506 | |
---|
507 | proc QuotientMain(poly f, poly g) |
---|
508 | "USAGE: QuotientMain(f, g); where f,g are polynomials |
---|
509 | PURPOSE: compute the quotient q and remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
510 | RETURN: list of polynomials |
---|
511 | @format |
---|
512 | _[1] = quotient q |
---|
513 | _[2] = remainder r |
---|
514 | @end format |
---|
515 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
516 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
517 | EXAMPLE: example Quotient; shows an example |
---|
518 | " |
---|
519 | { |
---|
520 | if(g == 0) { ERROR("Division by zero !");} |
---|
521 | |
---|
522 | def QMB = basering; |
---|
523 | def QMR = NewBaseRing(); |
---|
524 | setring QMR; |
---|
525 | poly f, g, h; |
---|
526 | h = imap(QMB, f) / imap(QMB, g); |
---|
527 | setring QMB; |
---|
528 | return(list(imap(QMR, h), 0)); |
---|
529 | } |
---|
530 | |
---|
531 | /////////////////////////////////////////////////////////////////////////////// |
---|
532 | |
---|
533 | proc Remainder(poly f, poly g) |
---|
534 | "USAGE: Remainder(f, g); where f,g are polynomials |
---|
535 | PURPOSE: compute the remainder of the division of f by g, i.e. a polynomial r |
---|
536 | s.t. f = g*q + r, deg(r) < deg(g). |
---|
537 | RETURN: poly |
---|
538 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
539 | " |
---|
540 | { |
---|
541 | def REMB = basering; |
---|
542 | def REMR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
543 | setring(REMR); |
---|
544 | export(REMR); |
---|
545 | poly f = imap(REMB, f); |
---|
546 | poly g = imap(REMB, g); |
---|
547 | poly h = RemainderMain(f, g); |
---|
548 | |
---|
549 | setring(REMB); |
---|
550 | poly r = imap(REMR, h); |
---|
551 | kill(REMR); |
---|
552 | return(r); |
---|
553 | } |
---|
554 | example |
---|
555 | {"EXAMPLE:"; echo = 2; |
---|
556 | ring R = (0,a), x, lp; |
---|
557 | minpoly = a2+1; |
---|
558 | poly f = x4 - 1; |
---|
559 | poly g = x3 - 1; |
---|
560 | Remainder(f, g); |
---|
561 | } |
---|
562 | |
---|
563 | proc RemainderMain(poly f, poly g) |
---|
564 | "USAGE: RemainderMain(f, g); where f,g are polynomials |
---|
565 | PURPOSE: compute the remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
566 | RETURN: poly |
---|
567 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
568 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
569 | " |
---|
570 | { |
---|
571 | int dg; |
---|
572 | intvec wt = 1,0;; |
---|
573 | poly lc, g1, r; |
---|
574 | |
---|
575 | if(deg(g, wt) == 0) { return(0); } |
---|
576 | |
---|
577 | lc = LeadTerm(g, 1)[3]; |
---|
578 | g1 = MultPolys(InvertNumberMain(lc), g); // make g monic |
---|
579 | |
---|
580 | return(SimplifyPoly(reduce(f, std(g1)))); |
---|
581 | } |
---|
582 | |
---|
583 | /////////////////////////////////////////////////////////////////////////////// |
---|
584 | |
---|
585 | proc EGCD(poly f, poly g) |
---|
586 | "USAGE: EGCD(f, g); where f,g are polynomials |
---|
587 | PURPOSE: compute the polynomial gcd of f and g over Q(a)[x] |
---|
588 | RETURN: polynomial h s.t. h is a greatest common divisor of f and g (not nec. |
---|
589 | monic) |
---|
590 | ASSUME: basering = Q(a)[t] |
---|
591 | EXAMPLE: example EGCD; shows an example |
---|
592 | " |
---|
593 | { |
---|
594 | def GCDB = basering; |
---|
595 | def GCDR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
596 | setring GCDR; |
---|
597 | export(GCDR); |
---|
598 | poly f = imap(GCDB, f); |
---|
599 | poly g = imap(GCDB, g); |
---|
600 | poly h = EGCDMain(f, g); // squarefree norm of f |
---|
601 | |
---|
602 | setring(GCDB); |
---|
603 | poly h = imap(GCDR, h); |
---|
604 | kill(GCDR); |
---|
605 | return(h); |
---|
606 | } |
---|
607 | example |
---|
608 | {"EXAMPLE:"; echo = 2; |
---|
609 | ring R = (0,a), x, lp; |
---|
610 | minpoly = a2+1; |
---|
611 | poly f = x4 - 1; |
---|
612 | poly g = x2 - 2*a*x - 1; |
---|
613 | EGCD(f, g); |
---|
614 | } |
---|
615 | |
---|
616 | proc EGCDMain(poly f, poly g) |
---|
617 | "USAGE: EGCDMain(f, g); where f,g are polynomials |
---|
618 | PURPOSE: compute the polynomial gcd of f and g over Q(a)[x] |
---|
619 | RETURN: poly |
---|
620 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
621 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
622 | EXAMPLE: example EGCD; shows an example |
---|
623 | " |
---|
624 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gcd |
---|
625 | { |
---|
626 | int i = 1; |
---|
627 | poly r1, r2, r; |
---|
628 | |
---|
629 | r1 = f; |
---|
630 | r2 = g; |
---|
631 | |
---|
632 | while(r2 != 0) { |
---|
633 | r = RemainderMain(r1, r2); |
---|
634 | r1 = r2; |
---|
635 | r2 = r; |
---|
636 | } |
---|
637 | return(r1); |
---|
638 | } |
---|
639 | |
---|
640 | /////////////////////////////////////////////////////////////////////////////// |
---|
641 | |
---|
642 | proc MEGCD(poly f, poly g, int varIndex) |
---|
643 | "USAGE: MEGCD(f, g, i); poly f, g; int i |
---|
644 | PURPOSE: compute the polynomial gcd of f and g in the i'th variable |
---|
645 | RETURN: poly |
---|
646 | ASSUME: f, g are polynomials in var(i), last variable is the algebraic number |
---|
647 | EXAMPLE: example MEGCD; shows an example |
---|
648 | " |
---|
649 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gc |
---|
650 | // not used ! |
---|
651 | { |
---|
652 | string @str, @sf, @sg, @mp, @parName; |
---|
653 | |
---|
654 | def @RGCDB = basering; |
---|
655 | |
---|
656 | @sf = string(f); |
---|
657 | @sg = string(g); |
---|
658 | @mp = string(minpoly); |
---|
659 | |
---|
660 | if(npars(basering) == 0) { @parName = "0";} |
---|
661 | else { @parName = "(0, " + parstr(basering) + ")"; } |
---|
662 | @str = "ring @RGCD = " + @parName + ", " + string(var(varIndex)) + ", dp;"; |
---|
663 | execute(@str); |
---|
664 | if(@mp != "0") { execute ("minpoly = " + @mp + ";"); } |
---|
665 | execute("poly @f = " + @sf + ";"); |
---|
666 | execute("poly @g = " + @sg + ";"); |
---|
667 | export(@RGCD); |
---|
668 | poly @h = EGCD(@f, @g); |
---|
669 | setring(@RGCDB); |
---|
670 | poly h = imap(@RGCD, @h); |
---|
671 | kill(@RGCD); |
---|
672 | return(h); |
---|
673 | } |
---|
674 | |
---|
675 | /////////////////////////////////////////////////////////////////////////////// |
---|
676 | |
---|
677 | proc SQFRNorm(poly f) |
---|
678 | "USAGE: SQFRNorm(f); where f is a polynomial |
---|
679 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
680 | RETURN: list with 3 entries |
---|
681 | @format |
---|
682 | _[1] = squarefree norm of g (poly) |
---|
683 | _[2] = g (= f(x - s*a)) (poly) |
---|
684 | _[3] = s (int) |
---|
685 | @end format |
---|
686 | ASSUME: f must be squarefree, basering = Q(a)[x] and minpoly != 0. |
---|
687 | NOTE: the norm is an element of Q[x] |
---|
688 | EXAMPLE: example SQFRNorm; shows an example |
---|
689 | " |
---|
690 | { |
---|
691 | def SNB = basering; |
---|
692 | def SNR = TransferRing(SNB); // new ring with parameter 'a' |
---|
693 | // replaced by a variable |
---|
694 | setring SNR; |
---|
695 | poly f = imap(SNB, f); |
---|
696 | list result = SQFRNormMain(f); // squarefree norm of f |
---|
697 | |
---|
698 | setring SNB; |
---|
699 | list result = imap(SNR, result); |
---|
700 | kill SNR; |
---|
701 | return(result); |
---|
702 | } |
---|
703 | example |
---|
704 | {"EXAMPLE:"; echo = 2; |
---|
705 | ring R = (0,a), x, lp; |
---|
706 | minpoly = a2+1; |
---|
707 | poly f = x4 - 2*x + 1; |
---|
708 | SQFRNorm(f); |
---|
709 | } |
---|
710 | |
---|
711 | proc SQFRNormMain(poly f) |
---|
712 | "USAGE: SQFRNorm(f); where f is a polynomial |
---|
713 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
714 | RETURN: list with 3 entries |
---|
715 | @format |
---|
716 | _[1] = squarefree norm of g (poly) |
---|
717 | _[2] = g (= f(x - s*a)) (poly) |
---|
718 | _[3] = s (int) |
---|
719 | @end format |
---|
720 | ASSUME: f must be squarefree, basering = Q[x,a] and ideal mpoly is equal to |
---|
721 | 'minpoly',this represents the ring Q(a)[x] together with 'minpoly'. |
---|
722 | NOTE: the norm is an element of Q[x] |
---|
723 | EXAMPLE: example SqfrNorm; shows an example |
---|
724 | " |
---|
725 | { |
---|
726 | def SNRMB = basering; |
---|
727 | int s = 0; |
---|
728 | intvec wt = 1,0; |
---|
729 | ideal mapId; |
---|
730 | // list result; |
---|
731 | poly g, N, N1, h; |
---|
732 | string ringSTR; |
---|
733 | |
---|
734 | mapId[1] = var(1) - var(2); // linear transformation |
---|
735 | mapId[2] = var(2); |
---|
736 | map Fs = SNRMB, mapId; |
---|
737 | |
---|
738 | N = resultant(f, mpoly[1], var(2)); // norm of f |
---|
739 | N1 = diff(N, var(1)); |
---|
740 | g = f; |
---|
741 | |
---|
742 | ringSTR = "ring SNRM1 = 0, " + string(var(1)) + ", dp;"; // univariate ring |
---|
743 | execute(ringSTR); |
---|
744 | poly N, N1, h; // N, N1 do not contain 'a', use built-in gcd |
---|
745 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
746 | setring(SNRMB); |
---|
747 | h = imap(SNRM1, h); |
---|
748 | while(deg(h, wt) != 0) { // while norm is not squarefree |
---|
749 | s = s + 1; |
---|
750 | g = reduce(Fs(g), mpoly); |
---|
751 | N = reduce(resultant(g, mpoly[1], var(2)), mpoly); // norm of g |
---|
752 | N1 = reduce(diff(N, var(1)), mpoly); |
---|
753 | setring(SNRM1); |
---|
754 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
755 | setring(SNRMB); |
---|
756 | h = imap(SNRM1, h); |
---|
757 | } |
---|
758 | return(list(N, g, s)); |
---|
759 | } |
---|
760 | |
---|
761 | /////////////////////////////////////////////////////////////////////////////// |
---|
762 | |
---|
763 | proc Factor(poly f) |
---|
764 | "USAGE: Factor(f); where f is a polynomial |
---|
765 | PURPOSE: compute the factorization of the squarefree poly f over Q(a)[t] |
---|
766 | RETURN: list with two entries |
---|
767 | @format |
---|
768 | _[1] = factors (monic), first entry is the leading coefficient |
---|
769 | _[2] = multiplicities (not yet implemented) |
---|
770 | @end format |
---|
771 | ASSUME: basering must be the univariate polynomial ring over a field, which |
---|
772 | is Q or a simple extension of Q given by a minpoly. |
---|
773 | NOTE: if basering = Q[t] then this is the built-in @code{factorize} |
---|
774 | EXAMPLE: example Factor; shows an example |
---|
775 | " |
---|
776 | { |
---|
777 | def FRB = basering; |
---|
778 | def FRR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
779 | setring FRR; |
---|
780 | export(FRR); |
---|
781 | poly f = imap(FRB, f); |
---|
782 | list result = FactorMain(f); // squarefree norm of f |
---|
783 | |
---|
784 | setring(FRB); |
---|
785 | list result = imap(FRR, result); |
---|
786 | kill(FRR); |
---|
787 | return(result); |
---|
788 | } |
---|
789 | example |
---|
790 | {"EXAMPLE:"; echo = 2; |
---|
791 | ring R = (0,a), x, lp; |
---|
792 | minpoly = a2+1; |
---|
793 | poly f = x4 - 1; |
---|
794 | list fl = Factor(f); |
---|
795 | fl; |
---|
796 | fl[1][1]*fl[1][2]*fl[1][3]*fl[1][4]*fl[1][5] - f; |
---|
797 | } |
---|
798 | |
---|
799 | /////////////////////////////////////////////////////////////////////////////// |
---|
800 | |
---|
801 | proc FactorMain(poly f) |
---|
802 | "USAGE: FactorMain(f); where f is a polynomial |
---|
803 | PURPOSE: compute the factorization of the squarefree poly f over Q(a)[t], |
---|
804 | minpoly = p(a). |
---|
805 | RETURN: list with 2 entries |
---|
806 | @format |
---|
807 | _[1] = factors, first is a constant |
---|
808 | _[2] = multiplicities (not yet implemented) |
---|
809 | @end format |
---|
810 | ASSUME: basering = Q[x,a], representing Q(a)[x]. An ideal mpoly must |
---|
811 | be defined, representing the minimal polynomial (it might be 0!). |
---|
812 | EXAMPLE: example Factor; shows an example |
---|
813 | " |
---|
814 | // extend this by a squarefree factorization !! |
---|
815 | // multiplicities are not valid !! |
---|
816 | { |
---|
817 | int i, s; |
---|
818 | list normList, factorList, quo_rem; |
---|
819 | poly f1, h, h1, H, g, leadCoef, invCoeff; |
---|
820 | ideal fac1, fac2; |
---|
821 | map F; |
---|
822 | |
---|
823 | // if no minimal polynomial is defined then use 'factorize' |
---|
824 | // FactorOverQ is wrapped around 'factorize' |
---|
825 | |
---|
826 | if(mpoly[1] == 0) { |
---|
827 | // print(" factorize : deg = " + string(deg(f, intvec(1,0)))); |
---|
828 | factorList = factorize(f); // FactorOverQ(f); |
---|
829 | return(factorList); |
---|
830 | } |
---|
831 | |
---|
832 | // if mpoly != 0 and f does not contain the algebraic number, a root of |
---|
833 | // f might be contained in Q(a). Hence one must not use 'factorize'. |
---|
834 | |
---|
835 | fac1[1] = 1; |
---|
836 | fac2[1] = 1; |
---|
837 | normList = SQFRNormMain(f); |
---|
838 | // print(" factorize : deg = " + string(deg(normList[1], intvec(1,0)))); |
---|
839 | factorList = factorize(normList[1]); // factor squarefree norm of f over Q[x] |
---|
840 | g = normList[2]; |
---|
841 | s = normList[3]; |
---|
842 | F[1] = var(1) + s*var(2); // inverse transformation |
---|
843 | F[2] = var(2); |
---|
844 | fac1[1] = factorList[1][1]; |
---|
845 | fac2[1] = factorList[2][1]; |
---|
846 | for(i = 2; i <= size(factorList[1]); i = i + 1) { |
---|
847 | H = factorList[1][i]; |
---|
848 | h = EGCDMain(H, g); |
---|
849 | quo_rem = QuotientMain(g, h); |
---|
850 | g = quo_rem[1]; |
---|
851 | fac1[i] = SimplifyPoly(F(h)); |
---|
852 | fac2[i] = 1; // to be changed later |
---|
853 | } |
---|
854 | return(list(fac1, fac2)); |
---|
855 | } |
---|
856 | |
---|
857 | /////////////////////////////////////////////////////////////////////////////// |
---|
858 | |
---|
859 | proc ZeroSetMain(ideal I, int primDecQ) |
---|
860 | "USAGE: ZeroSetMain(ideal I, int opt); ideal I, int opt |
---|
861 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a simple extension |
---|
862 | of the ground field. |
---|
863 | RETURN: list |
---|
864 | - 'f' is the polynomial f in Q(a) (a' being substituted by newA) |
---|
865 | _[1] = zero-set (list), is the list of the zero-set of the ideal I, |
---|
866 | each entry is an ideal. |
---|
867 | _[2] = 'newA'; if the ground field is Q(a') and the extension field |
---|
868 | is Q(a), then 'newA' is the representation of a' in Q(a). |
---|
869 | If the basering contains a parameter 'a' and the minpoly |
---|
870 | remains unchanged then 'newA' = 'a'. If the basering does not |
---|
871 | contain a parameter then 'newA' = 'a' (default). |
---|
872 | _[3] = 'mpoly' (ideal), the minimal polynomial of the simple extension |
---|
873 | of the ground field. |
---|
874 | ASSUME: basering = K[x_1,x_2,...,x_n] where K = Q or a simple extension of Q |
---|
875 | given by a minpoly; dim(I) = 0. |
---|
876 | NOTE: opt = 0 no primary decomposition |
---|
877 | opt > 0 use a primary decomposition |
---|
878 | EXAMPLE: example ZeroSet; shows an example |
---|
879 | " |
---|
880 | { |
---|
881 | // main work is done in ZeroSetMainWork, here the zero-set of each ideal from the |
---|
882 | // primary decompostion is coputed by menas of ZeroSetMainWork, and then the |
---|
883 | // minpoly and the parameter representing the algebraic extension are |
---|
884 | // transformed according to 'newA', i.e., only bookeeping is done. |
---|
885 | |
---|
886 | def altring=basering; |
---|
887 | int i, j, n, noMP, dbPrt; |
---|
888 | intvec w; |
---|
889 | list currentSol, result, idealList, primDecList, zeroSet; |
---|
890 | ideal J; |
---|
891 | map Fa; |
---|
892 | poly newA, oldMinPoly; |
---|
893 | |
---|
894 | dbPrt = printlevel-voice+2; |
---|
895 | dbprint(dbPrt, "ZeroSet of " + string(I) + ", minpoly = " + string(minpoly)); |
---|
896 | |
---|
897 | n = nvars(basering) - 1; |
---|
898 | for(i = 1; i <= n; i++) { w[i] = 1;} |
---|
899 | w[n + 1] = 0; |
---|
900 | |
---|
901 | if(primDecQ == 0) { return(ZeroSetMainWork(I, w, 0)); } |
---|
902 | |
---|
903 | newA = var(n + 1); |
---|
904 | if(mpoly[1] == 0) { noMP = 1;} |
---|
905 | else {noMP = 0;} |
---|
906 | |
---|
907 | primDecList = primdecGTZ(I); // primary decomposition |
---|
908 | dbprint(dbPrt, "primary decomposition consists of " + string(size(primDecList)) + " primary ideals "); |
---|
909 | // idealList = PDSort(idealList); // high degrees first |
---|
910 | |
---|
911 | for(i = 1; i <= size(primDecList); i = i + 1) { |
---|
912 | idealList[i] = primDecList[i][2]; // use prime component |
---|
913 | dbprint(dbPrt, string(i) + " " + string(idealList[i])); |
---|
914 | } |
---|
915 | |
---|
916 | // compute the zero-set of each primary ideal and join them. |
---|
917 | // If necessary, change the ground field and transform the zero-set |
---|
918 | |
---|
919 | dbprint(dbPrt, " |
---|
920 | find the zero-set of each primary ideal, form the union |
---|
921 | and keep track of the minimal polynomials "); |
---|
922 | |
---|
923 | for(i = 1; i <= size(idealList); i = i + 1) { |
---|
924 | J = idealList[i]; |
---|
925 | idealList[i] = 0; |
---|
926 | oldMinPoly = mpoly[1]; |
---|
927 | dbprint(dbPrt, " ideal#" + string(i) + " of " + string(size(idealList)) + " = " + string(J)); |
---|
928 | currentSol = ZeroSetMainWork(J, w, 0); |
---|
929 | |
---|
930 | if(oldMinPoly != currentSol[3]) { // change minpoly and transform solutions |
---|
931 | dbprint(dbPrt, " change minpoly to " + string(currentSol[3][1])); |
---|
932 | dbprint(dbPrt, " new representation of algebraic number = " + string(currentSol[2])); |
---|
933 | if(!noMP) { // transform the algebraic number a |
---|
934 | Fa = basering, maxideal(1); |
---|
935 | Fa[n + 1] = currentSol[2]; |
---|
936 | newA = SimplifyPoly(Fa(newA)); // new representation of a |
---|
937 | if(size(zeroSet) > 0) {zeroSet = SimplifyZeroset(Fa(zeroSet)); } |
---|
938 | if(i < size(idealList)) { idealList = SimplifyZeroset(Fa(idealList)); } |
---|
939 | } |
---|
940 | else { noMP = 0;} |
---|
941 | } |
---|
942 | zeroSet = zeroSet + currentSol[1]; // add new elements |
---|
943 | } |
---|
944 | return(list(zeroSet, newA, mpoly)); |
---|
945 | } |
---|
946 | |
---|
947 | /////////////////////////////////////////////////////////////////////////////// |
---|
948 | |
---|
949 | proc ZeroSetMainWork(ideal id, intvec wt, int sVars) |
---|
950 | "USAGE: ZeroSetMainWork(I, wt, sVars); |
---|
951 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
---|
952 | of the ground field (without multiplicities). |
---|
953 | RETURN: list, all entries are polynomials |
---|
954 | _[1] = zeros, each entry is an ideal |
---|
955 | _[2] = newA; if the ground field is Q(a') this is the rep. of a' w.r.t. a |
---|
956 | _[3] = minpoly of the algebraic extension of the ground field (ideal) |
---|
957 | _[4] = name of algebraic number (default = 'a') |
---|
958 | ASSUME: basering = Q[x_1,x_2,...,x_n,a] |
---|
959 | ideal mpoly must be defined, it might be 0! |
---|
960 | NOTE: might change 'mpoly' !! |
---|
961 | EXAMPLE: example IdealSolve; shows an example |
---|
962 | " |
---|
963 | { |
---|
964 | def altring=basering; |
---|
965 | int i, j, k, nrSols, n, noMP; |
---|
966 | ideal I, generators, gens, solid, partsolid; |
---|
967 | list linSol, linearSolution, nLinSol, nonlinSolutions, partSol, sol, solutions, result; |
---|
968 | list linIndex, nlinIndex, index; |
---|
969 | map Fa, Fsubs; |
---|
970 | poly oldMinPoly, newA; |
---|
971 | |
---|
972 | if(mpoly[1] == 0) { noMP = 1;} |
---|
973 | else { noMP = 0;} |
---|
974 | n = nvars(basering) - 1; |
---|
975 | newA = var(n + 1); |
---|
976 | |
---|
977 | I = std(id); |
---|
978 | |
---|
979 | // find linear solutions of univariate generators |
---|
980 | |
---|
981 | linSol = LinearZeroSetMain(I, wt); |
---|
982 | generators = linSol[3]; // they are a standardbasis |
---|
983 | linIndex = linSol[2]; |
---|
984 | linearSolution = linSol[1]; |
---|
985 | if(size(linIndex) + sVars == n) { // all variables solved |
---|
986 | solid = SubsMapIdeal(linearSolution, linIndex, 0); |
---|
987 | result[1] = list(solid); |
---|
988 | result[2] = var(n + 1); |
---|
989 | result[3] = mpoly; |
---|
990 | return(result); |
---|
991 | } |
---|
992 | |
---|
993 | // find roots of the nonlinear univariate polynomials of generators |
---|
994 | // if necessary, transform linear solutions w.r.t. newA |
---|
995 | |
---|
996 | oldMinPoly = mpoly[1]; |
---|
997 | nLinSol = NonLinearZeroSetMain(generators, wt); // find solutions of univariate generators |
---|
998 | nonlinSolutions = nLinSol[1]; // store solutions |
---|
999 | nlinIndex = nLinSol[4]; // and index of solved variables |
---|
1000 | generators = nLinSol[5]; // new generators |
---|
1001 | |
---|
1002 | // change minpoly if necessary and transform the ideal and the partial solutions |
---|
1003 | |
---|
1004 | if(oldMinPoly != nLinSol[3]) { |
---|
1005 | newA = nLinSol[2]; |
---|
1006 | if(!noMP && size(linearSolution) > 0) { // transform the algebraic number a |
---|
1007 | Fa = basering, maxideal(1); |
---|
1008 | Fa[n + 1] = newA; |
---|
1009 | linearSolution = SimplifyData(Fa(linearSolution)); // ... |
---|
1010 | } |
---|
1011 | } |
---|
1012 | |
---|
1013 | // check if all variables are solved. |
---|
1014 | |
---|
1015 | if(size(linIndex) + size(nlinIndex) == n - sVars) { |
---|
1016 | solutions = MergeSolutions(linearSolution, linIndex, nonlinSolutions, nlinIndex, list(), n); |
---|
1017 | } |
---|
1018 | |
---|
1019 | else { |
---|
1020 | |
---|
1021 | // some variables are not solved. |
---|
1022 | // substitute each partial solution in generators and find the |
---|
1023 | // zero set of the resulting ideal by recursive application |
---|
1024 | // of ZeroSetMainWork ! |
---|
1025 | |
---|
1026 | index = linIndex + nlinIndex; |
---|
1027 | nrSols = 0; |
---|
1028 | for(i = 1; i <= size(nonlinSolutions); i = i + 1) { |
---|
1029 | sol = linearSolution + nonlinSolutions[i]; |
---|
1030 | solid = SubsMapIdeal(sol, index, 1); |
---|
1031 | Fsubs = basering, solid; |
---|
1032 | gens = std(SimplifyData(Fsubs(generators))); // substitute partial solution |
---|
1033 | oldMinPoly = mpoly[1]; |
---|
1034 | partSol = ZeroSetMainWork(gens, wt, size(index) + sVars); |
---|
1035 | |
---|
1036 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
---|
1037 | Fa = basering, maxideal(1); |
---|
1038 | Fa[n + 1] = partSol[2]; // a -> p(a), representation of a w.r.t. new minpoly |
---|
1039 | newA = reduce(Fa(newA), mpoly); |
---|
1040 | generators = std(SimplifyData(Fa(generators))); |
---|
1041 | if(size(linearSolution) > 0) { linearSolution = SimplifyData(Fa(linearSolution));} |
---|
1042 | if(size(nonlinSolutions) > 0) { |
---|
1043 | nonlinSolutions = SimplifyZeroset(Fa(nonlinSolutions)); |
---|
1044 | } |
---|
1045 | sol = linearSolution + nonlinSolutions[i]; |
---|
1046 | } |
---|
1047 | |
---|
1048 | for(j = 1; j <= size(partSol[1]); j++) { // for all partial solutions |
---|
1049 | partsolid = partSol[1][j]; |
---|
1050 | for(k = 1; k <= size(index); k++) { |
---|
1051 | partsolid[index[k]] = sol[k]; |
---|
1052 | } |
---|
1053 | nrSols++; |
---|
1054 | solutions[nrSols] = partsolid; |
---|
1055 | } |
---|
1056 | } |
---|
1057 | |
---|
1058 | } // end else |
---|
1059 | return(list(solutions, newA, mpoly)); |
---|
1060 | } |
---|
1061 | |
---|
1062 | /////////////////////////////////////////////////////////////////////////////// |
---|
1063 | |
---|
1064 | proc LinearZeroSetMain(ideal I, intvec wt) |
---|
1065 | "USAGE: LinearZeroSetMain(I, wt) |
---|
1066 | PURPOSE: solve the univariate linear polys in I |
---|
1067 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
1068 | RETURN: list |
---|
1069 | _[1] = partial solution of I |
---|
1070 | _[2] = index of solved vars |
---|
1071 | _[3] = new generators (standardbasis) |
---|
1072 | " |
---|
1073 | { |
---|
1074 | def altring=basering; |
---|
1075 | int i, ok, n, found, nrSols; |
---|
1076 | ideal generators, newGens; |
---|
1077 | list result, index, totalIndex, vars, sol, temp; |
---|
1078 | map F; |
---|
1079 | poly f; |
---|
1080 | |
---|
1081 | result[1] = index; // sol[1] should be the empty list |
---|
1082 | n = nvars(basering) - 1; |
---|
1083 | generators = I; // might be wrong, use index ! |
---|
1084 | ok = 1; |
---|
1085 | nrSols = 0; |
---|
1086 | while(ok) { |
---|
1087 | found = 0; |
---|
1088 | for(i = 1; i <= size(generators); i = i + 1) { |
---|
1089 | f = generators[i]; |
---|
1090 | vars = Variables(f, n); |
---|
1091 | if(size(vars) == 1 && deg(f, wt) == 1) { // univariate,linear |
---|
1092 | nrSols++; found++; |
---|
1093 | index[nrSols] = vars[1]; |
---|
1094 | sol[nrSols] = var(vars[1]) - MultPolys(InvertNumberMain(LeadTerm(f, vars[1])[3]), f); |
---|
1095 | } |
---|
1096 | } |
---|
1097 | if(found > 0) { |
---|
1098 | F = basering, SubsMapIdeal(sol, index, 1); |
---|
1099 | newGens = std(SimplifyData(F(generators))); // substitute, simplify alg. number |
---|
1100 | if(size(newGens) == 0) {ok = 0;} |
---|
1101 | generators = newGens; |
---|
1102 | } |
---|
1103 | else { |
---|
1104 | ok = 0; |
---|
1105 | } |
---|
1106 | } |
---|
1107 | if(nrSols > 0) { result[1] = sol;} |
---|
1108 | result[2] = index; |
---|
1109 | result[3] = generators; |
---|
1110 | return(result); |
---|
1111 | } |
---|
1112 | |
---|
1113 | /////////////////////////////////////////////////////////////////////////////// |
---|
1114 | |
---|
1115 | proc NonLinearZeroSetMain(ideal I, intvec wt) |
---|
1116 | "USAGE: ZeroSetMainWork(I, wt, sVars); |
---|
1117 | PURPOSE: solves the (nonlinear) univariate polynomials in I |
---|
1118 | of the ground field (without multiplicities). |
---|
1119 | RETURN: list, all entries are polynomials |
---|
1120 | _[1] = list of solutions |
---|
1121 | _[2] = newA |
---|
1122 | _[3] = minpoly |
---|
1123 | _[4] - index of solved variables |
---|
1124 | _[5] = new representation of I |
---|
1125 | ASSUME: basering = Q[x_1,x_2,...,x_n,a], ideal 'mpoly' must be defined, |
---|
1126 | it might be 0 ! |
---|
1127 | NOTE: might change 'mpoly' !! |
---|
1128 | " |
---|
1129 | { |
---|
1130 | int i, nrSols, ok, n; |
---|
1131 | ideal generators; |
---|
1132 | list result, sols, index, vars, partSol; |
---|
1133 | map F; |
---|
1134 | poly f, newA; |
---|
1135 | string ringSTR; |
---|
1136 | |
---|
1137 | def NLZR = basering; |
---|
1138 | export(NLZR); |
---|
1139 | |
---|
1140 | n = nvars(basering) - 1; |
---|
1141 | |
---|
1142 | generators = I; |
---|
1143 | newA = var(n + 1); |
---|
1144 | result[2] = newA; // default |
---|
1145 | nrSols = 0; |
---|
1146 | ok = 1; |
---|
1147 | i = 1; |
---|
1148 | while(ok) { |
---|
1149 | |
---|
1150 | // test if the i-th generator of I is univariate |
---|
1151 | |
---|
1152 | f = generators[i]; |
---|
1153 | vars = Variables(f, n); |
---|
1154 | if(size(vars) == 1) { |
---|
1155 | generators[i] = 0; |
---|
1156 | generators = simplify(generators, 2); // remove 0 |
---|
1157 | nrSols++; |
---|
1158 | index[nrSols] = vars[1]; // store index of solved variable |
---|
1159 | |
---|
1160 | // create univariate ring |
---|
1161 | |
---|
1162 | ringSTR = "ring RIS1 = 0, (" + string(var(vars[1])) + ", " + string(var(n+1)) + "), lp;"; |
---|
1163 | execute(ringSTR); |
---|
1164 | ideal mpoly = std(imap(NLZR, mpoly)); |
---|
1165 | list roots; |
---|
1166 | poly f = imap(NLZR, f); |
---|
1167 | export(RIS1); |
---|
1168 | export(mpoly); |
---|
1169 | roots = RootsMain(f); |
---|
1170 | |
---|
1171 | // get "old" basering with new minpoly |
---|
1172 | |
---|
1173 | setring(NLZR); |
---|
1174 | partSol = imap(RIS1, roots); |
---|
1175 | kill(RIS1); |
---|
1176 | if(mpoly[1] != partSol[3]) { // change minpoly |
---|
1177 | mpoly = std(partSol[3]); |
---|
1178 | F = NLZR, maxideal(1); |
---|
1179 | F[n + 1] = partSol[2]; |
---|
1180 | if(size(sols) > 0) {sols = SimplifyZeroset(F(sols)); } |
---|
1181 | newA = reduce(F(newA), mpoly); // normal form |
---|
1182 | result[2] = newA; |
---|
1183 | generators = SimplifyData(F(generators)); // does not remove 0's |
---|
1184 | } |
---|
1185 | sols = ExtendSolutions(sols, partSol[1]); |
---|
1186 | } // end univariate |
---|
1187 | else { |
---|
1188 | i = i + 1; |
---|
1189 | } |
---|
1190 | if(i > size(generators)) { ok = 0;} |
---|
1191 | } |
---|
1192 | result[1] = sols; |
---|
1193 | result[3] = mpoly; |
---|
1194 | result[4] = index; |
---|
1195 | result[5] = std(generators); |
---|
1196 | |
---|
1197 | kill(NLZR); |
---|
1198 | return(result); |
---|
1199 | } |
---|
1200 | |
---|
1201 | /////////////////////////////////////////////////////////////////////////////// |
---|
1202 | |
---|
1203 | static proc ExtendSolutions(list solutions, list newSolutions) |
---|
1204 | "USAGE: ExtendSolutions(sols, newSols); list sols, newSols; |
---|
1205 | PURPOSE: extend the entries of 'sols' by the entries of 'newSols', |
---|
1206 | each entry of 'newSols' is a number. |
---|
1207 | RETURN: list |
---|
1208 | ASSUME: basering = Q[x_1,...,x_n,a], ideal 'mpoly' must be defined, |
---|
1209 | it might be 0 ! |
---|
1210 | NOTE: used by 'NonLinearZeroSetMain' |
---|
1211 | " |
---|
1212 | { |
---|
1213 | int i, j, k, n, nrSols; |
---|
1214 | list newSols, temp; |
---|
1215 | |
---|
1216 | nrSols = size(solutions); |
---|
1217 | if(nrSols > 0) {n = size(solutions[1]);} |
---|
1218 | else { |
---|
1219 | n = 0; |
---|
1220 | nrSols = 1; |
---|
1221 | } |
---|
1222 | k = 0; |
---|
1223 | for(i = 1; i <= nrSols; i++) { |
---|
1224 | for(j = 1; j <= size(newSolutions); j++) { |
---|
1225 | k++; |
---|
1226 | if(n == 0) { temp[1] = newSolutions[j];} |
---|
1227 | else { |
---|
1228 | temp = solutions[i]; |
---|
1229 | temp[n + 1] = newSolutions[j]; |
---|
1230 | } |
---|
1231 | newSols[k] = temp; |
---|
1232 | } |
---|
1233 | } |
---|
1234 | return(newSols); |
---|
1235 | } |
---|
1236 | |
---|
1237 | /////////////////////////////////////////////////////////////////////////////// |
---|
1238 | |
---|
1239 | static proc MergeSolutions(list sol1, list index1, list sol2, list index2) |
---|
1240 | "USAGE: MergeSolutions(sol1, index1, sol2, index2); all parameters are lists |
---|
1241 | RETURN: list |
---|
1242 | PURPOSE: create a list of solutions of size n, each entry of 'sol2' must |
---|
1243 | have size n. 'sol1' is one partial solution (from 'LinearZeroSetMain') |
---|
1244 | 'sol2' is a list of partial solutions (from 'NonLinearZeroSetMain') |
---|
1245 | ASSUME: 'sol2' is not empty |
---|
1246 | NOTE: used by 'ZeroSetMainWork' |
---|
1247 | { |
---|
1248 | int i, j, k, m; |
---|
1249 | ideal sol; |
---|
1250 | list newSols; |
---|
1251 | |
---|
1252 | m = 0; |
---|
1253 | for(i = 1; i <= size(sol2); i++) { |
---|
1254 | m++; |
---|
1255 | newSols[m] = SubsMapIdeal(sol1 + sol2[i], index1 + index2, 0); |
---|
1256 | } |
---|
1257 | return(newSols); |
---|
1258 | } |
---|
1259 | |
---|
1260 | /////////////////////////////////////////////////////////////////////////////// |
---|
1261 | |
---|
1262 | static proc SubsMapIdeal(list sol, list index, int opt) |
---|
1263 | "USAGE: SubsMapIdeal(sol,index,opt); list sol, index; int opt; |
---|
1264 | PURPOSE: built an ideal I as follows. |
---|
1265 | if i is contained in 'index' then set I[i] = sol[i] |
---|
1266 | if i is not contained in 'index' then |
---|
1267 | - opt = 0: set I[i] = 0 |
---|
1268 | - opt = 1: set I[i] = var(i) |
---|
1269 | if opt = 1 and n = nvars(basering) then set I[n] = var(n). |
---|
1270 | RETURN: ideal |
---|
1271 | ASSUME: size(sol) = size(index) <= nvars(basering) |
---|
1272 | " |
---|
1273 | { |
---|
1274 | int k = 0; |
---|
1275 | ideal I; |
---|
1276 | for(int i = 1; i <= nvars(basering) - 1; i = i + 1) { // built subs. map |
---|
1277 | if(ContainedQ(index, i)) { |
---|
1278 | k++; |
---|
1279 | I[index[k]] = sol[k]; |
---|
1280 | } |
---|
1281 | else { |
---|
1282 | if(opt) { I[i] = var(i); } |
---|
1283 | else { I[i] = 0; } |
---|
1284 | } |
---|
1285 | } |
---|
1286 | if(opt) {I[nvars(basering)] = var(nvars(basering));} |
---|
1287 | return(I); |
---|
1288 | } |
---|
1289 | |
---|
1290 | /////////////////////////////////////////////////////////////////////////////// |
---|
1291 | |
---|
1292 | proc SimplifyZeroset(data) |
---|
1293 | "USAGE: SimplifyZeroset(data); list data |
---|
1294 | PURPOSE: reduce the entries of the elements of 'data' w.r.t. the ideal 'mpoly' |
---|
1295 | 'data' is a list of ideals/lists. |
---|
1296 | RETURN: list |
---|
1297 | ASSUME: basering = Q[x_1,...,x_n,a], order = lp |
---|
1298 | 'data' is a list of ideals |
---|
1299 | ideal 'mpoly' must be defined, it might be 0 ! |
---|
1300 | " |
---|
1301 | { |
---|
1302 | int i; |
---|
1303 | list result; |
---|
1304 | |
---|
1305 | for(i = 1; i <= size(data); i++) { |
---|
1306 | result[i] = SimplifyData(data[i]); |
---|
1307 | } |
---|
1308 | return(result); |
---|
1309 | } |
---|
1310 | |
---|
1311 | /////////////////////////////////////////////////////////////////////////////// |
---|
1312 | |
---|
1313 | proc Variables(poly f, int n) |
---|
1314 | "USAGE: Variables(f,n); poly f; int n; |
---|
1315 | PURPOSE: list of variables among var(1),...,var(n) which occur in f. |
---|
1316 | RETURN: list |
---|
1317 | ASSUME: n <= nvars(basering) |
---|
1318 | " |
---|
1319 | { |
---|
1320 | int i, nrV; |
---|
1321 | list index; |
---|
1322 | |
---|
1323 | nrV = 0; |
---|
1324 | for(i = 1; i <= n; i = i + 1) { |
---|
1325 | if(diff(f, var(i)) != 0) { nrV++; index[nrV] = i; } |
---|
1326 | } |
---|
1327 | return(index); |
---|
1328 | } |
---|
1329 | |
---|
1330 | /////////////////////////////////////////////////////////////////////////////// |
---|
1331 | |
---|
1332 | proc ContainedQ(data, f, list #) |
---|
1333 | "USAGE: ContainedQ(data, f [, opt]); data=list; f=any type, opt=integer |
---|
1334 | PURPOSE: test if f is an element of data. |
---|
1335 | RETURN: int |
---|
1336 | 0 if f not contained in data |
---|
1337 | 1 if f contained in data |
---|
1338 | OPTIONS: opt = 0 : use '==' for comparing f with elements from data@* |
---|
1339 | opt = 1 : use @code{SameQ} for comparing f with elements from data |
---|
1340 | " |
---|
1341 | { |
---|
1342 | int opt, i, found; |
---|
1343 | if(size(#) > 0) { opt = #[1];} |
---|
1344 | else { opt = 0; } |
---|
1345 | i = 1; |
---|
1346 | found = 0; |
---|
1347 | |
---|
1348 | while((!found) && (i <= size(data))) { |
---|
1349 | if(opt == 0) { |
---|
1350 | if(f == data[i]) { found = 1;} |
---|
1351 | else {i = i + 1;} |
---|
1352 | } |
---|
1353 | else { |
---|
1354 | if(SameQ(f, data[i])) { found = 1;} |
---|
1355 | else {i = i + 1;} |
---|
1356 | } |
---|
1357 | } |
---|
1358 | return(found); |
---|
1359 | } |
---|
1360 | |
---|
1361 | ////////////////////////////////////////////////////////////////////////////// |
---|
1362 | |
---|
1363 | proc SameQ(a, b) |
---|
1364 | "USAGE: SameQ(a, b); a,b=list/intvec |
---|
1365 | PURPOSE: test a == b elementwise, i.e., a[i] = b[i]. |
---|
1366 | RETURN: int |
---|
1367 | 0 if a != b |
---|
1368 | 1 if a == b |
---|
1369 | " |
---|
1370 | { |
---|
1371 | if(typeof(a) == typeof(b)) { |
---|
1372 | if(typeof(a) == "list" || typeof(a) == "intvec") { |
---|
1373 | if(size(a) == size(b)) { |
---|
1374 | int i = 1; |
---|
1375 | int ok = 1; |
---|
1376 | while(ok && (i <= size(a))) { |
---|
1377 | if(a[i] == b[i]) { i = i + 1;} |
---|
1378 | else {ok = 0;} |
---|
1379 | } |
---|
1380 | return(ok); |
---|
1381 | } |
---|
1382 | else { return(0); } |
---|
1383 | } |
---|
1384 | else { return(a == b);} |
---|
1385 | } |
---|
1386 | else { return(0);} |
---|
1387 | } |
---|
1388 | |
---|
1389 | /////////////////////////////////////////////////////////////////////////////// |
---|
1390 | |
---|
1391 | static proc SimplifyPoly(poly f) |
---|
1392 | "USAGE: SimplifyPoly(f); poly f |
---|
1393 | PURPOSE: reduces the coefficients of f w.r.t. the ideal 'moly' if they contain |
---|
1394 | the algebraic number 'a'. |
---|
1395 | RETURN: poly |
---|
1396 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
1397 | ideal mpoly must be defined, it might be 0 ! |
---|
1398 | " |
---|
1399 | { |
---|
1400 | matrix coMx; |
---|
1401 | poly f1, vp; |
---|
1402 | |
---|
1403 | vp = 1; |
---|
1404 | for(int i = 1; i < nvars(basering); i++) { vp = vp * var(i);} |
---|
1405 | |
---|
1406 | coMx = coef(f, vp); |
---|
1407 | f1 = 0; |
---|
1408 | for(i = 1; i <= ncols(coMx); i++) { |
---|
1409 | f1 = f1 + coMx[1, i] * reduce(coMx[2, i], mpoly); |
---|
1410 | } |
---|
1411 | return(f1); |
---|
1412 | } |
---|
1413 | |
---|
1414 | /////////////////////////////////////////////////////////////////////////////// |
---|
1415 | |
---|
1416 | static proc SimplifyData(data) |
---|
1417 | "USAGE: SimplifyData(data); ideal/list data; |
---|
1418 | PURPOSE: reduces the entries of 'data' w.r.t. the ideal 'mpoly' if they contain |
---|
1419 | the algebraic number 'a' |
---|
1420 | RETURN: ideal/list |
---|
1421 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
1422 | ideal 'mpoly' must be defined, it might be 0 ! |
---|
1423 | " |
---|
1424 | { |
---|
1425 | def altring=basering; |
---|
1426 | int n; |
---|
1427 | poly f; |
---|
1428 | |
---|
1429 | if(typeof(data) == "ideal") { n = ncols(data); } |
---|
1430 | else { n = size(data);} |
---|
1431 | |
---|
1432 | for(int i = 1; i <= n; i++) { |
---|
1433 | f = data[i]; |
---|
1434 | data[i] = SimplifyPoly(f); |
---|
1435 | } |
---|
1436 | return(data); |
---|
1437 | } |
---|
1438 | |
---|
1439 | /////////////////////////////////////////////////////////////////////////////// |
---|
1440 | |
---|
1441 | static proc TransferRing(R) |
---|
1442 | "USAGE: TransferRing(R); |
---|
1443 | PURPOSE: creates a new ring containing the same variables as R, but without |
---|
1444 | parameters. If R contains a parameter then this parameter is added |
---|
1445 | as the last variable and 'minpoly' is represented by the ideal 'mpoly' |
---|
1446 | If the basering does not contain a parameter then 'a' is added and |
---|
1447 | 'mpoly' = 0. |
---|
1448 | RETURN: ring |
---|
1449 | ASSUME: R = K[x_1,...,x_n] where K = Q or K = Q(a). |
---|
1450 | NOTE: Creates the ring needed for all prodecures with name 'proc-name'Main |
---|
1451 | " |
---|
1452 | { |
---|
1453 | def altring=basering; |
---|
1454 | string ringSTR, parName, minPoly; |
---|
1455 | |
---|
1456 | setring(R); |
---|
1457 | |
---|
1458 | if(npars(basering) == 0) { |
---|
1459 | parName = "a"; |
---|
1460 | minPoly = "0"; |
---|
1461 | } |
---|
1462 | else { |
---|
1463 | parName = parstr(basering); |
---|
1464 | minPoly = string(minpoly); |
---|
1465 | } |
---|
1466 | ringSTR = "ring TR = 0, (" + varstr(basering) + "," + parName + "), lp;"; |
---|
1467 | |
---|
1468 | execute(ringSTR); |
---|
1469 | execute("ideal mpoly = std(" + minPoly + ");"); |
---|
1470 | export(mpoly); |
---|
1471 | setring altring; |
---|
1472 | return(TR); |
---|
1473 | } |
---|
1474 | |
---|
1475 | /////////////////////////////////////////////////////////////////////////////// |
---|
1476 | |
---|
1477 | static proc NewBaseRing() |
---|
1478 | "USAGE: NewBaseRing(); |
---|
1479 | PURPOSE: creates a new ring, the last variable is added as a parameter. |
---|
1480 | minpoly is set to mpoly[1]. |
---|
1481 | RETURN: ring |
---|
1482 | ASSUME: basering = Q[x_1,...,x_n, a], 'mpoly' must be defined |
---|
1483 | " |
---|
1484 | { |
---|
1485 | int n = nvars(basering); |
---|
1486 | int MP; |
---|
1487 | string ringSTR, parName, varString; |
---|
1488 | |
---|
1489 | def BR = basering; |
---|
1490 | if(mpoly[1] != 0) { |
---|
1491 | parName = "(0, " + string(var(n)) + ")"; |
---|
1492 | MP = 1; |
---|
1493 | } |
---|
1494 | else { |
---|
1495 | parName = "0"; |
---|
1496 | MP = 0; |
---|
1497 | } |
---|
1498 | |
---|
1499 | |
---|
1500 | for(int i = 1; i < n - 1; i++) { |
---|
1501 | varString = varString + string(var(i)) + ","; |
---|
1502 | } |
---|
1503 | varString = varString + string(var(n-1)); |
---|
1504 | |
---|
1505 | ringSTR = "ring TR = " + parName + ", (" + varString + "), lp;"; |
---|
1506 | execute(ringSTR); |
---|
1507 | if(MP) { minpoly = number(imap(BR, mpoly)[1]); } |
---|
1508 | setring BR; |
---|
1509 | return(TR); |
---|
1510 | } |
---|
1511 | |
---|
1512 | /////////////////////////////////////////////////////////////////////////////// |
---|
1513 | |
---|
1514 | /* |
---|
1515 | Examples: |
---|
1516 | |
---|
1517 | |
---|
1518 | // order = 20; |
---|
1519 | ring S1 = 0, (s(1..3)), lp; |
---|
1520 | ideal I = s(2)*s(3), s(1)^2*s(2)+s(1)^2*s(3)-1, s(1)^2*s(3)^2-s(3), s(2)^4-s(3)^4+s(1)^2, s(1)^4+s(2)^3-s(3)^3, s(3)^5-s(1)^2*s(3); |
---|
1521 | ideal mpoly = std(0); |
---|
1522 | |
---|
1523 | // order = 10 |
---|
1524 | ring S2 = 0, (s(1..5)), lp; |
---|
1525 | ideal I = s(2)+s(3)-s(5), s(4)^2-s(5), s(1)*s(5)+s(3)*s(4)-s(4)*s(5), s(1)*s(4)+s(3)-s(5), s(3)^2-2*s(3)*s(5), s(1)*s(3)-s(1)*s(5)+s(4)*s(5), s(1)^2+s(4)^2-2*s(5), -s(1)+s(5)^3, s(3)*s(5)^2+s(4)-s(5)^3, s(1)*s(5)^2-1; |
---|
1526 | ideal mpoly = std(0); |
---|
1527 | |
---|
1528 | //order = 126 |
---|
1529 | ring S3 = 0, (s(1..5)), lp; |
---|
1530 | ideal I = s(3)*s(4), s(2)*s(4), s(1)*s(3), s(1)*s(2), s(3)^3+s(4)^3-1, s(2)^3+s(4)^3-1, s(1)^3-s(4)^3, s(4)^4-s(4), s(1)*s(4)^3-s(1), s(5)^7-1; |
---|
1531 | ideal mpoly = std(0); |
---|
1532 | |
---|
1533 | // order = 192 |
---|
1534 | ring S4 = 0, (s(1..4)), lp; |
---|
1535 | ideal I = s(2)*s(3)^2*s(4)+s(1)*s(3)*s(4)^2, s(2)^2*s(3)*s(4)+s(1)*s(2)*s(4)^2, s(1)*s(3)^3+s(2)*s(4)^3, s(1)*s(2)*s(3)^2+s(1)^2*s(3)*s(4), s(1)^2*s(3)^2-s(2)^2*s(4)^2, s(1)*s(2)^2*s(3)+s(1)^2*s(2)*s(4), s(1)^3*s(3)+s(2)^3*s(4), s(2)^4-s(3)^4, s(1)*s(2)^3+s(3)*s(4)^3, s(1)^2*s(2)^2-s(3)^2*s(4)^2, s(1)^3*s(2)+s(3)^3*s(4), s(1)^4-s(4)^4, s(3)^5*s(4)-s(3)*s(4)^5, s(3)^8+14*s(3)^4*s(4)^4+s(4)^8-1, 15*s(2)*s(3)*s(4)^7-s(1)*s(4)^8+s(1), 15*s(3)^4*s(4)^5+s(4)^9-s(4), 16*s(3)*s(4)^9-s(3)*s(4), 16*s(2)*s(4)^9-s(2)*s(4), 16*s(1)*s(3)*s(4)^8-s(1)*s(3), 16*s(1)*s(2)*s(4)^8-s(1)*s(2), 16*s(1)*s(4)^10-15*s(2)*s(3)*s(4)-16*s(1)*s(4)^2, 16*s(1)^2*s(4)^9-15*s(1)*s(2)*s(3)-16*s(1)^2*s(4), 16*s(4)^13+15*s(3)^4*s(4)-16*s(4)^5; |
---|
1536 | ideal mpoly = std(0); |
---|
1537 | |
---|
1538 | ring R = (0,a), (x,y,z), lp; |
---|
1539 | minpoly = a2 + 1; |
---|
1540 | ideal I1 = x2 - 1/2, a*z - 1, y - 2; |
---|
1541 | ideal I2 = x3 - 1/2, a*z2 - 3, y - 2*a; |
---|
1542 | |
---|
1543 | */ |
---|