1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: dmod.lib,v 1.22 2007-11-27 11:08:51 Singular Exp $"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: dmod.lib Algorithms for algebraic D-modules |
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6 | AUTHORS: Viktor Levandovskyy, levandov@risc.uni-linz.ac.at |
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7 | @* Jorge Martin Morales, jorge@unizar.es |
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8 | |
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9 | THEORY: Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R, |
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10 | @* one is interested in the R[1/F]-module of rank one, generated by F^s |
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11 | @* for a natural number s. |
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12 | @* In fact, the module R[1/F]*F^s has a structure of a D(R)[s]-module, where D(R) |
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13 | @* is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1> and |
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14 | @* D(R)[s] = D(R) tensored with K[s] over K. |
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15 | @* Constructively, one needs to find a left ideal I = I(F^s) in D(R), such |
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16 | @* that K[x_1,...,x_n,1/F]*F^s is isomorphic to D(R)/I as a D(R)-module. |
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17 | @* We often write just D for D(R) and D[s] for D(R)[s]. |
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18 | @* One is interested in the following data: |
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19 | @* - Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output |
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20 | @* - global Bernstein polynomial in K[s], denoted by bs, its minimal integer root s0 and |
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21 | @* the list of all roots of bs, which are rational, with their multiplicities is denoted by BS |
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22 | @* - Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output (LD0 is a holonomic ideal in D(R)) |
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23 | @* - Ann^(1) F^s in D(R)[s], denoted by LD1 (logarithmic derivations) |
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24 | @* - an operator in D(R)[s], denoted by PS, such that PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F^s]. |
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25 | |
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26 | @* We provide the following implementations: |
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27 | @* OT) the classical Ann F^s algorithm from Oaku and Takayama (J. Pure |
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28 | Applied Math., 1999), |
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29 | @* LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (unpublished) |
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30 | @* BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur |
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31 | l'ideal de Bernstein associe a des polynomes, preprint, 2002) |
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32 | |
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33 | GUIDE: |
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34 | @* - Ann F^s = I = I(F^s) = LD in D(R)[s] can be computed by SannfsBM, SannfsOT, SannfsLOT |
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35 | @* - Ann^(1) F^s in D(R)[s] can be computed by Sannfslog |
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36 | @* - global Bernstein polynomial bs resp. BS in K[s] can be computed by bernsteinBM |
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37 | @* - Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfsBM, annfsOT, annfsLOT |
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38 | @* - all the relevant data (LD, LD0, bs, PS) are computed by operatorBM |
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39 | |
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40 | MAIN PROCEDURES: |
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41 | |
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42 | annfs(F[,S,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F |
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43 | Sannfs(F[,S,eng]); compute Ann F^s in D[s] for a poly F |
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44 | Sannfslog(F[,eng]); compute Ann^(1) F^s in D[s] for a poly F |
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45 | bernsteinBM(F[,eng]); compute global Bernstein poly for a poly F (algorithm of Briancon-Maisonobe) |
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46 | operatorBM(F[,eng]); compute Ann F^s, Ann F^s0, BS and PS for a poly F (algorithm of Briancon-Maisonobe) |
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47 | annfsParamBM(F[,eng]); compute the generic Ann F^s (algorithm by Briancon and Maisonobe) and exceptional parametric constellations for a poly F with parametric coefficients |
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48 | annfsBMI(F[,eng]); compute Ann F^s and Bernstein ideal for a poly F=f1*..*fP (multivariate algorithm of Briancon-Maisonobe) |
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49 | checkRoot(F,a[,S,eng]); check if a given rational is a root of the global Bernstein polynomial of F and compute its multiplicity |
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50 | |
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51 | SECONDARY PROCEDURES FOR D-MODULES: |
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52 | |
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53 | annfsBM(F[,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F (algorithm of Briancon-Maisonobe) |
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54 | annfsLOT(F[,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F (Levandovskyy modification of the Oaku-Takayama algorithm) |
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55 | annfsOT(F[,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F (algorithm of Oaku-Takayama) |
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56 | SannfsBM(F[,eng]); compute Ann F^s in D[s] for a poly F (algorithm of Briancon-Maisonobe) |
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57 | SannfsLOT(F[,eng]); compute Ann F^s in D[s] for a poly F (Levandovskyy modification of the Oaku-Takayama algorithm) |
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58 | SannfsOT(F[,eng]); compute Ann F^s in D[s] for a poly F (algorithm of Oaku-Takayama) |
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59 | annfs0(I,F [,eng]); compute Ann F^s0 in D and Bernstein poly from the known Ann F^s in D[s] |
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60 | checkRoot1(I,F,a[,eng]); check whether a rational is a root of the global Bernstein polynomial of F from the known Ann F^s in D[s] |
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61 | checkRoot2(I,F,a[,eng]); check whether a rational is a root of the global Bernstein polynomial of F and compute its multiplicity from the known Ann F^s in D[s] |
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62 | checkFactor(I,F,qs[,eng]); check whether a polynomial qs in K[s] is a factor of the global Bernstein polynomial of F from the known Ann F^s in D[s] |
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63 | |
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64 | AUXILIARY PROCEDURES: |
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65 | |
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66 | arrange(p); create a poly, describing a full hyperplane arrangement |
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67 | reiffen(p,q); create a poly, describing a Reiffen curve |
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68 | isHolonomic(M); check whether a module is holonomic |
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69 | convloc(L); replace global orderings with local in the ringlist L |
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70 | minIntRoot(P,fact); minimal integer root among the roots of a maximal ideal P |
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71 | varnum(s); the number of the variable with the name s |
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72 | |
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73 | SEE ALSO: gmssing_lib |
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74 | "; |
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75 | |
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76 | LIB "nctools.lib"; |
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77 | LIB "elim.lib"; |
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78 | LIB "qhmoduli.lib"; // for Max |
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79 | LIB "gkdim.lib"; |
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80 | LIB "gmssing.lib"; |
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81 | LIB "control.lib"; // for genericity |
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82 | |
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83 | proc tstdmodlib() |
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84 | { |
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85 | /* tests all procs for consistency */ |
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86 | /* adding the new proc, add it here */ |
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87 | |
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88 | "MAIN PROCEDURES:"; |
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89 | example annfs; |
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90 | example annfs0; |
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91 | example Sannfs; |
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92 | example Sannfslog; |
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93 | example bernsteinBM; |
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94 | example operatorBM; |
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95 | example annfspecial; |
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96 | example annfsParamBM; |
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97 | example annfsBMI; |
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98 | example checkRoot; |
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99 | "SECONDARY D-MOD PROCEDURES:"; |
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100 | example annfsBM; |
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101 | example annfsLOT; |
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102 | example annfsOT; |
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103 | example SannfsBM; |
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104 | example SannfsLOT; |
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105 | example SannfsOT; |
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106 | example annfs0; |
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107 | example checkRoot1; |
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108 | example checkRoot2; |
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109 | example checkFactor; |
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110 | } |
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111 | |
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112 | static proc engine(ideal I, int i) |
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113 | { |
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114 | /* std - slimgb mix */ |
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115 | ideal J; |
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116 | if (i==0) |
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117 | { |
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118 | J = slimgb(I); |
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119 | } |
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120 | else |
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121 | { |
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122 | // without options -> strange! (ringlist?) |
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123 | option(redSB); |
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124 | option(redTail); |
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125 | J = std(I); |
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126 | } |
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127 | return(J); |
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128 | } |
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129 | |
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130 | |
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131 | // new top-level procedures |
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132 | proc annfs(poly F, list #) |
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133 | "USAGE: annfs(f [,S,eng]); f a poly, S a string, eng an optional int |
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134 | RETURN: ring |
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135 | PURPOSE: compute the D-module structure of basering[1/f]*f^s with the algorithm given in S and with the Groebner basis engine given in 'eng' |
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136 | NOTE: activate the output ring with the @code{setring} command. |
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137 | @* The value of a string S can be |
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138 | @* 'bm' (default) - for the algorithm of Briancon and Maisonobe, |
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139 | @* 'ot' - for the algorithm of Oaku and Takayama, |
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140 | @* 'lot' - for the Levandovskyy's modification of the algorithm of Oaku and Takayama. |
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141 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
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142 | @* otherwise and by default @code{slimgb} is used. |
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143 | @* In the output ring: |
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144 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
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145 | @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
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146 | @* If @code{printlevel}=1, progress debug messages will be printed, |
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147 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
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148 | EXAMPLE: example annfs; shows examples |
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149 | " |
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150 | { |
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151 | int eng = 0; |
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152 | int chs = 0; // choice |
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153 | string algo = "bm"; |
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154 | string st; |
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155 | if ( size(#)>0 ) |
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156 | { |
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157 | if ( typeof(#[1]) == "string" ) |
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158 | { |
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159 | st = string(#[1]); |
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160 | if ( (st == "BM") || (st == "Bm") || (st == "bM") ||(st == "bm")) |
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161 | { |
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162 | algo = "bm"; |
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163 | chs = 1; |
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164 | } |
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165 | if ( (st == "OT") || (st == "Ot") || (st == "oT") || (st == "ot")) |
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166 | { |
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167 | algo = "ot"; |
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168 | chs = 1; |
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169 | } |
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170 | if ( (st == "LOT") || (st == "lOT") || (st == "Lot") || (st == "lot")) |
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171 | { |
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172 | algo = "lot"; |
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173 | chs = 1; |
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174 | } |
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175 | if (chs != 1) |
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176 | { |
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177 | // incorrect value of S |
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178 | print("Incorrect algorithm given, proceed with the default BM"); |
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179 | algo = "bm"; |
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180 | } |
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181 | // second arg |
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182 | if (size(#)>1) |
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183 | { |
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184 | // exists 2nd arg |
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185 | if ( typeof(#[2]) == "int" ) |
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186 | { |
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187 | // the case: given alg, given engine |
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188 | eng = int(#[2]); |
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189 | } |
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190 | else |
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191 | { |
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192 | eng = 0; |
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193 | } |
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194 | } |
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195 | else |
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196 | { |
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197 | // no second arg |
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198 | eng = 0; |
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199 | } |
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200 | } |
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201 | else |
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202 | { |
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203 | if ( typeof(#[1]) == "int" ) |
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204 | { |
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205 | // the case: default alg, engine |
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206 | eng = int(#[1]); |
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207 | // algo = "bm"; //is already set |
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208 | } |
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209 | else |
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210 | { |
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211 | // incorr. 1st arg |
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212 | algo = "bm"; |
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213 | } |
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214 | } |
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215 | } |
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216 | |
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217 | // size(#)=0, i.e. there is no algorithm and no engine given |
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218 | // eng = 0; algo = "bm"; //are already set |
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219 | // int ppl = printlevel-voice+2; |
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220 | |
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221 | int old_printlevel = printlevel; |
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222 | printlevel=printlevel+1; |
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223 | def save = basering; |
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224 | if ( algo=="ot") |
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225 | { |
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226 | def @A = annfsOT(F,eng); |
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227 | } |
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228 | else |
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229 | { |
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230 | if ( algo=="lot") |
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231 | { |
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232 | def @A = annfsLOT(F,eng); |
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233 | } |
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234 | else |
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235 | { |
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236 | // bm = default |
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237 | def @A = annfsBM(F,eng); |
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238 | } |
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239 | } |
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240 | printlevel = old_printlevel; |
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241 | return(@A); |
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242 | } |
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243 | example |
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244 | { |
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245 | "EXAMPLE:"; echo = 2; |
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246 | ring r = 0,(x,y,z),Dp; |
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247 | poly F = z*x^2+y^3; |
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248 | printlevel = 0; |
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249 | def A = annfs(F); // here, the default BM algorithm will be used |
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250 | setring A; |
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251 | LD; |
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252 | BS; |
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253 | } |
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254 | |
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255 | |
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256 | proc Sannfs(poly F, list #) |
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257 | "USAGE: Sannfs(f [,S,eng]); f a poly, S a string, eng an optional int |
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258 | RETURN: ring |
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259 | PURPOSE: compute the D-module structure of basering[f^s] with the algorithm given in S and with the Groebner basis engine given in eng |
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260 | NOTE: activate the output ring with the @code{setring} command. |
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261 | @* The value of a string S can be |
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262 | @* 'bm' (default) - for the algorithm of Briancon and Maisonobe, |
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263 | @* 'lot' - for the Levandovskyy's modification of the algorithm of Oaku and Takayama, |
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264 | @* 'ot' - for the algorithm of Oaku and Takayama. |
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265 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
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266 | @* otherwise, and by default @code{slimgb} is used. |
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267 | @* In the output ring: |
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268 | @* - the ideal LD is the needed D-module structure. |
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269 | @* If @code{printlevel}=1, progress debug messages will be printed, |
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270 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
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271 | EXAMPLE: example Sannfs; shows examples |
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272 | " |
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273 | { |
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274 | int eng = 0; |
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275 | int chs = 0; // choice |
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276 | string algo = "bm"; |
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277 | string st; |
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278 | if ( size(#)>0 ) |
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279 | { |
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280 | if ( typeof(#[1]) == "string" ) |
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281 | { |
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282 | st = string(#[1]); |
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283 | if ( (st == "BM") || (st == "Bm") || (st == "bM") ||(st == "bm")) |
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284 | { |
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285 | algo = "bm"; |
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286 | chs = 1; |
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287 | } |
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288 | if ( (st == "OT") || (st == "Ot") || (st == "oT") || (st == "ot")) |
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289 | { |
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290 | algo = "ot"; |
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291 | chs = 1; |
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292 | } |
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293 | if ( (st == "LOT") || (st == "lOT") || (st == "Lot") || (st == "lot")) |
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294 | { |
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295 | algo = "lot"; |
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296 | chs = 1; |
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297 | } |
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298 | if (chs != 1) |
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299 | { |
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300 | // incorrect value of S |
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301 | print("Incorrect algorithm given, proceed with the default BM"); |
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302 | algo = "bm"; |
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303 | } |
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304 | // second arg |
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305 | if (size(#)>1) |
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306 | { |
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307 | // exists 2nd arg |
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308 | if ( typeof(#[2]) == "int" ) |
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309 | { |
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310 | // the case: given alg, given engine |
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311 | eng = int(#[2]); |
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312 | } |
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313 | else |
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314 | { |
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315 | eng = 0; |
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316 | } |
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317 | } |
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318 | else |
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319 | { |
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320 | // no second arg |
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321 | eng = 0; |
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322 | } |
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323 | } |
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324 | else |
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325 | { |
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326 | if ( typeof(#[1]) == "int" ) |
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327 | { |
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328 | // the case: default alg, engine |
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329 | eng = int(#[1]); |
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330 | // algo = "bm"; //is already set |
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331 | } |
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332 | else |
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333 | { |
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334 | // incorr. 1st arg |
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335 | algo = "bm"; |
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336 | } |
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337 | } |
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338 | } |
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339 | // size(#)=0, i.e. there is no algorithm and no engine given |
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340 | // eng = 0; algo = "bm"; //are already set |
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341 | // int ppl = printlevel-voice+2; |
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342 | printlevel=printlevel+1; |
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343 | def save = basering; |
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344 | if ( algo=="ot") |
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345 | { |
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346 | def @A = SannfsOT(F,eng); |
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347 | } |
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348 | else |
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349 | { |
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350 | if ( algo=="lot") |
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351 | { |
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352 | def @A = SannfsLOT(F,eng); |
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353 | } |
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354 | else |
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355 | { |
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356 | // bm = default |
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357 | def @A = SannfsBM(F,eng); |
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358 | } |
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359 | } |
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360 | printlevel=printlevel-1; |
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361 | return(@A); |
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362 | } |
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363 | example |
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364 | { |
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365 | "EXAMPLE:"; echo = 2; |
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366 | ring r = 0,(x,y,z),Dp; |
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367 | poly F = x^3+y^3+z^3; |
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368 | printlevel = 0; |
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369 | def A = Sannfs(F); // here, the default BM algorithm will be used |
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370 | setring A; |
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371 | LD; |
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372 | } |
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373 | |
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374 | proc Sannfslog (poly F, list #) |
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375 | "USAGE: Sannfslog(f [,eng]); f a poly, eng an optional int |
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376 | RETURN: ring |
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377 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, where D is the Weyl algebra |
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378 | NOTE: activate this ring with the @code{setring} command. |
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379 | @* In the ring D[s], the ideal LD1 is generated by the elements in Ann F^s in D[s] coming from logarithmic derivations. |
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380 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
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381 | @* otherwise, and by default @code{slimgb} is used. |
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382 | @* If printlevel=1, progress debug messages will be printed, |
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383 | @* if printlevel>=2, all the debug messages will be printed. |
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384 | EXAMPLE: example Sannfslog; shows examples |
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385 | " |
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386 | { |
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387 | int eng = 0; |
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388 | if ( size(#)>0 ) |
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389 | { |
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390 | if ( typeof(#[1]) == "int" ) |
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391 | { |
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392 | eng = int(#[1]); |
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393 | } |
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394 | } |
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395 | int ppl = printlevel-voice+2; |
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396 | def save = basering; |
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397 | int N = nvars(basering); |
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398 | int Nnew = 2*N+1; |
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399 | int i; |
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400 | string s; |
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401 | list RL = ringlist(basering); |
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402 | list L, Lord; |
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403 | list tmp; |
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404 | intvec iv; |
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405 | L[1] = RL[1]; // char |
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406 | L[4] = RL[4]; // char, minpoly |
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407 | // check whether vars have admissible names |
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408 | list Name = RL[2]; |
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409 | for (i=1; i<=N; i++) |
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410 | { |
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411 | if (Name[i] == "s") |
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412 | { |
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413 | ERROR("Variable names should not include s"); |
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414 | } |
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415 | } |
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416 | // the ideal I |
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417 | ideal I = -F, jacob(F); |
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418 | dbprint(ppl,"// -1-1- starting the computation of syz(-F,_Dx(F))"); |
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419 | dbprint(ppl-1, I); |
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420 | matrix M = syz(I); |
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421 | M = transpose(M); // it is more usefull working with columns |
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422 | dbprint(ppl,"// -1-2- the module syz(-F,_Dx(F)) has been computed"); |
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423 | dbprint(ppl-1, M); |
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424 | // ------------ the ring @R ------------ |
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425 | // _x, _Dx, s; elim.ord for _x,_Dx. |
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426 | // now, create the names for new vars |
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427 | list DName; |
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428 | for (i=1; i<=N; i++) |
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429 | { |
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430 | DName[i] = "D"+Name[i]; // concat |
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431 | } |
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432 | tmp[1] = "s"; |
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433 | list NName; |
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434 | for (i=1; i<=N; i++) |
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435 | { |
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436 | NName[2*i-1] = Name[i]; |
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437 | NName[2*i] = DName[i]; |
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438 | //NName[2*i-1] = DName[i]; |
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439 | //NName[2*i] = Name[i]; |
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440 | } |
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441 | NName[Nnew] = tmp[1]; |
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442 | L[2] = NName; |
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443 | tmp = 0; |
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444 | // block ord (a(1,1),a(0,0,1,1),...,dp); |
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445 | //list("a",intvec(1,1)), list("a",intvec(0,0,1,1)), ... |
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446 | tmp[1] = "a"; // string |
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447 | for (i=1; i<=N; i++) |
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448 | { |
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449 | iv[2*i-1] = 1; |
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450 | iv[2*i] = 1; |
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451 | tmp[2] = iv; iv = 0; // intvec |
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452 | Lord[i] = tmp; |
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453 | } |
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454 | //list("dp",intvec(1,1,1,1,1,...)) |
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455 | s = "iv="; |
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456 | for (i=1; i<=Nnew; i++) |
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457 | { |
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458 | s = s+"1,"; |
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459 | } |
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460 | s[size(s)]=";"; |
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461 | execute(s); |
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462 | kill s; |
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463 | tmp[1] = "dp"; // string |
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464 | tmp[2] = iv; // intvec |
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465 | Lord[N+1] = tmp; |
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466 | //list("C",intvec(0)) |
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467 | tmp[1] = "C"; // string |
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468 | iv = 0; |
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469 | tmp[2] = iv; // intvec |
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470 | Lord[N+2] = tmp; |
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471 | tmp = 0; |
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472 | L[3] = Lord; |
---|
473 | // we are done with the list. Now add a Plural part |
---|
474 | def @R@ = ring(L); |
---|
475 | setring @R@; |
---|
476 | matrix @D[Nnew][Nnew]; |
---|
477 | for (i=1; i<=N; i++) |
---|
478 | { |
---|
479 | @D[2*i-1,2*i]=1; |
---|
480 | //@D[2*i-1,2*i]=-1; |
---|
481 | } |
---|
482 | def @R = nc_algebra(1,@D); |
---|
483 | setring @R; |
---|
484 | kill @R@; |
---|
485 | dbprint(ppl,"// -2-1- the ring @R(_x,_Dx,s) is ready"); |
---|
486 | dbprint(ppl-1, @R); |
---|
487 | matrix M = imap(save,M); |
---|
488 | // now, create the vector [-s,_Dx] |
---|
489 | vector v = [-s]; // now s is a variable |
---|
490 | for (i=1; i<=N; i++) |
---|
491 | { |
---|
492 | v = v + var(2*i)*gen(i+1); |
---|
493 | //v = v + var(2*i-1)*gen(i+1); |
---|
494 | } |
---|
495 | ideal J = ideal(M*v); |
---|
496 | // make leadcoeffs positive |
---|
497 | for (i=1; i<= ncols(J); i++) |
---|
498 | { |
---|
499 | if ( leadcoef(J[i])<0 ) |
---|
500 | { |
---|
501 | J[i] = -J[i]; |
---|
502 | } |
---|
503 | } |
---|
504 | ideal LD1 = J; |
---|
505 | kill J; |
---|
506 | export LD1; |
---|
507 | return(@R); |
---|
508 | } |
---|
509 | example |
---|
510 | { |
---|
511 | "EXAMPLE:"; echo = 2; |
---|
512 | ring r = 0,(x,y),Dp; |
---|
513 | poly F = x^4+y^5+x*y^4; |
---|
514 | printlevel = 0; |
---|
515 | def A = Sannfslog(F); |
---|
516 | setring A; |
---|
517 | LD1; |
---|
518 | } |
---|
519 | |
---|
520 | |
---|
521 | // alternative code for SannfsBM, renamed from annfsBM to ALTannfsBM |
---|
522 | // is superfluos - will not be included in the official documentation |
---|
523 | proc ALTannfsBM (poly F, list #) |
---|
524 | "USAGE: ALTannfsBM(f [,eng]); f a poly, eng an optional int |
---|
525 | RETURN: ring |
---|
526 | PURPOSE: compute the annihilator ideal of f^s in D[s], where D is the Weyl Algebra, according to the algorithm by Briancon and Maisonobe |
---|
527 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
528 | @* - the ideal LD is the annihilator of f^s. |
---|
529 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
530 | @* otherwise, and by default @code{slimgb} is used. |
---|
531 | @* If printlevel=1, progress debug messages will be printed, |
---|
532 | @* if printlevel>=2, all the debug messages will be printed. |
---|
533 | EXAMPLE: example ALTannfsBM; shows examples |
---|
534 | " |
---|
535 | { |
---|
536 | int eng = 0; |
---|
537 | if ( size(#)>0 ) |
---|
538 | { |
---|
539 | if ( typeof(#[1]) == "int" ) |
---|
540 | { |
---|
541 | eng = int(#[1]); |
---|
542 | } |
---|
543 | } |
---|
544 | // returns a list with a ring and an ideal LD in it |
---|
545 | int ppl = printlevel-voice+2; |
---|
546 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
547 | def save = basering; |
---|
548 | int N = nvars(basering); |
---|
549 | int Nnew = 2*N+2; |
---|
550 | int i,j; |
---|
551 | string s; |
---|
552 | list RL = ringlist(basering); |
---|
553 | list L, Lord; |
---|
554 | list tmp; |
---|
555 | intvec iv; |
---|
556 | L[1] = RL[1]; //char |
---|
557 | L[4] = RL[4]; //char, minpoly |
---|
558 | // check whether vars have admissible names |
---|
559 | list Name = RL[2]; |
---|
560 | list RName; |
---|
561 | RName[1] = "t"; |
---|
562 | RName[2] = "s"; |
---|
563 | for (i=1; i<=N; i++) |
---|
564 | { |
---|
565 | for(j=1; j<=size(RName); j++) |
---|
566 | { |
---|
567 | if (Name[i] == RName[j]) |
---|
568 | { |
---|
569 | ERROR("Variable names should not include t,s"); |
---|
570 | } |
---|
571 | } |
---|
572 | } |
---|
573 | // now, create the names for new vars |
---|
574 | list DName; |
---|
575 | for (i=1; i<=N; i++) |
---|
576 | { |
---|
577 | DName[i] = "D"+Name[i]; //concat |
---|
578 | } |
---|
579 | tmp[1] = "t"; |
---|
580 | tmp[2] = "s"; |
---|
581 | list NName = tmp + Name + DName; |
---|
582 | L[2] = NName; |
---|
583 | // Name, Dname will be used further |
---|
584 | kill NName; |
---|
585 | // block ord (lp(2),dp); |
---|
586 | tmp[1] = "lp"; // string |
---|
587 | iv = 1,1; |
---|
588 | tmp[2] = iv; //intvec |
---|
589 | Lord[1] = tmp; |
---|
590 | // continue with dp 1,1,1,1... |
---|
591 | tmp[1] = "dp"; // string |
---|
592 | s = "iv="; |
---|
593 | for (i=1; i<=Nnew; i++) |
---|
594 | { |
---|
595 | s = s+"1,"; |
---|
596 | } |
---|
597 | s[size(s)]= ";"; |
---|
598 | execute(s); |
---|
599 | kill s; |
---|
600 | tmp[2] = iv; |
---|
601 | Lord[2] = tmp; |
---|
602 | tmp[1] = "C"; |
---|
603 | iv = 0; |
---|
604 | tmp[2] = iv; |
---|
605 | Lord[3] = tmp; |
---|
606 | tmp = 0; |
---|
607 | L[3] = Lord; |
---|
608 | // we are done with the list |
---|
609 | def @R@ = ring(L); |
---|
610 | setring @R@; |
---|
611 | matrix @D[Nnew][Nnew]; |
---|
612 | @D[1,2]=t; |
---|
613 | for(i=1; i<=N; i++) |
---|
614 | { |
---|
615 | @D[2+i,N+2+i]=1; |
---|
616 | } |
---|
617 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
618 | // L[6] = @D; |
---|
619 | def @R = nc_algebra(1,@D); |
---|
620 | setring @R; |
---|
621 | kill @R@; |
---|
622 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
623 | dbprint(ppl-1, @R); |
---|
624 | // create the ideal I |
---|
625 | poly F = imap(save,F); |
---|
626 | ideal I = t*F+s; |
---|
627 | poly p; |
---|
628 | for(i=1; i<=N; i++) |
---|
629 | { |
---|
630 | p = t; //t |
---|
631 | p = diff(F,var(2+i))*p; |
---|
632 | I = I, var(N+2+i) + p; |
---|
633 | } |
---|
634 | // -------- the ideal I is ready ---------- |
---|
635 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
636 | dbprint(ppl-1, I); |
---|
637 | ideal J = engine(I,eng); |
---|
638 | ideal K = nselect(J,1); |
---|
639 | kill I,J; |
---|
640 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
641 | dbprint(ppl-1, K); //K is without t |
---|
642 | // create Dn[s], where Dn is the ordinary Weyl Algebra, and put the result into it, |
---|
643 | // thus creating the ring @R2 |
---|
644 | // keep: N, i,j,s, tmp, RL |
---|
645 | setring save; |
---|
646 | Nnew = 2*N+1; |
---|
647 | // list RL = ringlist(save); //is defined earlier |
---|
648 | kill Lord, tmp, iv; |
---|
649 | L = 0; |
---|
650 | list Lord, tmp; |
---|
651 | intvec iv; |
---|
652 | L[1] = RL[1]; |
---|
653 | L[4] = RL[4]; //char, minpoly |
---|
654 | // check whether vars have admissible names -> done earlier |
---|
655 | // list Name = RL[2] |
---|
656 | // DName is defined earlier |
---|
657 | tmp[1] = "s"; |
---|
658 | list NName = Name + DName + tmp; |
---|
659 | L[2] = NName; |
---|
660 | // dp ordering; |
---|
661 | string s = "iv="; |
---|
662 | for (i=1; i<=Nnew; i++) |
---|
663 | { |
---|
664 | s = s+"1,"; |
---|
665 | } |
---|
666 | s[size(s)] = ";"; |
---|
667 | execute(s); |
---|
668 | kill s; |
---|
669 | tmp = 0; |
---|
670 | tmp[1] = "dp"; //string |
---|
671 | tmp[2] = iv; //intvec |
---|
672 | Lord[1] = tmp; |
---|
673 | tmp[1] = "C"; |
---|
674 | iv = 0; |
---|
675 | tmp[2] = iv; |
---|
676 | Lord[2] = tmp; |
---|
677 | tmp = 0; |
---|
678 | L[3] = Lord; |
---|
679 | // we are done with the list |
---|
680 | // Add: Plural part |
---|
681 | def @R2@ = ring(L); |
---|
682 | setring @R2@; |
---|
683 | matrix @D[Nnew][Nnew]; |
---|
684 | for (i=1; i<=N; i++) |
---|
685 | { |
---|
686 | @D[i,N+i]=1; |
---|
687 | } |
---|
688 | def @R2 = nc_algebra(1,@D); |
---|
689 | setring @R2; |
---|
690 | kill @R2@; |
---|
691 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
692 | dbprint(ppl-1, @R2); |
---|
693 | ideal K = imap(@R,K); |
---|
694 | option(redSB); |
---|
695 | //dbprint(ppl,"// -2-2- the final cosmetic std"); |
---|
696 | //K = engine(K,eng); //std does the job too |
---|
697 | // total cleanup |
---|
698 | kill @R; |
---|
699 | ideal LD = K; |
---|
700 | export LD; |
---|
701 | return(@R2); |
---|
702 | } |
---|
703 | example |
---|
704 | { |
---|
705 | "EXAMPLE:"; echo = 2; |
---|
706 | ring r = 0,(x,y,z,w),Dp; |
---|
707 | poly F = x^3+y^3+z^2*w; |
---|
708 | printlevel = 0; |
---|
709 | def A = ALTannfsBM(F); |
---|
710 | setring A; |
---|
711 | LD; |
---|
712 | } |
---|
713 | |
---|
714 | proc bernsteinBM(poly F, list #) |
---|
715 | "USAGE: bernsteinBM(f [,eng]); f a poly, eng an optional int |
---|
716 | RETURN: list of roots of the Bernstein polynomial b and its multiplicies |
---|
717 | PURPOSE: compute the global Bernstein-Sato polynomial for a hypersurface, defined by f, according to the algorithm by Briancon and Maisonobe |
---|
718 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
719 | @* otherwise, and by default @code{slimgb} is used. |
---|
720 | @* If printlevel=1, progress debug messages will be printed, |
---|
721 | @* if printlevel>=2, all the debug messages will be printed. |
---|
722 | EXAMPLE: example bernsteinBM; shows examples |
---|
723 | " |
---|
724 | { |
---|
725 | int eng = 0; |
---|
726 | if ( size(#)>0 ) |
---|
727 | { |
---|
728 | if ( typeof(#[1]) == "int" ) |
---|
729 | { |
---|
730 | eng = int(#[1]); |
---|
731 | } |
---|
732 | } |
---|
733 | // returns a list with a ring and an ideal LD in it |
---|
734 | int ppl = printlevel-voice+2; |
---|
735 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
736 | def save = basering; |
---|
737 | int N = nvars(basering); |
---|
738 | int Nnew = 2*N+2; |
---|
739 | int i,j; |
---|
740 | string s; |
---|
741 | list RL = ringlist(basering); |
---|
742 | list L, Lord; |
---|
743 | list tmp; |
---|
744 | intvec iv; |
---|
745 | L[1] = RL[1]; //char |
---|
746 | L[4] = RL[4]; //char, minpoly |
---|
747 | // check whether vars have admissible names |
---|
748 | list Name = RL[2]; |
---|
749 | list RName; |
---|
750 | RName[1] = "t"; |
---|
751 | RName[2] = "s"; |
---|
752 | for (i=1; i<=N; i++) |
---|
753 | { |
---|
754 | for(j=1; j<=size(RName); j++) |
---|
755 | { |
---|
756 | if (Name[i] == RName[j]) |
---|
757 | { |
---|
758 | ERROR("Variable names should not include t,s"); |
---|
759 | } |
---|
760 | } |
---|
761 | } |
---|
762 | // now, create the names for new vars |
---|
763 | list DName; |
---|
764 | for (i=1; i<=N; i++) |
---|
765 | { |
---|
766 | DName[i] = "D"+Name[i]; //concat |
---|
767 | } |
---|
768 | tmp[1] = "t"; |
---|
769 | tmp[2] = "s"; |
---|
770 | list NName = tmp + Name + DName; |
---|
771 | L[2] = NName; |
---|
772 | // Name, Dname will be used further |
---|
773 | kill NName; |
---|
774 | // block ord (lp(2),dp); |
---|
775 | tmp[1] = "lp"; // string |
---|
776 | iv = 1,1; |
---|
777 | tmp[2] = iv; //intvec |
---|
778 | Lord[1] = tmp; |
---|
779 | // continue with dp 1,1,1,1... |
---|
780 | tmp[1] = "dp"; // string |
---|
781 | s = "iv="; |
---|
782 | for (i=1; i<=Nnew; i++) |
---|
783 | { |
---|
784 | s = s+"1,"; |
---|
785 | } |
---|
786 | s[size(s)]= ";"; |
---|
787 | execute(s); |
---|
788 | kill s; |
---|
789 | tmp[2] = iv; |
---|
790 | Lord[2] = tmp; |
---|
791 | tmp[1] = "C"; |
---|
792 | iv = 0; |
---|
793 | tmp[2] = iv; |
---|
794 | Lord[3] = tmp; |
---|
795 | tmp = 0; |
---|
796 | L[3] = Lord; |
---|
797 | // we are done with the list |
---|
798 | def @R@ = ring(L); |
---|
799 | setring @R@; |
---|
800 | matrix @D[Nnew][Nnew]; |
---|
801 | @D[1,2]=t; |
---|
802 | for(i=1; i<=N; i++) |
---|
803 | { |
---|
804 | @D[2+i,N+2+i]=1; |
---|
805 | } |
---|
806 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
807 | // L[6] = @D; |
---|
808 | def @R = nc_algebra(1,@D); |
---|
809 | setring @R; |
---|
810 | kill @R@; |
---|
811 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
812 | dbprint(ppl-1, @R); |
---|
813 | // create the ideal I |
---|
814 | poly F = imap(save,F); |
---|
815 | ideal I = t*F+s; |
---|
816 | poly p; |
---|
817 | for(i=1; i<=N; i++) |
---|
818 | { |
---|
819 | p = t; //t |
---|
820 | p = diff(F,var(2+i))*p; |
---|
821 | I = I, var(N+2+i) + p; |
---|
822 | } |
---|
823 | // -------- the ideal I is ready ---------- |
---|
824 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
825 | dbprint(ppl-1, I); |
---|
826 | ideal J = engine(I,eng); |
---|
827 | ideal K = nselect(J,1); |
---|
828 | kill I,J; |
---|
829 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
830 | dbprint(ppl-1, K); //K is without t |
---|
831 | // ----------- the ring @R2 ------------ |
---|
832 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
833 | // keep: N, i,j,s, tmp, RL |
---|
834 | setring save; |
---|
835 | Nnew = 2*N+1; |
---|
836 | kill Lord, tmp, iv, RName; |
---|
837 | list Lord, tmp; |
---|
838 | intvec iv; |
---|
839 | L[1] = RL[1]; |
---|
840 | L[4] = RL[4]; //char, minpoly |
---|
841 | // check whether vars hava admissible names -> done earlier |
---|
842 | // now, create the names for new var |
---|
843 | tmp[1] = "s"; |
---|
844 | // DName is defined earlier |
---|
845 | list NName = Name + DName + tmp; |
---|
846 | L[2] = NName; |
---|
847 | tmp = 0; |
---|
848 | // block ord (dp(N),dp); |
---|
849 | string s = "iv="; |
---|
850 | for (i=1; i<=Nnew-1; i++) |
---|
851 | { |
---|
852 | s = s+"1,"; |
---|
853 | } |
---|
854 | s[size(s)]=";"; |
---|
855 | execute(s); |
---|
856 | tmp[1] = "dp"; //string |
---|
857 | tmp[2] = iv; //intvec |
---|
858 | Lord[1] = tmp; |
---|
859 | // continue with dp 1,1,1,1... |
---|
860 | tmp[1] = "dp"; //string |
---|
861 | s[size(s)] = ","; |
---|
862 | s = s+"1;"; |
---|
863 | execute(s); |
---|
864 | kill s; |
---|
865 | kill NName; |
---|
866 | tmp[2] = iv; |
---|
867 | Lord[2] = tmp; |
---|
868 | tmp[1] = "C"; |
---|
869 | iv = 0; |
---|
870 | tmp[2] = iv; |
---|
871 | Lord[3] = tmp; |
---|
872 | tmp = 0; |
---|
873 | L[3] = Lord; |
---|
874 | // we are done with the list. Now add a Plural part |
---|
875 | def @R2@ = ring(L); |
---|
876 | setring @R2@; |
---|
877 | matrix @D[Nnew][Nnew]; |
---|
878 | for (i=1; i<=N; i++) |
---|
879 | { |
---|
880 | @D[i,N+i]=1; |
---|
881 | } |
---|
882 | def @R2 = nc_algebra(1,@D); |
---|
883 | setring @R2; |
---|
884 | kill @R2@; |
---|
885 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
886 | dbprint(ppl-1, @R2); |
---|
887 | ideal MM = maxideal(1); |
---|
888 | MM = 0,s,MM; |
---|
889 | map R01 = @R, MM; |
---|
890 | ideal K = R01(K); |
---|
891 | kill @R, R01; |
---|
892 | poly F = imap(save,F); |
---|
893 | K = K,F; |
---|
894 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
895 | dbprint(ppl-1, K); |
---|
896 | ideal M = engine(K,eng); |
---|
897 | ideal K2 = nselect(M,1,Nnew-1); |
---|
898 | kill K,M; |
---|
899 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
900 | dbprint(ppl-1, K2); |
---|
901 | // the ring @R3 and the search for minimal negative int s |
---|
902 | ring @R3 = 0,s,dp; |
---|
903 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
904 | ideal K3 = imap(@R2,K2); |
---|
905 | kill @R2; |
---|
906 | poly p = K3[1]; |
---|
907 | dbprint(ppl,"// -3-2- factorization"); |
---|
908 | list P = factorize(p); //with constants and multiplicities |
---|
909 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
910 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
911 | { |
---|
912 | bs[i-1] = P[1][i]; |
---|
913 | m[i-1] = P[2][i]; |
---|
914 | } |
---|
915 | // "--------- b-function factorizes into ---------"; P; |
---|
916 | //int sP = minIntRoot(bs,1); |
---|
917 | //dbprint(ppl,"// -3-3- minimal integer root found"); |
---|
918 | //dbprint(ppl-1, sP); |
---|
919 | // convert factors to a list of their roots and multiplicities |
---|
920 | bs = normalize(bs); |
---|
921 | bs = -subst(bs,s,0); |
---|
922 | setring save; |
---|
923 | ideal bs = imap(@R3,bs); |
---|
924 | kill @R3; |
---|
925 | list BS = bs,m; |
---|
926 | return(BS); |
---|
927 | } |
---|
928 | example |
---|
929 | { |
---|
930 | "EXAMPLE:"; echo = 2; |
---|
931 | ring r = 0,(x,y,z,w),Dp; |
---|
932 | poly F = x^3+y^3+z^2*w; |
---|
933 | printlevel = 0; |
---|
934 | bernsteinBM(F); |
---|
935 | } |
---|
936 | |
---|
937 | // some changes |
---|
938 | proc annfsBM (poly F, list #) |
---|
939 | "USAGE: annfsBM(f [,eng]); f a poly, eng an optional int |
---|
940 | RETURN: ring |
---|
941 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according |
---|
942 | to the algorithm by Briancon and Maisonobe |
---|
943 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
944 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
---|
945 | @* which is obtained by substituting the minimal integer root of a Bernstein |
---|
946 | @* polynomial into the s-parametric ideal; |
---|
947 | @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
---|
948 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
949 | @* otherwise, and by default @code{slimgb} is used. |
---|
950 | @* If printlevel=1, progress debug messages will be printed, |
---|
951 | @* if printlevel>=2, all the debug messages will be printed. |
---|
952 | EXAMPLE: example annfsBM; shows examples |
---|
953 | " |
---|
954 | { |
---|
955 | int eng = 0; |
---|
956 | if ( size(#)>0 ) |
---|
957 | { |
---|
958 | if ( typeof(#[1]) == "int" ) |
---|
959 | { |
---|
960 | eng = int(#[1]); |
---|
961 | } |
---|
962 | } |
---|
963 | // returns a list with a ring and an ideal LD in it |
---|
964 | int ppl = printlevel-voice+2; |
---|
965 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
966 | def save = basering; |
---|
967 | int N = nvars(basering); |
---|
968 | int Nnew = 2*N+2; |
---|
969 | int i,j; |
---|
970 | string s; |
---|
971 | list RL = ringlist(basering); |
---|
972 | list L, Lord; |
---|
973 | list tmp; |
---|
974 | intvec iv; |
---|
975 | L[1] = RL[1]; //char |
---|
976 | L[4] = RL[4]; //char, minpoly |
---|
977 | // check whether vars have admissible names |
---|
978 | list Name = RL[2]; |
---|
979 | list RName; |
---|
980 | RName[1] = "t"; |
---|
981 | RName[2] = "s"; |
---|
982 | for (i=1; i<=N; i++) |
---|
983 | { |
---|
984 | for(j=1; j<=size(RName); j++) |
---|
985 | { |
---|
986 | if (Name[i] == RName[j]) |
---|
987 | { |
---|
988 | ERROR("Variable names should not include t,s"); |
---|
989 | } |
---|
990 | } |
---|
991 | } |
---|
992 | // now, create the names for new vars |
---|
993 | list DName; |
---|
994 | for (i=1; i<=N; i++) |
---|
995 | { |
---|
996 | DName[i] = "D"+Name[i]; //concat |
---|
997 | } |
---|
998 | tmp[1] = "t"; |
---|
999 | tmp[2] = "s"; |
---|
1000 | list NName = tmp + Name + DName; |
---|
1001 | L[2] = NName; |
---|
1002 | // Name, Dname will be used further |
---|
1003 | kill NName; |
---|
1004 | // block ord (lp(2),dp); |
---|
1005 | tmp[1] = "lp"; // string |
---|
1006 | iv = 1,1; |
---|
1007 | tmp[2] = iv; //intvec |
---|
1008 | Lord[1] = tmp; |
---|
1009 | // continue with dp 1,1,1,1... |
---|
1010 | tmp[1] = "dp"; // string |
---|
1011 | s = "iv="; |
---|
1012 | for (i=1; i<=Nnew; i++) |
---|
1013 | { |
---|
1014 | s = s+"1,"; |
---|
1015 | } |
---|
1016 | s[size(s)]= ";"; |
---|
1017 | execute(s); |
---|
1018 | kill s; |
---|
1019 | tmp[2] = iv; |
---|
1020 | Lord[2] = tmp; |
---|
1021 | tmp[1] = "C"; |
---|
1022 | iv = 0; |
---|
1023 | tmp[2] = iv; |
---|
1024 | Lord[3] = tmp; |
---|
1025 | tmp = 0; |
---|
1026 | L[3] = Lord; |
---|
1027 | // we are done with the list |
---|
1028 | def @R@ = ring(L); |
---|
1029 | setring @R@; |
---|
1030 | matrix @D[Nnew][Nnew]; |
---|
1031 | @D[1,2]=t; |
---|
1032 | for(i=1; i<=N; i++) |
---|
1033 | { |
---|
1034 | @D[2+i,N+2+i]=1; |
---|
1035 | } |
---|
1036 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1037 | // L[6] = @D; |
---|
1038 | def @R = nc_algebra(1,@D); |
---|
1039 | setring @R; |
---|
1040 | kill @R@; |
---|
1041 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
1042 | dbprint(ppl-1, @R); |
---|
1043 | // create the ideal I |
---|
1044 | poly F = imap(save,F); |
---|
1045 | ideal I = t*F+s; |
---|
1046 | poly p; |
---|
1047 | for(i=1; i<=N; i++) |
---|
1048 | { |
---|
1049 | p = t; //t |
---|
1050 | p = diff(F,var(2+i))*p; |
---|
1051 | I = I, var(N+2+i) + p; |
---|
1052 | } |
---|
1053 | // -------- the ideal I is ready ---------- |
---|
1054 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
1055 | dbprint(ppl-1, I); |
---|
1056 | ideal J = engine(I,eng); |
---|
1057 | ideal K = nselect(J,1); |
---|
1058 | kill I,J; |
---|
1059 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
1060 | dbprint(ppl-1, K); //K is without t |
---|
1061 | setring save; |
---|
1062 | // ----------- the ring @R2 ------------ |
---|
1063 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
1064 | // keep: N, i,j,s, tmp, RL |
---|
1065 | Nnew = 2*N+1; |
---|
1066 | kill Lord, tmp, iv, RName; |
---|
1067 | list Lord, tmp; |
---|
1068 | intvec iv; |
---|
1069 | L[1] = RL[1]; |
---|
1070 | L[4] = RL[4]; //char, minpoly |
---|
1071 | // check whether vars hava admissible names -> done earlier |
---|
1072 | // now, create the names for new var |
---|
1073 | tmp[1] = "s"; |
---|
1074 | // DName is defined earlier |
---|
1075 | list NName = Name + DName + tmp; |
---|
1076 | L[2] = NName; |
---|
1077 | tmp = 0; |
---|
1078 | // block ord (dp(N),dp); |
---|
1079 | string s = "iv="; |
---|
1080 | for (i=1; i<=Nnew-1; i++) |
---|
1081 | { |
---|
1082 | s = s+"1,"; |
---|
1083 | } |
---|
1084 | s[size(s)]=";"; |
---|
1085 | execute(s); |
---|
1086 | tmp[1] = "dp"; //string |
---|
1087 | tmp[2] = iv; //intvec |
---|
1088 | Lord[1] = tmp; |
---|
1089 | // continue with dp 1,1,1,1... |
---|
1090 | tmp[1] = "dp"; //string |
---|
1091 | s[size(s)] = ","; |
---|
1092 | s = s+"1;"; |
---|
1093 | execute(s); |
---|
1094 | kill s; |
---|
1095 | kill NName; |
---|
1096 | tmp[2] = iv; |
---|
1097 | Lord[2] = tmp; |
---|
1098 | tmp[1] = "C"; |
---|
1099 | iv = 0; |
---|
1100 | tmp[2] = iv; |
---|
1101 | Lord[3] = tmp; |
---|
1102 | tmp = 0; |
---|
1103 | L[3] = Lord; |
---|
1104 | // we are done with the list. Now add a Plural part |
---|
1105 | def @R2@ = ring(L); |
---|
1106 | setring @R2@; |
---|
1107 | matrix @D[Nnew][Nnew]; |
---|
1108 | for (i=1; i<=N; i++) |
---|
1109 | { |
---|
1110 | @D[i,N+i]=1; |
---|
1111 | } |
---|
1112 | def @R2 = nc_algebra(1,@D); |
---|
1113 | setring @R2; |
---|
1114 | kill @R2@; |
---|
1115 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
1116 | dbprint(ppl-1, @R2); |
---|
1117 | ideal MM = maxideal(1); |
---|
1118 | MM = 0,s,MM; |
---|
1119 | map R01 = @R, MM; |
---|
1120 | ideal K = R01(K); |
---|
1121 | poly F = imap(save,F); |
---|
1122 | K = K,F; |
---|
1123 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
1124 | dbprint(ppl-1, K); |
---|
1125 | ideal M = engine(K,eng); |
---|
1126 | ideal K2 = nselect(M,1,Nnew-1); |
---|
1127 | kill K,M; |
---|
1128 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
1129 | dbprint(ppl-1, K2); |
---|
1130 | // the ring @R3 and the search for minimal negative int s |
---|
1131 | ring @R3 = 0,s,dp; |
---|
1132 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
1133 | ideal K3 = imap(@R2,K2); |
---|
1134 | poly p = K3[1]; |
---|
1135 | dbprint(ppl,"// -3-2- factorization"); |
---|
1136 | list P = factorize(p); //with constants and multiplicities |
---|
1137 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
1138 | for (i=2; i<= size(P[1]); i++) //we ignore P[1][1] (constant) and P[2][1] (its mult.) |
---|
1139 | { |
---|
1140 | bs[i-1] = P[1][i]; |
---|
1141 | m[i-1] = P[2][i]; |
---|
1142 | } |
---|
1143 | // "--------- b-function factorizes into ---------"; P; |
---|
1144 | int sP = minIntRoot(bs,1); |
---|
1145 | dbprint(ppl,"// -3-3- minimal integer root found"); |
---|
1146 | dbprint(ppl-1, sP); |
---|
1147 | // convert factors to a list of their roots |
---|
1148 | bs = normalize(bs); |
---|
1149 | bs = -subst(bs,s,0); |
---|
1150 | list BS = bs,m; |
---|
1151 | //TODO: sort BS! |
---|
1152 | // --------- substitute s found in the ideal --------- |
---|
1153 | // --------- going back to @R and substitute --------- |
---|
1154 | setring @R; |
---|
1155 | ideal K2 = subst(K,s,sP); |
---|
1156 | kill K; |
---|
1157 | // create the ordinary Weyl algebra and put the result into it, |
---|
1158 | // thus creating the ring @R4 |
---|
1159 | // keep: N, i,j,s, tmp, RL |
---|
1160 | setring save; |
---|
1161 | Nnew = 2*N; |
---|
1162 | // list RL = ringlist(save); //is defined earlier |
---|
1163 | kill Lord, tmp, iv; |
---|
1164 | L = 0; |
---|
1165 | list Lord, tmp; |
---|
1166 | intvec iv; |
---|
1167 | L[1] = RL[1]; |
---|
1168 | L[4] = RL[4]; //char, minpoly |
---|
1169 | // check whether vars have admissible names -> done earlier |
---|
1170 | // list Name = RL[2]M |
---|
1171 | // DName is defined earlier |
---|
1172 | list NName = Name + DName; |
---|
1173 | L[2] = NName; |
---|
1174 | // dp ordering; |
---|
1175 | string s = "iv="; |
---|
1176 | for (i=1; i<=Nnew; i++) |
---|
1177 | { |
---|
1178 | s = s+"1,"; |
---|
1179 | } |
---|
1180 | s[size(s)] = ";"; |
---|
1181 | execute(s); |
---|
1182 | kill s; |
---|
1183 | tmp = 0; |
---|
1184 | tmp[1] = "dp"; //string |
---|
1185 | tmp[2] = iv; //intvec |
---|
1186 | Lord[1] = tmp; |
---|
1187 | tmp[1] = "C"; |
---|
1188 | iv = 0; |
---|
1189 | tmp[2] = iv; |
---|
1190 | Lord[2] = tmp; |
---|
1191 | tmp = 0; |
---|
1192 | L[3] = Lord; |
---|
1193 | // we are done with the list |
---|
1194 | // Add: Plural part |
---|
1195 | def @R4@ = ring(L); |
---|
1196 | setring @R4@; |
---|
1197 | matrix @D[Nnew][Nnew]; |
---|
1198 | for (i=1; i<=N; i++) |
---|
1199 | { |
---|
1200 | @D[i,N+i]=1; |
---|
1201 | } |
---|
1202 | def @R4 = nc_algebra(1,@D); |
---|
1203 | setring @R4; |
---|
1204 | kill @R4@; |
---|
1205 | dbprint(ppl,"// -4-1- the ring @R4(_x,_Dx) is ready"); |
---|
1206 | dbprint(ppl-1, @R4); |
---|
1207 | ideal K4 = imap(@R,K2); |
---|
1208 | option(redSB); |
---|
1209 | dbprint(ppl,"// -4-2- the final cosmetic std"); |
---|
1210 | K4 = engine(K4,eng); //std does the job too |
---|
1211 | // total cleanup |
---|
1212 | kill @R; |
---|
1213 | kill @R2; |
---|
1214 | list BS = imap(@R3,BS); |
---|
1215 | export BS; |
---|
1216 | kill @R3; |
---|
1217 | ideal LD = K4; |
---|
1218 | export LD; |
---|
1219 | return(@R4); |
---|
1220 | } |
---|
1221 | example |
---|
1222 | { |
---|
1223 | "EXAMPLE:"; echo = 2; |
---|
1224 | ring r = 0,(x,y,z),Dp; |
---|
1225 | poly F = z*x^2+y^3; |
---|
1226 | printlevel = 0; |
---|
1227 | def A = annfsBM(F); |
---|
1228 | setring A; |
---|
1229 | LD; |
---|
1230 | BS; |
---|
1231 | } |
---|
1232 | |
---|
1233 | proc operatorBM(poly F, list #) |
---|
1234 | "USAGE: operatorBM(f [,eng]); f a poly, eng an optional int |
---|
1235 | RETURN: ring |
---|
1236 | PURPOSE: compute the B-operator and other relevant data for Ann F^s, according to the algorithm by Briancon and Maisonobe |
---|
1237 | NOTE: activate this ring with the @code{setring} command. In this ring D[s] |
---|
1238 | @* - the polynomial F is the same as the input, |
---|
1239 | @* - the ideal LD is the annihilator of f^s in Dn[s], |
---|
1240 | @* - the ideal LD0 is the needed D-mod structure, where LD0 = LD|s=s0, |
---|
1241 | @* - the polynomial bs is the global Bernstein polynomial of f in the variable s, |
---|
1242 | @* - the list BS contains all the roots with multiplicities of the global Bernstein polynomial of f, |
---|
1243 | @* - the polynomial PS is an operator in Dn[s] such that PS*f^(s+1) = bs*f^s. |
---|
1244 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
1245 | @* otherwise and by default @code{slimgb} is used. |
---|
1246 | @* If printlevel=1, progress debug messages will be printed, |
---|
1247 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1248 | EXAMPLE: example operatorBM; shows examples |
---|
1249 | " |
---|
1250 | { |
---|
1251 | int eng = 0; |
---|
1252 | if ( size(#)>0 ) |
---|
1253 | { |
---|
1254 | if ( typeof(#[1]) == "int" ) |
---|
1255 | { |
---|
1256 | eng = int(#[1]); |
---|
1257 | } |
---|
1258 | } |
---|
1259 | // returns a list with a ring and an ideal LD in it |
---|
1260 | int ppl = printlevel-voice+2; |
---|
1261 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
1262 | def save = basering; |
---|
1263 | int N = nvars(basering); |
---|
1264 | int Nnew = 2*N+2; |
---|
1265 | int i,j; |
---|
1266 | string s; |
---|
1267 | list RL = ringlist(basering); |
---|
1268 | list L, Lord; |
---|
1269 | list tmp; |
---|
1270 | intvec iv; |
---|
1271 | L[1] = RL[1]; //char |
---|
1272 | L[4] = RL[4]; //char, minpoly |
---|
1273 | // check whether vars have admissible names |
---|
1274 | list Name = RL[2]; |
---|
1275 | list RName; |
---|
1276 | RName[1] = "t"; |
---|
1277 | RName[2] = "s"; |
---|
1278 | for (i=1; i<=N; i++) |
---|
1279 | { |
---|
1280 | for(j=1; j<=size(RName); j++) |
---|
1281 | { |
---|
1282 | if (Name[i] == RName[j]) |
---|
1283 | { |
---|
1284 | ERROR("Variable names should not include t,s"); |
---|
1285 | } |
---|
1286 | } |
---|
1287 | } |
---|
1288 | // now, create the names for new vars |
---|
1289 | list DName; |
---|
1290 | for (i=1; i<=N; i++) |
---|
1291 | { |
---|
1292 | DName[i] = "D"+Name[i]; //concat |
---|
1293 | } |
---|
1294 | tmp[1] = "t"; |
---|
1295 | tmp[2] = "s"; |
---|
1296 | list NName = tmp + Name + DName; |
---|
1297 | L[2] = NName; |
---|
1298 | // Name, Dname will be used further |
---|
1299 | kill NName; |
---|
1300 | // block ord (lp(2),dp); |
---|
1301 | tmp[1] = "lp"; // string |
---|
1302 | iv = 1,1; |
---|
1303 | tmp[2] = iv; //intvec |
---|
1304 | Lord[1] = tmp; |
---|
1305 | // continue with dp 1,1,1,1... |
---|
1306 | tmp[1] = "dp"; // string |
---|
1307 | s = "iv="; |
---|
1308 | for (i=1; i<=Nnew; i++) |
---|
1309 | { |
---|
1310 | s = s+"1,"; |
---|
1311 | } |
---|
1312 | s[size(s)]= ";"; |
---|
1313 | execute(s); |
---|
1314 | kill s; |
---|
1315 | tmp[2] = iv; |
---|
1316 | Lord[2] = tmp; |
---|
1317 | tmp[1] = "C"; |
---|
1318 | iv = 0; |
---|
1319 | tmp[2] = iv; |
---|
1320 | Lord[3] = tmp; |
---|
1321 | tmp = 0; |
---|
1322 | L[3] = Lord; |
---|
1323 | // we are done with the list |
---|
1324 | def @R@ = ring(L); |
---|
1325 | setring @R@; |
---|
1326 | matrix @D[Nnew][Nnew]; |
---|
1327 | @D[1,2]=t; |
---|
1328 | for(i=1; i<=N; i++) |
---|
1329 | { |
---|
1330 | @D[2+i,N+2+i]=1; |
---|
1331 | } |
---|
1332 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1333 | // L[6] = @D; |
---|
1334 | def @R = nc_algebra(1,@D); |
---|
1335 | setring @R; |
---|
1336 | kill @R@; |
---|
1337 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
1338 | dbprint(ppl-1, @R); |
---|
1339 | // create the ideal I |
---|
1340 | poly F = imap(save,F); |
---|
1341 | ideal I = t*F+s; |
---|
1342 | poly p; |
---|
1343 | for(i=1; i<=N; i++) |
---|
1344 | { |
---|
1345 | p = t; //t |
---|
1346 | p = diff(F,var(2+i))*p; |
---|
1347 | I = I, var(N+2+i) + p; |
---|
1348 | } |
---|
1349 | // -------- the ideal I is ready ---------- |
---|
1350 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
1351 | dbprint(ppl-1, I); |
---|
1352 | ideal J = engine(I,eng); |
---|
1353 | ideal K = nselect(J,1); |
---|
1354 | kill I,J; |
---|
1355 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
1356 | dbprint(ppl-1, K); //K is without t |
---|
1357 | setring save; |
---|
1358 | // ----------- the ring @R2 ------------ |
---|
1359 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
1360 | // keep: N, i,j,s, tmp, RL |
---|
1361 | Nnew = 2*N+1; |
---|
1362 | kill Lord, tmp, iv, RName; |
---|
1363 | list Lord, tmp; |
---|
1364 | intvec iv; |
---|
1365 | L[1] = RL[1]; |
---|
1366 | L[4] = RL[4]; //char, minpoly |
---|
1367 | // check whether vars hava admissible names -> done earlier |
---|
1368 | // now, create the names for new var |
---|
1369 | tmp[1] = "s"; |
---|
1370 | // DName is defined earlier |
---|
1371 | list NName = Name + DName + tmp; |
---|
1372 | L[2] = NName; |
---|
1373 | tmp = 0; |
---|
1374 | // block ord (dp(N),dp); |
---|
1375 | string s = "iv="; |
---|
1376 | for (i=1; i<=Nnew-1; i++) |
---|
1377 | { |
---|
1378 | s = s+"1,"; |
---|
1379 | } |
---|
1380 | s[size(s)]=";"; |
---|
1381 | execute(s); |
---|
1382 | tmp[1] = "dp"; //string |
---|
1383 | tmp[2] = iv; //intvec |
---|
1384 | Lord[1] = tmp; |
---|
1385 | // continue with dp 1,1,1,1... |
---|
1386 | tmp[1] = "dp"; //string |
---|
1387 | s[size(s)] = ","; |
---|
1388 | s = s+"1;"; |
---|
1389 | execute(s); |
---|
1390 | kill s; |
---|
1391 | kill NName; |
---|
1392 | tmp[2] = iv; |
---|
1393 | Lord[2] = tmp; |
---|
1394 | tmp[1] = "C"; |
---|
1395 | iv = 0; |
---|
1396 | tmp[2] = iv; |
---|
1397 | Lord[3] = tmp; |
---|
1398 | tmp = 0; |
---|
1399 | L[3] = Lord; |
---|
1400 | // we are done with the list. Now add a Plural part |
---|
1401 | def @R2@ = ring(L); |
---|
1402 | setring @R2@; |
---|
1403 | matrix @D[Nnew][Nnew]; |
---|
1404 | for (i=1; i<=N; i++) |
---|
1405 | { |
---|
1406 | @D[i,N+i]=1; |
---|
1407 | } |
---|
1408 | def @R2 = nc_algebra(1,@D); |
---|
1409 | setring @R2; |
---|
1410 | kill @R2@; |
---|
1411 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
1412 | dbprint(ppl-1, @R2); |
---|
1413 | ideal MM = maxideal(1); |
---|
1414 | MM = 0,s,MM; |
---|
1415 | map R01 = @R, MM; |
---|
1416 | ideal K = R01(K); |
---|
1417 | poly F = imap(save,F); |
---|
1418 | K = K,F; |
---|
1419 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
1420 | dbprint(ppl-1, K); |
---|
1421 | ideal M = engine(K,eng); |
---|
1422 | ideal K2 = nselect(M,1,Nnew-1); |
---|
1423 | kill K,M; |
---|
1424 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
1425 | dbprint(ppl-1, K2); |
---|
1426 | // the ring @R3 and the search for minimal negative int s |
---|
1427 | ring @R3 = 0,s,dp; |
---|
1428 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
1429 | ideal K3 = imap(@R2,K2); |
---|
1430 | kill @R2; |
---|
1431 | poly p = K3[1]; |
---|
1432 | dbprint(ppl,"// -3-2- factorization"); |
---|
1433 | list P = factorize(p); //with constants and multiplicities |
---|
1434 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
1435 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
1436 | { |
---|
1437 | bs[i-1] = P[1][i]; |
---|
1438 | m[i-1] = P[2][i]; |
---|
1439 | } |
---|
1440 | // "--------- b-function factorizes into ---------"; P; |
---|
1441 | int sP = minIntRoot(bs,1); |
---|
1442 | dbprint(ppl,"// -3-3- minimal integer root found"); |
---|
1443 | dbprint(ppl-1, sP); |
---|
1444 | // convert factors to a list of their roots with multiplicities |
---|
1445 | bs = normalize(bs); |
---|
1446 | bs = -subst(bs,s,0); |
---|
1447 | list BS = bs,m; |
---|
1448 | //TODO: sort BS! |
---|
1449 | // --------- substitute s found in the ideal --------- |
---|
1450 | // --------- going back to @R and substitute --------- |
---|
1451 | setring @R; |
---|
1452 | ideal K2 = subst(K,s,sP); |
---|
1453 | // create Dn[s], where Dn is the ordinary Weyl algebra, and put the result into it, |
---|
1454 | // thus creating the ring @R4 |
---|
1455 | // keep: N, i,j,s, tmp, RL |
---|
1456 | setring save; |
---|
1457 | Nnew = 2*N+1; |
---|
1458 | // list RL = ringlist(save); //is defined earlier |
---|
1459 | kill Lord, tmp, iv; |
---|
1460 | L = 0; |
---|
1461 | list Lord, tmp; |
---|
1462 | intvec iv; |
---|
1463 | L[1] = RL[1]; |
---|
1464 | L[4] = RL[4]; //char, minpoly |
---|
1465 | // check whether vars have admissible names -> done earlier |
---|
1466 | // list Name = RL[2] |
---|
1467 | // DName is defined earlier |
---|
1468 | tmp[1] = "s"; |
---|
1469 | list NName = Name + DName + tmp; |
---|
1470 | L[2] = NName; |
---|
1471 | // dp ordering; |
---|
1472 | string s = "iv="; |
---|
1473 | for (i=1; i<=Nnew; i++) |
---|
1474 | { |
---|
1475 | s = s+"1,"; |
---|
1476 | } |
---|
1477 | s[size(s)] = ";"; |
---|
1478 | execute(s); |
---|
1479 | kill s; |
---|
1480 | tmp = 0; |
---|
1481 | tmp[1] = "dp"; //string |
---|
1482 | tmp[2] = iv; //intvec |
---|
1483 | Lord[1] = tmp; |
---|
1484 | tmp[1] = "C"; |
---|
1485 | iv = 0; |
---|
1486 | tmp[2] = iv; |
---|
1487 | Lord[2] = tmp; |
---|
1488 | tmp = 0; |
---|
1489 | L[3] = Lord; |
---|
1490 | // we are done with the list |
---|
1491 | // Add: Plural part |
---|
1492 | def @R4@ = ring(L); |
---|
1493 | setring @R4@; |
---|
1494 | matrix @D[Nnew][Nnew]; |
---|
1495 | for (i=1; i<=N; i++) |
---|
1496 | { |
---|
1497 | @D[i,N+i]=1; |
---|
1498 | } |
---|
1499 | def @R4 = nc_algebra(1,@D); |
---|
1500 | setring @R4; |
---|
1501 | kill @R4@; |
---|
1502 | dbprint(ppl,"// -4-1- the ring @R4(_x,_Dx,s) is ready"); |
---|
1503 | dbprint(ppl-1, @R4); |
---|
1504 | ideal LD0 = imap(@R,K2); |
---|
1505 | ideal LD = imap(@R,K); |
---|
1506 | kill @R; |
---|
1507 | poly bs = imap(@R3,p); |
---|
1508 | list BS = imap(@R3,BS); |
---|
1509 | kill @R3; |
---|
1510 | bs = normalize(bs); |
---|
1511 | poly F = imap(save,F); |
---|
1512 | dbprint(ppl,"// -4-2- starting the computation of PS via lift"); |
---|
1513 | //better liftstd, I didn't knot it works also for Plural, liftslimgb? |
---|
1514 | // liftstd may give extra coeffs in the resulting ideal |
---|
1515 | matrix T = lift(F+LD,bs); |
---|
1516 | poly PS = T[1,1]; |
---|
1517 | dbprint(ppl,"// -4-3- an operator PS found, PS*f^(s+1) = b(s)*f^s"); |
---|
1518 | dbprint(ppl-1,PS); |
---|
1519 | option(redSB); |
---|
1520 | dbprint(ppl,"// -4-4- the final cosmetic std"); |
---|
1521 | LD0 = engine(LD0,eng); //std does the job too |
---|
1522 | LD = engine(LD,eng); |
---|
1523 | export F,LD,LD0,bs,BS,PS; |
---|
1524 | return(@R4); |
---|
1525 | } |
---|
1526 | example |
---|
1527 | { |
---|
1528 | "EXAMPLE:"; echo = 2; |
---|
1529 | // ring r = 0,(x,y,z,w),Dp; |
---|
1530 | // poly F = x^3+y^3+z^2*w; |
---|
1531 | ring r = 0,(x,y,z),Dp; |
---|
1532 | poly F = x^3+y^3+z^3; |
---|
1533 | printlevel = 0; |
---|
1534 | def A = operatorBM(F); |
---|
1535 | setring A; |
---|
1536 | F; // the original polynomial itself |
---|
1537 | LD; // generic annihilator |
---|
1538 | LD0; // annihilator |
---|
1539 | bs; // normalized Bernstein poly |
---|
1540 | BS; // root and multiplicities of the Bernstein poly |
---|
1541 | PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD |
---|
1542 | reduce(PS*F-bs,LD); // check the property of PS |
---|
1543 | } |
---|
1544 | |
---|
1545 | proc annfsParamBM (poly F, list #) |
---|
1546 | "USAGE: annfsParamBM(f [,eng]); f a poly, eng an optional int |
---|
1547 | RETURN: ring |
---|
1548 | PURPOSE: compute the generic Ann F^s and exceptional parametric constellations of a polynomial with parametric coefficients, according to the algorithm by Briancon and Maisonobe |
---|
1549 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
1550 | @* - the ideal LD is the D-module structure oa Ann F^s |
---|
1551 | @* - the ideal Param contains the list of the special parameters. |
---|
1552 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
1553 | @* otherwise, and by default @code{slimgb} is used. |
---|
1554 | @* If printlevel=1, progress debug messages will be printed, |
---|
1555 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1556 | EXAMPLE: example annfsParamBM; shows examples |
---|
1557 | " |
---|
1558 | { |
---|
1559 | //PURPOSE: compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients, according to the algorithm by Briancon and Maisonobe |
---|
1560 | // @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
---|
1561 | // ***** not implented yet **** |
---|
1562 | int eng = 0; |
---|
1563 | if ( size(#)>0 ) |
---|
1564 | { |
---|
1565 | if ( typeof(#[1]) == "int" ) |
---|
1566 | { |
---|
1567 | eng = int(#[1]); |
---|
1568 | } |
---|
1569 | } |
---|
1570 | // returns a list with a ring and an ideal LD in it |
---|
1571 | int ppl = printlevel-voice+2; |
---|
1572 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
1573 | def save = basering; |
---|
1574 | int N = nvars(basering); |
---|
1575 | int Nnew = 2*N+2; |
---|
1576 | int i,j; |
---|
1577 | string s; |
---|
1578 | list RL = ringlist(basering); |
---|
1579 | list L, Lord; |
---|
1580 | list tmp; |
---|
1581 | intvec iv; |
---|
1582 | L[1] = RL[1]; //char |
---|
1583 | L[4] = RL[4]; //char, minpoly |
---|
1584 | // check whether vars have admissible names |
---|
1585 | list Name = RL[2]; |
---|
1586 | list RName; |
---|
1587 | RName[1] = "t"; |
---|
1588 | RName[2] = "s"; |
---|
1589 | for (i=1; i<=N; i++) |
---|
1590 | { |
---|
1591 | for(j=1; j<=size(RName); j++) |
---|
1592 | { |
---|
1593 | if (Name[i] == RName[j]) |
---|
1594 | { |
---|
1595 | ERROR("Variable names should not include t,s"); |
---|
1596 | } |
---|
1597 | } |
---|
1598 | } |
---|
1599 | // now, create the names for new vars |
---|
1600 | list DName; |
---|
1601 | for (i=1; i<=N; i++) |
---|
1602 | { |
---|
1603 | DName[i] = "D"+Name[i]; //concat |
---|
1604 | } |
---|
1605 | tmp[1] = "t"; |
---|
1606 | tmp[2] = "s"; |
---|
1607 | list NName = tmp + Name + DName; |
---|
1608 | L[2] = NName; |
---|
1609 | // Name, Dname will be used further |
---|
1610 | kill NName; |
---|
1611 | // block ord (lp(2),dp); |
---|
1612 | tmp[1] = "lp"; // string |
---|
1613 | iv = 1,1; |
---|
1614 | tmp[2] = iv; //intvec |
---|
1615 | Lord[1] = tmp; |
---|
1616 | // continue with dp 1,1,1,1... |
---|
1617 | tmp[1] = "dp"; // string |
---|
1618 | s = "iv="; |
---|
1619 | for (i=1; i<=Nnew; i++) |
---|
1620 | { |
---|
1621 | s = s+"1,"; |
---|
1622 | } |
---|
1623 | s[size(s)]= ";"; |
---|
1624 | execute(s); |
---|
1625 | kill s; |
---|
1626 | tmp[2] = iv; |
---|
1627 | Lord[2] = tmp; |
---|
1628 | tmp[1] = "C"; |
---|
1629 | iv = 0; |
---|
1630 | tmp[2] = iv; |
---|
1631 | Lord[3] = tmp; |
---|
1632 | tmp = 0; |
---|
1633 | L[3] = Lord; |
---|
1634 | // we are done with the list |
---|
1635 | def @R@ = ring(L); |
---|
1636 | setring @R@; |
---|
1637 | matrix @D[Nnew][Nnew]; |
---|
1638 | @D[1,2]=t; |
---|
1639 | for(i=1; i<=N; i++) |
---|
1640 | { |
---|
1641 | @D[2+i,N+2+i]=1; |
---|
1642 | } |
---|
1643 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1644 | // L[6] = @D; |
---|
1645 | def @R = nc_algebra(1,@D); |
---|
1646 | setring @R; |
---|
1647 | kill @R@; |
---|
1648 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
1649 | dbprint(ppl-1, @R); |
---|
1650 | // create the ideal I |
---|
1651 | poly F = imap(save,F); |
---|
1652 | ideal I = t*F+s; |
---|
1653 | poly p; |
---|
1654 | for(i=1; i<=N; i++) |
---|
1655 | { |
---|
1656 | p = t; //t |
---|
1657 | p = diff(F,var(2+i))*p; |
---|
1658 | I = I, var(N+2+i) + p; |
---|
1659 | } |
---|
1660 | // -------- the ideal I is ready ---------- |
---|
1661 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
1662 | dbprint(ppl-1, I); |
---|
1663 | ideal J = engine(I,eng); |
---|
1664 | ideal K = nselect(J,1); |
---|
1665 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
1666 | dbprint(ppl-1, K); //K is without t |
---|
1667 | // ----- looking for special parameters ----- |
---|
1668 | dbprint(ppl,"// -2-1- starting the computation of the transformation matrix (via lift)"); |
---|
1669 | J = normalize(J); |
---|
1670 | matrix T = lift(I,J); //try also with liftstd |
---|
1671 | kill I,J; |
---|
1672 | dbprint(ppl,"// -2-2- the transformation matrix has been computed"); |
---|
1673 | dbprint(ppl-1, T); //T is the transformation matrix |
---|
1674 | dbprint(ppl,"// -2-3- genericity does the job"); |
---|
1675 | list lParam = genericity(T); |
---|
1676 | int ip = size(lParam); |
---|
1677 | int cip; |
---|
1678 | string sParam; |
---|
1679 | if (sParam[1]=="-") { sParam=""; } //genericity returns "-" |
---|
1680 | // if no parameters exist in a basering |
---|
1681 | for (cip=1; cip <= ip; cip++) |
---|
1682 | { |
---|
1683 | sParam = sParam + "," +lParam[cip]; |
---|
1684 | } |
---|
1685 | if (size(sParam) >=2) |
---|
1686 | { |
---|
1687 | sParam = sParam[2..size(sParam)]; // removes the 1st colon |
---|
1688 | } |
---|
1689 | export sParam; |
---|
1690 | kill T; |
---|
1691 | dbprint(ppl,"// -2-4- the special parameters has been computed"); |
---|
1692 | dbprint(ppl, sParam); |
---|
1693 | // create Dn[s], where Dn is the ordinary Weyl Algebra, and put the result into it, |
---|
1694 | // thus creating the ring @R2 |
---|
1695 | // keep: N, i,j,s, tmp, RL |
---|
1696 | setring save; |
---|
1697 | Nnew = 2*N+1; |
---|
1698 | // list RL = ringlist(save); //is defined earlier |
---|
1699 | kill Lord, tmp, iv; |
---|
1700 | L = 0; |
---|
1701 | list Lord, tmp; |
---|
1702 | intvec iv; |
---|
1703 | L[1] = RL[1]; |
---|
1704 | L[4] = RL[4]; //char, minpoly |
---|
1705 | // check whether vars have admissible names -> done earlier |
---|
1706 | // list Name = RL[2]M |
---|
1707 | // DName is defined earlier |
---|
1708 | tmp[1] = "s"; |
---|
1709 | list NName = Name + DName + tmp; |
---|
1710 | L[2] = NName; |
---|
1711 | // dp ordering; |
---|
1712 | string s = "iv="; |
---|
1713 | for (i=1; i<=Nnew; i++) |
---|
1714 | { |
---|
1715 | s = s+"1,"; |
---|
1716 | } |
---|
1717 | s[size(s)] = ";"; |
---|
1718 | execute(s); |
---|
1719 | kill s; |
---|
1720 | tmp = 0; |
---|
1721 | tmp[1] = "dp"; //string |
---|
1722 | tmp[2] = iv; //intvec |
---|
1723 | Lord[1] = tmp; |
---|
1724 | tmp[1] = "C"; |
---|
1725 | iv = 0; |
---|
1726 | tmp[2] = iv; |
---|
1727 | Lord[2] = tmp; |
---|
1728 | tmp = 0; |
---|
1729 | L[3] = Lord; |
---|
1730 | // we are done with the list |
---|
1731 | // Add: Plural part |
---|
1732 | def @R2@ = ring(L); |
---|
1733 | setring @R2@; |
---|
1734 | matrix @D[Nnew][Nnew]; |
---|
1735 | for (i=1; i<=N; i++) |
---|
1736 | { |
---|
1737 | @D[i,N+i]=1; |
---|
1738 | } |
---|
1739 | def @R2 = nc_algebra(1,@D); |
---|
1740 | setring @R2; |
---|
1741 | kill @R2@; |
---|
1742 | dbprint(ppl,"// -3-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
1743 | dbprint(ppl-1, @R2); |
---|
1744 | ideal K = imap(@R,K); |
---|
1745 | kill @R; |
---|
1746 | option(redSB); |
---|
1747 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
1748 | K = engine(K,eng); //std does the job too |
---|
1749 | ideal LD = K; |
---|
1750 | export LD; |
---|
1751 | if (sParam[1] == ",") |
---|
1752 | { |
---|
1753 | sParam = sParam[2..size(sParam)]; |
---|
1754 | } |
---|
1755 | // || ((sParam[1] == " ") && (sParam[2] == ","))) |
---|
1756 | execute("ideal Param ="+sParam+";"); |
---|
1757 | export Param; |
---|
1758 | kill sParam; |
---|
1759 | return(@R2); |
---|
1760 | } |
---|
1761 | example |
---|
1762 | { |
---|
1763 | "EXAMPLE:"; echo = 2; |
---|
1764 | ring r = (0,a,b),(x,y),Dp; |
---|
1765 | poly F = x^2 - (y-a)*(y-b); |
---|
1766 | printlevel = 0; |
---|
1767 | def A = annfsParamBM(F); setring A; |
---|
1768 | LD; |
---|
1769 | Param; |
---|
1770 | setring r; |
---|
1771 | poly G = x2-(y-a)^2; // try the exceptional value b=a of parameters |
---|
1772 | def B = annfsParamBM(G); setring B; |
---|
1773 | LD; |
---|
1774 | Param; |
---|
1775 | } |
---|
1776 | |
---|
1777 | // *** the following example is nice, but too complicated for the documentation *** |
---|
1778 | // ring r = (0,a),(x,y,z),Dp; |
---|
1779 | // poly F = x^4+y^4+z^2+a*x*y*z; |
---|
1780 | // printlevel = 2; //0 |
---|
1781 | // def A = annfsParamBM(F); |
---|
1782 | // setring A; |
---|
1783 | // LD; |
---|
1784 | // Param; |
---|
1785 | |
---|
1786 | |
---|
1787 | proc annfsBMI(ideal F, list #) |
---|
1788 | "USAGE: annfsBMI(F [,eng]); F an ideal, eng an optional int |
---|
1789 | RETURN: ring |
---|
1790 | PURPOSE: compute the D-module structure of basering[1/f]*f^s where f = F[1]*..*F[P], |
---|
1791 | according to the algorithm by Briancon and Maisonobe. |
---|
1792 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
1793 | @* - the ideal LD is the needed D-mod structure, |
---|
1794 | @* - the list BS is the Bernstein ideal of a polynomial f = F[1]*..*F[P]. |
---|
1795 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
1796 | @* otherwise, and by default @code{slimgb} is used. |
---|
1797 | @* If printlevel=1, progress debug messages will be printed, |
---|
1798 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1799 | EXAMPLE: example annfsBMI; shows examples |
---|
1800 | " |
---|
1801 | { |
---|
1802 | int eng = 0; |
---|
1803 | if ( size(#)>0 ) |
---|
1804 | { |
---|
1805 | if ( typeof(#[1]) == "int" ) |
---|
1806 | { |
---|
1807 | eng = int(#[1]); |
---|
1808 | } |
---|
1809 | } |
---|
1810 | // returns a list with a ring and an ideal LD in it |
---|
1811 | int ppl = printlevel-voice+2; |
---|
1812 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
1813 | def save = basering; |
---|
1814 | int N = nvars(basering); |
---|
1815 | int P = size(F); //if F has some generators which are zero, int P = ncols(I); |
---|
1816 | int Nnew = 2*N+2*P; |
---|
1817 | int i,j; |
---|
1818 | string s; |
---|
1819 | list RL = ringlist(basering); |
---|
1820 | list L, Lord; |
---|
1821 | list tmp; |
---|
1822 | intvec iv; |
---|
1823 | L[1] = RL[1]; //char |
---|
1824 | L[4] = RL[4]; //char, minpoly |
---|
1825 | // check whether vars have admissible names |
---|
1826 | list Name = RL[2]; |
---|
1827 | list RName; |
---|
1828 | for (j=1; j<=P; j++) |
---|
1829 | { |
---|
1830 | RName[j] = "t("+string(j)+")"; |
---|
1831 | RName[j+P] = "s("+string(j)+")"; |
---|
1832 | } |
---|
1833 | for(i=1; i<=N; i++) |
---|
1834 | { |
---|
1835 | for(j=1; j<=size(RName); j++) |
---|
1836 | { |
---|
1837 | if (Name[i] == RName[j]) |
---|
1838 | { ERROR("Variable names should not include t(i),s(i)"); } |
---|
1839 | } |
---|
1840 | } |
---|
1841 | // now, create the names for new vars |
---|
1842 | list DName; |
---|
1843 | for(i=1; i<=N; i++) |
---|
1844 | { |
---|
1845 | DName[i] = "D"+Name[i]; //concat |
---|
1846 | } |
---|
1847 | list NName = RName + Name + DName; |
---|
1848 | L[2] = NName; |
---|
1849 | // Name, Dname will be used further |
---|
1850 | kill NName; |
---|
1851 | // block ord (lp(P),dp); |
---|
1852 | tmp[1] = "lp"; //string |
---|
1853 | s = "iv="; |
---|
1854 | for (i=1; i<=2*P; i++) |
---|
1855 | { |
---|
1856 | s = s+"1,"; |
---|
1857 | } |
---|
1858 | s[size(s)]= ";"; |
---|
1859 | execute(s); |
---|
1860 | tmp[2] = iv; //intvec |
---|
1861 | Lord[1] = tmp; |
---|
1862 | // continue with dp 1,1,1,1... |
---|
1863 | tmp[1] = "dp"; //string |
---|
1864 | s = "iv="; |
---|
1865 | for (i=1; i<=Nnew; i++) //actually i<=2*N |
---|
1866 | { |
---|
1867 | s = s+"1,"; |
---|
1868 | } |
---|
1869 | s[size(s)]= ";"; |
---|
1870 | execute(s); |
---|
1871 | kill s; |
---|
1872 | tmp[2] = iv; |
---|
1873 | Lord[2] = tmp; |
---|
1874 | tmp[1] = "C"; |
---|
1875 | iv = 0; |
---|
1876 | tmp[2] = iv; |
---|
1877 | Lord[3] = tmp; |
---|
1878 | tmp = 0; |
---|
1879 | L[3] = Lord; |
---|
1880 | // we are done with the list |
---|
1881 | def @R@ = ring(L); |
---|
1882 | setring @R@; |
---|
1883 | matrix @D[Nnew][Nnew]; |
---|
1884 | for (i=1; i<=P; i++) |
---|
1885 | { |
---|
1886 | @D[i,i+P] = t(i); |
---|
1887 | } |
---|
1888 | for(i=1; i<=N; i++) |
---|
1889 | { |
---|
1890 | @D[2*P+i,2*P+N+i] = 1; |
---|
1891 | } |
---|
1892 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1893 | // L[6] = @D; |
---|
1894 | def @R = nc_algebra(1,@D); |
---|
1895 | setring @R; |
---|
1896 | kill @R@; |
---|
1897 | dbprint(ppl,"// -1-1- the ring @R(_t,_s,_x,_Dx) is ready"); |
---|
1898 | dbprint(ppl-1, @R); |
---|
1899 | // create the ideal I |
---|
1900 | ideal F = imap(save,F); |
---|
1901 | ideal I = t(1)*F[1]+s(1); |
---|
1902 | for (j=2; j<=P; j++) |
---|
1903 | { |
---|
1904 | I = I, t(j)*F[j]+s(j); |
---|
1905 | } |
---|
1906 | poly p,q; |
---|
1907 | for (i=1; i<=N; i++) |
---|
1908 | { |
---|
1909 | p=0; |
---|
1910 | for (j=1; j<=P; j++) |
---|
1911 | { |
---|
1912 | q = t(j); |
---|
1913 | q = diff(F[j],var(2*P+i))*q; |
---|
1914 | p = p + q; |
---|
1915 | } |
---|
1916 | I = I, var(2*P+N+i) + p; |
---|
1917 | } |
---|
1918 | // -------- the ideal I is ready ---------- |
---|
1919 | dbprint(ppl,"// -1-2- starting the elimination of "+string(t(1..P))+" in @R"); |
---|
1920 | dbprint(ppl-1, I); |
---|
1921 | ideal J = engine(I,eng); |
---|
1922 | ideal K = nselect(J,1,P); |
---|
1923 | kill I,J; |
---|
1924 | dbprint(ppl,"// -1-3- all t(i) are eliminated"); |
---|
1925 | dbprint(ppl-1, K); //K is without t(i) |
---|
1926 | // ----------- the ring @R2 ------------ |
---|
1927 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
1928 | // keep: N, i,j,s, tmp, RL |
---|
1929 | setring save; |
---|
1930 | Nnew = 2*N+P; |
---|
1931 | kill Lord, tmp, iv, RName; |
---|
1932 | list Lord, tmp; |
---|
1933 | intvec iv; |
---|
1934 | L[1] = RL[1]; //char |
---|
1935 | L[4] = RL[4]; //char, minpoly |
---|
1936 | // check whether vars hava admissible names -> done earlier |
---|
1937 | // now, create the names for new var |
---|
1938 | for (j=1; j<=P; j++) |
---|
1939 | { |
---|
1940 | tmp[j] = "s("+string(j)+")"; |
---|
1941 | } |
---|
1942 | // DName is defined earlier |
---|
1943 | list NName = Name + DName + tmp; |
---|
1944 | L[2] = NName; |
---|
1945 | tmp = 0; |
---|
1946 | // block ord (dp(N),dp); |
---|
1947 | string s = "iv="; |
---|
1948 | for (i=1; i<=Nnew-P; i++) |
---|
1949 | { |
---|
1950 | s = s+"1,"; |
---|
1951 | } |
---|
1952 | s[size(s)]=";"; |
---|
1953 | execute(s); |
---|
1954 | tmp[1] = "dp"; //string |
---|
1955 | tmp[2] = iv; //intvec |
---|
1956 | Lord[1] = tmp; |
---|
1957 | // continue with dp 1,1,1,1... |
---|
1958 | tmp[1] = "dp"; //string |
---|
1959 | s[size(s)] = ","; |
---|
1960 | for (j=1; j<=P; j++) |
---|
1961 | { |
---|
1962 | s = s+"1,"; |
---|
1963 | } |
---|
1964 | s[size(s)]=";"; |
---|
1965 | execute(s); |
---|
1966 | kill s; |
---|
1967 | kill NName; |
---|
1968 | tmp[2] = iv; |
---|
1969 | Lord[2] = tmp; |
---|
1970 | tmp[1] = "C"; |
---|
1971 | iv = 0; |
---|
1972 | tmp[2] = iv; |
---|
1973 | Lord[3] = tmp; |
---|
1974 | tmp = 0; |
---|
1975 | L[3] = Lord; |
---|
1976 | // we are done with the list. Now add a Plural part |
---|
1977 | def @R2@ = ring(L); |
---|
1978 | setring @R2@; |
---|
1979 | matrix @D[Nnew][Nnew]; |
---|
1980 | for (i=1; i<=N; i++) |
---|
1981 | { |
---|
1982 | @D[i,N+i]=1; |
---|
1983 | } |
---|
1984 | def @R2 = nc_algebra(1,@D); |
---|
1985 | setring @R2; |
---|
1986 | kill @R2@; |
---|
1987 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,_s) is ready"); |
---|
1988 | dbprint(ppl-1, @R2); |
---|
1989 | // ideal MM = maxideal(1); |
---|
1990 | // MM = 0,s,MM; |
---|
1991 | // map R01 = @R, MM; |
---|
1992 | // ideal K = R01(K); |
---|
1993 | ideal F = imap(save,F); // maybe ideal F = R01(I); ? |
---|
1994 | ideal K = imap(@R,K); // maybe ideal K = R01(I); ? |
---|
1995 | poly f=1; |
---|
1996 | for (j=1; j<=P; j++) |
---|
1997 | { |
---|
1998 | f = f*F[j]; |
---|
1999 | } |
---|
2000 | K = K,f; // to compute B (Bernstein-Sato ideal) |
---|
2001 | //j=2; // for example |
---|
2002 | //K = K,F[j]; // to compute Bj (see "On the computation of Bernstein-Sato ideals"; Castro, Ucha) |
---|
2003 | //K = K,F; // to compute Bsigma (see "On the computation of Bernstein-Sato ideals"; Castro, Ucha) |
---|
2004 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
2005 | dbprint(ppl-1, K); |
---|
2006 | ideal M = engine(K,eng); |
---|
2007 | ideal K2 = nselect(M,1,Nnew-P); |
---|
2008 | kill K,M; |
---|
2009 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
2010 | dbprint(ppl-1, K2); |
---|
2011 | // the ring @R3 and factorize |
---|
2012 | ring @R3 = 0,s(1..P),dp; |
---|
2013 | dbprint(ppl,"// -3-1- the ring @R3(_s) is ready"); |
---|
2014 | ideal K3 = imap(@R2,K2); |
---|
2015 | if (size(K3)==1) |
---|
2016 | { |
---|
2017 | poly p = K3[1]; |
---|
2018 | dbprint(ppl,"// -3-2- factorization"); |
---|
2019 | // Warning: now P is an integer |
---|
2020 | list Q = factorize(p); //with constants and multiplicities |
---|
2021 | ideal bs; intvec m; |
---|
2022 | for (i=2; i<=size(Q[1]); i++) //we delete Q[1][1] and Q[2][1] |
---|
2023 | { |
---|
2024 | bs[i-1] = Q[1][i]; |
---|
2025 | m[i-1] = Q[2][i]; |
---|
2026 | } |
---|
2027 | // "--------- Q-ideal factorizes into ---------"; list(bs,m); |
---|
2028 | list BS = bs,m; |
---|
2029 | } |
---|
2030 | else |
---|
2031 | { |
---|
2032 | // conjecture: the Bernstein ideal is principal |
---|
2033 | dbprint(ppl,"// -3-2- the Bernstein ideal is not principal"); |
---|
2034 | ideal BS = K3; |
---|
2035 | } |
---|
2036 | // create the ring @R4(_x,_Dx,_s) and put the result into it, |
---|
2037 | // _x, _Dx,s; ord "dp". |
---|
2038 | // keep: N, i,j,s, tmp, RL |
---|
2039 | setring save; |
---|
2040 | Nnew = 2*N+P; |
---|
2041 | // list RL = ringlist(save); //is defined earlier |
---|
2042 | kill Lord, tmp, iv; |
---|
2043 | L = 0; |
---|
2044 | list Lord, tmp; |
---|
2045 | intvec iv; |
---|
2046 | L[1] = RL[1]; //char |
---|
2047 | L[4] = RL[4]; //char, minpoly |
---|
2048 | // check whether vars hava admissible names -> done earlier |
---|
2049 | // now, create the names for new var |
---|
2050 | for (j=1; j<=P; j++) |
---|
2051 | { |
---|
2052 | tmp[j] = "s("+string(j)+")"; |
---|
2053 | } |
---|
2054 | // DName is defined earlier |
---|
2055 | list NName = Name + DName + tmp; |
---|
2056 | L[2] = NName; |
---|
2057 | tmp = 0; |
---|
2058 | // dp ordering; |
---|
2059 | string s = "iv="; |
---|
2060 | for (i=1; i<=Nnew; i++) |
---|
2061 | { |
---|
2062 | s = s+"1,"; |
---|
2063 | } |
---|
2064 | s[size(s)]=";"; |
---|
2065 | execute(s); |
---|
2066 | kill s; |
---|
2067 | kill NName; |
---|
2068 | tmp[1] = "dp"; //string |
---|
2069 | tmp[2] = iv; //intvec |
---|
2070 | Lord[1] = tmp; |
---|
2071 | tmp[1] = "C"; |
---|
2072 | iv = 0; |
---|
2073 | tmp[2] = iv; |
---|
2074 | Lord[2] = tmp; |
---|
2075 | tmp = 0; |
---|
2076 | L[3] = Lord; |
---|
2077 | // we are done with the list. Now add a Plural part |
---|
2078 | def @R4@ = ring(L); |
---|
2079 | setring @R4@; |
---|
2080 | matrix @D[Nnew][Nnew]; |
---|
2081 | for (i=1; i<=N; i++) |
---|
2082 | { |
---|
2083 | @D[i,N+i]=1; |
---|
2084 | } |
---|
2085 | def @R4 = nc_algebra(1,@D); |
---|
2086 | setring @R4; |
---|
2087 | kill @R4@; |
---|
2088 | dbprint(ppl,"// -4-1- the ring @R4i(_x,_Dx,_s) is ready"); |
---|
2089 | dbprint(ppl-1, @R4); |
---|
2090 | ideal K4 = imap(@R,K); |
---|
2091 | option(redSB); |
---|
2092 | dbprint(ppl,"// -4-2- the final cosmetic std"); |
---|
2093 | K4 = engine(K4,eng); //std does the job too |
---|
2094 | // total cleanup |
---|
2095 | kill @R; |
---|
2096 | kill @R2; |
---|
2097 | def BS = imap(@R3,BS); |
---|
2098 | export BS; |
---|
2099 | kill @R3; |
---|
2100 | ideal LD = K4; |
---|
2101 | export LD; |
---|
2102 | return(@R4); |
---|
2103 | } |
---|
2104 | example |
---|
2105 | { |
---|
2106 | "EXAMPLE:"; echo = 2; |
---|
2107 | ring r = 0,(x,y),Dp; |
---|
2108 | ideal F = x,y,x+y; |
---|
2109 | printlevel = 0; |
---|
2110 | def A = annfsBMI(F); |
---|
2111 | setring A; |
---|
2112 | LD; |
---|
2113 | BS; |
---|
2114 | } |
---|
2115 | |
---|
2116 | proc annfsOT(poly F, list #) |
---|
2117 | "USAGE: annfsOT(f [,eng]); f a poly, eng an optional int |
---|
2118 | RETURN: ring |
---|
2119 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according |
---|
2120 | to the algorithm by Oaku and Takayama |
---|
2121 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
2122 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
---|
2123 | @* which is obtained by substituting the minimal integer root of a Bernstein |
---|
2124 | @* polynomial into the s-parametric ideal; |
---|
2125 | @* - the list BS contains roots with multiplicities of a Bernstein polynomial of f. |
---|
2126 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2127 | @* otherwise, and by default @code{slimgb} is used. |
---|
2128 | @* If printlevel=1, progress debug messages will be printed, |
---|
2129 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2130 | EXAMPLE: example annfsOT; shows examples |
---|
2131 | " |
---|
2132 | { |
---|
2133 | int eng = 0; |
---|
2134 | if ( size(#)>0 ) |
---|
2135 | { |
---|
2136 | if ( typeof(#[1]) == "int" ) |
---|
2137 | { |
---|
2138 | eng = int(#[1]); |
---|
2139 | } |
---|
2140 | } |
---|
2141 | // returns a list with a ring and an ideal LD in it |
---|
2142 | int ppl = printlevel-voice+2; |
---|
2143 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
2144 | def save = basering; |
---|
2145 | int N = nvars(basering); |
---|
2146 | int Nnew = 2*(N+2); |
---|
2147 | int i,j; |
---|
2148 | string s; |
---|
2149 | list RL = ringlist(basering); |
---|
2150 | list L, Lord; |
---|
2151 | list tmp; |
---|
2152 | intvec iv; |
---|
2153 | L[1] = RL[1]; // char |
---|
2154 | L[4] = RL[4]; // char, minpoly |
---|
2155 | // check whether vars have admissible names |
---|
2156 | list Name = RL[2]; |
---|
2157 | list RName; |
---|
2158 | RName[1] = "u"; |
---|
2159 | RName[2] = "v"; |
---|
2160 | RName[3] = "t"; |
---|
2161 | RName[4] = "Dt"; |
---|
2162 | for(i=1;i<=N;i++) |
---|
2163 | { |
---|
2164 | for(j=1; j<=size(RName);j++) |
---|
2165 | { |
---|
2166 | if (Name[i] == RName[j]) |
---|
2167 | { |
---|
2168 | ERROR("Variable names should not include u,v,t,Dt"); |
---|
2169 | } |
---|
2170 | } |
---|
2171 | } |
---|
2172 | // now, create the names for new vars |
---|
2173 | tmp[1] = "u"; |
---|
2174 | tmp[2] = "v"; |
---|
2175 | list UName = tmp; |
---|
2176 | list DName; |
---|
2177 | for(i=1;i<=N;i++) |
---|
2178 | { |
---|
2179 | DName[i] = "D"+Name[i]; // concat |
---|
2180 | } |
---|
2181 | tmp = 0; |
---|
2182 | tmp[1] = "t"; |
---|
2183 | tmp[2] = "Dt"; |
---|
2184 | list NName = UName + tmp + Name + DName; |
---|
2185 | L[2] = NName; |
---|
2186 | tmp = 0; |
---|
2187 | // Name, Dname will be used further |
---|
2188 | kill UName; |
---|
2189 | kill NName; |
---|
2190 | // block ord (a(1,1),dp); |
---|
2191 | tmp[1] = "a"; // string |
---|
2192 | iv = 1,1; |
---|
2193 | tmp[2] = iv; //intvec |
---|
2194 | Lord[1] = tmp; |
---|
2195 | // continue with dp 1,1,1,1... |
---|
2196 | tmp[1] = "dp"; // string |
---|
2197 | s = "iv="; |
---|
2198 | for(i=1;i<=Nnew;i++) |
---|
2199 | { |
---|
2200 | s = s+"1,"; |
---|
2201 | } |
---|
2202 | s[size(s)]= ";"; |
---|
2203 | execute(s); |
---|
2204 | tmp[2] = iv; |
---|
2205 | Lord[2] = tmp; |
---|
2206 | tmp[1] = "C"; |
---|
2207 | iv = 0; |
---|
2208 | tmp[2] = iv; |
---|
2209 | Lord[3] = tmp; |
---|
2210 | tmp = 0; |
---|
2211 | L[3] = Lord; |
---|
2212 | // we are done with the list |
---|
2213 | def @R@ = ring(L); |
---|
2214 | setring @R@; |
---|
2215 | matrix @D[Nnew][Nnew]; |
---|
2216 | @D[3,4]=1; |
---|
2217 | for(i=1; i<=N; i++) |
---|
2218 | { |
---|
2219 | @D[4+i,N+4+i]=1; |
---|
2220 | } |
---|
2221 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
2222 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
2223 | // L[6] = @D; |
---|
2224 | def @R = nc_algebra(1,@D); |
---|
2225 | setring @R; |
---|
2226 | kill @R@; |
---|
2227 | dbprint(ppl,"// -1-1- the ring @R(u,v,t,Dt,_x,_Dx) is ready"); |
---|
2228 | dbprint(ppl-1, @R); |
---|
2229 | // create the ideal I |
---|
2230 | poly F = imap(save,F); |
---|
2231 | ideal I = u*F-t,u*v-1; |
---|
2232 | poly p; |
---|
2233 | for(i=1; i<=N; i++) |
---|
2234 | { |
---|
2235 | p = u*Dt; // u*Dt |
---|
2236 | p = diff(F,var(4+i))*p; |
---|
2237 | I = I, var(N+4+i) + p; |
---|
2238 | } |
---|
2239 | // -------- the ideal I is ready ---------- |
---|
2240 | dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
2241 | dbprint(ppl-1, I); |
---|
2242 | ideal J = engine(I,eng); |
---|
2243 | ideal K = nselect(J,1,2); |
---|
2244 | dbprint(ppl,"// -1-3- u,v are eliminated"); |
---|
2245 | dbprint(ppl-1, K); // K is without u,v |
---|
2246 | setring save; |
---|
2247 | // ------------ new ring @R2 ------------------ |
---|
2248 | // without u,v and with the elim.ord for t,Dt |
---|
2249 | // tensored with the K[s] |
---|
2250 | // keep: N, i,j,s, tmp, RL |
---|
2251 | Nnew = 2*N+2+1; |
---|
2252 | // list RL = ringlist(save); // is defined earlier |
---|
2253 | L = 0; // kill L; |
---|
2254 | kill Lord, tmp, iv, RName; |
---|
2255 | list Lord, tmp; |
---|
2256 | intvec iv; |
---|
2257 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
2258 | // check whether vars have admissible names -> done earlier |
---|
2259 | // list Name = RL[2]; |
---|
2260 | list RName; |
---|
2261 | RName[1] = "t"; |
---|
2262 | RName[2] = "Dt"; |
---|
2263 | // now, create the names for new var (here, s only) |
---|
2264 | tmp[1] = "s"; |
---|
2265 | // DName is defined earlier |
---|
2266 | list NName = RName + Name + DName + tmp; |
---|
2267 | L[2] = NName; |
---|
2268 | tmp = 0; |
---|
2269 | // block ord (a(1,1),dp); |
---|
2270 | tmp[1] = "a"; iv = 1,1; tmp[2] = iv; //intvec |
---|
2271 | Lord[1] = tmp; |
---|
2272 | // continue with a(1,1,1,1)... |
---|
2273 | tmp[1] = "dp"; s = "iv="; |
---|
2274 | for(i=1; i<= Nnew; i++) |
---|
2275 | { |
---|
2276 | s = s+"1,"; |
---|
2277 | } |
---|
2278 | s[size(s)]= ";"; execute(s); |
---|
2279 | kill NName; |
---|
2280 | tmp[2] = iv; |
---|
2281 | Lord[2] = tmp; |
---|
2282 | // extra block for s |
---|
2283 | // tmp[1] = "dp"; iv = 1; |
---|
2284 | // s[size(s)]= ","; s = s + "1,1,1;"; execute(s); tmp[2] = iv; |
---|
2285 | // Lord[3] = tmp; |
---|
2286 | kill s; |
---|
2287 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
2288 | Lord[3] = tmp; tmp = 0; |
---|
2289 | L[3] = Lord; |
---|
2290 | // we are done with the list. Now, add a Plural part |
---|
2291 | def @R2@ = ring(L); |
---|
2292 | setring @R2@; |
---|
2293 | matrix @D[Nnew][Nnew]; |
---|
2294 | @D[1,2] = 1; |
---|
2295 | for(i=1; i<=N; i++) |
---|
2296 | { |
---|
2297 | @D[2+i,2+N+i] = 1; |
---|
2298 | } |
---|
2299 | def @R2 = nc_algebra(1,@D); |
---|
2300 | setring @R2; |
---|
2301 | kill @R2@; |
---|
2302 | dbprint(ppl,"// -2-1- the ring @R2(t,Dt,_x,_Dx,s) is ready"); |
---|
2303 | dbprint(ppl-1, @R2); |
---|
2304 | ideal MM = maxideal(1); |
---|
2305 | MM = 0,0,MM; |
---|
2306 | map R01 = @R, MM; |
---|
2307 | ideal K = R01(K); |
---|
2308 | // ideal K = imap(@R,K); // names of vars are important! |
---|
2309 | poly G = t*Dt+s+1; // s is a variable here |
---|
2310 | K = NF(K,std(G)),G; |
---|
2311 | // -------- the ideal K_(@R2) is ready ---------- |
---|
2312 | dbprint(ppl,"// -2-2- starting the elimination of t,Dt in @R2"); |
---|
2313 | dbprint(ppl-1, K); |
---|
2314 | ideal M = engine(K,eng); |
---|
2315 | ideal K2 = nselect(M,1,2); |
---|
2316 | dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
2317 | dbprint(ppl-1, K2); |
---|
2318 | // dbprint(ppl-1+1," -2-4- std of K2"); |
---|
2319 | // option(redSB); option(redTail); K2 = std(K2); |
---|
2320 | // K2; // without t,Dt, and with s |
---|
2321 | // -------- the ring @R3 ---------- |
---|
2322 | // _x, _Dx, s; elim.ord for _x,_Dx. |
---|
2323 | // keep: N, i,j,s, tmp, RL |
---|
2324 | setring save; |
---|
2325 | Nnew = 2*N+1; |
---|
2326 | // list RL = ringlist(save); // is defined earlier |
---|
2327 | // kill L; |
---|
2328 | kill Lord, tmp, iv, RName; |
---|
2329 | list Lord, tmp; |
---|
2330 | intvec iv; |
---|
2331 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
2332 | // check whether vars have admissible names -> done earlier |
---|
2333 | // list Name = RL[2]; |
---|
2334 | // now, create the names for new var (here, s only) |
---|
2335 | tmp[1] = "s"; |
---|
2336 | // DName is defined earlier |
---|
2337 | list NName = Name + DName + tmp; |
---|
2338 | L[2] = NName; |
---|
2339 | tmp = 0; |
---|
2340 | // block ord (a(1,1...),dp); |
---|
2341 | string s = "iv="; |
---|
2342 | for(i=1; i<=Nnew-1; i++) |
---|
2343 | { |
---|
2344 | s = s+"1,"; |
---|
2345 | } |
---|
2346 | s[size(s)]= ";"; |
---|
2347 | execute(s); |
---|
2348 | tmp[1] = "a"; // string |
---|
2349 | tmp[2] = iv; //intvec |
---|
2350 | Lord[1] = tmp; |
---|
2351 | // continue with dp 1,1,1,1... |
---|
2352 | tmp[1] = "dp"; // string |
---|
2353 | s[size(s)]=","; s= s+"1;"; |
---|
2354 | execute(s); |
---|
2355 | kill s; |
---|
2356 | kill NName; |
---|
2357 | tmp[2] = iv; |
---|
2358 | Lord[2] = tmp; |
---|
2359 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
2360 | Lord[3] = tmp; tmp = 0; |
---|
2361 | L[3] = Lord; |
---|
2362 | // we are done with the list. Now add a Plural part |
---|
2363 | def @R3@ = ring(L); |
---|
2364 | setring @R3@; |
---|
2365 | matrix @D[Nnew][Nnew]; |
---|
2366 | for(i=1; i<=N; i++) |
---|
2367 | { |
---|
2368 | @D[i,N+i]=1; |
---|
2369 | } |
---|
2370 | def @R3 = nc_algebra(1,@D); |
---|
2371 | setring @R3; |
---|
2372 | kill @R3@; |
---|
2373 | dbprint(ppl,"// -3-1- the ring @R3(_x,_Dx,s) is ready"); |
---|
2374 | dbprint(ppl-1, @R3); |
---|
2375 | ideal MM = maxideal(1); |
---|
2376 | MM = 0,0,MM; |
---|
2377 | map R12 = @R2, MM; |
---|
2378 | ideal K = R12(K2); |
---|
2379 | poly F = imap(save,F); |
---|
2380 | K = K,F; |
---|
2381 | dbprint(ppl,"// -3-2- starting the elimination of _x,_Dx in @R3"); |
---|
2382 | dbprint(ppl-1, K); |
---|
2383 | ideal M = engine(K,eng); |
---|
2384 | ideal K3 = nselect(M,1,Nnew-1); |
---|
2385 | dbprint(ppl,"// -3-3- _x,_Dx are eliminated in @R3"); |
---|
2386 | dbprint(ppl-1, K3); |
---|
2387 | // the ring @R4 and the search for minimal negative int s |
---|
2388 | ring @R4 = 0,(s),dp; |
---|
2389 | dbprint(ppl,"// -4-1- the ring @R4 is ready"); |
---|
2390 | ideal K4 = imap(@R3,K3); |
---|
2391 | poly p = K4[1]; |
---|
2392 | dbprint(ppl,"// -4-2- factorization"); |
---|
2393 | //// ideal P = factorize(p,1); // without constants and multiplicities |
---|
2394 | list P = factorize(p); // with constants and multiplicities |
---|
2395 | ideal bs; intvec m; // the Bernstein polynomial is monic, so we are not interested in constants |
---|
2396 | for (i=2; i<=size(P[1]); i++) // we delete P[1][1] and P[2][1] |
---|
2397 | { |
---|
2398 | bs[i-1] = P[1][i]; |
---|
2399 | m[i-1] = P[2][i]; |
---|
2400 | } |
---|
2401 | // "------ b-function factorizes into ----------"; P; |
---|
2402 | //// int sP = minIntRoot(P, 1); |
---|
2403 | int sP = minIntRoot(bs,1); |
---|
2404 | dbprint(ppl,"// -4-3- minimal integer root found"); |
---|
2405 | dbprint(ppl-1, sP); |
---|
2406 | // convert factors to a list of their roots |
---|
2407 | // assume all factors are linear |
---|
2408 | //// ideal BS = normalize(P); |
---|
2409 | //// BS = subst(BS,s,0); |
---|
2410 | //// BS = -BS; |
---|
2411 | bs = normalize(bs); |
---|
2412 | bs = subst(bs,s,0); |
---|
2413 | bs = -bs; |
---|
2414 | list BS = bs,m; |
---|
2415 | // TODO: sort BS! |
---|
2416 | // ------ substitute s found in the ideal ------ |
---|
2417 | // ------- going back to @R2 and substitute -------- |
---|
2418 | setring @R2; |
---|
2419 | ideal K3 = subst(K2,s,sP); |
---|
2420 | // create the ordinary Weyl algebra and put the result into it, |
---|
2421 | // thus creating the ring @R5 |
---|
2422 | // keep: N, i,j,s, tmp, RL |
---|
2423 | setring save; |
---|
2424 | Nnew = 2*N; |
---|
2425 | // list RL = ringlist(save); // is defined earlier |
---|
2426 | kill Lord, tmp, iv; |
---|
2427 | L = 0; |
---|
2428 | list Lord, tmp; |
---|
2429 | intvec iv; |
---|
2430 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
2431 | // check whether vars have admissible names -> done earlier |
---|
2432 | // list Name = RL[2]; |
---|
2433 | // DName is defined earlier |
---|
2434 | list NName = Name + DName; |
---|
2435 | L[2] = NName; |
---|
2436 | // dp ordering; |
---|
2437 | string s = "iv="; |
---|
2438 | for(i=1;i<=Nnew;i++) |
---|
2439 | { |
---|
2440 | s = s+"1,"; |
---|
2441 | } |
---|
2442 | s[size(s)]= ";"; |
---|
2443 | execute(s); |
---|
2444 | tmp = 0; |
---|
2445 | tmp[1] = "dp"; // string |
---|
2446 | tmp[2] = iv; //intvec |
---|
2447 | Lord[1] = tmp; |
---|
2448 | kill s; |
---|
2449 | tmp[1] = "C"; |
---|
2450 | iv = 0; |
---|
2451 | tmp[2] = iv; |
---|
2452 | Lord[2] = tmp; |
---|
2453 | tmp = 0; |
---|
2454 | L[3] = Lord; |
---|
2455 | // we are done with the list |
---|
2456 | // Add: Plural part |
---|
2457 | def @R5@ = ring(L); |
---|
2458 | setring @R5@; |
---|
2459 | matrix @D[Nnew][Nnew]; |
---|
2460 | for(i=1; i<=N; i++) |
---|
2461 | { |
---|
2462 | @D[i,N+i]=1; |
---|
2463 | } |
---|
2464 | def @R5 = nc_algebra(1,@D); |
---|
2465 | setring @R5; |
---|
2466 | kill @R5@; |
---|
2467 | dbprint(ppl,"// -5-1- the ring @R5 is ready"); |
---|
2468 | dbprint(ppl-1, @R5); |
---|
2469 | ideal K5 = imap(@R2,K3); |
---|
2470 | option(redSB); |
---|
2471 | dbprint(ppl,"// -5-2- the final cosmetic std"); |
---|
2472 | K5 = engine(K5,eng); // std does the job too |
---|
2473 | // total cleanup |
---|
2474 | kill @R; |
---|
2475 | kill @R2; |
---|
2476 | kill @R3; |
---|
2477 | //// ideal BS = imap(@R4,BS); |
---|
2478 | list BS = imap(@R4,BS); |
---|
2479 | export BS; |
---|
2480 | ideal LD = K5; |
---|
2481 | kill @R4; |
---|
2482 | export LD; |
---|
2483 | return(@R5); |
---|
2484 | } |
---|
2485 | example |
---|
2486 | { |
---|
2487 | "EXAMPLE:"; echo = 2; |
---|
2488 | ring r = 0,(x,y,z),Dp; |
---|
2489 | poly F = x^2+y^3+z^5; |
---|
2490 | printlevel = 0; |
---|
2491 | def A = annfsOT(F); |
---|
2492 | setring A; |
---|
2493 | LD; |
---|
2494 | BS; |
---|
2495 | } |
---|
2496 | |
---|
2497 | |
---|
2498 | proc SannfsOT(poly F, list #) |
---|
2499 | "USAGE: SannfsOT(f [,eng]); f a poly, eng an optional int |
---|
2500 | RETURN: ring |
---|
2501 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the 1st step of the algorithm by Oaku and Takayama in the ring D[s], where D is the Weyl algebra |
---|
2502 | NOTE: activate this ring with the @code{setring} command. |
---|
2503 | @* In the ring D[s], the ideal LD (which is NOT a Groebner basis) is the needed D-module structure. |
---|
2504 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2505 | @* otherwise, and by default @code{slimgb} is used. |
---|
2506 | @* If printlevel=1, progress debug messages will be printed, |
---|
2507 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2508 | EXAMPLE: example SannfsOT; shows examples |
---|
2509 | " |
---|
2510 | { |
---|
2511 | int eng = 0; |
---|
2512 | if ( size(#)>0 ) |
---|
2513 | { |
---|
2514 | if ( typeof(#[1]) == "int" ) |
---|
2515 | { |
---|
2516 | eng = int(#[1]); |
---|
2517 | } |
---|
2518 | } |
---|
2519 | // returns a list with a ring and an ideal LD in it |
---|
2520 | int ppl = printlevel-voice+2; |
---|
2521 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
2522 | def save = basering; |
---|
2523 | int N = nvars(basering); |
---|
2524 | int Nnew = 2*(N+2); |
---|
2525 | int i,j; |
---|
2526 | string s; |
---|
2527 | list RL = ringlist(basering); |
---|
2528 | list L, Lord; |
---|
2529 | list tmp; |
---|
2530 | intvec iv; |
---|
2531 | L[1] = RL[1]; // char |
---|
2532 | L[4] = RL[4]; // char, minpoly |
---|
2533 | // check whether vars have admissible names |
---|
2534 | list Name = RL[2]; |
---|
2535 | list RName; |
---|
2536 | RName[1] = "u"; |
---|
2537 | RName[2] = "v"; |
---|
2538 | RName[3] = "t"; |
---|
2539 | RName[4] = "Dt"; |
---|
2540 | for(i=1;i<=N;i++) |
---|
2541 | { |
---|
2542 | for(j=1; j<=size(RName);j++) |
---|
2543 | { |
---|
2544 | if (Name[i] == RName[j]) |
---|
2545 | { |
---|
2546 | ERROR("Variable names should not include u,v,t,Dt"); |
---|
2547 | } |
---|
2548 | } |
---|
2549 | } |
---|
2550 | // now, create the names for new vars |
---|
2551 | tmp[1] = "u"; |
---|
2552 | tmp[2] = "v"; |
---|
2553 | list UName = tmp; |
---|
2554 | list DName; |
---|
2555 | for(i=1;i<=N;i++) |
---|
2556 | { |
---|
2557 | DName[i] = "D"+Name[i]; // concat |
---|
2558 | } |
---|
2559 | tmp = 0; |
---|
2560 | tmp[1] = "t"; |
---|
2561 | tmp[2] = "Dt"; |
---|
2562 | list NName = UName + tmp + Name + DName; |
---|
2563 | L[2] = NName; |
---|
2564 | tmp = 0; |
---|
2565 | // Name, Dname will be used further |
---|
2566 | kill UName; |
---|
2567 | kill NName; |
---|
2568 | // block ord (a(1,1),dp); |
---|
2569 | tmp[1] = "a"; // string |
---|
2570 | iv = 1,1; |
---|
2571 | tmp[2] = iv; //intvec |
---|
2572 | Lord[1] = tmp; |
---|
2573 | // continue with dp 1,1,1,1... |
---|
2574 | tmp[1] = "dp"; // string |
---|
2575 | s = "iv="; |
---|
2576 | for(i=1;i<=Nnew;i++) |
---|
2577 | { |
---|
2578 | s = s+"1,"; |
---|
2579 | } |
---|
2580 | s[size(s)]= ";"; |
---|
2581 | execute(s); |
---|
2582 | tmp[2] = iv; |
---|
2583 | Lord[2] = tmp; |
---|
2584 | tmp[1] = "C"; |
---|
2585 | iv = 0; |
---|
2586 | tmp[2] = iv; |
---|
2587 | Lord[3] = tmp; |
---|
2588 | tmp = 0; |
---|
2589 | L[3] = Lord; |
---|
2590 | // we are done with the list |
---|
2591 | def @R@ = ring(L); |
---|
2592 | setring @R@; |
---|
2593 | matrix @D[Nnew][Nnew]; |
---|
2594 | @D[3,4]=1; |
---|
2595 | for(i=1; i<=N; i++) |
---|
2596 | { |
---|
2597 | @D[4+i,N+4+i]=1; |
---|
2598 | } |
---|
2599 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
2600 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
2601 | // L[6] = @D; |
---|
2602 | def @R = nc_algebra(1,@D); |
---|
2603 | setring @R; |
---|
2604 | kill @R@; |
---|
2605 | dbprint(ppl,"// -1-1- the ring @R(u,v,t,Dt,_x,_Dx) is ready"); |
---|
2606 | dbprint(ppl-1, @R); |
---|
2607 | // create the ideal I |
---|
2608 | poly F = imap(save,F); |
---|
2609 | ideal I = u*F-t,u*v-1; |
---|
2610 | poly p; |
---|
2611 | for(i=1; i<=N; i++) |
---|
2612 | { |
---|
2613 | p = u*Dt; // u*Dt |
---|
2614 | p = diff(F,var(4+i))*p; |
---|
2615 | I = I, var(N+4+i) + p; |
---|
2616 | } |
---|
2617 | // -------- the ideal I is ready ---------- |
---|
2618 | dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
2619 | dbprint(ppl-1, I); |
---|
2620 | ideal J = engine(I,eng); |
---|
2621 | ideal K = nselect(J,1,2); |
---|
2622 | dbprint(ppl,"// -1-3- u,v are eliminated"); |
---|
2623 | dbprint(ppl-1, K); // K is without u,v |
---|
2624 | |
---|
2625 | |
---|
2626 | setring save; |
---|
2627 | // ------------ new ring @R2 ------------------ |
---|
2628 | // without u,v and with the elim.ord for t,Dt |
---|
2629 | // tensored with the K[s] |
---|
2630 | // keep: N, i,j,s, tmp, RL |
---|
2631 | Nnew = 2*N+2+1; |
---|
2632 | // list RL = ringlist(save); // is defined earlier |
---|
2633 | L = 0; // kill L; |
---|
2634 | kill Lord, tmp, iv, RName; |
---|
2635 | list Lord, tmp; |
---|
2636 | intvec iv; |
---|
2637 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
2638 | // check whether vars have admissible names -> done earlier |
---|
2639 | // list Name = RL[2]; |
---|
2640 | list RName; |
---|
2641 | RName[1] = "t"; |
---|
2642 | RName[2] = "Dt"; |
---|
2643 | // now, create the names for new var (here, s only) |
---|
2644 | tmp[1] = "s"; |
---|
2645 | // DName is defined earlier |
---|
2646 | list NName = RName + Name + DName + tmp; |
---|
2647 | L[2] = NName; |
---|
2648 | tmp = 0; |
---|
2649 | // block ord (a(1,1),dp); |
---|
2650 | tmp[1] = "a"; iv = 1,1; tmp[2] = iv; //intvec |
---|
2651 | Lord[1] = tmp; |
---|
2652 | // continue with a(1,1,1,1)... |
---|
2653 | tmp[1] = "dp"; s = "iv="; |
---|
2654 | for(i=1; i<= Nnew; i++) |
---|
2655 | { |
---|
2656 | s = s+"1,"; |
---|
2657 | } |
---|
2658 | s[size(s)]= ";"; execute(s); |
---|
2659 | kill NName; |
---|
2660 | tmp[2] = iv; |
---|
2661 | Lord[2] = tmp; |
---|
2662 | // extra block for s |
---|
2663 | // tmp[1] = "dp"; iv = 1; |
---|
2664 | // s[size(s)]= ","; s = s + "1,1,1;"; execute(s); tmp[2] = iv; |
---|
2665 | // Lord[3] = tmp; |
---|
2666 | kill s; |
---|
2667 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
2668 | Lord[3] = tmp; tmp = 0; |
---|
2669 | L[3] = Lord; |
---|
2670 | // we are done with the list. Now, add a Plural part |
---|
2671 | def @R2@ = ring(L); |
---|
2672 | setring @R2@; |
---|
2673 | matrix @D[Nnew][Nnew]; |
---|
2674 | @D[1,2] = 1; |
---|
2675 | for(i=1; i<=N; i++) |
---|
2676 | { |
---|
2677 | @D[2+i,2+N+i] = 1; |
---|
2678 | } |
---|
2679 | def @R2 = nc_algebra(1,@D); |
---|
2680 | setring @R2; |
---|
2681 | kill @R2@; |
---|
2682 | dbprint(ppl,"// -2-1- the ring @R2(t,Dt,_x,_Dx,s) is ready"); |
---|
2683 | dbprint(ppl-1, @R2); |
---|
2684 | ideal MM = maxideal(1); |
---|
2685 | MM = 0,0,MM; |
---|
2686 | map R01 = @R, MM; |
---|
2687 | ideal K = R01(K); |
---|
2688 | // ideal K = imap(@R,K); // names of vars are important! |
---|
2689 | poly G = t*Dt+s+1; // s is a variable here |
---|
2690 | K = NF(K,std(G)),G; |
---|
2691 | // -------- the ideal K_(@R2) is ready ---------- |
---|
2692 | dbprint(ppl,"// -2-2- starting the elimination of t,Dt in @R2"); |
---|
2693 | dbprint(ppl-1, K); |
---|
2694 | ideal M = engine(K,eng); |
---|
2695 | ideal K2 = nselect(M,1,2); |
---|
2696 | dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
2697 | dbprint(ppl-1, K2); |
---|
2698 | K2 = engine(K2,eng); |
---|
2699 | setring save; |
---|
2700 | // ----------- the ring @R3 ------------ |
---|
2701 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
2702 | // keep: N, i,j,s, tmp, RL |
---|
2703 | Nnew = 2*N+1; |
---|
2704 | kill Lord, tmp, iv, RName; |
---|
2705 | list Lord, tmp; |
---|
2706 | intvec iv; |
---|
2707 | L[1] = RL[1]; |
---|
2708 | L[4] = RL[4]; // char, minpoly |
---|
2709 | // check whether vars hava admissible names -> done earlier |
---|
2710 | // now, create the names for new var |
---|
2711 | tmp[1] = "s"; |
---|
2712 | // DName is defined earlier |
---|
2713 | list NName = Name + DName + tmp; |
---|
2714 | L[2] = NName; |
---|
2715 | tmp = 0; |
---|
2716 | // block ord (dp(N),dp); |
---|
2717 | string s = "iv="; |
---|
2718 | for (i=1; i<=Nnew-1; i++) |
---|
2719 | { |
---|
2720 | s = s+"1,"; |
---|
2721 | } |
---|
2722 | s[size(s)]=";"; |
---|
2723 | execute(s); |
---|
2724 | tmp[1] = "dp"; // string |
---|
2725 | tmp[2] = iv; // intvec |
---|
2726 | Lord[1] = tmp; |
---|
2727 | // continue with dp 1,1,1,1... |
---|
2728 | tmp[1] = "dp"; // string |
---|
2729 | s[size(s)] = ","; |
---|
2730 | s = s+"1;"; |
---|
2731 | execute(s); |
---|
2732 | kill s; |
---|
2733 | kill NName; |
---|
2734 | tmp[2] = iv; |
---|
2735 | Lord[2] = tmp; |
---|
2736 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
2737 | Lord[3] = tmp; tmp = 0; |
---|
2738 | L[3] = Lord; |
---|
2739 | // we are done with the list. Now add a Plural part |
---|
2740 | def @R3@ = ring(L); |
---|
2741 | setring @R3@; |
---|
2742 | matrix @D[Nnew][Nnew]; |
---|
2743 | for (i=1; i<=N; i++) |
---|
2744 | { |
---|
2745 | @D[i,N+i]=1; |
---|
2746 | } |
---|
2747 | def @R3 = nc_algebra(1,@D); |
---|
2748 | setring @R3; |
---|
2749 | kill @R3@; |
---|
2750 | dbprint(ppl,"// -3-1- the ring @R3(_x,_Dx,s) is ready"); |
---|
2751 | dbprint(ppl-1, @R3); |
---|
2752 | ideal MM = maxideal(1); |
---|
2753 | MM = 0,s,MM; |
---|
2754 | map R01 = @R2, MM; |
---|
2755 | ideal K2 = R01(K2); |
---|
2756 | // total cleanup |
---|
2757 | ideal LD = K2; |
---|
2758 | // make leadcoeffs positive |
---|
2759 | for (i=1; i<= ncols(LD); i++) |
---|
2760 | { |
---|
2761 | if (leadcoef(LD[i]) <0 ) |
---|
2762 | { |
---|
2763 | LD[i] = -LD[i]; |
---|
2764 | } |
---|
2765 | } |
---|
2766 | export LD; |
---|
2767 | kill @R; |
---|
2768 | kill @R2; |
---|
2769 | return(@R3); |
---|
2770 | } |
---|
2771 | example |
---|
2772 | { |
---|
2773 | "EXAMPLE:"; echo = 2; |
---|
2774 | ring r = 0,(x,y,z),Dp; |
---|
2775 | poly F = x^3+y^3+z^3; |
---|
2776 | printlevel = 0; |
---|
2777 | def A = SannfsOT(F); |
---|
2778 | setring A; |
---|
2779 | LD; |
---|
2780 | } |
---|
2781 | |
---|
2782 | proc SannfsBM(poly F, list #) |
---|
2783 | "USAGE: SannfsBM(f [,eng]); f a poly, eng an optional int |
---|
2784 | RETURN: ring |
---|
2785 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the 1st step of the algorithm by Briancon and Maisonobe in the ring D[s], where D is the Weyl algebra |
---|
2786 | NOTE: activate this ring with the @code{setring} command. |
---|
2787 | @* In the ring D[s], the ideal LD (which is NOT a Groebner basis) is the needed D-module structure, |
---|
2788 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2789 | @* otherwise, and by default @code{slimgb} is used. |
---|
2790 | @* If printlevel=1, progress debug messages will be printed, |
---|
2791 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2792 | EXAMPLE: example SannfsBM; shows examples |
---|
2793 | " |
---|
2794 | { |
---|
2795 | int eng = 0; |
---|
2796 | if ( size(#)>0 ) |
---|
2797 | { |
---|
2798 | if ( typeof(#[1]) == "int" ) |
---|
2799 | { |
---|
2800 | eng = int(#[1]); |
---|
2801 | } |
---|
2802 | } |
---|
2803 | // returns a list with a ring and an ideal LD in it |
---|
2804 | int ppl = printlevel-voice+2; |
---|
2805 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
2806 | def save = basering; |
---|
2807 | int N = nvars(basering); |
---|
2808 | int Nnew = 2*N+2; |
---|
2809 | int i,j; |
---|
2810 | string s; |
---|
2811 | list RL = ringlist(basering); |
---|
2812 | list L, Lord; |
---|
2813 | list tmp; |
---|
2814 | intvec iv; |
---|
2815 | L[1] = RL[1]; // char |
---|
2816 | L[4] = RL[4]; // char, minpoly |
---|
2817 | // check whether vars have admissible names |
---|
2818 | list Name = RL[2]; |
---|
2819 | list RName; |
---|
2820 | RName[1] = "t"; |
---|
2821 | RName[2] = "s"; |
---|
2822 | for(i=1;i<=N;i++) |
---|
2823 | { |
---|
2824 | for(j=1; j<=size(RName);j++) |
---|
2825 | { |
---|
2826 | if (Name[i] == RName[j]) |
---|
2827 | { |
---|
2828 | ERROR("Variable names should not include t,s"); |
---|
2829 | } |
---|
2830 | } |
---|
2831 | } |
---|
2832 | // now, create the names for new vars |
---|
2833 | list DName; |
---|
2834 | for(i=1;i<=N;i++) |
---|
2835 | { |
---|
2836 | DName[i] = "D"+Name[i]; // concat |
---|
2837 | } |
---|
2838 | tmp[1] = "t"; |
---|
2839 | tmp[2] = "s"; |
---|
2840 | list NName = tmp + Name + DName; |
---|
2841 | L[2] = NName; |
---|
2842 | // Name, Dname will be used further |
---|
2843 | kill NName; |
---|
2844 | // block ord (lp(2),dp); |
---|
2845 | tmp[1] = "lp"; // string |
---|
2846 | iv = 1,1; |
---|
2847 | tmp[2] = iv; //intvec |
---|
2848 | Lord[1] = tmp; |
---|
2849 | // continue with dp 1,1,1,1... |
---|
2850 | tmp[1] = "dp"; // string |
---|
2851 | s = "iv="; |
---|
2852 | for(i=1;i<=Nnew;i++) |
---|
2853 | { |
---|
2854 | s = s+"1,"; |
---|
2855 | } |
---|
2856 | s[size(s)]= ";"; |
---|
2857 | execute(s); |
---|
2858 | kill s; |
---|
2859 | tmp[2] = iv; |
---|
2860 | Lord[2] = tmp; |
---|
2861 | tmp[1] = "C"; |
---|
2862 | iv = 0; |
---|
2863 | tmp[2] = iv; |
---|
2864 | Lord[3] = tmp; |
---|
2865 | tmp = 0; |
---|
2866 | L[3] = Lord; |
---|
2867 | // we are done with the list |
---|
2868 | def @R@ = ring(L); |
---|
2869 | setring @R@; |
---|
2870 | matrix @D[Nnew][Nnew]; |
---|
2871 | @D[1,2]=t; |
---|
2872 | for(i=1; i<=N; i++) |
---|
2873 | { |
---|
2874 | @D[2+i,N+2+i]=1; |
---|
2875 | } |
---|
2876 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
2877 | // L[6] = @D; |
---|
2878 | def @R = nc_algebra(1,@D); |
---|
2879 | setring @R; |
---|
2880 | kill @R@; |
---|
2881 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
2882 | dbprint(ppl-1, @R); |
---|
2883 | // create the ideal I |
---|
2884 | poly F = imap(save,F); |
---|
2885 | ideal I = t*F+s; |
---|
2886 | poly p; |
---|
2887 | for(i=1; i<=N; i++) |
---|
2888 | { |
---|
2889 | p = t; // t |
---|
2890 | p = diff(F,var(2+i))*p; |
---|
2891 | I = I, var(N+2+i) + p; |
---|
2892 | } |
---|
2893 | // -------- the ideal I is ready ---------- |
---|
2894 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
2895 | dbprint(ppl-1, I); |
---|
2896 | ideal J = engine(I,eng); |
---|
2897 | ideal K = nselect(J,1); |
---|
2898 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
2899 | dbprint(ppl-1, K); // K is without t |
---|
2900 | K = engine(K,eng); // std does the job too |
---|
2901 | // now, we must change the ordering |
---|
2902 | // and create a ring without t, Dt |
---|
2903 | // setring S; |
---|
2904 | // ----------- the ring @R3 ------------ |
---|
2905 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
2906 | // keep: N, i,j,s, tmp, RL |
---|
2907 | Nnew = 2*N+1; |
---|
2908 | kill Lord, tmp, iv, RName; |
---|
2909 | list Lord, tmp; |
---|
2910 | intvec iv; |
---|
2911 | list L=imap(save,L); |
---|
2912 | list RL=imap(save,RL); |
---|
2913 | L[1] = RL[1]; |
---|
2914 | L[4] = RL[4]; // char, minpoly |
---|
2915 | // check whether vars hava admissible names -> done earlier |
---|
2916 | // now, create the names for new var |
---|
2917 | tmp[1] = "s"; |
---|
2918 | // DName is defined earlier |
---|
2919 | list NName = Name + DName + tmp; |
---|
2920 | L[2] = NName; |
---|
2921 | tmp = 0; |
---|
2922 | // block ord (dp(N),dp); |
---|
2923 | string s = "iv="; |
---|
2924 | for (i=1; i<=Nnew-1; i++) |
---|
2925 | { |
---|
2926 | s = s+"1,"; |
---|
2927 | } |
---|
2928 | s[size(s)]=";"; |
---|
2929 | execute(s); |
---|
2930 | tmp[1] = "dp"; // string |
---|
2931 | tmp[2] = iv; // intvec |
---|
2932 | Lord[1] = tmp; |
---|
2933 | // continue with dp 1,1,1,1... |
---|
2934 | tmp[1] = "dp"; // string |
---|
2935 | s[size(s)] = ","; |
---|
2936 | s = s+"1;"; |
---|
2937 | execute(s); |
---|
2938 | kill s; |
---|
2939 | kill NName; |
---|
2940 | tmp[2] = iv; |
---|
2941 | Lord[2] = tmp; |
---|
2942 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
2943 | Lord[3] = tmp; tmp = 0; |
---|
2944 | L[3] = Lord; |
---|
2945 | // we are done with the list. Now add a Plural part |
---|
2946 | def @R2@ = ring(L); |
---|
2947 | setring @R2@; |
---|
2948 | matrix @D[Nnew][Nnew]; |
---|
2949 | for (i=1; i<=N; i++) |
---|
2950 | { |
---|
2951 | @D[i,N+i]=1; |
---|
2952 | } |
---|
2953 | def @R2 = nc_algebra(1,@D); |
---|
2954 | setring @R2; |
---|
2955 | kill @R2@; |
---|
2956 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
2957 | dbprint(ppl-1, @R2); |
---|
2958 | ideal MM = maxideal(1); |
---|
2959 | MM = 0,s,MM; |
---|
2960 | map R01 = @R, MM; |
---|
2961 | ideal K = R01(K); |
---|
2962 | // total cleanup |
---|
2963 | ideal LD = K; |
---|
2964 | // make leadcoeffs positive |
---|
2965 | for (i=1; i<= ncols(LD); i++) |
---|
2966 | { |
---|
2967 | if (leadcoef(LD[i]) <0 ) |
---|
2968 | { |
---|
2969 | LD[i] = -LD[i]; |
---|
2970 | } |
---|
2971 | } |
---|
2972 | export LD; |
---|
2973 | kill @R; |
---|
2974 | return(@R2); |
---|
2975 | } |
---|
2976 | example |
---|
2977 | { |
---|
2978 | "EXAMPLE:"; echo = 2; |
---|
2979 | ring r = 0,(x,y,z),Dp; |
---|
2980 | poly F = x^3+y^3+z^3; |
---|
2981 | printlevel = 0; |
---|
2982 | def A = SannfsBM(F); |
---|
2983 | setring A; |
---|
2984 | LD; |
---|
2985 | } |
---|
2986 | |
---|
2987 | proc SannfsLOT(poly F, list #) |
---|
2988 | "USAGE: SannfsLOT(f [,eng]); f a poly, eng an optional int |
---|
2989 | RETURN: ring |
---|
2990 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the Levandovskyy's modification of the algorithm by Oaku and Takayama in the ring D[s], where D is the Weyl algebra |
---|
2991 | NOTE: activate this ring with the @code{setring} command. |
---|
2992 | @* In the ring D[s], the ideal LD (which is NOT a Groebner basis) is the needed D-module structure. |
---|
2993 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2994 | @* otherwise, and by default @code{slimgb} is used. |
---|
2995 | @* If printlevel=1, progress debug messages will be printed, |
---|
2996 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2997 | EXAMPLE: example SannfsLOT; shows examples |
---|
2998 | " |
---|
2999 | { |
---|
3000 | int eng = 0; |
---|
3001 | if ( size(#)>0 ) |
---|
3002 | { |
---|
3003 | if ( typeof(#[1]) == "int" ) |
---|
3004 | { |
---|
3005 | eng = int(#[1]); |
---|
3006 | } |
---|
3007 | } |
---|
3008 | // returns a list with a ring and an ideal LD in it |
---|
3009 | int ppl = printlevel-voice+2; |
---|
3010 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
3011 | def save = basering; |
---|
3012 | int N = nvars(basering); |
---|
3013 | // int Nnew = 2*(N+2); |
---|
3014 | int Nnew = 2*(N+1)+1; //removed u,v; added s |
---|
3015 | int i,j; |
---|
3016 | string s; |
---|
3017 | list RL = ringlist(basering); |
---|
3018 | list L, Lord; |
---|
3019 | list tmp; |
---|
3020 | intvec iv; |
---|
3021 | L[1] = RL[1]; // char |
---|
3022 | L[4] = RL[4]; // char, minpoly |
---|
3023 | // check whether vars have admissible names |
---|
3024 | list Name = RL[2]; |
---|
3025 | list RName; |
---|
3026 | // RName[1] = "u"; |
---|
3027 | // RName[2] = "v"; |
---|
3028 | RName[1] = "t"; |
---|
3029 | RName[2] = "Dt"; |
---|
3030 | for(i=1;i<=N;i++) |
---|
3031 | { |
---|
3032 | for(j=1; j<=size(RName);j++) |
---|
3033 | { |
---|
3034 | if (Name[i] == RName[j]) |
---|
3035 | { |
---|
3036 | ERROR("Variable names should not include t,Dt"); |
---|
3037 | } |
---|
3038 | } |
---|
3039 | } |
---|
3040 | // now, create the names for new vars |
---|
3041 | // tmp[1] = "u"; |
---|
3042 | // tmp[2] = "v"; |
---|
3043 | // list UName = tmp; |
---|
3044 | list DName; |
---|
3045 | for(i=1;i<=N;i++) |
---|
3046 | { |
---|
3047 | DName[i] = "D"+Name[i]; // concat |
---|
3048 | } |
---|
3049 | tmp = 0; |
---|
3050 | tmp[1] = "t"; |
---|
3051 | tmp[2] = "Dt"; |
---|
3052 | list SName ; SName[1] = "s"; |
---|
3053 | // list NName = UName + tmp + Name + DName; |
---|
3054 | list NName = tmp + Name + DName + SName; |
---|
3055 | L[2] = NName; |
---|
3056 | tmp = 0; |
---|
3057 | // Name, Dname will be used further |
---|
3058 | // kill UName; |
---|
3059 | kill NName; |
---|
3060 | // block ord (a(1,1),dp); |
---|
3061 | tmp[1] = "a"; // string |
---|
3062 | iv = 1,1; |
---|
3063 | tmp[2] = iv; //intvec |
---|
3064 | Lord[1] = tmp; |
---|
3065 | // continue with dp 1,1,1,1... |
---|
3066 | tmp[1] = "dp"; // string |
---|
3067 | s = "iv="; |
---|
3068 | for(i=1;i<=Nnew;i++) |
---|
3069 | { |
---|
3070 | s = s+"1,"; |
---|
3071 | } |
---|
3072 | s[size(s)]= ";"; |
---|
3073 | execute(s); |
---|
3074 | tmp[2] = iv; |
---|
3075 | Lord[2] = tmp; |
---|
3076 | tmp[1] = "C"; |
---|
3077 | iv = 0; |
---|
3078 | tmp[2] = iv; |
---|
3079 | Lord[3] = tmp; |
---|
3080 | tmp = 0; |
---|
3081 | L[3] = Lord; |
---|
3082 | // we are done with the list |
---|
3083 | def @R@ = ring(L); |
---|
3084 | setring @R@; |
---|
3085 | matrix @D[Nnew][Nnew]; |
---|
3086 | @D[1,2]=1; |
---|
3087 | for(i=1; i<=N; i++) |
---|
3088 | { |
---|
3089 | @D[2+i,N+2+i]=1; |
---|
3090 | } |
---|
3091 | // ADD [s,t]=-t, [s,Dt]=Dt |
---|
3092 | @D[1,Nnew] = -var(1); |
---|
3093 | @D[2,Nnew] = var(2); |
---|
3094 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
3095 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
3096 | // L[6] = @D; |
---|
3097 | def @R = nc_algebra(1,@D); |
---|
3098 | setring @R; |
---|
3099 | kill @R@; |
---|
3100 | dbprint(ppl,"// -1-1- the ring @R(t,Dt,_x,_Dx,s) is ready"); |
---|
3101 | dbprint(ppl-1, @R); |
---|
3102 | // create the ideal I |
---|
3103 | poly F = imap(save,F); |
---|
3104 | // ideal I = u*F-t,u*v-1; |
---|
3105 | ideal I = F-t; |
---|
3106 | poly p; |
---|
3107 | for(i=1; i<=N; i++) |
---|
3108 | { |
---|
3109 | // p = u*Dt; // u*Dt |
---|
3110 | p = Dt; |
---|
3111 | p = diff(F,var(2+i))*p; |
---|
3112 | I = I, var(N+2+i) + p; |
---|
3113 | } |
---|
3114 | // I = I, var(1)*var(2) + var(Nnew) +1; // reduce it with t-f!!! |
---|
3115 | // t*Dt + s +1 reduced with t-f gives f*Dt + s |
---|
3116 | I = I, F*var(2) + var(Nnew); |
---|
3117 | // -------- the ideal I is ready ---------- |
---|
3118 | dbprint(ppl,"// -1-2- starting the elimination of t,Dt in @R"); |
---|
3119 | dbprint(ppl-1, I); |
---|
3120 | ideal J = engine(I,eng); |
---|
3121 | ideal K = nselect(J,1,2); |
---|
3122 | dbprint(ppl,"// -1-3- t,Dt are eliminated"); |
---|
3123 | dbprint(ppl-1, K); // K is without t, Dt |
---|
3124 | K = engine(K,eng); // std does the job too |
---|
3125 | // now, we must change the ordering |
---|
3126 | // and create a ring without t, Dt |
---|
3127 | setring save; |
---|
3128 | // ----------- the ring @R3 ------------ |
---|
3129 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
3130 | // keep: N, i,j,s, tmp, RL |
---|
3131 | Nnew = 2*N+1; |
---|
3132 | kill Lord, tmp, iv, RName; |
---|
3133 | list Lord, tmp; |
---|
3134 | intvec iv; |
---|
3135 | L[1] = RL[1]; |
---|
3136 | L[4] = RL[4]; // char, minpoly |
---|
3137 | // check whether vars hava admissible names -> done earlier |
---|
3138 | // now, create the names for new var |
---|
3139 | tmp[1] = "s"; |
---|
3140 | // DName is defined earlier |
---|
3141 | list NName = Name + DName + tmp; |
---|
3142 | L[2] = NName; |
---|
3143 | tmp = 0; |
---|
3144 | // block ord (dp(N),dp); |
---|
3145 | // string s is already defined |
---|
3146 | s = "iv="; |
---|
3147 | for (i=1; i<=Nnew-1; i++) |
---|
3148 | { |
---|
3149 | s = s+"1,"; |
---|
3150 | } |
---|
3151 | s[size(s)]=";"; |
---|
3152 | execute(s); |
---|
3153 | tmp[1] = "dp"; // string |
---|
3154 | tmp[2] = iv; // intvec |
---|
3155 | Lord[1] = tmp; |
---|
3156 | // continue with dp 1,1,1,1... |
---|
3157 | tmp[1] = "dp"; // string |
---|
3158 | s[size(s)] = ","; |
---|
3159 | s = s+"1;"; |
---|
3160 | execute(s); |
---|
3161 | kill s; |
---|
3162 | kill NName; |
---|
3163 | tmp[2] = iv; |
---|
3164 | Lord[2] = tmp; |
---|
3165 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
3166 | Lord[3] = tmp; tmp = 0; |
---|
3167 | L[3] = Lord; |
---|
3168 | // we are done with the list. Now add a Plural part |
---|
3169 | def @R2@ = ring(L); |
---|
3170 | setring @R2@; |
---|
3171 | matrix @D[Nnew][Nnew]; |
---|
3172 | for (i=1; i<=N; i++) |
---|
3173 | { |
---|
3174 | @D[i,N+i]=1; |
---|
3175 | } |
---|
3176 | def @R2 = nc_algebra(1,@D); |
---|
3177 | setring @R2; |
---|
3178 | kill @R2@; |
---|
3179 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
3180 | dbprint(ppl-1, @R2); |
---|
3181 | ideal MM = maxideal(1); |
---|
3182 | MM = 0,s,MM; |
---|
3183 | map R01 = @R, MM; |
---|
3184 | ideal K = R01(K); |
---|
3185 | // total cleanup |
---|
3186 | ideal LD = K; |
---|
3187 | // make leadcoeffs positive |
---|
3188 | for (i=1; i<= ncols(LD); i++) |
---|
3189 | { |
---|
3190 | if (leadcoef(LD[i]) <0 ) |
---|
3191 | { |
---|
3192 | LD[i] = -LD[i]; |
---|
3193 | } |
---|
3194 | } |
---|
3195 | export LD; |
---|
3196 | kill @R; |
---|
3197 | return(@R2); |
---|
3198 | } |
---|
3199 | example |
---|
3200 | { |
---|
3201 | "EXAMPLE:"; echo = 2; |
---|
3202 | ring r = 0,(x,y,z),Dp; |
---|
3203 | poly F = x^3+y^3+z^3; |
---|
3204 | printlevel = 0; |
---|
3205 | def A = SannfsLOT(F); |
---|
3206 | setring A; |
---|
3207 | LD; |
---|
3208 | } |
---|
3209 | |
---|
3210 | |
---|
3211 | proc annfsLOT(poly F, list #) |
---|
3212 | "USAGE: annfsLOT(F [,eng]); F a poly, eng an optional int |
---|
3213 | RETURN: ring |
---|
3214 | PURPOSE: compute the D-module structure of basering[1/f]*f^s, according to the Levandovskyy's modification of the algorithm by Oaku and Takayama |
---|
3215 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
3216 | @* - the ideal LD (which is a Groebner basis) is the needed D-module structure, |
---|
3217 | @* which is obtained by substituting the minimal integer root of a Bernstein |
---|
3218 | @* polynomial into the s-parametric ideal; |
---|
3219 | @* - the list BS contains the roots with multiplicities of a Bernstein polynomial of f. |
---|
3220 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
3221 | @* otherwise and by default @code{slimgb} is used. |
---|
3222 | @* If printlevel=1, progress debug messages will be printed, |
---|
3223 | @* if printlevel>=2, all the debug messages will be printed. |
---|
3224 | EXAMPLE: example annfsLOT; shows examples |
---|
3225 | " |
---|
3226 | { |
---|
3227 | int eng = 0; |
---|
3228 | if ( size(#)>0 ) |
---|
3229 | { |
---|
3230 | if ( typeof(#[1]) == "int" ) |
---|
3231 | { |
---|
3232 | eng = int(#[1]); |
---|
3233 | } |
---|
3234 | } |
---|
3235 | printlevel=printlevel+1; |
---|
3236 | def save = basering; |
---|
3237 | def @A = SannfsLOT(F,eng); |
---|
3238 | setring @A; |
---|
3239 | poly F = imap(save,F); |
---|
3240 | def B = annfs0(LD,F,eng); |
---|
3241 | return(B); |
---|
3242 | } |
---|
3243 | example |
---|
3244 | { |
---|
3245 | "EXAMPLE:"; echo = 2; |
---|
3246 | ring r = 0,(x,y,z),Dp; |
---|
3247 | poly F = z*x^2+y^3; |
---|
3248 | printlevel = 0; |
---|
3249 | def A = annfsLOT(F); |
---|
3250 | setring A; |
---|
3251 | LD; |
---|
3252 | BS; |
---|
3253 | } |
---|
3254 | |
---|
3255 | proc annfs0(ideal I, poly F, list #) |
---|
3256 | "USAGE: annfs0(I, F [,eng]); I an ideal, F a poly, eng an optional int |
---|
3257 | RETURN: ring |
---|
3258 | PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, based on the |
---|
3259 | output of procedures SannfsBM, SannfsOT or SannfsLOT |
---|
3260 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
3261 | @* - the ideal LD (which is a Groebner basis) is the annihilator of f^s, |
---|
3262 | @* - the list BS contains the roots with multiplicities of a Bernstein polynomial of f. |
---|
3263 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
3264 | @* otherwise and by default @code{slimgb} is used. |
---|
3265 | @* If printlevel=1, progress debug messages will be printed, |
---|
3266 | @* if printlevel>=2, all the debug messages will be printed. |
---|
3267 | EXAMPLE: example annfs0; shows examples |
---|
3268 | " |
---|
3269 | { |
---|
3270 | int eng = 0; |
---|
3271 | if ( size(#)>0 ) |
---|
3272 | { |
---|
3273 | if ( typeof(#[1]) == "int" ) |
---|
3274 | { |
---|
3275 | eng = int(#[1]); |
---|
3276 | } |
---|
3277 | } |
---|
3278 | def @R2 = basering; |
---|
3279 | // we're in D_n[s], where the elim ord for s is set |
---|
3280 | ideal J = NF(I,std(F)); |
---|
3281 | // make leadcoeffs positive |
---|
3282 | int i; |
---|
3283 | for (i=1; i<= ncols(J); i++) |
---|
3284 | { |
---|
3285 | if (leadcoef(J[i]) <0 ) |
---|
3286 | { |
---|
3287 | J[i] = -J[i]; |
---|
3288 | } |
---|
3289 | } |
---|
3290 | J = J,F; |
---|
3291 | ideal M = engine(J,eng); |
---|
3292 | int Nnew = nvars(@R2); |
---|
3293 | ideal K2 = nselect(M,1,Nnew-1); |
---|
3294 | int ppl = printlevel-voice+2; |
---|
3295 | dbprint(ppl,"// -1-1- _x,_Dx are eliminated in basering"); |
---|
3296 | dbprint(ppl-1, K2); |
---|
3297 | // the ring @R3 and the search for minimal negative int s |
---|
3298 | ring @R3 = 0,s,dp; |
---|
3299 | dbprint(ppl,"// -2-1- the ring @R3 i.e. K[s] is ready"); |
---|
3300 | ideal K3 = imap(@R2,K2); |
---|
3301 | poly p = K3[1]; |
---|
3302 | dbprint(ppl,"// -2-2- factorization"); |
---|
3303 | // ideal P = factorize(p,1); //without constants and multiplicities |
---|
3304 | // "--------- b-function factorizes into ---------"; P; |
---|
3305 | // convert factors to the list of their roots with mults |
---|
3306 | // assume all factors are linear |
---|
3307 | // ideal BS = normalize(P); |
---|
3308 | // BS = subst(BS,s,0); |
---|
3309 | // BS = -BS; |
---|
3310 | list P = factorize(p); //with constants and multiplicities |
---|
3311 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
3312 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
3313 | { |
---|
3314 | bs[i-1] = P[1][i]; |
---|
3315 | m[i-1] = P[2][i]; |
---|
3316 | } |
---|
3317 | int sP = minIntRoot(bs,1); |
---|
3318 | bs = normalize(bs); |
---|
3319 | bs = -subst(bs,s,0); |
---|
3320 | dbprint(ppl,"// -2-3- minimal integer root found"); |
---|
3321 | dbprint(ppl-1, sP); |
---|
3322 | //TODO: sort BS! |
---|
3323 | // --------- substitute s found in the ideal --------- |
---|
3324 | // --------- going back to @R and substitute --------- |
---|
3325 | setring @R2; |
---|
3326 | K2 = subst(I,s,sP); |
---|
3327 | // create the ordinary Weyl algebra and put the result into it, |
---|
3328 | // thus creating the ring @R5 |
---|
3329 | // keep: N, i,j,s, tmp, RL |
---|
3330 | Nnew = Nnew - 1; // former 2*N; |
---|
3331 | // list RL = ringlist(save); // is defined earlier |
---|
3332 | // kill Lord, tmp, iv; |
---|
3333 | list L = 0; |
---|
3334 | list Lord, tmp; |
---|
3335 | intvec iv; |
---|
3336 | list RL = ringlist(basering); |
---|
3337 | L[1] = RL[1]; |
---|
3338 | L[4] = RL[4]; //char, minpoly |
---|
3339 | // check whether vars have admissible names -> done earlier |
---|
3340 | // list Name = RL[2]M |
---|
3341 | // DName is defined earlier |
---|
3342 | list NName; // = RL[2]; // skip the last var 's' |
---|
3343 | for (i=1; i<=Nnew; i++) |
---|
3344 | { |
---|
3345 | NName[i] = RL[2][i]; |
---|
3346 | } |
---|
3347 | L[2] = NName; |
---|
3348 | // dp ordering; |
---|
3349 | string s = "iv="; |
---|
3350 | for (i=1; i<=Nnew; i++) |
---|
3351 | { |
---|
3352 | s = s+"1,"; |
---|
3353 | } |
---|
3354 | s[size(s)] = ";"; |
---|
3355 | execute(s); |
---|
3356 | tmp = 0; |
---|
3357 | tmp[1] = "dp"; // string |
---|
3358 | tmp[2] = iv; // intvec |
---|
3359 | Lord[1] = tmp; |
---|
3360 | kill s; |
---|
3361 | tmp[1] = "C"; |
---|
3362 | iv = 0; |
---|
3363 | tmp[2] = iv; |
---|
3364 | Lord[2] = tmp; |
---|
3365 | tmp = 0; |
---|
3366 | L[3] = Lord; |
---|
3367 | // we are done with the list |
---|
3368 | // Add: Plural part |
---|
3369 | def @R4@ = ring(L); |
---|
3370 | setring @R4@; |
---|
3371 | int N = Nnew/2; |
---|
3372 | matrix @D[Nnew][Nnew]; |
---|
3373 | for (i=1; i<=N; i++) |
---|
3374 | { |
---|
3375 | @D[i,N+i]=1; |
---|
3376 | } |
---|
3377 | def @R4 = nc_algebra(1,@D); |
---|
3378 | setring @R4; |
---|
3379 | kill @R4@; |
---|
3380 | dbprint(ppl,"// -3-1- the ring @R4 is ready"); |
---|
3381 | dbprint(ppl-1, @R4); |
---|
3382 | ideal K4 = imap(@R2,K2); |
---|
3383 | option(redSB); |
---|
3384 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
3385 | K4 = engine(K4,eng); // std does the job too |
---|
3386 | // total cleanup |
---|
3387 | ideal bs = imap(@R3,bs); |
---|
3388 | kill @R3; |
---|
3389 | list BS = bs,m; |
---|
3390 | export BS; |
---|
3391 | ideal LD = K4; |
---|
3392 | export LD; |
---|
3393 | return(@R4); |
---|
3394 | } |
---|
3395 | example |
---|
3396 | { "EXAMPLE:"; echo = 2; |
---|
3397 | ring r = 0,(x,y,z),Dp; |
---|
3398 | poly F = x^3+y^3+z^3; |
---|
3399 | printlevel = 0; |
---|
3400 | def A = SannfsBM(F); |
---|
3401 | // alternatively, one can use SannfsOT or SannfsLOT |
---|
3402 | setring A; |
---|
3403 | LD; |
---|
3404 | poly F = imap(r,F); |
---|
3405 | def B = annfs0(LD,F); |
---|
3406 | setring B; |
---|
3407 | LD; |
---|
3408 | BS; |
---|
3409 | } |
---|
3410 | |
---|
3411 | // proc annfsgms(poly F, list #) |
---|
3412 | // "USAGE: annfsgms(f [,eng]); f a poly, eng an optional int |
---|
3413 | // ASSUME: f has an isolated critical point at 0 |
---|
3414 | // RETURN: ring |
---|
3415 | // PURPOSE: compute the D-module structure of basering[1/f]*f^s |
---|
3416 | // NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
3417 | // @* - the ideal LD is the needed D-mod structure, |
---|
3418 | // @* - the ideal BS is the list of roots of a Bernstein polynomial of f. |
---|
3419 | // @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
3420 | // @* otherwise (and by default) @code{slimgb} is used. |
---|
3421 | // @* If printlevel=1, progress debug messages will be printed, |
---|
3422 | // @* if printlevel>=2, all the debug messages will be printed. |
---|
3423 | // EXAMPLE: example annfsgms; shows examples |
---|
3424 | // " |
---|
3425 | // { |
---|
3426 | // LIB "gmssing.lib"; |
---|
3427 | // int eng = 0; |
---|
3428 | // if ( size(#)>0 ) |
---|
3429 | // { |
---|
3430 | // if ( typeof(#[1]) == "int" ) |
---|
3431 | // { |
---|
3432 | // eng = int(#[1]); |
---|
3433 | // } |
---|
3434 | // } |
---|
3435 | // int ppl = printlevel-voice+2; |
---|
3436 | // // returns a ring with the ideal LD in it |
---|
3437 | // def save = basering; |
---|
3438 | // // compute the Bernstein poly from gmssing.lib |
---|
3439 | // list RL = ringlist(basering); |
---|
3440 | // // in the descr. of the ordering, replace "p" by "s" |
---|
3441 | // list NL = convloc(RL); |
---|
3442 | // // create a ring with the ordering, converted to local |
---|
3443 | // def @LR = ring(NL); |
---|
3444 | // setring @LR; |
---|
3445 | // poly F = imap(save, F); |
---|
3446 | // ideal B = bernstein(F)[1]; |
---|
3447 | // // since B may not contain (s+1) [following gmssing.lib] |
---|
3448 | // // add it! |
---|
3449 | // B = B,-1; |
---|
3450 | // B = simplify(B,2+4); // erase zero and repeated entries |
---|
3451 | // // find the minimal integer value |
---|
3452 | // int S = minIntRoot(B,0); |
---|
3453 | // dbprint(ppl,"// -0- minimal integer root found"); |
---|
3454 | // dbprint(ppl-1,S); |
---|
3455 | // setring save; |
---|
3456 | // int N = nvars(basering); |
---|
3457 | // int Nnew = 2*(N+2); |
---|
3458 | // int i,j; |
---|
3459 | // string s; |
---|
3460 | // // list RL = ringlist(basering); |
---|
3461 | // list L, Lord; |
---|
3462 | // list tmp; |
---|
3463 | // intvec iv; |
---|
3464 | // L[1] = RL[1]; // char |
---|
3465 | // L[4] = RL[4]; // char, minpoly |
---|
3466 | // // check whether vars have admissible names |
---|
3467 | // list Name = RL[2]; |
---|
3468 | // list RName; |
---|
3469 | // RName[1] = "u"; |
---|
3470 | // RName[2] = "v"; |
---|
3471 | // RName[3] = "t"; |
---|
3472 | // RName[4] = "Dt"; |
---|
3473 | // for(i=1;i<=N;i++) |
---|
3474 | // { |
---|
3475 | // for(j=1; j<=size(RName);j++) |
---|
3476 | // { |
---|
3477 | // if (Name[i] == RName[j]) |
---|
3478 | // { |
---|
3479 | // ERROR("Variable names should not include u,v,t,Dt"); |
---|
3480 | // } |
---|
3481 | // } |
---|
3482 | // } |
---|
3483 | // // now, create the names for new vars |
---|
3484 | // // tmp[1] = "u"; tmp[2] = "v"; tmp[3] = "t"; tmp[4] = "Dt"; |
---|
3485 | // list UName = RName; |
---|
3486 | // list DName; |
---|
3487 | // for(i=1;i<=N;i++) |
---|
3488 | // { |
---|
3489 | // DName[i] = "D"+Name[i]; // concat |
---|
3490 | // } |
---|
3491 | // list NName = UName + Name + DName; |
---|
3492 | // L[2] = NName; |
---|
3493 | // tmp = 0; |
---|
3494 | // // Name, Dname will be used further |
---|
3495 | // kill UName; |
---|
3496 | // kill NName; |
---|
3497 | // // block ord (a(1,1),dp); |
---|
3498 | // tmp[1] = "a"; // string |
---|
3499 | // iv = 1,1; |
---|
3500 | // tmp[2] = iv; //intvec |
---|
3501 | // Lord[1] = tmp; |
---|
3502 | // // continue with dp 1,1,1,1... |
---|
3503 | // tmp[1] = "dp"; // string |
---|
3504 | // s = "iv="; |
---|
3505 | // for(i=1; i<=Nnew; i++) // need really all vars! |
---|
3506 | // { |
---|
3507 | // s = s+"1,"; |
---|
3508 | // } |
---|
3509 | // s[size(s)]= ";"; |
---|
3510 | // execute(s); |
---|
3511 | // tmp[2] = iv; |
---|
3512 | // Lord[2] = tmp; |
---|
3513 | // tmp[1] = "C"; |
---|
3514 | // iv = 0; |
---|
3515 | // tmp[2] = iv; |
---|
3516 | // Lord[3] = tmp; |
---|
3517 | // tmp = 0; |
---|
3518 | // L[3] = Lord; |
---|
3519 | // // we are done with the list |
---|
3520 | // def @R = ring(L); |
---|
3521 | // setring @R; |
---|
3522 | // matrix @D[Nnew][Nnew]; |
---|
3523 | // @D[3,4] = 1; // t,Dt |
---|
3524 | // for(i=1; i<=N; i++) |
---|
3525 | // { |
---|
3526 | // @D[4+i,4+N+i]=1; |
---|
3527 | // } |
---|
3528 | // // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
3529 | // // L[6] = @D; |
---|
3530 | // nc_algebra(1,@D); |
---|
3531 | // dbprint(ppl,"// -1-1- the ring @R is ready"); |
---|
3532 | // dbprint(ppl-1,@R); |
---|
3533 | // // create the ideal |
---|
3534 | // poly F = imap(save,F); |
---|
3535 | // ideal I = u*F-t,u*v-1; |
---|
3536 | // poly p; |
---|
3537 | // for(i=1; i<=N; i++) |
---|
3538 | // { |
---|
3539 | // p = u*Dt; // u*Dt |
---|
3540 | // p = diff(F,var(4+i))*p; |
---|
3541 | // I = I, var(N+4+i) + p; // Dx, Dy |
---|
3542 | // } |
---|
3543 | // // add the relations between t,Dt and s |
---|
3544 | // // I = I, t*Dt+1+S; |
---|
3545 | // // -------- the ideal I is ready ---------- |
---|
3546 | // dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
3547 | // ideal J = engine(I,eng); |
---|
3548 | // ideal K = nselect(J,1,2); |
---|
3549 | // dbprint(ppl,"// -1-3- u,v are eliminated in @R"); |
---|
3550 | // dbprint(ppl-1,K); // without u,v: not yet our answer |
---|
3551 | // //----- create a ring with elim.ord for t,Dt ------- |
---|
3552 | // setring save; |
---|
3553 | // // ------------ new ring @R2 ------------------ |
---|
3554 | // // without u,v and with the elim.ord for t,Dt |
---|
3555 | // // keep: N, i,j,s, tmp, RL |
---|
3556 | // Nnew = 2*N+2; |
---|
3557 | // // list RL = ringlist(save); // is defined earlier |
---|
3558 | // kill Lord,tmp,iv, RName; |
---|
3559 | // L = 0; |
---|
3560 | // list Lord, tmp; |
---|
3561 | // intvec iv; |
---|
3562 | // L[1] = RL[1]; // char |
---|
3563 | // L[4] = RL[4]; // char, minpoly |
---|
3564 | // // check whether vars have admissible names -> done earlier |
---|
3565 | // // list Name = RL[2]; |
---|
3566 | // list RName; |
---|
3567 | // RName[1] = "t"; |
---|
3568 | // RName[2] = "Dt"; |
---|
3569 | // // DName is defined earlier |
---|
3570 | // list NName = RName + Name + DName; |
---|
3571 | // L[2] = NName; |
---|
3572 | // tmp = 0; |
---|
3573 | // // block ord (a(1,1),dp); |
---|
3574 | // tmp[1] = "a"; // string |
---|
3575 | // iv = 1,1; |
---|
3576 | // tmp[2] = iv; //intvec |
---|
3577 | // Lord[1] = tmp; |
---|
3578 | // // continue with dp 1,1,1,1... |
---|
3579 | // tmp[1] = "dp"; // string |
---|
3580 | // s = "iv="; |
---|
3581 | // for(i=1;i<=Nnew;i++) |
---|
3582 | // { |
---|
3583 | // s = s+"1,"; |
---|
3584 | // } |
---|
3585 | // s[size(s)]= ";"; |
---|
3586 | // execute(s); |
---|
3587 | // kill s; |
---|
3588 | // kill NName; |
---|
3589 | // tmp[2] = iv; |
---|
3590 | // Lord[2] = tmp; |
---|
3591 | // tmp[1] = "C"; |
---|
3592 | // iv = 0; |
---|
3593 | // tmp[2] = iv; |
---|
3594 | // Lord[3] = tmp; |
---|
3595 | // tmp = 0; |
---|
3596 | // L[3] = Lord; |
---|
3597 | // // we are done with the list |
---|
3598 | // // Add: Plural part |
---|
3599 | // def @R2 = ring(L); |
---|
3600 | // setring @R2; |
---|
3601 | // matrix @D[Nnew][Nnew]; |
---|
3602 | // @D[1,2]=1; |
---|
3603 | // for(i=1; i<=N; i++) |
---|
3604 | // { |
---|
3605 | // @D[2+i,2+N+i]=1; |
---|
3606 | // } |
---|
3607 | // nc_algebra(1,@D); |
---|
3608 | // dbprint(ppl,"// -2-1- the ring @R2 is ready"); |
---|
3609 | // dbprint(ppl-1,@R2); |
---|
3610 | // ideal MM = maxideal(1); |
---|
3611 | // MM = 0,0,MM; |
---|
3612 | // map R01 = @R, MM; |
---|
3613 | // ideal K2 = R01(K); |
---|
3614 | // // add the relations between t,Dt and s |
---|
3615 | // // K2 = K2, t*Dt+1+S; |
---|
3616 | // poly G = t*Dt+S+1; |
---|
3617 | // K2 = NF(K2,std(G)),G; |
---|
3618 | // dbprint(ppl,"// -2-2- starting elimination for t,Dt in @R2"); |
---|
3619 | // ideal J = engine(K2,eng); |
---|
3620 | // ideal K = nselect(J,1,2); |
---|
3621 | // dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
3622 | // dbprint(ppl-1,K); |
---|
3623 | // // "------- produce a final result --------"; |
---|
3624 | // // ----- create the ordinary Weyl algebra and put |
---|
3625 | // // ----- the result into it |
---|
3626 | // // ------ create the ring @R5 |
---|
3627 | // // keep: N, i,j,s, tmp, RL |
---|
3628 | // setring save; |
---|
3629 | // Nnew = 2*N; |
---|
3630 | // // list RL = ringlist(save); // is defined earlier |
---|
3631 | // kill Lord, tmp, iv; |
---|
3632 | // // kill L; |
---|
3633 | // L = 0; |
---|
3634 | // list Lord, tmp; |
---|
3635 | // intvec iv; |
---|
3636 | // L[1] = RL[1]; // char |
---|
3637 | // L[4] = RL[4]; // char, minpoly |
---|
3638 | // // check whether vars have admissible names -> done earlier |
---|
3639 | // // list Name = RL[2]; |
---|
3640 | // // DName is defined earlier |
---|
3641 | // list NName = Name + DName; |
---|
3642 | // L[2] = NName; |
---|
3643 | // // dp ordering; |
---|
3644 | // string s = "iv="; |
---|
3645 | // for(i=1;i<=2*N;i++) |
---|
3646 | // { |
---|
3647 | // s = s+"1,"; |
---|
3648 | // } |
---|
3649 | // s[size(s)]= ";"; |
---|
3650 | // execute(s); |
---|
3651 | // tmp = 0; |
---|
3652 | // tmp[1] = "dp"; // string |
---|
3653 | // tmp[2] = iv; //intvec |
---|
3654 | // Lord[1] = tmp; |
---|
3655 | // kill s; |
---|
3656 | // tmp[1] = "C"; |
---|
3657 | // iv = 0; |
---|
3658 | // tmp[2] = iv; |
---|
3659 | // Lord[2] = tmp; |
---|
3660 | // tmp = 0; |
---|
3661 | // L[3] = Lord; |
---|
3662 | // // we are done with the list |
---|
3663 | // // Add: Plural part |
---|
3664 | // def @R5 = ring(L); |
---|
3665 | // setring @R5; |
---|
3666 | // matrix @D[Nnew][Nnew]; |
---|
3667 | // for(i=1; i<=N; i++) |
---|
3668 | // { |
---|
3669 | // @D[i,N+i]=1; |
---|
3670 | // } |
---|
3671 | // nc_algebra(1,@D); |
---|
3672 | // dbprint(ppl,"// -3-1- the ring @R5 is ready"); |
---|
3673 | // dbprint(ppl-1,@R5); |
---|
3674 | // ideal K5 = imap(@R2,K); |
---|
3675 | // option(redSB); |
---|
3676 | // dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
3677 | // K5 = engine(K5,eng); // std does the job too |
---|
3678 | // // total cleanup |
---|
3679 | // kill @R; |
---|
3680 | // kill @R2; |
---|
3681 | // ideal LD = K5; |
---|
3682 | // ideal BS = imap(@LR,B); |
---|
3683 | // kill @LR; |
---|
3684 | // export BS; |
---|
3685 | // export LD; |
---|
3686 | // return(@R5); |
---|
3687 | // } |
---|
3688 | // example |
---|
3689 | // { |
---|
3690 | // "EXAMPLE:"; echo = 2; |
---|
3691 | // ring r = 0,(x,y,z),Dp; |
---|
3692 | // poly F = x^2+y^3+z^5; |
---|
3693 | // def A = annfsgms(F); |
---|
3694 | // setring A; |
---|
3695 | // LD; |
---|
3696 | // print(matrix(BS)); |
---|
3697 | // } |
---|
3698 | |
---|
3699 | |
---|
3700 | proc convloc(list @NL) |
---|
3701 | "USAGE: convloc(L); L a list |
---|
3702 | RETURN: list |
---|
3703 | PURPOSE: convert a ringlist L into another ringlist, |
---|
3704 | where all the 'p' orderings are replaced with the 's' orderings. |
---|
3705 | ASSUME: L is a result of a ringlist command |
---|
3706 | EXAMPLE: example convloc; shows examples |
---|
3707 | " |
---|
3708 | { |
---|
3709 | list NL = @NL; |
---|
3710 | // given a ringlist, returns a new ringlist, |
---|
3711 | // where all the p-orderings are replaced with s-ord's |
---|
3712 | int i,j,k,found; |
---|
3713 | int nblocks = size(NL[3]); |
---|
3714 | for(i=1; i<=nblocks; i++) |
---|
3715 | { |
---|
3716 | for(j=1; j<=size(NL[3][i]); j++) |
---|
3717 | { |
---|
3718 | if (typeof(NL[3][i][j]) == "string" ) |
---|
3719 | { |
---|
3720 | found = 0; |
---|
3721 | for (k=1; k<=size(NL[3][i][j]); k++) |
---|
3722 | { |
---|
3723 | if (NL[3][i][j][k]=="p") |
---|
3724 | { |
---|
3725 | NL[3][i][j][k]="s"; |
---|
3726 | found = 1; |
---|
3727 | // printf("replaced at %s,%s,%s",i,j,k); |
---|
3728 | } |
---|
3729 | } |
---|
3730 | } |
---|
3731 | } |
---|
3732 | } |
---|
3733 | return(NL); |
---|
3734 | } |
---|
3735 | example |
---|
3736 | { |
---|
3737 | "EXAMPLE:"; echo = 2; |
---|
3738 | ring r = 0,(x,y,z),(Dp(2),dp(1)); |
---|
3739 | list L = ringlist(r); |
---|
3740 | list N = convloc(L); |
---|
3741 | def rs = ring(N); |
---|
3742 | setring rs; |
---|
3743 | rs; |
---|
3744 | } |
---|
3745 | |
---|
3746 | proc annfspecial(ideal I, poly F, int mir, number n) |
---|
3747 | "USAGE: annfspecial(I,F,mir,s0); I an ideal, F a poly, int mir, number s0 |
---|
3748 | RETURN: ideal |
---|
3749 | PURPOSE: compute the annihilator ideal of f^s0 in the Weyl Algebra for a rational number s0 |
---|
3750 | ASSUME: the basering contains 's' as a variable |
---|
3751 | NOTE: We assume that the basering is D[s], |
---|
3752 | @* ideal I is the Ann F^s in D[s] (obtained with e.g. SannfsBM, SannfsLOT, SannfsOT) |
---|
3753 | @* integer 'mir' is the minimal integer root of the Bernstein polynomial of F |
---|
3754 | @* and the number s0 is rational. |
---|
3755 | @* We compute the real annihilator for any value of s0 (both generic and exceptional). |
---|
3756 | @* The implementation goes along the lines of Saito-Sturmfels-Takayama, Alg. 5.3.15 |
---|
3757 | @* If printlevel=1, progress debug messages will be printed, |
---|
3758 | @* if printlevel>=2, all the debug messages will be printed. |
---|
3759 | EXAMPLE: example annfspecial; shows examples |
---|
3760 | " |
---|
3761 | { |
---|
3762 | int ppl = printlevel-voice+2; |
---|
3763 | // int mir = minIntRoot(L[1],0); |
---|
3764 | int ns = varnum("s"); |
---|
3765 | if (!ns) |
---|
3766 | { |
---|
3767 | ERROR("Variable s expected in the ideal AnnFs"); |
---|
3768 | } |
---|
3769 | int d; |
---|
3770 | ideal P = subst(I,var(ns),n); |
---|
3771 | if (denominator(n) == 1) |
---|
3772 | { |
---|
3773 | // n is fraction-free |
---|
3774 | d = int(numerator(n)); |
---|
3775 | if ( (!d) && (n!=0)) |
---|
3776 | { |
---|
3777 | ERROR("no parametric special values are allowed"); |
---|
3778 | } |
---|
3779 | d = d - mir; |
---|
3780 | if (d>0) |
---|
3781 | { |
---|
3782 | dbprint(ppl,"// -1-1- starting syzygy computations"); |
---|
3783 | matrix J[1][1] = F^d; |
---|
3784 | dbprint(ppl-1,"// -1-1-1- of the polynomial ideal"); |
---|
3785 | dbprint(ppl-1,J); |
---|
3786 | matrix K[1][size(I)] = subst(I,var(ns),mir); |
---|
3787 | dbprint(ppl-1,"// -1-1-2- modulo ideal:"); |
---|
3788 | dbprint(ppl-1, K); |
---|
3789 | module M = modulo(J,K); |
---|
3790 | dbprint(ppl-1,"// -1-1-3- getting the result:"); |
---|
3791 | dbprint(ppl-1, M); |
---|
3792 | P = P, ideal(M); |
---|
3793 | dbprint(ppl,"// -1-2- finished syzygy computations"); |
---|
3794 | } |
---|
3795 | else |
---|
3796 | { |
---|
3797 | dbprint(ppl,"// -1-1- d<=0, no syzygy computations needed"); |
---|
3798 | dbprint(ppl-1,"// -1-2- for d ="); |
---|
3799 | dbprint(ppl-1, d); |
---|
3800 | } |
---|
3801 | } |
---|
3802 | // also the else case: d<=0 or n is not an integer |
---|
3803 | dbprint(ppl,"// -2-1- starting final Groebner basis"); |
---|
3804 | P = groebner(P); |
---|
3805 | dbprint(ppl,"// -2-2- finished final Groebner basis"); |
---|
3806 | return(P); |
---|
3807 | } |
---|
3808 | example |
---|
3809 | { |
---|
3810 | "EXAMPLE:"; echo = 2; |
---|
3811 | ring r = 0,(x,y),dp; |
---|
3812 | poly F = x3-y2; |
---|
3813 | def B = annfs(F); setring B; |
---|
3814 | minIntRoot(BS[1],0); |
---|
3815 | // So, the minimal integer root is -1 |
---|
3816 | setring r; |
---|
3817 | def A = SannfsBM(F); |
---|
3818 | setring A; |
---|
3819 | poly F = x3-y2; |
---|
3820 | annfspecial(LD,F,-1,3/4); // generic root |
---|
3821 | annfspecial(LD,F,-1,-2); // integer but still generic root |
---|
3822 | annfspecial(LD,F,-1,1); // exceptional root |
---|
3823 | } |
---|
3824 | |
---|
3825 | static proc new_ex_annfspecial() |
---|
3826 | { |
---|
3827 | //another example for annfspecial: x3+y3+z3 |
---|
3828 | ring r = 0,(x,y,z),dp; |
---|
3829 | poly F = x3+y3+z3; |
---|
3830 | def B = annfs(F); setring B; |
---|
3831 | minIntRoot(BS[1],0); |
---|
3832 | // So, the minimal integer root is -1 |
---|
3833 | setring r; |
---|
3834 | def A = SannfsBM(F); |
---|
3835 | setring A; |
---|
3836 | poly F = x3+y3+z3; |
---|
3837 | annfspecial(LD,F,-1,3/4); // generic root |
---|
3838 | annfspecial(LD,F,-1,-2); // integer but still generic root |
---|
3839 | annfspecial(LD,F,-1,1); // exceptional root |
---|
3840 | } |
---|
3841 | |
---|
3842 | proc minIntRoot(ideal P, int fact) |
---|
3843 | "USAGE: minIntRoot(P, fact); P an ideal, fact an int |
---|
3844 | RETURN: int |
---|
3845 | PURPOSE: minimal integer root of a maximal ideal P |
---|
3846 | NOTE: if fact==1, P is the result of some 'factorize' call, |
---|
3847 | @* else P is treated as the result of bernstein::gmssing.lib |
---|
3848 | @* in both cases without constants and multiplicities |
---|
3849 | EXAMPLE: example minIntRoot; shows examples |
---|
3850 | " |
---|
3851 | { |
---|
3852 | // ideal P = factorize(p,1); |
---|
3853 | // or ideal P = bernstein(F)[1]; |
---|
3854 | intvec vP; |
---|
3855 | number nP; |
---|
3856 | int i; |
---|
3857 | if ( fact ) |
---|
3858 | { |
---|
3859 | // the result comes from "factorize" |
---|
3860 | P = normalize(P); // now leadcoef = 1 |
---|
3861 | // TODO: hunt for units and kill then !!! |
---|
3862 | P = lead(P)-P; |
---|
3863 | // nP = leadcoef(P[i]-lead(P[i])); // for 1 var only, extract the coeff |
---|
3864 | } |
---|
3865 | else |
---|
3866 | { |
---|
3867 | // bernstein deletes -1 usually |
---|
3868 | P = P, -1; |
---|
3869 | } |
---|
3870 | // for both situations: |
---|
3871 | // now we have an ideal of fractions of type "number" |
---|
3872 | int sP = size(P); |
---|
3873 | for(i=1; i<=sP; i++) |
---|
3874 | { |
---|
3875 | nP = leadcoef(P[i]); |
---|
3876 | if ( (nP - int(nP)) == 0 ) |
---|
3877 | { |
---|
3878 | vP = vP,int(nP); |
---|
3879 | } |
---|
3880 | } |
---|
3881 | if ( size(vP)>=2 ) |
---|
3882 | { |
---|
3883 | vP = vP[2..size(vP)]; |
---|
3884 | } |
---|
3885 | sP = -Max(-vP); |
---|
3886 | if (sP == 0) |
---|
3887 | { |
---|
3888 | "Warning: zero root!"; |
---|
3889 | } |
---|
3890 | return(sP); |
---|
3891 | } |
---|
3892 | example |
---|
3893 | { |
---|
3894 | "EXAMPLE:"; echo = 2; |
---|
3895 | ring r = 0,(x,y),ds; |
---|
3896 | poly f1 = x*y*(x+y); |
---|
3897 | ideal I1 = bernstein(f1)[1]; // a local Bernstein poly |
---|
3898 | minIntRoot(I1,0); |
---|
3899 | poly f2 = x2-y3; |
---|
3900 | ideal I2 = bernstein(f2)[1]; |
---|
3901 | minIntRoot(I2,0); |
---|
3902 | // now we illustrate the behaviour of factorize |
---|
3903 | // together with a global ordering |
---|
3904 | ring r2 = 0,x,dp; |
---|
3905 | poly f3 = 9*(x+2/3)*(x+1)*(x+4/3); //global b-poly of f1=x*y*(x+y) |
---|
3906 | ideal I3 = factorize(f3,1); |
---|
3907 | minIntRoot(I3,1); |
---|
3908 | // and a more interesting situation |
---|
3909 | ring s = 0,(x,y,z),ds; |
---|
3910 | poly f = x3 + y3 + z3; |
---|
3911 | ideal I = bernstein(f)[1]; |
---|
3912 | minIntRoot(I,0); |
---|
3913 | } |
---|
3914 | |
---|
3915 | proc isHolonomic(def M) |
---|
3916 | "USAGE: isHolonomic(M); M an ideal/module/matrix |
---|
3917 | RETURN: int, 1 if M is holonomic and 0 otherwise |
---|
3918 | PURPOSE: check the modules for the property of holonomy |
---|
3919 | NOTE: M is holonomic, if 2*dim(M) = dim(R), where R is a |
---|
3920 | ground ring; dim stays for Gelfand-Kirillov dimension |
---|
3921 | EXAMPLE: example isHolonomic; shows examples |
---|
3922 | " |
---|
3923 | { |
---|
3924 | if ( (typeof(M) != "ideal") && (typeof(M) != "module") && (typeof(M) != "matrix") ) |
---|
3925 | { |
---|
3926 | // print(typeof(M)); |
---|
3927 | ERROR("an argument of type ideal/module/matrix expected"); |
---|
3928 | } |
---|
3929 | if (attrib(M,"isSB")!=1) |
---|
3930 | { |
---|
3931 | M = std(M); |
---|
3932 | } |
---|
3933 | int dimR = gkdim(std(0)); |
---|
3934 | int dimM = gkdim(M); |
---|
3935 | return( (dimR==2*dimM) ); |
---|
3936 | } |
---|
3937 | example |
---|
3938 | { |
---|
3939 | "EXAMPLE:"; echo = 2; |
---|
3940 | ring R = 0,(x,y),dp; |
---|
3941 | poly F = x*y*(x+y); |
---|
3942 | def A = annfsBM(F,0); |
---|
3943 | setring A; |
---|
3944 | LD; |
---|
3945 | isHolonomic(LD); |
---|
3946 | ideal I = std(LD[1]); |
---|
3947 | I; |
---|
3948 | isHolonomic(I); |
---|
3949 | } |
---|
3950 | |
---|
3951 | proc reiffen(int p, int q) |
---|
3952 | "USAGE: reiffen(p, q); int p, int q |
---|
3953 | RETURN: ring |
---|
3954 | PURPOSE: set up the polynomial, describing a Reiffen curve |
---|
3955 | NOTE: activate this ring with the @code{setring} command and find the |
---|
3956 | curve as a polynomial RC |
---|
3957 | @* a Reiffen curve is defined as F = x^p + y^q + xy^{q-1}, q >= p+1 >= 5 |
---|
3958 | ASSUME: q >= p+1 >= 5. Otherwise an error message is returned |
---|
3959 | EXAMPLE: example reiffen; shows examples |
---|
3960 | " |
---|
3961 | { |
---|
3962 | // a Reiffen curve is defined as |
---|
3963 | // F = x^p + y^q +x*y^{q-1}, q \geq p+1 \geq 5 |
---|
3964 | |
---|
3965 | if ( (p<4) || (q<5) || (q-p<1) ) |
---|
3966 | { |
---|
3967 | ERROR("Some of conditions p>=4, q>=5 or q>=p+1 is not satisfied!"); |
---|
3968 | } |
---|
3969 | ring @r = 0,(x,y),dp; |
---|
3970 | poly RC = y^q +x^p + x*y^(q-1); |
---|
3971 | export RC; |
---|
3972 | return(@r); |
---|
3973 | } |
---|
3974 | example |
---|
3975 | { |
---|
3976 | "EXAMPLE:"; echo = 2; |
---|
3977 | def r = reiffen(4,5); |
---|
3978 | setring r; |
---|
3979 | RC; |
---|
3980 | } |
---|
3981 | |
---|
3982 | proc arrange(int p) |
---|
3983 | "USAGE: arrange(p); int p |
---|
3984 | RETURN: ring |
---|
3985 | PURPOSE: set up the polynomial, describing a hyperplane arrangement |
---|
3986 | NOTE: must be executed in a ring |
---|
3987 | ASSUME: basering is present |
---|
3988 | EXAMPLE: example arrange; shows examples |
---|
3989 | " |
---|
3990 | { |
---|
3991 | int UseBasering = 0 ; |
---|
3992 | if (p<2) |
---|
3993 | { |
---|
3994 | ERROR("p>=2 is required!"); |
---|
3995 | } |
---|
3996 | if ( nameof(basering)!="basering" ) |
---|
3997 | { |
---|
3998 | if (p > nvars(basering)) |
---|
3999 | { |
---|
4000 | ERROR("too big p"); |
---|
4001 | } |
---|
4002 | else |
---|
4003 | { |
---|
4004 | def @r = basering; |
---|
4005 | UseBasering = 1; |
---|
4006 | } |
---|
4007 | } |
---|
4008 | else |
---|
4009 | { |
---|
4010 | // no basering found |
---|
4011 | ERROR("no basering found!"); |
---|
4012 | // ring @r = 0,(x(1..p)),dp; |
---|
4013 | } |
---|
4014 | int i,j,sI; |
---|
4015 | poly q = 1; |
---|
4016 | list ar; |
---|
4017 | ideal tmp; |
---|
4018 | tmp = ideal(var(1)); |
---|
4019 | ar[1] = tmp; |
---|
4020 | for (i = 2; i<=p; i++) |
---|
4021 | { |
---|
4022 | // add i-nary sums to the product |
---|
4023 | ar = indAR(ar,i); |
---|
4024 | } |
---|
4025 | for (i = 1; i<=p; i++) |
---|
4026 | { |
---|
4027 | tmp = ar[i]; sI = size(tmp); |
---|
4028 | for (j = 1; j<=sI; j++) |
---|
4029 | { |
---|
4030 | q = q*tmp[j]; |
---|
4031 | } |
---|
4032 | } |
---|
4033 | if (UseBasering) |
---|
4034 | { |
---|
4035 | return(q); |
---|
4036 | } |
---|
4037 | // poly AR = q; export AR; |
---|
4038 | // return(@r); |
---|
4039 | } |
---|
4040 | example |
---|
4041 | { |
---|
4042 | "EXAMPLE:"; echo = 2; |
---|
4043 | ring X = 0,(x,y,z,t),dp; |
---|
4044 | poly q = arrange(3); |
---|
4045 | factorize(q,1); |
---|
4046 | } |
---|
4047 | |
---|
4048 | proc checkRoot(poly F, number a, list #) |
---|
4049 | "USAGE: checkRoot(f,alpha [,S,eng]); f a poly, alpha a number, S a string , eng an optional int |
---|
4050 | RETURN: int |
---|
4051 | PURPOSE: check whether a rational is a root of the global Bernstein polynomial of f (and compute its multiplicity) |
---|
4052 | with the algorithm given in S and with the Groebner basis engine given in eng |
---|
4053 | NOTE: The annihilator of f^s in D[s] is needed and it is computed according to the algorithm by Briancon and Maisonobe |
---|
4054 | @* The value of a string S can be |
---|
4055 | @* 'alg1' (default) - for the algorithm 1 of J. Martin-Morales (unpublished) |
---|
4056 | @* 'alg2' - for the algorithm 2 of J. Martin-Morales (unpublished) |
---|
4057 | @* The output int is: |
---|
4058 | @* - if the algorithm 1 is chosen: 1 if -alpha is a root of the global Bernstein polynomial and 0 otherwise |
---|
4059 | @* - if the algorithm 2 is chosen: the multiplicity of -alpha as a root of the global Bernstein polynomial of f. |
---|
4060 | @* (If -alpha is not a root, the output is 0) |
---|
4061 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
4062 | @* otherwise (and by default) @code{slimgb} is used. |
---|
4063 | @* If printlevel=1, progress debug messages will be printed, |
---|
4064 | @* if printlevel>=2, all the debug messages will be printed. |
---|
4065 | EXAMPLE: example checkRoot; shows examples |
---|
4066 | " |
---|
4067 | { |
---|
4068 | int eng = 0; |
---|
4069 | int chs = 0; // choice |
---|
4070 | string algo = "alg1"; |
---|
4071 | string st; |
---|
4072 | if ( size(#)>0 ) |
---|
4073 | { |
---|
4074 | if ( typeof(#[1]) == "string" ) |
---|
4075 | { |
---|
4076 | st = string(#[1]); |
---|
4077 | if ( (st == "alg1") || (st == "ALG1") || (st == "Alg1") ||(st == "aLG1")) |
---|
4078 | { |
---|
4079 | algo = "alg1"; |
---|
4080 | chs = 1; |
---|
4081 | } |
---|
4082 | if ( (st == "alg2") || (st == "ALG2") || (st == "Alg2") || (st == "aLG2")) |
---|
4083 | { |
---|
4084 | algo = "alg2"; |
---|
4085 | chs = 1; |
---|
4086 | } |
---|
4087 | if (chs != 1) |
---|
4088 | { |
---|
4089 | // incorrect value of S |
---|
4090 | print("Incorrect algorithm given, proceed with the default alg1 of J. MartÃn-Morales"); |
---|
4091 | algo = "alg1"; |
---|
4092 | } |
---|
4093 | // second arg |
---|
4094 | if (size(#)>1) |
---|
4095 | { |
---|
4096 | // exists 2nd arg |
---|
4097 | if ( typeof(#[2]) == "int" ) |
---|
4098 | { |
---|
4099 | // the case: given alg, given engine |
---|
4100 | eng = int(#[2]); |
---|
4101 | } |
---|
4102 | else |
---|
4103 | { |
---|
4104 | eng = 0; |
---|
4105 | } |
---|
4106 | } |
---|
4107 | else |
---|
4108 | { |
---|
4109 | // no second arg |
---|
4110 | eng = 0; |
---|
4111 | } |
---|
4112 | } |
---|
4113 | else |
---|
4114 | { |
---|
4115 | if ( typeof(#[1]) == "int" ) |
---|
4116 | { |
---|
4117 | // the case: default alg, engine |
---|
4118 | eng = int(#[1]); |
---|
4119 | // algo = "alg1"; //is already set |
---|
4120 | } |
---|
4121 | else |
---|
4122 | { |
---|
4123 | // incorr. 1st arg |
---|
4124 | algo = "alg1"; |
---|
4125 | } |
---|
4126 | } |
---|
4127 | } |
---|
4128 | // size(#)=0, i.e. there is no algorithm and no engine given |
---|
4129 | // eng = 0; algo = "alg1"; //are already set |
---|
4130 | // int ppl = printlevel-voice+2; |
---|
4131 | printlevel=printlevel+1; |
---|
4132 | def save = basering; |
---|
4133 | def @A = SannfsBM(F); |
---|
4134 | setring @A; |
---|
4135 | poly F = imap(save,F); |
---|
4136 | number a = imap(save,a); |
---|
4137 | if ( algo=="alg1") |
---|
4138 | { |
---|
4139 | int output = checkRoot1(LD,F,a,eng); |
---|
4140 | } |
---|
4141 | else |
---|
4142 | { |
---|
4143 | if ( algo=="alg2") |
---|
4144 | { |
---|
4145 | int output = checkRoot2(LD,F,a,eng); |
---|
4146 | } |
---|
4147 | } |
---|
4148 | printlevel=printlevel-1; |
---|
4149 | return(output); |
---|
4150 | } |
---|
4151 | example |
---|
4152 | { |
---|
4153 | "EXAMPLE:"; echo = 2; |
---|
4154 | printlevel=0; |
---|
4155 | ring r = 0,(x,y),Dp; |
---|
4156 | poly F = x^4+y^5+x*y^4; |
---|
4157 | checkRoot(F,11/20); // -11/20 is a root of bf |
---|
4158 | poly G = x*y; |
---|
4159 | checkRoot(G,1,"alg2"); // -1 is a root of bg with multiplicity 2 |
---|
4160 | } |
---|
4161 | |
---|
4162 | proc checkRoot1(ideal I, poly F, number a, list #) |
---|
4163 | "USAGE: checkRoot1(I,f,alpha [,eng]); I an ideal, f a poly, alpha a number, eng an optional int |
---|
4164 | ASSUME: I is the annihilator of f^s in D[s], f is a polynomial in K[_x] |
---|
4165 | RETURN: int, 1 if -alpha is a root of the global Bernstein polynomial of f and 0 otherwise |
---|
4166 | PURPOSE: check whether a rational is a root of the global Bernstein polynomial of f |
---|
4167 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
4168 | @* otherwise (and by default) @code{slimgb} is used. |
---|
4169 | @* If printlevel=1, progress debug messages will be printed, |
---|
4170 | @* if printlevel>=2, all the debug messages will be printed. |
---|
4171 | EXAMPLE: example checkRoot1; shows examples |
---|
4172 | " |
---|
4173 | { |
---|
4174 | int eng = 0; |
---|
4175 | if ( size(#)>0 ) |
---|
4176 | { |
---|
4177 | if ( typeof(#[1]) == "int" ) |
---|
4178 | { |
---|
4179 | eng = int(#[1]); |
---|
4180 | } |
---|
4181 | } |
---|
4182 | int ppl = printlevel-voice+2; |
---|
4183 | dbprint(ppl,"// -0-1- starting the procedure checkRoot1"); |
---|
4184 | def save = basering; |
---|
4185 | int N = nvars(basering); |
---|
4186 | int Nnew = N-1; |
---|
4187 | int n = Nnew / 2; |
---|
4188 | int i; |
---|
4189 | string s; |
---|
4190 | list RL = ringlist(basering); |
---|
4191 | list L, Lord; |
---|
4192 | list tmp; |
---|
4193 | intvec iv; |
---|
4194 | L[1] = RL[1]; // char |
---|
4195 | L[4] = RL[4]; // char, minpoly |
---|
4196 | // check whether basering is D[s]=K(_x,_Dx,s) |
---|
4197 | list Name = RL[2]; |
---|
4198 | for (i=1; i<=n; i++) |
---|
4199 | { |
---|
4200 | if ( bracket(var(i+n),var(i))!=1 ) |
---|
4201 | { |
---|
4202 | ERROR("basering should be D[s]=K(_x,_Dx,s)"); |
---|
4203 | } |
---|
4204 | } |
---|
4205 | if ( Name[N]!="s" ) |
---|
4206 | { |
---|
4207 | ERROR("the last variable of basering should be s"); |
---|
4208 | } |
---|
4209 | // now, create the new vars |
---|
4210 | list NName; |
---|
4211 | for (i=1; i<=Nnew; i++) |
---|
4212 | { |
---|
4213 | NName[i] = Name[i]; |
---|
4214 | } |
---|
4215 | L[2] = NName; |
---|
4216 | kill Name,NName; |
---|
4217 | // block ord (dp); |
---|
4218 | tmp[1] = "dp"; // string |
---|
4219 | s = "iv="; |
---|
4220 | for (i=1; i<=Nnew; i++) |
---|
4221 | { |
---|
4222 | s = s+"1,"; |
---|
4223 | } |
---|
4224 | s[size(s)]=";"; |
---|
4225 | execute(s); |
---|
4226 | kill s; |
---|
4227 | tmp[2] = iv; |
---|
4228 | Lord[1] = tmp; |
---|
4229 | tmp[1] = "C"; |
---|
4230 | iv = 0; |
---|
4231 | tmp[2] = iv; |
---|
4232 | Lord[2] = tmp; |
---|
4233 | tmp = 0; |
---|
4234 | L[3] = Lord; |
---|
4235 | // we are done with the list |
---|
4236 | def @R@ = ring(L); |
---|
4237 | setring @R@; |
---|
4238 | matrix @D[Nnew][Nnew]; |
---|
4239 | for (i=1; i<=n; i++) |
---|
4240 | { |
---|
4241 | @D[i,i+n]=1; |
---|
4242 | } |
---|
4243 | def @R = nc_algebra(1,@D); |
---|
4244 | setring @R; |
---|
4245 | kill @R@; |
---|
4246 | dbprint(ppl,"// -1-1- the ring @R(_x,_Dx) is ready"); |
---|
4247 | dbprint(ppl-1, S); |
---|
4248 | // create the ideal K = ann_D[s](f^s)_{s=-alpha} + < f > |
---|
4249 | setring save; |
---|
4250 | ideal K = subst(I,s,-a); |
---|
4251 | dbprint(ppl,"// -1-2- the variable s has been substituted by "+string(-a)); |
---|
4252 | dbprint(ppl-1, K); |
---|
4253 | K = NF(K,std(F)); |
---|
4254 | // make leadcoeffs positive |
---|
4255 | for (i=1; i<=ncols(K); i++) |
---|
4256 | { |
---|
4257 | if ( leadcoef(K[i])<0 ) |
---|
4258 | { |
---|
4259 | K[i] = -K[i]; |
---|
4260 | } |
---|
4261 | } |
---|
4262 | K = K,F; |
---|
4263 | // ------------ the ideal K is ready ------------ |
---|
4264 | setring @R; |
---|
4265 | ideal K = imap(save,K); |
---|
4266 | dbprint(ppl,"// -1-3- starting the computation of a Groebner basis of K in @R"); |
---|
4267 | dbprint(ppl-1, K); |
---|
4268 | ideal G = engine(K,eng); |
---|
4269 | dbprint(ppl,"// -1-4- the Groebner basis has been computed"); |
---|
4270 | dbprint(ppl-1, G); |
---|
4271 | return(G[1]!=1); |
---|
4272 | } |
---|
4273 | example |
---|
4274 | { |
---|
4275 | "EXAMPLE:"; echo = 2; |
---|
4276 | ring r = 0,(x,y),Dp; |
---|
4277 | poly F = x^4+y^5+x*y^4; |
---|
4278 | printlevel = 0; |
---|
4279 | def A = Sannfs(F); |
---|
4280 | setring A; |
---|
4281 | poly F = imap(r,F); |
---|
4282 | checkRoot1(LD,F,31/20); // -31/20 is not a root of bs |
---|
4283 | checkRoot1(LD,F,11/20); // -11/20 is a root of bs |
---|
4284 | } |
---|
4285 | |
---|
4286 | proc checkRoot2 (ideal I, poly F, number a, list #) |
---|
4287 | "USAGE: checkRoot2(I,f,alpha [,eng]); I an ideal, f a poly, alpha a number, eng an optional int |
---|
4288 | ASSUME: I is the annihilator of f^s in D[s], f is a polynomial in K[_x] |
---|
4289 | RETURN: int, the multiplicity of -alpha as a root of the global Bernstein polynomial of f. If -alpha is not a root, the output is 0 |
---|
4290 | PURPOSE: check whether a rational is a root of the global Bernstein polynomial of f and compute its multiplicity from the known Ann F^s in D[s] |
---|
4291 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
4292 | @* otherwise (and by default) @code{slimgb} is used. |
---|
4293 | @* If printlevel=1, progress debug messages will be printed, |
---|
4294 | @* if printlevel>=2, all the debug messages will be printed. |
---|
4295 | EXAMPLE: example checkRoot2; shows examples |
---|
4296 | " |
---|
4297 | { |
---|
4298 | int eng = 0; |
---|
4299 | if ( size(#)>0 ) |
---|
4300 | { |
---|
4301 | if ( typeof(#[1]) == "int" ) |
---|
4302 | { |
---|
4303 | eng = int(#[1]); |
---|
4304 | } |
---|
4305 | } |
---|
4306 | int ppl = printlevel-voice+2; |
---|
4307 | dbprint(ppl,"// -0-1- starting the procedure checkRoot2"); |
---|
4308 | def save = basering; |
---|
4309 | int N = nvars(basering); |
---|
4310 | int n = (N-1) / 2; |
---|
4311 | int i; |
---|
4312 | string s; |
---|
4313 | list RL = ringlist(basering); |
---|
4314 | list L, Lord; |
---|
4315 | list tmp; |
---|
4316 | intvec iv; |
---|
4317 | L[1] = RL[1]; // char |
---|
4318 | L[4] = RL[4]; // char, minpoly |
---|
4319 | // check whether basering is D[s]=K(_x,_Dx,s) |
---|
4320 | list Name = RL[2]; |
---|
4321 | for (i=1; i<=n; i++) |
---|
4322 | { |
---|
4323 | if ( bracket(var(i+n),var(i))!=1 ) |
---|
4324 | { |
---|
4325 | ERROR("basering should be D[s]=K(_x,_Dx,s)"); |
---|
4326 | } |
---|
4327 | } |
---|
4328 | if ( Name[N]!="s" ) |
---|
4329 | { |
---|
4330 | ERROR("the last variable of basering should be s"); |
---|
4331 | } |
---|
4332 | // now, create the new vars |
---|
4333 | L[2] = Name; |
---|
4334 | kill Name; |
---|
4335 | // block ord (dp); |
---|
4336 | tmp[1] = "dp"; // string |
---|
4337 | s = "iv="; |
---|
4338 | for (i=1; i<=N; i++) |
---|
4339 | { |
---|
4340 | s = s+"1,"; |
---|
4341 | } |
---|
4342 | s[size(s)]=";"; |
---|
4343 | execute(s); |
---|
4344 | kill s; |
---|
4345 | tmp[2] = iv; |
---|
4346 | Lord[1] = tmp; |
---|
4347 | tmp[1] = "C"; |
---|
4348 | iv = 0; |
---|
4349 | tmp[2] = iv; |
---|
4350 | Lord[2] = tmp; |
---|
4351 | tmp = 0; |
---|
4352 | L[3] = Lord; |
---|
4353 | // we are done with the list |
---|
4354 | def @R@ = ring(L); |
---|
4355 | setring @R@; |
---|
4356 | matrix @D[N][N]; |
---|
4357 | for (i=1; i<=n; i++) |
---|
4358 | { |
---|
4359 | @D[i,i+n]=1; |
---|
4360 | } |
---|
4361 | def @R = nc_algebra(1,@D); |
---|
4362 | setring @R; |
---|
4363 | kill @R@; |
---|
4364 | dbprint(ppl,"// -1-1- the ring @R(_x,_Dx,s) is ready"); |
---|
4365 | dbprint(ppl-1, @R); |
---|
4366 | // now, continue with the algorithm |
---|
4367 | ideal I = imap(save,I); |
---|
4368 | poly F = imap(save,F); |
---|
4369 | number a = imap(save,a); |
---|
4370 | ideal II = NF(I,std(F)); |
---|
4371 | // make leadcoeffs positive |
---|
4372 | for (i=1; i<=ncols(II); i++) |
---|
4373 | { |
---|
4374 | if ( leadcoef(II[i])<0 ) |
---|
4375 | { |
---|
4376 | II[i] = -II[i]; |
---|
4377 | } |
---|
4378 | } |
---|
4379 | ideal J,G; |
---|
4380 | int m; // the output (multiplicity) |
---|
4381 | dbprint(ppl,"// -2- starting the bucle"); |
---|
4382 | for (i=0; i<=n; i++) // the multiplicity has to be <= n |
---|
4383 | { |
---|
4384 | // create the ideal Ji = ann_D[s](f^s) + < f, (s+alpha)^{i+1} > |
---|
4385 | // (s+alpha)^i in Ji <==> -alpha is a root with multiplicity >= i |
---|
4386 | J = II,F,(s+a)^(i+1); |
---|
4387 | // ------------ the ideal Ji is ready ----------- |
---|
4388 | dbprint(ppl,"// -2-"+string(i+1)+"-1- starting the computation of a Groebner basis of J"+string(i)+" in @R"); |
---|
4389 | dbprint(ppl-1, J); |
---|
4390 | G = engine(J,eng); |
---|
4391 | dbprint(ppl,"// -2-"+string(i+1)+"-2- the Groebner basis has been computed"); |
---|
4392 | dbprint(ppl-1, G); |
---|
4393 | if ( NF((s+a)^i,G)==0 ) |
---|
4394 | { |
---|
4395 | dbprint(ppl,"// -2-"+string(i+1)+"-3- the number "+string(-a)+" has not multiplicity "+string(i+1)); |
---|
4396 | m = i; |
---|
4397 | break; |
---|
4398 | } |
---|
4399 | dbprint(ppl,"// -2-"+string(i+1)+"-3- the number "+string(-a)+" has multiplicity at least "+string(i+1)); |
---|
4400 | } |
---|
4401 | dbprint(ppl,"// -3- the bucle has finished"); |
---|
4402 | return(m); |
---|
4403 | } |
---|
4404 | example |
---|
4405 | { |
---|
4406 | "EXAMPLE:"; echo = 2; |
---|
4407 | ring r = 0,(x,y,z),Dp; |
---|
4408 | poly F = x*y*z; |
---|
4409 | printlevel = 0; |
---|
4410 | def A = Sannfs(F); |
---|
4411 | setring A; |
---|
4412 | poly F = imap(r,F); |
---|
4413 | checkRoot2(LD,F,1); // -1 is a root of bs with multiplicity 3 |
---|
4414 | checkRoot2(LD,F,1/3); // -1/3 is not a root |
---|
4415 | } |
---|
4416 | |
---|
4417 | proc checkFactor(ideal I, poly F, poly q, list #) |
---|
4418 | "USAGE: checkFactor(I,f,qs [,eng]); I an ideal, f a poly, qs a poly, eng an optional int |
---|
4419 | ASSUME: I is the output of Sannfs, SannfsBM, SannfsLOT or SannfsOT, |
---|
4420 | f is a polynomial in K[_x], qs is a polynomial in K[s] |
---|
4421 | RETURN: int, 1 if qs is a factor of the global Bernstein polynomial of f and 0 otherwise |
---|
4422 | PURPOSE: check whether a rational is a root of the global Bernstein polynomial of f and compute its multiplicity from the known Ann F^s in D[s] |
---|
4423 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
4424 | @* otherwise (and by default) @code{slimgb} is used. |
---|
4425 | @* If printlevel=1, progress debug messages will be printed, |
---|
4426 | @* if printlevel>=2, all the debug messages will be printed. |
---|
4427 | EXAMPLE: example checkFactor; shows examples |
---|
4428 | " |
---|
4429 | { |
---|
4430 | int eng = 0; |
---|
4431 | if ( size(#)>0 ) |
---|
4432 | { |
---|
4433 | if ( typeof(#[1]) == "int" ) |
---|
4434 | { |
---|
4435 | eng = int(#[1]); |
---|
4436 | } |
---|
4437 | } |
---|
4438 | int ppl = printlevel-voice+2; |
---|
4439 | def @R2 = basering; |
---|
4440 | int N = nvars(@R2); |
---|
4441 | int i; |
---|
4442 | // we're in D_n[s], where the elim ord for s is set |
---|
4443 | dbprint(ppl,"// -0-1- starting the procedure checkFactor"); |
---|
4444 | dbprint(ppl,"// -1-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
4445 | dbprint(ppl-1, @R2); |
---|
4446 | // create the ideal J = ann_D[s](f^s) + < f,q > |
---|
4447 | ideal J = NF(I,std(F)); |
---|
4448 | // make leadcoeffs positive |
---|
4449 | for (i=1; i<=ncols(J); i++) |
---|
4450 | { |
---|
4451 | if ( leadcoef(J[i])<0 ) |
---|
4452 | { |
---|
4453 | J[i] = -J[i]; |
---|
4454 | } |
---|
4455 | } |
---|
4456 | J = J,F,q; |
---|
4457 | // ------------ the ideal J is ready ----------- |
---|
4458 | dbprint(ppl,"// -1-2- starting the elimination of _x,_Dx in @R2"); |
---|
4459 | dbprint(ppl-1, J); |
---|
4460 | ideal G = engine(J,eng); |
---|
4461 | ideal K = nselect(G,1,N-1); |
---|
4462 | kill J,G; |
---|
4463 | dbprint(ppl,"// -1-3- _x,_Dx are eliminated"); |
---|
4464 | dbprint(ppl-1, K); |
---|
4465 | //q is a factor of bs iff K = < q > |
---|
4466 | //K = normalize(K); |
---|
4467 | //q = normalize(q); |
---|
4468 | //return( (K[1]==q) ); |
---|
4469 | return( NF(K[1],std(q))==0 ); |
---|
4470 | } |
---|
4471 | example |
---|
4472 | { |
---|
4473 | "EXAMPLE:"; echo = 2; |
---|
4474 | ring r = 0,(x,y),Dp; |
---|
4475 | poly F = x^4+y^5+x*y^4; |
---|
4476 | printlevel = 0; |
---|
4477 | def A = Sannfs(F); |
---|
4478 | setring A; |
---|
4479 | poly F = imap(r,F); |
---|
4480 | checkFactor(LD,F,20*s+31); // -31/20 is not a root of bs |
---|
4481 | checkFactor(LD,F,20*s+11); // -11/20 is a root of bs |
---|
4482 | checkFactor(LD,F,(20*s+11)^2); // the multiplicity of -11/20 is 1 |
---|
4483 | } |
---|
4484 | |
---|
4485 | proc varnum(string s) |
---|
4486 | "USAGE: varnum(s); string s |
---|
4487 | RETURN: int |
---|
4488 | PURPOSE: returns the number of the variable with the name s |
---|
4489 | among the variables of basering or 0 if there is no such variable |
---|
4490 | EXAMPLE: example varnum; shows examples |
---|
4491 | " |
---|
4492 | { |
---|
4493 | int i; |
---|
4494 | for (i=1; i<= nvars(basering); i++) |
---|
4495 | { |
---|
4496 | if ( string(var(i)) == s ) |
---|
4497 | { |
---|
4498 | return(i); |
---|
4499 | } |
---|
4500 | } |
---|
4501 | return(0); |
---|
4502 | } |
---|
4503 | example |
---|
4504 | { |
---|
4505 | "EXAMPLE:"; echo = 2; |
---|
4506 | ring X = 0,(x,y1,z(0),tTa),dp; |
---|
4507 | varnum("z(0)"); |
---|
4508 | varnum("tTa"); |
---|
4509 | varnum("xyz"); |
---|
4510 | } |
---|
4511 | |
---|
4512 | static proc indAR(list L, int n) |
---|
4513 | "USAGE: indAR(L,n); list L, int n |
---|
4514 | RETURN: list |
---|
4515 | PURPOSE: computes arrangement inductively, using L and var(n) as the |
---|
4516 | next variable |
---|
4517 | ASSUME: L has a structure of an arrangement |
---|
4518 | EXAMPLE: example indAR; shows examples |
---|
4519 | " |
---|
4520 | { |
---|
4521 | if ( (n<2) || (n>nvars(basering)) ) |
---|
4522 | { |
---|
4523 | ERROR("incorrect n"); |
---|
4524 | } |
---|
4525 | int sl = size(L); |
---|
4526 | list K; |
---|
4527 | ideal tmp; |
---|
4528 | poly @t = L[sl][1] + var(n); //1 elt |
---|
4529 | K[sl+1] = ideal(@t); |
---|
4530 | tmp = L[1]+var(n); |
---|
4531 | K[1] = tmp; tmp = 0; |
---|
4532 | int i,j,sI; |
---|
4533 | ideal I; |
---|
4534 | for(i=sl; i>=2; i--) |
---|
4535 | { |
---|
4536 | I = L[i-1]; sI = size(I); |
---|
4537 | for(j=1; j<=sI; j++) |
---|
4538 | { |
---|
4539 | I[j] = I[j] + var(n); |
---|
4540 | } |
---|
4541 | tmp = L[i],I; |
---|
4542 | K[i] = tmp; |
---|
4543 | I = 0; tmp = 0; |
---|
4544 | } |
---|
4545 | kill I; kill tmp; |
---|
4546 | return(K); |
---|
4547 | } |
---|
4548 | example |
---|
4549 | { |
---|
4550 | "EXAMPLE:"; echo = 2; |
---|
4551 | ring r = 0,(x,y,z,t,v),dp; |
---|
4552 | list L; |
---|
4553 | L[1] = ideal(x); |
---|
4554 | list K = indAR(L,2); |
---|
4555 | K; |
---|
4556 | list M = indAR(K,3); |
---|
4557 | M; |
---|
4558 | M = indAR(M,4); |
---|
4559 | M; |
---|
4560 | } |
---|
4561 | |
---|
4562 | static proc exCheckGenericity() |
---|
4563 | { |
---|
4564 | LIB "control.lib"; |
---|
4565 | ring r = (0,a,b,c),x,dp; |
---|
4566 | poly p = (x-a)*(x-b)*(x-c); |
---|
4567 | def A = annfsBM(p); |
---|
4568 | setring A; |
---|
4569 | ideal J = slimgb(LD); |
---|
4570 | matrix T = lift(LD,J); |
---|
4571 | T = normalize(T); |
---|
4572 | genericity(T); |
---|
4573 | // Ann =x^3*Dx+3*x^2*t*Dt+(-a-b-c)*x^2*Dx+(-2*a-2*b-2*c)*x*t*Dt+3*x^2+(a*b+a*c+b*c)*x*Dx+(a*b+a*c+b*c)*t*Dt+(-2*a-2*b-2*c)*x+(-a*b*c)*Dx+(a*b+a*c+b*c) |
---|
4574 | // genericity: g = a2-ab-ac+b2-bc+c2 =0 |
---|
4575 | // g = (a -(b+c)/2)^2 + (3/4)*(b-c)^2; |
---|
4576 | // g ==0 <=> a=b=c |
---|
4577 | // indeed, Ann = (x-a)^2*(x*Dx+3*t*Dt+(-a)*Dx+3) |
---|
4578 | // -------------------------------------------- |
---|
4579 | // BUT a direct computation shows |
---|
4580 | // when a=b=c, |
---|
4581 | // Ann = x*Dx+3*t*Dt+(-a)*Dx+3 |
---|
4582 | } |
---|
4583 | |
---|
4584 | static proc exOT_17() |
---|
4585 | { |
---|
4586 | // Oaku-Takayama, p.208 |
---|
4587 | ring R = 0,(x,y),dp; |
---|
4588 | poly F = x^3-y^2; // x^2+x*y+y^2; |
---|
4589 | option(prot); |
---|
4590 | option(mem); |
---|
4591 | // option(redSB); |
---|
4592 | def A = annfsOT(F,0); |
---|
4593 | setring A; |
---|
4594 | LD; |
---|
4595 | gkdim(LD); // a holonomic check |
---|
4596 | // poly F = x^3-y^2; // = x^7 - y^5; // x^3-y^4; // x^5 - y^4; |
---|
4597 | } |
---|
4598 | |
---|
4599 | static proc exOT_16() |
---|
4600 | { |
---|
4601 | // Oaku-Takayama, p.208 |
---|
4602 | ring R = 0,(x),dp; |
---|
4603 | poly F = x*(1-x); |
---|
4604 | option(prot); |
---|
4605 | option(mem); |
---|
4606 | // option(redSB); |
---|
4607 | def A = annfsOT(F,0); |
---|
4608 | setring A; |
---|
4609 | LD; |
---|
4610 | gkdim(LD); // a holonomic check |
---|
4611 | } |
---|
4612 | |
---|
4613 | static proc ex_bcheck() |
---|
4614 | { |
---|
4615 | ring R = 0,(x,y),dp; |
---|
4616 | poly F = x*y*(x+y); |
---|
4617 | option(prot); |
---|
4618 | option(mem); |
---|
4619 | int eng = 0; |
---|
4620 | // option(redSB); |
---|
4621 | def A = annfsOT(F,eng); |
---|
4622 | setring A; |
---|
4623 | LD; |
---|
4624 | } |
---|
4625 | |
---|
4626 | static proc ex_bcheck2() |
---|
4627 | { |
---|
4628 | ring R = 0,(x,y),dp; |
---|
4629 | poly F = x*y*(x+y); |
---|
4630 | int eng = 0; |
---|
4631 | def A = annfsBM(F,eng); |
---|
4632 | setring A; |
---|
4633 | LD; |
---|
4634 | } |
---|
4635 | |
---|
4636 | static proc ex_BMI() |
---|
4637 | { |
---|
4638 | // a hard example |
---|
4639 | ring r = 0,(x,y),Dp; |
---|
4640 | poly F1 = (x2-y3)*(x3-y2); |
---|
4641 | poly F2 = (x2-y3)*(xy4+y5+x4); |
---|
4642 | ideal F = F1,F2; |
---|
4643 | def A = annfsBMI(F); |
---|
4644 | setring A; |
---|
4645 | LD; |
---|
4646 | BS; |
---|
4647 | } |
---|
4648 | |
---|
4649 | static proc ex2_BMI() |
---|
4650 | { |
---|
4651 | // this example was believed to be intractable in 2005 by Gago-Vargas, Castro and Ucha |
---|
4652 | ring r = 0,(x,y),Dp; |
---|
4653 | option(prot); |
---|
4654 | option(mem); |
---|
4655 | ideal F = x2+y3,x3+y2; |
---|
4656 | printlevel = 2; |
---|
4657 | def A = annfsBMI(F); |
---|
4658 | setring A; |
---|
4659 | LD; |
---|
4660 | BS; |
---|
4661 | } |
---|
4662 | |
---|
4663 | static proc ex_operatorBM() |
---|
4664 | { |
---|
4665 | ring r = 0,(x,y,z,w),Dp; |
---|
4666 | poly F = x^3+y^3+z^2*w; |
---|
4667 | printlevel = 0; |
---|
4668 | def A = operatorBM(F); |
---|
4669 | setring A; |
---|
4670 | F; // the original polynomial itself |
---|
4671 | LD; // generic annihilator |
---|
4672 | LD0; // annihilator |
---|
4673 | bs; // normalized Bernstein poly |
---|
4674 | BS; // root and multiplicities of the Bernstein poly |
---|
4675 | PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD |
---|
4676 | reduce(PS*F-bs,LD); // check the property of PS |
---|
4677 | } |
---|