1 | // Last change 12.02.2001 (Eric Westenberger) |
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2 | /////////////////////////////////////////////////////////////////////////////// |
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3 | version="$Id: zeroset.lib,v 1.18 2008-10-09 09:31:58 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 Address: 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 scope 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 | Quotient(f, g) quotient q of f w.r.t. g (in f = q*g + remainder) |
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26 | Remainder(f,g) remainder of the division of f by g |
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27 | Roots(f) computes all roots of f in an extension field of Q |
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28 | SQFRNorm(f) norm of f (f must be squarefree) |
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29 | ZeroSet(I) zero-set of the 0-dim. ideal I |
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30 | |
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31 | AUXILIARY PROCEDURES: |
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32 | EGCDMain(f, g) gcd over an algebraic extension field of Q |
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33 | FactorMain(f) factorization of f over an algebraic extension field |
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34 | InvertNumberMain(c) inverts an element of an algebraic extension field |
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35 | QuotientMain(f, g) quotient of f w.r.t. g |
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36 | RemainderMain(f,g) remainder of the division of f by g |
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37 | RootsMain(f) computes all roots of f, might extend the ground field |
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38 | SQFRNormMain(f) norm of f (f must be squarefree) |
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39 | ContainedQ(data, f) f in data ? |
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40 | SameQ(a, b) a == b (list a,b) |
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41 | "; |
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42 | |
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43 | LIB "primitiv.lib"; |
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44 | LIB "primdec.lib"; |
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45 | |
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46 | // note : return a ring : ring need not be exported !!! |
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47 | |
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48 | // Artihmetic in Q(a)[x] without built-in procedures |
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49 | // assume basering = Q[x,a] and minpoly is represented by mpoly(a). |
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50 | // the algorithms are taken from "Polynomial Algorithms in Computer Algebra", |
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51 | // F. Winkler, Springer Verlag Wien, 1996. |
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52 | |
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53 | |
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54 | // To do : |
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55 | // squarefree factorization |
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56 | // multiplicities |
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57 | |
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58 | // Improvement : |
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59 | // a main problem is the growth of the coefficients. Try Roots(x7 - 1) |
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60 | // return ideal mpoly ! |
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61 | // mpoly is not monic, comes from primitive_extra |
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62 | |
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63 | // IMPLEMENTATION |
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64 | // |
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65 | // In procedures with name 'proc-name'Main a polynomial ring over a simple |
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66 | // extension field is represented as Q[x...,a] together with the ideal |
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67 | // 'mpoly' (attribute "isSB"). The arithmetic in the extension field is |
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68 | // implemented in the procedures in the procedures 'MultPolys' (multiplication) |
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69 | // and 'InvertNumber' (inversion). After addition and substraction one should |
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70 | // apply 'SimplifyPoly' to the result to reduce the result w.r.t. 'mpoly'. |
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71 | // This is done by reducing each coefficient seperately, which is more |
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72 | // efficient for polynomials with many terms. |
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73 | |
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74 | |
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75 | /////////////////////////////////////////////////////////////////////////////// |
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76 | |
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77 | proc Roots(poly f) |
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78 | "USAGE: Roots(f); where f is a polynomial |
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79 | PURPOSE: compute all roots of f in a finite extension of the ground field |
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80 | without multiplicities. |
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81 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
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82 | containing a list 'roots' and polynomials 'newA' and 'f': |
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83 | @format |
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84 | - 'roots' is the list of roots of the polynomial f (no multiplicities) |
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85 | - if the ground field is Q(a') and the extension field is Q(a), then |
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86 | 'newA' is the representation of a' in Q(a). |
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87 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
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88 | then 'newA' = 'a'. |
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89 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
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90 | - 'f' is the polynomial f in Q(a) (a' being substituted by 'newA') |
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91 | @end format |
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92 | ASSUME: ground field to be Q or a simple extension of Q given by a minpoly |
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93 | EXAMPLE: example Roots; shows an example |
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94 | " |
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95 | { |
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96 | int dbPrt = printlevel-voice+3; |
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97 | |
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98 | // create a new ring where par(1) is replaced by the variable |
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99 | // with the same name or, if basering does not contain a parameter, |
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100 | // with a new variable 'a'. |
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101 | |
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102 | def ROB = basering; |
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103 | def ROR = TransferRing(basering); |
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104 | setring ROR; |
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105 | export(ROR); |
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106 | |
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107 | // get the polynomial f and find the roots |
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108 | |
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109 | poly f = imap(ROB, f); |
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110 | list result = RootsMain(f); // find roots of f |
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111 | |
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112 | // store the roots and the new representation of 'a' and transform |
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113 | // the coefficients of f. |
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114 | |
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115 | list roots = result[1]; |
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116 | poly newA = result[2]; |
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117 | map F = basering, maxideal(1); |
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118 | F[nvars(basering)] = newA; |
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119 | poly fn = SimplifyPoly(F(f)); |
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120 | |
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121 | // create a new ring with minploy = mpoly[1] (from ROR) |
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122 | |
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123 | def RON = NewBaseRing(); |
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124 | setring(RON); |
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125 | list roots = imap(ROR, roots); |
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126 | poly newA = imap(ROR, newA); |
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127 | poly f = imap(ROR, fn); |
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128 | kill ROR; |
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129 | export(roots); |
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130 | export(newA); |
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131 | export(f); dbprint(dbPrt," |
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132 | // 'Roots' created a new ring which contains the list 'roots' and |
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133 | // the polynomials 'f' and 'newA' |
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134 | // To access the roots, newA and the new representation of f, type |
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135 | def R = Roots(f); setring R; roots; newA; f; |
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136 | "); |
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137 | return(RON); |
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138 | } |
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139 | example |
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140 | {"EXAMPLE:"; echo = 2; |
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141 | ring R = (0,a), x, lp; |
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142 | minpoly = a2+1; |
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143 | poly f = x3 - a; |
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144 | def R1 = Roots(f); |
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145 | setring R1; |
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146 | minpoly; |
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147 | newA; |
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148 | f; |
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149 | roots; |
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150 | map F; |
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151 | F[1] = roots[1]; |
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152 | F(f); |
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153 | } |
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154 | |
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155 | /////////////////////////////////////////////////////////////////////////////// |
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156 | |
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157 | proc RootsMain(poly f) |
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158 | "USAGE: RootsMain(f); where f is a polynomial |
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159 | PURPOSE: compute all roots of f in a finite extension of the ground field |
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160 | without multiplicities. |
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161 | RETURN: list, all entries are polynomials |
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162 | @format |
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163 | _[1] = roots of f, each entry is a polynomial |
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164 | _[2] = 'newA' - if the ground field is Q(a') and the extension field |
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165 | is Q(a), then 'newA' is the representation of a' in Q(a) |
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166 | _[3] = minpoly of the algebraic extension of the ground field |
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167 | @end format |
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168 | ASSUME: basering = Q[x,a] ideal mpoly must be defined, it might be 0! |
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169 | NOTE: might change the ideal mpoly!! |
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170 | EXAMPLE: example Roots; shows an example |
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171 | " |
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172 | { |
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173 | int i, linFactors, nlinFactors, dbPrt; |
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174 | intvec wt = 1,0; // deg(a) = 0 |
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175 | list factorList, nlFactors, nlMult, roots, result; |
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176 | poly fa, lc; |
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177 | |
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178 | dbPrt = printlevel-voice+3; |
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179 | |
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180 | // factor f in Q(a)[t] to obtain the roots lying in Q(a) |
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181 | // firstly, find roots of the linear factors, |
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182 | // nonlinear factors are processed later |
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183 | |
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184 | dbprint(dbPrt, "Roots of " + string(f) + ", minimal polynomial = " + string(mpoly[1])); |
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185 | factorList = FactorMain(f); // Factorize f |
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186 | dbprint(dbPrt, (" prime factors of f are : " + string(factorList[1]))); |
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187 | |
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188 | linFactors = 0; |
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189 | nlinFactors = 0; |
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190 | for(i = 2; i <= size(factorList[1]); i = i + 1) { // find linear and nonlinear factors |
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191 | fa = factorList[1][i]; |
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192 | if(deg(fa, wt) == 1) { |
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193 | linFactors++; // get the root from the linear factor |
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194 | lc = LeadTerm(fa, 1)[3]; |
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195 | fa = MultPolys(InvertNumberMain(lc), fa); // make factor monic |
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196 | roots[linFactors] = var(1) - fa; // fa is monic !! |
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197 | } |
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198 | else { // ignore nonlinear factors |
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199 | nlinFactors++; |
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200 | nlFactors[nlinFactors] = factorList[1][i]; |
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201 | nlMult[nlinFactors] = factorList[2][i]; |
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202 | } |
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203 | } |
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204 | if(linFactors == size(factorList[1]) - 1) { // all roots of f are contained in the ground field |
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205 | result[1] = roots; |
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206 | result[2] = var(2); |
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207 | result[3] = mpoly[1]; |
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208 | return(result); |
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209 | } |
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210 | |
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211 | // process the nonlinear factors, i.e., extend the ground field |
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212 | // where a nonlinear factor (irreducible) is a minimal polynomial |
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213 | // compute the primitive element of this extension |
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214 | |
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215 | ideal primElem, minPolys, Fid; |
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216 | list partSol; |
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217 | map F, Xchange; |
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218 | poly f1, newA, mp, oldMinPoly; |
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219 | |
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220 | Fid = mpoly; |
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221 | F[1] = var(1); |
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222 | Xchange[1] = var(2); // the variables have to be exchanged |
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223 | Xchange[2] = var(1); // for the use of 'primitive' |
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224 | |
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225 | if(nlinFactors == 1) { // one nl factor |
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226 | |
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227 | // compute the roots of the nonlinear (irreducible, monic) factor f1 of f |
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228 | // by extending the basefield by a' with minimal polynomial f1 |
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229 | // Then call Roots(f1) to find the roots of f1 over the new base field |
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230 | |
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231 | f1 = nlFactors[1]; |
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232 | if(mpoly[1] != 0) { |
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233 | mp = mpoly[1]; |
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234 | minPolys = Xchange(mp), Xchange(f1); |
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235 | primElem = primitive_extra(minPolys); // no random coord. change |
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236 | mpoly = std(primElem[1]); |
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237 | F = basering, maxideal(1); |
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238 | F[2] = primElem[2]; // transfer all to the new representation |
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239 | newA = primElem[2]; // new representation of a |
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240 | f1 = SimplifyPoly(F(f1)); //reduce(F(f1), mpoly); |
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241 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
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242 | } |
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243 | else { |
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244 | mpoly = std(Xchange(f1)); |
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245 | newA = var(2); |
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246 | } |
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247 | result[3] = mpoly[1]; |
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248 | oldMinPoly = mpoly[1]; |
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249 | partSol = RootsMain(f1); // find roots of f1 over extended field |
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250 | |
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251 | if(oldMinPoly != partSol[3]) { // minpoly has changed ? |
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252 | // all previously computed roots must be transformed |
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253 | // because the minpoly has changed |
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254 | result[3] = partSol[3]; // new minpoly |
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255 | F[2] = partSol[2]; // new representation of algebraic number |
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256 | if(size(roots) > 0) {roots = SimplifyData(F(roots)); } |
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257 | newA = SimplifyPoly(F(newA)); // F(newA); |
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258 | } |
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259 | roots = roots + partSol[1]; // add roots |
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260 | result[2] = newA; |
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261 | result[1] = roots; |
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262 | } |
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263 | else { // more than one nonlinear (irreducible) factor (f_1,...,f_r) |
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264 | // solve each of them by RootsMain(f_i), append their roots |
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265 | // change the minpoly and transform all previously computed |
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266 | // roots if necessary. |
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267 | // Note that the for-loop is more or less book-keeping |
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268 | |
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269 | newA = var(2); |
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270 | result[2] = newA; |
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271 | for(i = 1; i <= size(nlFactors); i = i + 1) { |
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272 | oldMinPoly = mpoly[1]; |
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273 | partSol = RootsMain(nlFactors[i]); // main work |
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274 | nlFactors[i] = 0; // delete factor |
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275 | result[3] = partSol[3]; // store minpoly |
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276 | |
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277 | // book-keeping starts here as in the case 1 nonlinear factor |
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278 | |
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279 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
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280 | F = basering, maxideal(1); |
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281 | F[2] = partSol[2]; // transfer all to the new representation |
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282 | newA = SimplifyPoly(F(newA)); // F(newA); new representation of a |
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283 | result[2] = newA; |
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284 | if(i < size(nlFactors)) { |
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285 | nlFactors = SimplifyData(F(nlFactors)); |
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286 | } // transform remaining factors |
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287 | if(size(roots) > 0) {roots = SimplifyData(F(roots));} |
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288 | } |
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289 | roots = roots + partSol[1]; // transform roots |
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290 | result[1] = roots; |
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291 | } // end more than one nl factor |
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292 | |
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293 | } |
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294 | return(result); |
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295 | } |
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296 | |
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297 | /////////////////////////////////////////////////////////////////////////////// |
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298 | |
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299 | proc ZeroSet(ideal I, list #) |
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300 | "USAGE: ZeroSet(I [,opt] ); I=ideal, opt=integer |
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301 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
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302 | of the ground field. |
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303 | RETURN: ring, a polynomial ring over an extension field of the ground field, |
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304 | containing a list 'zeroset', a polynomial 'newA', and an |
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305 | ideal 'id': |
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306 | @format |
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307 | - 'zeroset' is the list of the zeros of the ideal I, each zero is an ideal. |
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308 | - if the ground field is Q(a') and the extension field is Q(a), then |
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309 | 'newA' is the representation of a' in Q(a). |
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310 | If the basering contains a parameter 'a' and the minpoly remains unchanged |
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311 | then 'newA' = 'a'. |
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312 | If the basering does not contain a parameter then 'newA' = 'a' (default). |
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313 | - 'id' is the ideal I in Q(a)[x_1,...] (a' substituted by 'newA') |
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314 | @end format |
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315 | ASSUME: dim(I) = 0, and ground field to be Q or a simple extension of Q given |
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316 | by a minpoly. |
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317 | OPTIONS: opt = 0 no primary decomposition (default) |
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318 | opt > 0 primary decomposition |
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319 | NOTE: If I contains an algebraic number (parameter) then 'I' must be |
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320 | transformed w.r.t. 'newA' in the new ring. |
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321 | EXAMPLE: example ZeroSet; shows an example |
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322 | " |
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323 | { |
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324 | int primaryDecQ, dbPrt; |
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325 | list rp; |
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326 | |
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327 | dbPrt = printlevel-voice+2; |
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328 | |
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329 | if(size(#) > 0) { primaryDecQ = #[1]; } |
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330 | else { primaryDecQ = 0; } |
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331 | |
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332 | // create a new ring 'ZSR' with one additional variable instead of the |
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333 | // parameter |
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334 | // if the basering does not contain a parameter then 'a' is used as the |
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335 | // additional variable. |
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336 | |
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337 | def RZSB = basering; |
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338 | def ZSR = TransferRing(RZSB); |
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339 | setring ZSR; |
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340 | |
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341 | // get ideal I and find the zero-set |
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342 | |
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343 | ideal id = std(imap(RZSB, I)); |
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344 | // print(dim(id)); |
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345 | if(dim(id) > 1) { // new variable adjoined to ZSR |
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346 | ERROR(" ideal not zerodimensional "); |
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347 | } |
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348 | |
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349 | list result = ZeroSetMain(id, primaryDecQ); |
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350 | |
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351 | // store the zero-set, minimal polynomial and the new representative of 'a' |
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352 | |
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353 | list zeroset = result[1]; |
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354 | poly newA = result[2]; |
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355 | poly minPoly = result[3][1]; |
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356 | |
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357 | // transform the generators of the ideal I w.r.t. the new representation |
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358 | // of 'a' |
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359 | |
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360 | map F = basering, maxideal(1); |
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361 | F[nvars(basering)] = newA; |
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362 | id = SimplifyData(F(id)); |
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363 | |
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364 | // create a new ring with minpoly = minPoly |
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365 | |
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366 | def RZBN = NewBaseRing(); |
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367 | setring RZBN; |
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368 | |
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369 | list zeroset = imap(ZSR, zeroset); |
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370 | poly newA = imap(ZSR, newA); |
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371 | ideal id = imap(ZSR, id); |
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372 | kill ZSR; |
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373 | |
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374 | export(id); |
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375 | export(zeroset); |
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376 | export(newA); |
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377 | dbprint(dbPrt," |
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378 | // 'ZeroSet' created a new ring which contains the list 'zeroset', the ideal |
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379 | // 'id' and the polynomial 'newA'. 'id' is the ideal of the input transformed |
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380 | // w.r.t. 'newA'. |
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381 | // To access the zero-set, 'newA' and the new representation of the ideal, type |
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382 | def R = ZeroSet(I); setring R; zeroset; newA; id; |
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383 | "); |
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384 | setring RZSB; |
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385 | return(RZBN); |
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386 | } |
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387 | example |
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388 | {"EXAMPLE:"; echo = 2; |
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389 | ring R = (0,a), (x,y,z), lp; |
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390 | minpoly = a2 + 1; |
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391 | ideal I = x2 - 1/2, a*z - 1, y - 2; |
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392 | def T = ZeroSet(I); |
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393 | setring T; |
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394 | minpoly; |
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395 | newA; |
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396 | id; |
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397 | zeroset; |
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398 | map F1 = basering, zeroset[1]; |
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399 | map F2 = basering, zeroset[2]; |
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400 | F1(id); |
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401 | F2(id); |
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402 | } |
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403 | |
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404 | /////////////////////////////////////////////////////////////////////////////// |
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405 | |
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406 | proc InvertNumberMain(poly f) |
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407 | "USAGE: InvertNumberMain(f); where f is a polynomial |
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408 | PURPOSE: compute 1/f if f is a number in Q(a), i.e., f is represented by a |
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409 | polynomial in Q[a]. |
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410 | RETURN: poly 1/f |
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411 | ASSUME: basering = Q[x_1,...,x_n,a], ideal mpoly must be defined and != 0 ! |
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412 | NOTE: outdated, use / instead |
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413 | " |
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414 | { |
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415 | if(diff(f, var(1)) != 0) { ERROR("number must not contain variable !");} |
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416 | |
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417 | int n = nvars(basering); |
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418 | def RINB = basering; |
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419 | string ringSTR = "ring RINR = 0, " + string(var(n)) + ", dp;"; |
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420 | execute(ringSTR); // new ring = Q[a] |
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421 | |
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422 | list gcdList; |
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423 | poly f, g, inv; |
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424 | |
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425 | f = imap(RINB, f); |
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426 | g = imap(RINB, mpoly)[1]; |
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427 | |
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428 | if(diff(f, var(1)) != 0) { inv = extgcd(f, g)[2]; } // f contains var(1) |
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429 | else { inv = 1/f;} // f element in Q |
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430 | |
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431 | setring(RINB); |
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432 | return(imap(RINR, inv)); |
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433 | } |
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434 | |
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435 | /////////////////////////////////////////////////////////////////////////////// |
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436 | |
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437 | proc MultPolys(poly f, poly g) |
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438 | "USAGE: MultPolys(f, g); poly f,g |
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439 | PURPOSE: multiply the polynomials f and g and reduce them w.r.t. mpoly |
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440 | RETURN: poly f*g |
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441 | ASSUME: basering = Q[x,a], ideal mpoly must be defined, it might be 0 ! |
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442 | " |
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443 | { |
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444 | return(SimplifyPoly(f * g)); |
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445 | } |
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446 | |
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447 | /////////////////////////////////////////////////////////////////////////////// |
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448 | |
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449 | proc LeadTerm(poly f, int i) |
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450 | "USAGE: LeadTerm(f); poly f, int i |
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451 | PURPOSE: compute the leading coef and term of f w.r.t var(i), where the last |
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452 | ring variable is treated as a parameter. |
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453 | RETURN: list of polynomials |
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454 | _[1] = leading term |
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455 | _[2] = leading monomial |
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456 | _[3] = leading coefficient |
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457 | ASSUME: basering = Q[x_1,...,x_n,a] |
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458 | " |
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459 | { |
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460 | list result; |
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461 | matrix co = coef(f, var(i)); |
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462 | result[1] = co[1, 1]*co[2, 1]; |
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463 | result[2] = co[1, 1]; |
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464 | result[3] = co[2, 1]; |
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465 | return(result); |
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466 | } |
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467 | |
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468 | /////////////////////////////////////////////////////////////////////////////// |
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469 | |
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470 | proc Quotient(poly f, poly g) |
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471 | "USAGE: Quotient(f, g); where f,g are polynomials; |
---|
472 | PURPOSE: compute the quotient q and remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
473 | RETURN: list of polynomials |
---|
474 | @format |
---|
475 | _[1] = quotient q |
---|
476 | _[2] = remainder r |
---|
477 | @end format |
---|
478 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
479 | NOTE: outdated, use div/mod instead |
---|
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.th. 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 | NOTE: outdated, use div/mod instead |
---|
518 | EXAMPLE: example Quotient; shows an example |
---|
519 | " |
---|
520 | { |
---|
521 | if(g == 0) { ERROR("Division by zero !");} |
---|
522 | |
---|
523 | def QMB = basering; |
---|
524 | def QMR = NewBaseRing(); |
---|
525 | setring QMR; |
---|
526 | poly f, g, h; |
---|
527 | h = imap(QMB, f) / imap(QMB, g); |
---|
528 | setring QMB; |
---|
529 | return(list(imap(QMR, h), 0)); |
---|
530 | } |
---|
531 | |
---|
532 | /////////////////////////////////////////////////////////////////////////////// |
---|
533 | |
---|
534 | proc Remainder(poly f, poly g) |
---|
535 | "USAGE: Remainder(f, g); where f,g are polynomials |
---|
536 | PURPOSE: compute the remainder of the division of f by g, i.e. a polynomial r |
---|
537 | s.t. f = g*q + r, deg(r) < deg(g). |
---|
538 | RETURN: poly |
---|
539 | ASSUME: basering = Q[x] or Q(a)[x] |
---|
540 | NOTE: outdated, use mod/reduce instead |
---|
541 | " |
---|
542 | { |
---|
543 | def REMB = basering; |
---|
544 | def REMR = TransferRing(basering); // new ring with parameter 'a' replaced by a variable |
---|
545 | setring(REMR); |
---|
546 | export(REMR); |
---|
547 | poly f = imap(REMB, f); |
---|
548 | poly g = imap(REMB, g); |
---|
549 | poly h = RemainderMain(f, g); |
---|
550 | |
---|
551 | setring(REMB); |
---|
552 | poly r = imap(REMR, h); |
---|
553 | kill REMR; |
---|
554 | return(r); |
---|
555 | } |
---|
556 | example |
---|
557 | {"EXAMPLE:"; echo = 2; |
---|
558 | ring R = (0,a), x, lp; |
---|
559 | minpoly = a2+1; |
---|
560 | poly f = x4 - 1; |
---|
561 | poly g = x3 - 1; |
---|
562 | Remainder(f, g); |
---|
563 | } |
---|
564 | |
---|
565 | proc RemainderMain(poly f, poly g) |
---|
566 | "USAGE: RemainderMain(f, g); where f,g are polynomials |
---|
567 | PURPOSE: compute the remainder r s.t. f = g*q + r, deg(r) < deg(g) |
---|
568 | RETURN: poly |
---|
569 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
570 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
571 | NOTE: outdated, use mod/reduce instead |
---|
572 | " |
---|
573 | { |
---|
574 | int dg; |
---|
575 | intvec wt = 1,0;; |
---|
576 | poly lc, g1, r; |
---|
577 | |
---|
578 | if(deg(g, wt) == 0) { return(0); } |
---|
579 | |
---|
580 | lc = LeadTerm(g, 1)[3]; |
---|
581 | g1 = MultPolys(InvertNumberMain(lc), g); // make g monic |
---|
582 | |
---|
583 | return(SimplifyPoly(reduce(f, std(g1)))); |
---|
584 | } |
---|
585 | |
---|
586 | /////////////////////////////////////////////////////////////////////////////// |
---|
587 | |
---|
588 | proc EGCDMain(poly f, poly g) |
---|
589 | "USAGE: EGCDMain(f, g); where f,g are polynomials |
---|
590 | PURPOSE: compute the polynomial gcd of f and g over Q(a)[x] |
---|
591 | RETURN: poly |
---|
592 | ASSUME: basering = Q[x,a] and ideal mpoly is defined (it might be 0), |
---|
593 | this represents the ring Q(a)[x] together with its minimal polynomial. |
---|
594 | NOTE: outdated, use gcd instead |
---|
595 | EXAMPLE: example EGCD; shows an example |
---|
596 | " |
---|
597 | { |
---|
598 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gcd |
---|
599 | int i = 1; |
---|
600 | poly r1, r2, r; |
---|
601 | |
---|
602 | r1 = f; |
---|
603 | r2 = g; |
---|
604 | |
---|
605 | while(r2 != 0) { |
---|
606 | r = RemainderMain(r1, r2); |
---|
607 | r1 = r2; |
---|
608 | r2 = r; |
---|
609 | } |
---|
610 | return(r1); |
---|
611 | } |
---|
612 | |
---|
613 | /////////////////////////////////////////////////////////////////////////////// |
---|
614 | |
---|
615 | proc MEGCD(poly f, poly g, int varIndex) |
---|
616 | "USAGE: MEGCD(f, g, i); poly f, g; int i |
---|
617 | PURPOSE: compute the polynomial gcd of f and g in the i'th variable |
---|
618 | RETURN: poly |
---|
619 | ASSUME: f, g are polynomials in var(i), last variable is the algebraic number |
---|
620 | EXAMPLE: example MEGCD; shows an example |
---|
621 | " |
---|
622 | // might be extended to return s1, s2 s.t. f*s1 + g*s2 = gc |
---|
623 | // not used ! |
---|
624 | { |
---|
625 | string @str, @sf, @sg, @mp, @parName; |
---|
626 | |
---|
627 | def @RGCDB = basering; |
---|
628 | |
---|
629 | @sf = string(f); |
---|
630 | @sg = string(g); |
---|
631 | @mp = string(minpoly); |
---|
632 | |
---|
633 | if(npars(basering) == 0) { @parName = "0";} |
---|
634 | else { @parName = "(0, " + parstr(basering) + ")"; } |
---|
635 | @str = "ring @RGCD = " + @parName + ", " + string(var(varIndex)) + ", dp;"; |
---|
636 | execute(@str); |
---|
637 | if(@mp != "0") { execute ("minpoly = " + @mp + ";"); } |
---|
638 | execute("poly @f = " + @sf + ";"); |
---|
639 | execute("poly @g = " + @sg + ";"); |
---|
640 | export(@RGCD); |
---|
641 | poly @h = gcd(@f, @g); |
---|
642 | setring(@RGCDB); |
---|
643 | poly h = imap(@RGCD, @h); |
---|
644 | kill @RGCD; |
---|
645 | return(h); |
---|
646 | } |
---|
647 | |
---|
648 | /////////////////////////////////////////////////////////////////////////////// |
---|
649 | |
---|
650 | proc SQFRNorm(poly f) |
---|
651 | "USAGE: SQFRNorm(f); where f is a polynomial |
---|
652 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
653 | RETURN: list with 3 entries |
---|
654 | @format |
---|
655 | _[1] = squarefree norm of g (poly) |
---|
656 | _[2] = g (= f(x - s*a)) (poly) |
---|
657 | _[3] = s (int) |
---|
658 | @end format |
---|
659 | ASSUME: f must be squarefree, basering = Q(a)[x] and minpoly != 0. |
---|
660 | NOTE: the norm is an element of Q[x] |
---|
661 | EXAMPLE: example SQFRNorm; shows an example |
---|
662 | " |
---|
663 | { |
---|
664 | def SNB = basering; |
---|
665 | def SNR = TransferRing(SNB); // new ring with parameter 'a' |
---|
666 | // replaced by a variable |
---|
667 | setring SNR; |
---|
668 | poly f = imap(SNB, f); |
---|
669 | list result = SQFRNormMain(f); // squarefree norm of f |
---|
670 | |
---|
671 | setring SNB; |
---|
672 | list result = imap(SNR, result); |
---|
673 | kill SNR; |
---|
674 | return(result); |
---|
675 | } |
---|
676 | example |
---|
677 | {"EXAMPLE:"; echo = 2; |
---|
678 | ring R = (0,a), x, lp; |
---|
679 | minpoly = a2+1; |
---|
680 | poly f = x4 - 2*x + 1; |
---|
681 | SQFRNorm(f); |
---|
682 | } |
---|
683 | |
---|
684 | proc SQFRNormMain(poly f) |
---|
685 | "USAGE: SQFRNorm(f); where f is a polynomial |
---|
686 | PURPOSE: compute the norm of the squarefree polynomial f in Q(a)[x]. |
---|
687 | RETURN: list with 3 entries |
---|
688 | @format |
---|
689 | _[1] = squarefree norm of g (poly) |
---|
690 | _[2] = g (= f(x - s*a)) (poly) |
---|
691 | _[3] = s (int) |
---|
692 | @end format |
---|
693 | ASSUME: f must be squarefree, basering = Q[x,a] and ideal mpoly is equal to |
---|
694 | 'minpoly', this represents the ring Q(a)[x] together with 'minpoly'. |
---|
695 | NOTE: the norm is an element of Q[x] |
---|
696 | EXAMPLE: example SqfrNorm; shows an example |
---|
697 | " |
---|
698 | { |
---|
699 | def SNRMB = basering; |
---|
700 | int s = 0; |
---|
701 | intvec wt = 1,0; |
---|
702 | ideal mapId; |
---|
703 | // list result; |
---|
704 | poly g, N, N1, h; |
---|
705 | string ringSTR; |
---|
706 | |
---|
707 | mapId[1] = var(1) - var(2); // linear transformation |
---|
708 | mapId[2] = var(2); |
---|
709 | map Fs = SNRMB, mapId; |
---|
710 | |
---|
711 | N = resultant(f, mpoly[1], var(2)); // norm of f |
---|
712 | N1 = diff(N, var(1)); |
---|
713 | g = f; |
---|
714 | |
---|
715 | ringSTR = "ring SNRM1 = 0, " + string(var(1)) + ", dp;"; // univariate ring |
---|
716 | execute(ringSTR); |
---|
717 | poly N, N1, h; // N, N1 do not contain 'a', use built-in gcd |
---|
718 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
719 | setring(SNRMB); |
---|
720 | h = imap(SNRM1, h); |
---|
721 | while(deg(h, wt) != 0) { // while norm is not squarefree |
---|
722 | s = s + 1; |
---|
723 | g = reduce(Fs(g), mpoly); |
---|
724 | N = reduce(resultant(g, mpoly[1], var(2)), mpoly); // norm of g |
---|
725 | N1 = reduce(diff(N, var(1)), mpoly); |
---|
726 | setring(SNRM1); |
---|
727 | h = gcd(imap(SNRMB, N), imap(SNRMB, N1)); |
---|
728 | setring(SNRMB); |
---|
729 | h = imap(SNRM1, h); |
---|
730 | } |
---|
731 | return(list(N, g, s)); |
---|
732 | } |
---|
733 | |
---|
734 | /////////////////////////////////////////////////////////////////////////////// |
---|
735 | |
---|
736 | proc FactorMain(poly f) |
---|
737 | "USAGE: FactorMain(f); where f is a polynomial |
---|
738 | PURPOSE: compute the factorization of the squarefree poly f over Q(a)[t], |
---|
739 | minpoly = p(a). |
---|
740 | RETURN: list with 2 entries |
---|
741 | @format |
---|
742 | _[1] = factors, first is a constant |
---|
743 | _[2] = multiplicities (not yet implemented) |
---|
744 | @end format |
---|
745 | ASSUME: basering = Q[x,a], representing Q(a)[x]. An ideal mpoly must |
---|
746 | be defined, representing the minimal polynomial (it might be 0!). |
---|
747 | NOTE: outdated, use factorize instead |
---|
748 | EXAMPLE: example Factor; shows an example |
---|
749 | " |
---|
750 | { |
---|
751 | // extend this by a squarefree factorization !! |
---|
752 | // multiplicities are not valid !! |
---|
753 | int i, s; |
---|
754 | list normList, factorList, quo_rem; |
---|
755 | poly f1, h, h1, H, g, leadCoef, invCoeff; |
---|
756 | ideal fac1, fac2; |
---|
757 | map F; |
---|
758 | |
---|
759 | // if no minimal polynomial is defined then use 'factorize' |
---|
760 | // FactorOverQ is wrapped around 'factorize' |
---|
761 | |
---|
762 | if(mpoly[1] == 0) { |
---|
763 | // print(" factorize : deg = " + string(deg(f, intvec(1,0)))); |
---|
764 | factorList = factorize(f); // FactorOverQ(f); |
---|
765 | return(factorList); |
---|
766 | } |
---|
767 | |
---|
768 | // if mpoly != 0 and f does not contain the algebraic number, a root of |
---|
769 | // f might be contained in Q(a). Hence one must not use 'factorize'. |
---|
770 | |
---|
771 | fac1[1] = 1; |
---|
772 | fac2[1] = 1; |
---|
773 | normList = SQFRNormMain(f); |
---|
774 | // print(" factorize : deg = " + string(deg(normList[1], intvec(1,0)))); |
---|
775 | factorList = factorize(normList[1]); // factor squarefree norm of f over Q[x] |
---|
776 | g = normList[2]; |
---|
777 | s = normList[3]; |
---|
778 | F[1] = var(1) + s*var(2); // inverse transformation |
---|
779 | F[2] = var(2); |
---|
780 | fac1[1] = factorList[1][1]; |
---|
781 | fac2[1] = factorList[2][1]; |
---|
782 | for(i = 2; i <= size(factorList[1]); i = i + 1) { |
---|
783 | H = factorList[1][i]; |
---|
784 | h = EGCDMain(H, g); |
---|
785 | quo_rem = QuotientMain(g, h); |
---|
786 | g = quo_rem[1]; |
---|
787 | fac1[i] = SimplifyPoly(F(h)); |
---|
788 | fac2[i] = 1; // to be changed later |
---|
789 | } |
---|
790 | return(list(fac1, fac2)); |
---|
791 | } |
---|
792 | |
---|
793 | /////////////////////////////////////////////////////////////////////////////// |
---|
794 | |
---|
795 | proc ZeroSetMain(ideal I, int primDecQ) |
---|
796 | "USAGE: ZeroSetMain(ideal I, int opt); ideal I, int opt |
---|
797 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a simple extension |
---|
798 | of the ground field. |
---|
799 | RETURN: list |
---|
800 | - 'f' is the polynomial f in Q(a) (a' being substituted by newA) |
---|
801 | _[1] = zero-set (list), is the list of the zero-set of the ideal I, |
---|
802 | each entry is an ideal. |
---|
803 | _[2] = 'newA'; if the ground field is Q(a') and the extension field |
---|
804 | is Q(a), then 'newA' is the representation of a' in Q(a). |
---|
805 | If the basering contains a parameter 'a' and the minpoly |
---|
806 | remains unchanged then 'newA' = 'a'. If the basering does not |
---|
807 | contain a parameter then 'newA' = 'a' (default). |
---|
808 | _[3] = 'mpoly' (ideal), the minimal polynomial of the simple extension |
---|
809 | of the ground field. |
---|
810 | ASSUME: basering = K[x_1,x_2,...,x_n] where K = Q or a simple extension of Q |
---|
811 | given by a minpoly; dim(I) = 0. |
---|
812 | NOTE: opt = 0 no primary decomposition |
---|
813 | opt > 0 use a primary decomposition |
---|
814 | EXAMPLE: example ZeroSet; shows an example |
---|
815 | " |
---|
816 | { |
---|
817 | // main work is done in ZeroSetMainWork, here the zero-set of each ideal from the |
---|
818 | // primary decompostion is coputed by menas of ZeroSetMainWork, and then the |
---|
819 | // minpoly and the parameter representing the algebraic extension are |
---|
820 | // transformed according to 'newA', i.e., only bookeeping is done. |
---|
821 | |
---|
822 | def altring=basering; |
---|
823 | int i, j, n, noMP, dbPrt; |
---|
824 | intvec w; |
---|
825 | list currentSol, result, idealList, primDecList, zeroSet; |
---|
826 | ideal J; |
---|
827 | map Fa; |
---|
828 | poly newA, oldMinPoly; |
---|
829 | |
---|
830 | dbPrt = printlevel-voice+2; |
---|
831 | dbprint(dbPrt, "ZeroSet of " + string(I) + ", minpoly = " + string(minpoly)); |
---|
832 | |
---|
833 | n = nvars(basering) - 1; |
---|
834 | for(i = 1; i <= n; i++) { w[i] = 1;} |
---|
835 | w[n + 1] = 0; |
---|
836 | |
---|
837 | if(primDecQ == 0) { return(ZeroSetMainWork(I, w, 0)); } |
---|
838 | |
---|
839 | newA = var(n + 1); |
---|
840 | if(mpoly[1] == 0) { noMP = 1;} |
---|
841 | else {noMP = 0;} |
---|
842 | |
---|
843 | primDecList = primdecGTZ(I); // primary decomposition |
---|
844 | dbprint(dbPrt, "primary decomposition consists of " + string(size(primDecList)) + " primary ideals "); |
---|
845 | // idealList = PDSort(idealList); // high degrees first |
---|
846 | |
---|
847 | for(i = 1; i <= size(primDecList); i = i + 1) { |
---|
848 | idealList[i] = primDecList[i][2]; // use prime component |
---|
849 | dbprint(dbPrt, string(i) + " " + string(idealList[i])); |
---|
850 | } |
---|
851 | |
---|
852 | // compute the zero-set of each primary ideal and join them. |
---|
853 | // If necessary, change the ground field and transform the zero-set |
---|
854 | |
---|
855 | dbprint(dbPrt, " |
---|
856 | find the zero-set of each primary ideal, form the union |
---|
857 | and keep track of the minimal polynomials "); |
---|
858 | |
---|
859 | for(i = 1; i <= size(idealList); i = i + 1) { |
---|
860 | J = idealList[i]; |
---|
861 | idealList[i] = 0; |
---|
862 | oldMinPoly = mpoly[1]; |
---|
863 | dbprint(dbPrt, " ideal#" + string(i) + " of " + string(size(idealList)) + " = " + string(J)); |
---|
864 | currentSol = ZeroSetMainWork(J, w, 0); |
---|
865 | |
---|
866 | if(oldMinPoly != currentSol[3]) { // change minpoly and transform solutions |
---|
867 | dbprint(dbPrt, " change minpoly to " + string(currentSol[3][1])); |
---|
868 | dbprint(dbPrt, " new representation of algebraic number = " + string(currentSol[2])); |
---|
869 | if(!noMP) { // transform the algebraic number a |
---|
870 | Fa = basering, maxideal(1); |
---|
871 | Fa[n + 1] = currentSol[2]; |
---|
872 | newA = SimplifyPoly(Fa(newA)); // new representation of a |
---|
873 | if(size(zeroSet) > 0) {zeroSet = SimplifyZeroset(Fa(zeroSet)); } |
---|
874 | if(i < size(idealList)) { idealList = SimplifyZeroset(Fa(idealList)); } |
---|
875 | } |
---|
876 | else { noMP = 0;} |
---|
877 | } |
---|
878 | zeroSet = zeroSet + currentSol[1]; // add new elements |
---|
879 | } |
---|
880 | return(list(zeroSet, newA, mpoly)); |
---|
881 | } |
---|
882 | |
---|
883 | /////////////////////////////////////////////////////////////////////////////// |
---|
884 | |
---|
885 | proc ZeroSetMainWork(ideal id, intvec wt, int sVars) |
---|
886 | "USAGE: ZeroSetMainWork(I, wt, sVars); |
---|
887 | PURPOSE: compute the zero-set of the zero-dim. ideal I, in a finite extension |
---|
888 | of the ground field (without multiplicities). |
---|
889 | RETURN: list, all entries are polynomials |
---|
890 | _[1] = zeros, each entry is an ideal |
---|
891 | _[2] = newA; if the ground field is Q(a') this is the rep. of a' w.r.t. a |
---|
892 | _[3] = minpoly of the algebraic extension of the ground field (ideal) |
---|
893 | _[4] = name of algebraic number (default = 'a') |
---|
894 | ASSUME: basering = Q[x_1,x_2,...,x_n,a] |
---|
895 | ideal mpoly must be defined, it might be 0! |
---|
896 | NOTE: might change 'mpoly' !! |
---|
897 | EXAMPLE: example IdealSolve; shows an example |
---|
898 | " |
---|
899 | { |
---|
900 | def altring=basering; |
---|
901 | int i, j, k, nrSols, n, noMP; |
---|
902 | ideal I, generators, gens, solid, partsolid; |
---|
903 | list linSol, linearSolution, nLinSol, nonlinSolutions, partSol, sol, solutions, result; |
---|
904 | list linIndex, nlinIndex, index; |
---|
905 | map Fa, Fsubs; |
---|
906 | poly oldMinPoly, newA; |
---|
907 | |
---|
908 | if(mpoly[1] == 0) { noMP = 1;} |
---|
909 | else { noMP = 0;} |
---|
910 | n = nvars(basering) - 1; |
---|
911 | newA = var(n + 1); |
---|
912 | |
---|
913 | I = std(id); |
---|
914 | |
---|
915 | // find linear solutions of univariate generators |
---|
916 | |
---|
917 | linSol = LinearZeroSetMain(I, wt); |
---|
918 | generators = linSol[3]; // they are a standardbasis |
---|
919 | linIndex = linSol[2]; |
---|
920 | linearSolution = linSol[1]; |
---|
921 | if(size(linIndex) + sVars == n) { // all variables solved |
---|
922 | solid = SubsMapIdeal(linearSolution, linIndex, 0); |
---|
923 | result[1] = list(solid); |
---|
924 | result[2] = var(n + 1); |
---|
925 | result[3] = mpoly; |
---|
926 | return(result); |
---|
927 | } |
---|
928 | |
---|
929 | // find roots of the nonlinear univariate polynomials of generators |
---|
930 | // if necessary, transform linear solutions w.r.t. newA |
---|
931 | |
---|
932 | oldMinPoly = mpoly[1]; |
---|
933 | nLinSol = NonLinearZeroSetMain(generators, wt); // find solutions of univariate generators |
---|
934 | nonlinSolutions = nLinSol[1]; // store solutions |
---|
935 | nlinIndex = nLinSol[4]; // and index of solved variables |
---|
936 | generators = nLinSol[5]; // new generators |
---|
937 | |
---|
938 | // change minpoly if necessary and transform the ideal and the partial solutions |
---|
939 | |
---|
940 | if(oldMinPoly != nLinSol[3]) { |
---|
941 | newA = nLinSol[2]; |
---|
942 | if(!noMP && size(linearSolution) > 0) { // transform the algebraic number a |
---|
943 | Fa = basering, maxideal(1); |
---|
944 | Fa[n + 1] = newA; |
---|
945 | linearSolution = SimplifyData(Fa(linearSolution)); // ... |
---|
946 | } |
---|
947 | } |
---|
948 | |
---|
949 | // check if all variables are solved. |
---|
950 | |
---|
951 | if(size(linIndex) + size(nlinIndex) == n - sVars) { |
---|
952 | solutions = MergeSolutions(linearSolution, linIndex, nonlinSolutions, nlinIndex, list(), n); |
---|
953 | } |
---|
954 | |
---|
955 | else { |
---|
956 | |
---|
957 | // some variables are not solved. |
---|
958 | // substitute each partial solution in generators and find the |
---|
959 | // zero set of the resulting ideal by recursive application |
---|
960 | // of ZeroSetMainWork ! |
---|
961 | |
---|
962 | index = linIndex + nlinIndex; |
---|
963 | nrSols = 0; |
---|
964 | for(i = 1; i <= size(nonlinSolutions); i = i + 1) { |
---|
965 | sol = linearSolution + nonlinSolutions[i]; |
---|
966 | solid = SubsMapIdeal(sol, index, 1); |
---|
967 | Fsubs = basering, solid; |
---|
968 | gens = std(SimplifyData(Fsubs(generators))); // substitute partial solution |
---|
969 | oldMinPoly = mpoly[1]; |
---|
970 | partSol = ZeroSetMainWork(gens, wt, size(index) + sVars); |
---|
971 | |
---|
972 | if(oldMinPoly != partSol[3]) { // minpoly has changed |
---|
973 | Fa = basering, maxideal(1); |
---|
974 | Fa[n + 1] = partSol[2]; // a -> p(a), representation of a w.r.t. new minpoly |
---|
975 | newA = reduce(Fa(newA), mpoly); |
---|
976 | generators = std(SimplifyData(Fa(generators))); |
---|
977 | if(size(linearSolution) > 0) { linearSolution = SimplifyData(Fa(linearSolution));} |
---|
978 | if(size(nonlinSolutions) > 0) { |
---|
979 | nonlinSolutions = SimplifyZeroset(Fa(nonlinSolutions)); |
---|
980 | } |
---|
981 | sol = linearSolution + nonlinSolutions[i]; |
---|
982 | } |
---|
983 | |
---|
984 | for(j = 1; j <= size(partSol[1]); j++) { // for all partial solutions |
---|
985 | partsolid = partSol[1][j]; |
---|
986 | for(k = 1; k <= size(index); k++) { |
---|
987 | partsolid[index[k]] = sol[k]; |
---|
988 | } |
---|
989 | nrSols++; |
---|
990 | solutions[nrSols] = partsolid; |
---|
991 | } |
---|
992 | } |
---|
993 | |
---|
994 | } // end else |
---|
995 | return(list(solutions, newA, mpoly)); |
---|
996 | } |
---|
997 | |
---|
998 | /////////////////////////////////////////////////////////////////////////////// |
---|
999 | |
---|
1000 | proc LinearZeroSetMain(ideal I, intvec wt) |
---|
1001 | "USAGE: LinearZeroSetMain(I, wt) |
---|
1002 | PURPOSE: solve the univariate linear polys in I |
---|
1003 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
1004 | RETURN: list |
---|
1005 | _[1] = partial solution of I |
---|
1006 | _[2] = index of solved vars |
---|
1007 | _[3] = new generators (standardbasis) |
---|
1008 | " |
---|
1009 | { |
---|
1010 | def altring=basering; |
---|
1011 | int i, ok, n, found, nrSols; |
---|
1012 | ideal generators, newGens; |
---|
1013 | list result, index, totalIndex, vars, sol, temp; |
---|
1014 | map F; |
---|
1015 | poly f; |
---|
1016 | |
---|
1017 | result[1] = index; // sol[1] should be the empty list |
---|
1018 | n = nvars(basering) - 1; |
---|
1019 | generators = I; // might be wrong, use index ! |
---|
1020 | ok = 1; |
---|
1021 | nrSols = 0; |
---|
1022 | while(ok) { |
---|
1023 | found = 0; |
---|
1024 | for(i = 1; i <= size(generators); i = i + 1) { |
---|
1025 | f = generators[i]; |
---|
1026 | vars = Variables(f, n); |
---|
1027 | if(size(vars) == 1 && deg(f, wt) == 1) { // univariate,linear |
---|
1028 | nrSols++; found++; |
---|
1029 | index[nrSols] = vars[1]; |
---|
1030 | sol[nrSols] = var(vars[1]) - MultPolys(InvertNumberMain(LeadTerm(f, vars[1])[3]), f); |
---|
1031 | } |
---|
1032 | } |
---|
1033 | if(found > 0) { |
---|
1034 | F = basering, SubsMapIdeal(sol, index, 1); |
---|
1035 | newGens = std(SimplifyData(F(generators))); // substitute, simplify alg. number |
---|
1036 | if(size(newGens) == 0) {ok = 0;} |
---|
1037 | generators = newGens; |
---|
1038 | } |
---|
1039 | else { |
---|
1040 | ok = 0; |
---|
1041 | } |
---|
1042 | } |
---|
1043 | if(nrSols > 0) { result[1] = sol;} |
---|
1044 | result[2] = index; |
---|
1045 | result[3] = generators; |
---|
1046 | return(result); |
---|
1047 | } |
---|
1048 | |
---|
1049 | /////////////////////////////////////////////////////////////////////////////// |
---|
1050 | |
---|
1051 | proc NonLinearZeroSetMain(ideal I, intvec wt) |
---|
1052 | "USAGE: ZeroSetMainWork(I, wt, sVars); |
---|
1053 | PURPOSE: solves the (nonlinear) univariate polynomials in I |
---|
1054 | of the ground field (without multiplicities). |
---|
1055 | RETURN: list, all entries are polynomials |
---|
1056 | _[1] = list of solutions |
---|
1057 | _[2] = newA |
---|
1058 | _[3] = minpoly |
---|
1059 | _[4] - index of solved variables |
---|
1060 | _[5] = new representation of I |
---|
1061 | ASSUME: basering = Q[x_1,x_2,...,x_n,a], ideal 'mpoly' must be defined, |
---|
1062 | it might be 0 ! |
---|
1063 | NOTE: might change 'mpoly' !! |
---|
1064 | " |
---|
1065 | { |
---|
1066 | int i, nrSols, ok, n; |
---|
1067 | ideal generators; |
---|
1068 | list result, sols, index, vars, partSol; |
---|
1069 | map F; |
---|
1070 | poly f, newA; |
---|
1071 | string ringSTR; |
---|
1072 | |
---|
1073 | def NLZR = basering; |
---|
1074 | export(NLZR); |
---|
1075 | |
---|
1076 | n = nvars(basering) - 1; |
---|
1077 | |
---|
1078 | generators = I; |
---|
1079 | newA = var(n + 1); |
---|
1080 | result[2] = newA; // default |
---|
1081 | nrSols = 0; |
---|
1082 | ok = 1; |
---|
1083 | i = 1; |
---|
1084 | while(ok) { |
---|
1085 | |
---|
1086 | // test if the i-th generator of I is univariate |
---|
1087 | |
---|
1088 | f = generators[i]; |
---|
1089 | vars = Variables(f, n); |
---|
1090 | if(size(vars) == 1) { |
---|
1091 | generators[i] = 0; |
---|
1092 | generators = simplify(generators, 2); // remove 0 |
---|
1093 | nrSols++; |
---|
1094 | index[nrSols] = vars[1]; // store index of solved variable |
---|
1095 | |
---|
1096 | // create univariate ring |
---|
1097 | |
---|
1098 | ringSTR = "ring RIS1 = 0, (" + string(var(vars[1])) + ", " + string(var(n+1)) + "), lp;"; |
---|
1099 | execute(ringSTR); |
---|
1100 | ideal mpoly = std(imap(NLZR, mpoly)); |
---|
1101 | list roots; |
---|
1102 | poly f = imap(NLZR, f); |
---|
1103 | export(RIS1); |
---|
1104 | export(mpoly); |
---|
1105 | roots = RootsMain(f); |
---|
1106 | |
---|
1107 | // get "old" basering with new minpoly |
---|
1108 | |
---|
1109 | setring(NLZR); |
---|
1110 | partSol = imap(RIS1, roots); |
---|
1111 | kill RIS1; |
---|
1112 | if(mpoly[1] != partSol[3]) { // change minpoly |
---|
1113 | mpoly = std(partSol[3]); |
---|
1114 | F = NLZR, maxideal(1); |
---|
1115 | F[n + 1] = partSol[2]; |
---|
1116 | if(size(sols) > 0) {sols = SimplifyZeroset(F(sols)); } |
---|
1117 | newA = reduce(F(newA), mpoly); // normal form |
---|
1118 | result[2] = newA; |
---|
1119 | generators = SimplifyData(F(generators)); // does not remove 0's |
---|
1120 | } |
---|
1121 | sols = ExtendSolutions(sols, partSol[1]); |
---|
1122 | } // end univariate |
---|
1123 | else { |
---|
1124 | i = i + 1; |
---|
1125 | } |
---|
1126 | if(i > size(generators)) { ok = 0;} |
---|
1127 | } |
---|
1128 | result[1] = sols; |
---|
1129 | result[3] = mpoly; |
---|
1130 | result[4] = index; |
---|
1131 | result[5] = std(generators); |
---|
1132 | |
---|
1133 | kill NLZR; |
---|
1134 | return(result); |
---|
1135 | } |
---|
1136 | |
---|
1137 | /////////////////////////////////////////////////////////////////////////////// |
---|
1138 | |
---|
1139 | static proc ExtendSolutions(list solutions, list newSolutions) |
---|
1140 | "USAGE: ExtendSolutions(sols, newSols); list sols, newSols; |
---|
1141 | PURPOSE: extend the entries of 'sols' by the entries of 'newSols', |
---|
1142 | each entry of 'newSols' is a number. |
---|
1143 | RETURN: list |
---|
1144 | ASSUME: basering = Q[x_1,...,x_n,a], ideal 'mpoly' must be defined, |
---|
1145 | it might be 0 ! |
---|
1146 | NOTE: used by 'NonLinearZeroSetMain' |
---|
1147 | " |
---|
1148 | { |
---|
1149 | int i, j, k, n, nrSols; |
---|
1150 | list newSols, temp; |
---|
1151 | |
---|
1152 | nrSols = size(solutions); |
---|
1153 | if(nrSols > 0) {n = size(solutions[1]);} |
---|
1154 | else { |
---|
1155 | n = 0; |
---|
1156 | nrSols = 1; |
---|
1157 | } |
---|
1158 | k = 0; |
---|
1159 | for(i = 1; i <= nrSols; i++) { |
---|
1160 | for(j = 1; j <= size(newSolutions); j++) { |
---|
1161 | k++; |
---|
1162 | if(n == 0) { temp[1] = newSolutions[j];} |
---|
1163 | else { |
---|
1164 | temp = solutions[i]; |
---|
1165 | temp[n + 1] = newSolutions[j]; |
---|
1166 | } |
---|
1167 | newSols[k] = temp; |
---|
1168 | } |
---|
1169 | } |
---|
1170 | return(newSols); |
---|
1171 | } |
---|
1172 | |
---|
1173 | /////////////////////////////////////////////////////////////////////////////// |
---|
1174 | |
---|
1175 | static proc MergeSolutions(list sol1, list index1, list sol2, list index2) |
---|
1176 | "USAGE: MergeSolutions(sol1, index1, sol2, index2); all parameters are lists |
---|
1177 | RETURN: list |
---|
1178 | PURPOSE: create a list of solutions of size n, each entry of 'sol2' must |
---|
1179 | have size n. 'sol1' is one partial solution (from 'LinearZeroSetMain') |
---|
1180 | 'sol2' is a list of partial solutions (from 'NonLinearZeroSetMain') |
---|
1181 | ASSUME: 'sol2' is not empty |
---|
1182 | NOTE: used by 'ZeroSetMainWork' |
---|
1183 | { |
---|
1184 | int i, j, k, m; |
---|
1185 | ideal sol; |
---|
1186 | list newSols; |
---|
1187 | |
---|
1188 | m = 0; |
---|
1189 | for(i = 1; i <= size(sol2); i++) { |
---|
1190 | m++; |
---|
1191 | newSols[m] = SubsMapIdeal(sol1 + sol2[i], index1 + index2, 0); |
---|
1192 | } |
---|
1193 | return(newSols); |
---|
1194 | } |
---|
1195 | |
---|
1196 | /////////////////////////////////////////////////////////////////////////////// |
---|
1197 | |
---|
1198 | static proc SubsMapIdeal(list sol, list index, int opt) |
---|
1199 | "USAGE: SubsMapIdeal(sol,index,opt); list sol, index; int opt; |
---|
1200 | PURPOSE: built an ideal I as follows. |
---|
1201 | if i is contained in 'index' then set I[i] = sol[i] |
---|
1202 | if i is not contained in 'index' then |
---|
1203 | - opt = 0: set I[i] = 0 |
---|
1204 | - opt = 1: set I[i] = var(i) |
---|
1205 | if opt = 1 and n = nvars(basering) then set I[n] = var(n). |
---|
1206 | RETURN: ideal |
---|
1207 | ASSUME: size(sol) = size(index) <= nvars(basering) |
---|
1208 | " |
---|
1209 | { |
---|
1210 | int k = 0; |
---|
1211 | ideal I; |
---|
1212 | for(int i = 1; i <= nvars(basering) - 1; i = i + 1) { // built subs. map |
---|
1213 | if(ContainedQ(index, i)) { |
---|
1214 | k++; |
---|
1215 | I[index[k]] = sol[k]; |
---|
1216 | } |
---|
1217 | else { |
---|
1218 | if(opt) { I[i] = var(i); } |
---|
1219 | else { I[i] = 0; } |
---|
1220 | } |
---|
1221 | } |
---|
1222 | if(opt) {I[nvars(basering)] = var(nvars(basering));} |
---|
1223 | return(I); |
---|
1224 | } |
---|
1225 | |
---|
1226 | /////////////////////////////////////////////////////////////////////////////// |
---|
1227 | |
---|
1228 | proc SimplifyZeroset(data) |
---|
1229 | "USAGE: SimplifyZeroset(data); list data |
---|
1230 | PURPOSE: reduce the entries of the elements of 'data' w.r.t. the ideal 'mpoly' |
---|
1231 | 'data' is a list of ideals/lists. |
---|
1232 | RETURN: list |
---|
1233 | ASSUME: basering = Q[x_1,...,x_n,a], order = lp |
---|
1234 | 'data' is a list of ideals |
---|
1235 | ideal 'mpoly' must be defined, it might be 0 ! |
---|
1236 | " |
---|
1237 | { |
---|
1238 | int i; |
---|
1239 | list result; |
---|
1240 | |
---|
1241 | for(i = 1; i <= size(data); i++) { |
---|
1242 | result[i] = SimplifyData(data[i]); |
---|
1243 | } |
---|
1244 | return(result); |
---|
1245 | } |
---|
1246 | |
---|
1247 | /////////////////////////////////////////////////////////////////////////////// |
---|
1248 | |
---|
1249 | proc Variables(poly f, int n) |
---|
1250 | "USAGE: Variables(f,n); poly f; int n; |
---|
1251 | PURPOSE: list of variables among var(1),...,var(n) which occur in f. |
---|
1252 | RETURN: list |
---|
1253 | ASSUME: n <= nvars(basering) |
---|
1254 | " |
---|
1255 | { |
---|
1256 | int i, nrV; |
---|
1257 | list index; |
---|
1258 | |
---|
1259 | nrV = 0; |
---|
1260 | for(i = 1; i <= n; i = i + 1) { |
---|
1261 | if(diff(f, var(i)) != 0) { nrV++; index[nrV] = i; } |
---|
1262 | } |
---|
1263 | return(index); |
---|
1264 | } |
---|
1265 | |
---|
1266 | /////////////////////////////////////////////////////////////////////////////// |
---|
1267 | |
---|
1268 | proc ContainedQ(data, f, list #) |
---|
1269 | "USAGE: ContainedQ(data, f [, opt]); data=list; f=any type, opt=integer |
---|
1270 | PURPOSE: test if f is an element of data. |
---|
1271 | RETURN: int |
---|
1272 | 0 if f not contained in data |
---|
1273 | 1 if f contained in data |
---|
1274 | OPTIONS: opt = 0 : use '==' for comparing f with elements from data@* |
---|
1275 | opt = 1 : use @code{SameQ} for comparing f with elements from data |
---|
1276 | " |
---|
1277 | { |
---|
1278 | int opt, i, found; |
---|
1279 | if(size(#) > 0) { opt = #[1];} |
---|
1280 | else { opt = 0; } |
---|
1281 | i = 1; |
---|
1282 | found = 0; |
---|
1283 | |
---|
1284 | while((!found) && (i <= size(data))) { |
---|
1285 | if(opt == 0) { |
---|
1286 | if(f == data[i]) { found = 1;} |
---|
1287 | else {i = i + 1;} |
---|
1288 | } |
---|
1289 | else { |
---|
1290 | if(SameQ(f, data[i])) { found = 1;} |
---|
1291 | else {i = i + 1;} |
---|
1292 | } |
---|
1293 | } |
---|
1294 | return(found); |
---|
1295 | } |
---|
1296 | |
---|
1297 | ////////////////////////////////////////////////////////////////////////////// |
---|
1298 | |
---|
1299 | proc SameQ(a, b) |
---|
1300 | "USAGE: SameQ(a, b); a,b=list/intvec |
---|
1301 | PURPOSE: test a == b elementwise, i.e., a[i] = b[i]. |
---|
1302 | RETURN: int |
---|
1303 | 0 if a != b |
---|
1304 | 1 if a == b |
---|
1305 | " |
---|
1306 | { |
---|
1307 | if(typeof(a) == typeof(b)) { |
---|
1308 | if(typeof(a) == "list" || typeof(a) == "intvec") { |
---|
1309 | if(size(a) == size(b)) { |
---|
1310 | int i = 1; |
---|
1311 | int ok = 1; |
---|
1312 | while(ok && (i <= size(a))) { |
---|
1313 | if(a[i] == b[i]) { i = i + 1;} |
---|
1314 | else {ok = 0;} |
---|
1315 | } |
---|
1316 | return(ok); |
---|
1317 | } |
---|
1318 | else { return(0); } |
---|
1319 | } |
---|
1320 | else { return(a == b);} |
---|
1321 | } |
---|
1322 | else { return(0);} |
---|
1323 | } |
---|
1324 | |
---|
1325 | /////////////////////////////////////////////////////////////////////////////// |
---|
1326 | |
---|
1327 | static proc SimplifyPoly(poly f) |
---|
1328 | "USAGE: SimplifyPoly(f); poly f |
---|
1329 | PURPOSE: reduces the coefficients of f w.r.t. the ideal 'moly' if they contain |
---|
1330 | the algebraic number 'a'. |
---|
1331 | RETURN: poly |
---|
1332 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
1333 | ideal mpoly must be defined, it might be 0 ! |
---|
1334 | NOTE: outdated, use reduce instead |
---|
1335 | " |
---|
1336 | { |
---|
1337 | matrix coMx; |
---|
1338 | poly f1, vp; |
---|
1339 | |
---|
1340 | vp = 1; |
---|
1341 | for(int i = 1; i < nvars(basering); i++) { vp = vp * var(i);} |
---|
1342 | |
---|
1343 | coMx = coef(f, vp); |
---|
1344 | f1 = 0; |
---|
1345 | for(i = 1; i <= ncols(coMx); i++) { |
---|
1346 | f1 = f1 + coMx[1, i] * reduce(coMx[2, i], mpoly); |
---|
1347 | } |
---|
1348 | return(f1); |
---|
1349 | } |
---|
1350 | |
---|
1351 | /////////////////////////////////////////////////////////////////////////////// |
---|
1352 | |
---|
1353 | static proc SimplifyData(data) |
---|
1354 | "USAGE: SimplifyData(data); ideal/list data; |
---|
1355 | PURPOSE: reduces the entries of 'data' w.r.t. the ideal 'mpoly' if they contain |
---|
1356 | the algebraic number 'a' |
---|
1357 | RETURN: ideal/list |
---|
1358 | ASSUME: basering = Q[x_1,...,x_n,a] |
---|
1359 | ideal 'mpoly' must be defined, it might be 0 ! |
---|
1360 | " |
---|
1361 | { |
---|
1362 | def altring=basering; |
---|
1363 | int n; |
---|
1364 | poly f; |
---|
1365 | |
---|
1366 | if(typeof(data) == "ideal") { n = ncols(data); } |
---|
1367 | else { n = size(data);} |
---|
1368 | |
---|
1369 | for(int i = 1; i <= n; i++) { |
---|
1370 | f = data[i]; |
---|
1371 | data[i] = SimplifyPoly(f); |
---|
1372 | } |
---|
1373 | return(data); |
---|
1374 | } |
---|
1375 | |
---|
1376 | /////////////////////////////////////////////////////////////////////////////// |
---|
1377 | |
---|
1378 | static proc TransferRing(R) |
---|
1379 | "USAGE: TransferRing(R); |
---|
1380 | PURPOSE: creates a new ring containing the same variables as R, but without |
---|
1381 | parameters. If R contains a parameter then this parameter is added |
---|
1382 | as the last variable and 'minpoly' is represented by the ideal 'mpoly' |
---|
1383 | If the basering does not contain a parameter then 'a' is added and |
---|
1384 | 'mpoly' = 0. |
---|
1385 | RETURN: ring |
---|
1386 | ASSUME: R = K[x_1,...,x_n] where K = Q or K = Q(a). |
---|
1387 | NOTE: Creates the ring needed for all prodecures with name 'proc-name'Main |
---|
1388 | " |
---|
1389 | { |
---|
1390 | def altring=basering; |
---|
1391 | string ringSTR, parName, minPoly; |
---|
1392 | |
---|
1393 | setring(R); |
---|
1394 | |
---|
1395 | if(npars(basering) == 0) { |
---|
1396 | parName = "a"; |
---|
1397 | minPoly = "0"; |
---|
1398 | } |
---|
1399 | else { |
---|
1400 | parName = parstr(basering); |
---|
1401 | minPoly = string(minpoly); |
---|
1402 | } |
---|
1403 | ringSTR = "ring TR = 0, (" + varstr(basering) + "," + parName + "), lp;"; |
---|
1404 | |
---|
1405 | execute(ringSTR); |
---|
1406 | execute("ideal mpoly = std(" + minPoly + ");"); |
---|
1407 | export(mpoly); |
---|
1408 | setring altring; |
---|
1409 | return(TR); |
---|
1410 | } |
---|
1411 | |
---|
1412 | /////////////////////////////////////////////////////////////////////////////// |
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1413 | |
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1414 | static proc NewBaseRing() |
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1415 | "USAGE: NewBaseRing(); |
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1416 | PURPOSE: creates a new ring, the last variable is added as a parameter. |
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1417 | minpoly is set to mpoly[1]. |
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1418 | RETURN: ring |
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1419 | ASSUME: basering = Q[x_1,...,x_n, a], 'mpoly' must be defined |
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1420 | " |
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1421 | { |
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1422 | int n = nvars(basering); |
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1423 | int MP; |
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1424 | string ringSTR, parName, varString; |
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1425 | |
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1426 | def BR = basering; |
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1427 | if(mpoly[1] != 0) { |
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1428 | parName = "(0, " + string(var(n)) + ")"; |
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1429 | MP = 1; |
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1430 | } |
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1431 | else { |
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1432 | parName = "0"; |
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1433 | MP = 0; |
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1434 | } |
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1435 | |
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1436 | |
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1437 | for(int i = 1; i < n - 1; i++) { |
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1438 | varString = varString + string(var(i)) + ","; |
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1439 | } |
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1440 | varString = varString + string(var(n-1)); |
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1441 | |
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1442 | ringSTR = "ring TR = " + parName + ", (" + varString + "), lp;"; |
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1443 | execute(ringSTR); |
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1444 | if(MP) { minpoly = number(imap(BR, mpoly)[1]); } |
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1445 | setring BR; |
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1446 | return(TR); |
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1447 | } |
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1448 | |
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1449 | /////////////////////////////////////////////////////////////////////////////// |
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1450 | |
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1451 | /* |
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1452 | Examples: |
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1453 | |
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1454 | |
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1455 | // order = 20; |
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1456 | ring S1 = 0, (s(1..3)), lp; |
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1457 | 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); |
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1458 | ideal mpoly = std(0); |
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1459 | |
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1460 | // order = 10 |
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1461 | ring S2 = 0, (s(1..5)), lp; |
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1462 | 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; |
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1463 | ideal mpoly = std(0); |
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1464 | |
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1465 | //order = 126 |
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1466 | ring S3 = 0, (s(1..5)), lp; |
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1467 | 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; |
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1468 | ideal mpoly = std(0); |
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1469 | |
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1470 | // order = 192 |
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1471 | ring S4 = 0, (s(1..4)), lp; |
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1472 | 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; |
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1473 | ideal mpoly = std(0); |
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1474 | |
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1475 | ring R = (0,a), (x,y,z), lp; |
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1476 | minpoly = a2 + 1; |
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1477 | ideal I1 = x2 - 1/2, a*z - 1, y - 2; |
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1478 | ideal I2 = x3 - 1/2, a*z2 - 3, y - 2*a; |
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1479 | |
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1480 | */ |
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