1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: bfun.lib Algorithms for b-functions and Bernstein-Sato polynomial |
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6 | AUTHORS: Daniel Andres, daniel.andres@math.rwth-aachen.de |
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7 | @* Viktor Levandovskyy, levandov@math.rwth-aachen.de |
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8 | |
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9 | OVERVIEW: |
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10 | Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R, |
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11 | one is interested in the global b-function (also known as Bernstein-Sato |
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12 | polynomial) b(s) in K[s], defined to be the non-zero monic polynomial of minimal |
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13 | degree, satisfying a functional identity L * F^{s+1} = b(s) F^s, |
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14 | for some operator L in D[s] (* stands for the action of differential operator)@* |
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15 | By D one denotes the n-th Weyl algebra |
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16 | K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>. |
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17 | One is interested in the following data:@* |
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18 | - Bernstein-Sato polynomial b(s) in K[s],@* |
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19 | - the list of its roots, which are known to be rational@* |
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20 | - the multiplicities of the roots.@* |
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21 | |
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22 | There is a constructive definition of a b-function of a holonomic ideal I in D |
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23 | (that is, an ideal I in a Weyl algebra D, such that D/I is holonomic module) |
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24 | with respect to the given weight vector w: For a polynomial p in D, its initial |
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25 | form w.r.t. (-w,w) is defined as the sum of all terms of p which have |
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26 | maximal weighted total degree where the weight of x_i is -w_i and the weight |
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27 | of d_i is w_i. Let J be the initial ideal of I w.r.t. (-w,w), i.e. the |
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28 | K-vector space generated by all initial forms w.r.t (-w,w) of elements of I. |
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29 | Put s = w_1 x_1 d_1 + ... + w_n x_n d_n. Then the monic generator b_w(s) of |
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30 | the intersection of J with the PID K[s] is called the b-function of I w.r.t. w. |
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31 | Unlike Bernstein-Sato polynomial, general b-function with respect to |
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32 | arbitrary weights need not have rational roots at all. However, b-function |
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33 | of a holonomic ideal is known to be non-zero as well. |
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34 | |
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35 | REFERENCES: [SST] Saito, Sturmfels, Takayama: Groebner Deformations of |
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36 | Hypergeometric Differential Equations (2000),@* |
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37 | Noro: An Efficient Modular Algorithm for Computing the Global b-function, |
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38 | (2002). |
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39 | |
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40 | |
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41 | PROCEDURES: |
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42 | bfct(f[,s,t,v]); compute the BS polynomial of f with linear reductions |
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43 | bfctSyz(f[,r,s,t,u,v]); compute the BS polynomial of f with syzygy-solver |
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44 | bfctAnn(f[,s]); compute the BS polynomial of f via Ann f^s + f |
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45 | bfctOneGB(f[,s,t]); compute the BS polynomial of f by just one GB computation |
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46 | bfctIdeal(I,w[,s,t]); compute the b-function of ideal w.r.t. weight |
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47 | pIntersect(f,I[,s]); intersection of ideal with subalgebra K[f] for a polynomial f |
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48 | pIntersectSyz(f,I[,p,s,t]); intersection of ideal with subalgebra K[f] with syz-solver |
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49 | linReduce(f,I[,s]); reduce a polynomial by linear reductions w.r.t. ideal |
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50 | linReduceIdeal(I,[s,t]); interreduce generators of ideal by linear reductions |
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51 | linSyzSolve(I[,s]); compute a linear dependency of elements of ideal I |
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52 | |
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53 | allPositive(v); checks whether all entries of an intvec are positive |
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54 | scalarProd(v,w); computes the standard scalar product of two intvecs |
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55 | vec2poly(v[,i]); constructs an univariate polynomial with given coefficients |
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56 | |
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57 | |
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58 | SEE ALSO: dmod_lib, dmodapp_lib, dmodvar_lib, gmssing_lib |
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59 | |
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60 | KEYWORDS: D-module; global Bernstein-Sato polynomial; Bernstein-Sato polynomial; b-function; |
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61 | graded Weyl algebra; initial ideal; initial form; principal intersection; linear interreduction; |
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62 | initial ideal approach |
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63 | "; |
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64 | |
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65 | |
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66 | |
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67 | /* |
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68 | CHANGELOG |
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69 | |
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70 | 03.03.11: |
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71 | - simplified scalarProd |
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72 | - fixed bug in bfct when user used vars t,Dt |
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73 | - now bFactor is used by bfct, bfctAnn, i.e. the static procs |
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74 | addRoot, listofroots are superfluous |
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75 | - fixed printlevel/debug message issue in bfct, bfctAnn |
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76 | - fixed small issue for zero ideal input in linReduceIdeal |
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77 | */ |
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78 | |
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79 | |
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80 | |
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81 | LIB "qhmoduli.lib"; // for Max |
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82 | LIB "dmod.lib"; // for SannfsBFCT etc |
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83 | LIB "dmodapp.lib"; // for initialIdealW etc |
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84 | LIB "nctools.lib"; // for isWeyl etc |
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85 | LIB "presolve.lib"; // for valvars |
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86 | |
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87 | //--------------- auxiliary procedures ---------------------------------------- |
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88 | |
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89 | /* |
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90 | static proc gradedWeyl (intvec u,intvec v) |
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91 | "USAGE: gradedWeyl(u,v); u,v intvecs |
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92 | RETURN: a ring, the associated graded ring of the basering w.r.t. u and v |
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93 | PURPOSE: compute the associated graded ring of the basering w.r.t. u and v |
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94 | ASSUME: basering is a Weyl algebra |
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95 | EXAMPLE: example gradedWeyl; shows examples |
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96 | NOTE: u[i] is the weight of x(i), v[i] the weight of D(i). |
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97 | @* u+v has to be a non-negative intvec. |
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98 | " |
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99 | { |
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100 | // todo check assumption |
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101 | int i; |
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102 | def save = basering; |
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103 | int n = nvars(save)/2; |
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104 | if (nrows(u)<>n || nrows(v)<>n) |
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105 | { |
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106 | ERROR("weight vectors have wrong dimension"); |
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107 | } |
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108 | intvec uv,gr; |
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109 | uv = u+v; |
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110 | for (i=1; i<=n; i++) |
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111 | { |
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112 | if (uv[i]>=0) |
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113 | { |
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114 | if (uv[i]==0) |
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115 | { |
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116 | gr[i] = 0; |
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117 | } |
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118 | else |
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119 | { |
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120 | gr[i] = 1; |
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121 | } |
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122 | } |
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123 | else |
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124 | { |
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125 | ERROR("the sum of the weight vectors has to be a non-negative intvec"); |
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126 | } |
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127 | } |
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128 | list l = ringlist(save); |
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129 | list l2 = l[2]; |
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130 | matrix l6 = l[6]; |
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131 | for (i=1; i<=n; i++) |
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132 | { |
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133 | if (gr[i] == 1) |
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134 | { |
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135 | l2[n+i] = "xi("+string(i)+")"; |
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136 | l6[i,n+i] = 0; |
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137 | } |
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138 | } |
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139 | l[2] = l2; |
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140 | l[6] = l6; |
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141 | def G = ring(l); |
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142 | return(G); |
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143 | } |
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144 | example |
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145 | { |
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146 | "EXAMPLE:"; echo = 2; |
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147 | ring @D = 0,(x,y,z,Dx,Dy,Dz),dp; |
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148 | def D = Weyl(); |
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149 | setring D; |
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150 | intvec u = -1,-1,1; intvec v = 2,1,1; |
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151 | def G = gradedWeyl(u,v); |
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152 | setring G; G; |
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153 | } |
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154 | */ |
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155 | |
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156 | static proc safeVarName (string s) |
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157 | { |
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158 | string S = "," + charstr(basering) + "," + varstr(basering) + ","; |
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159 | s = "," + s + ","; |
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160 | while (find(S,s) <> 0) |
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161 | { |
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162 | s[1] = "@"; |
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163 | s = "," + s; |
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164 | } |
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165 | s = s[2..size(s)-1]; |
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166 | return(s) |
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167 | } |
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168 | |
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169 | proc allPositive (intvec v) |
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170 | "USAGE: allPositive(v); v an intvec |
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171 | RETURN: int, 1 if all components of v are positive, or 0 otherwise |
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172 | PURPOSE: check whether all components of an intvec are positive |
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173 | EXAMPLE: example allPositive; shows an example |
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174 | " |
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175 | { |
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176 | int i; |
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177 | for (i=1; i<=size(v); i++) |
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178 | { |
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179 | if (v[i]<=0) |
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180 | { |
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181 | return(0); |
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182 | break; |
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183 | } |
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184 | } |
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185 | return(1); |
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186 | } |
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187 | example |
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188 | { |
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189 | "EXAMPLE:"; echo = 2; |
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190 | intvec v = 1,2,3; |
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191 | allPositive(v); |
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192 | intvec w = 1,-2,3; |
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193 | allPositive(w); |
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194 | } |
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195 | |
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196 | static proc findFirst (list l, i) |
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197 | "USAGE: findFirst(l,i); l a list, i an argument of any type |
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198 | RETURN: int, the position of the first appearance of i in l, |
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199 | @* or 0 if i is not a member of l |
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200 | PURPOSE: check whether the second argument is a member of a list |
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201 | EXAMPLE: example findFirst; shows an example |
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202 | " |
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203 | { |
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204 | int j; |
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205 | for (j=1; j<=size(l); j++) |
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206 | { |
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207 | if (l[j]==i) |
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208 | { |
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209 | return(j); |
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210 | break; |
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211 | } |
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212 | } |
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213 | return(0); |
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214 | } |
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215 | // findFirst(list(1, 2, list(1)),2); // seems to be a bit buggy, |
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216 | // findFirst(list(1, 2, list(1)),3); // but works for the purposes |
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217 | // findFirst(list(1, 2, list(1)),list(1)); // of this library |
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218 | // findFirst(l,list(2)); |
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219 | example |
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220 | { |
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221 | "EXAMPLE:"; echo = 2; |
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222 | ring r = 0,x,dp; |
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223 | list l = 1,2,3; |
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224 | findFirst(l,4); |
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225 | findFirst(l,2); |
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226 | } |
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227 | |
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228 | proc scalarProd (intvec v, intvec w) |
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229 | "USAGE: scalarProd(v,w); v,w intvecs |
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230 | RETURN: int, the standard scalar product of v and w |
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231 | PURPOSE: computes the scalar product of two intvecs |
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232 | ASSUME: the arguments are of the same size |
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233 | EXAMPLE: example scalarProd; shows examples |
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234 | " |
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235 | { |
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236 | if (size(v)!=size(w)) |
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237 | { |
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238 | ERROR("non-matching dimensions"); |
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239 | } |
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240 | else |
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241 | { |
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242 | intvec u = transpose(v)*w; |
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243 | } |
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244 | return(u[1]); |
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245 | } |
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246 | example |
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247 | { |
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248 | "EXAMPLE:"; echo = 2; |
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249 | intvec v = 1,2,3; |
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250 | intvec w = 4,5,6; |
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251 | scalarProd(v,w); |
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252 | } |
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253 | |
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254 | //-------------- main procedures ------------------------------------------------------- |
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255 | proc linReduceIdeal(ideal I, list #) |
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256 | "USAGE: linReduceIdeal(I [,s,t,u]); I an ideal, s,t,u optional ints |
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257 | RETURN: ideal or list, linear reductum (over field) of I by its elements |
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258 | PURPOSE: reduces a list of polys only by linear reductions (no monomial |
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259 | @* multiplications) |
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260 | NOTE: If s<>0, a list consisting of the reduced ideal and the coefficient |
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261 | @* vectors of the used reductions given as module is returned. |
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262 | @* Otherwise (and by default), only the reduced ideal is returned. |
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263 | @* If t<>0 (and by default) all monomials are reduced (if possible), |
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264 | @* otherwise, only leading monomials are reduced. |
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265 | @* If u<>0 (and by default), the ideal is first sorted in increasing order. |
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266 | @* If u is set to 0 and the given ideal is not sorted in the way described, |
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267 | @* the result might not be as expected. |
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268 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
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269 | @* if printlevel>=2, all the debug messages will be printed. |
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270 | EXAMPLE: example linReduceIdeal; shows examples |
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271 | " |
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272 | { |
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273 | // #[1] = remembercoeffs |
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274 | // #[2] = redtail |
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275 | // #[3] = sortideal |
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276 | int ppl = printlevel - voice + 2; |
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277 | int remembercoeffs = 0; // default |
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278 | int redtail = 1; // default |
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279 | int sortideal = 1; // default |
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280 | if (size(#)>0) |
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281 | { |
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282 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
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283 | { |
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284 | remembercoeffs = #[1]; |
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285 | } |
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286 | if (size(#)>1) |
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287 | { |
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288 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
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289 | { |
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290 | redtail = #[2]; |
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291 | } |
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292 | if (size(#)>2) |
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293 | { |
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294 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
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295 | { |
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296 | sortideal = #[3]; |
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297 | } |
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298 | } |
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299 | } |
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300 | } |
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301 | int sI = ncols(I); |
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302 | int sZeros = sI - size(I); |
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303 | int i,j; |
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304 | ideal J,lmJ,ordJ; |
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305 | list lJ = sort(I); |
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306 | module M; // for the coefficients |
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307 | // step 1: prepare, e.g. sort I |
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308 | if (sortideal <> 0) |
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309 | { |
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310 | if (sZeros > 0) // I contains zeros |
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311 | { |
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312 | if (remembercoeffs <> 0) |
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313 | { |
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314 | j = 0; |
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315 | for (i=1; i<=sI; i++) |
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316 | { |
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317 | if (I[i] == 0) |
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318 | { |
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319 | j++; |
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320 | J[j] = 0; |
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321 | ordJ[j] = -1; |
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322 | M[j] = gen(i); |
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323 | } |
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324 | else |
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325 | { |
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326 | M[i+sZeros-j] = gen(lJ[2][i-j]+j); |
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327 | } |
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328 | } |
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329 | } |
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330 | else |
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331 | { |
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332 | for (i=1; i<=sZeros; i++) |
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333 | { |
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334 | J[i] = 0; |
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335 | ordJ[i] = -1; |
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336 | } |
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337 | } |
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338 | I = J,lJ[1]; |
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339 | } |
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340 | else // I contains no zeros |
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341 | { |
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342 | I = lJ[1]; |
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343 | if (remembercoeffs <> 0) |
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344 | { |
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345 | for (i=1; i<=size(lJ[1]); i++) { M[i+sZeros] = gen(lJ[2][i]); } |
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346 | } |
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347 | } |
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348 | } |
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349 | else // assume I is already sorted |
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350 | { |
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351 | if (remembercoeffs <> 0) |
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352 | { |
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353 | for (i=1; i<=ncols(I); i++) { M[i] = gen(i); } |
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354 | } |
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355 | } |
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356 | dbprint(ppl-1,"// initially sorted ideal:", I); |
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357 | if (remembercoeffs <> 0) { dbprint(ppl-1,"// used permutations:", M); } |
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358 | // step 2: reduce leading monomials by linear reductions |
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359 | poly lm,c,redpoly,maxlmJ; |
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360 | J[sZeros+1] = I[sZeros+1]; |
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361 | lm = I[sZeros+1]; |
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362 | maxlmJ = leadmonom(lm); |
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363 | lmJ[sZeros+1] = maxlmJ; |
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364 | int ordlm,reduction; |
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365 | ordJ[sZeros+1] = ord(lm); |
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366 | vector v; |
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367 | int noRedPast; |
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368 | for (i=sZeros+2; i<=sI; i++) |
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369 | { |
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370 | redpoly = I[i]; |
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371 | lm = leadmonom(redpoly); |
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372 | ordlm = ord(lm); |
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373 | if (remembercoeffs <> 0) { v = M[i]; } |
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374 | reduction = 1; |
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375 | while (reduction == 1) // while there was a reduction |
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376 | { |
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377 | noRedPast = i; |
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378 | reduction = 0; |
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379 | for (j=sZeros+1; j<noRedPast; j++) |
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380 | { |
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381 | if (lm == 0) { break; } // nothing more to reduce |
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382 | if (lm > maxlmJ) { break; } //lm is bigger than maximal monomial to reduce with |
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383 | if (ordlm == ordJ[j]) |
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384 | { |
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385 | if (lm == lmJ[j]) |
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386 | { |
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387 | dbprint(ppl-1,"// reducing " + string(redpoly)); |
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388 | dbprint(ppl-1,"// with " + string(J[j])); |
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389 | c = leadcoef(redpoly)/leadcoef(J[j]); |
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390 | redpoly = redpoly - c*J[j]; |
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391 | dbprint(ppl-1,"// to " + string(redpoly)); |
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392 | lm = leadmonom(redpoly); |
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393 | ordlm = ord(lm); |
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394 | if (remembercoeffs <> 0) { M[i] = M[i] - c * M[j]; } |
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395 | noRedPast = j; |
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396 | reduction = 1; |
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397 | } |
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398 | } |
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399 | } |
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400 | } |
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401 | for (j=sZeros+1; j<i; j++) |
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402 | { |
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403 | if (redpoly < J[j]) { break; } |
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404 | } |
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405 | J = insertGenerator(J,redpoly,j); |
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406 | lmJ = insertGenerator(lmJ,lm,j); |
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407 | ordJ = insertGenerator(ordJ,poly(ordlm),j); |
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408 | if (remembercoeffs <> 0) |
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409 | { |
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410 | v = M[i]; |
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411 | M = deleteGenerator(M,i); |
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412 | M = insertGenerator(M,v,j); |
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413 | } |
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414 | maxlmJ = lmJ[i]; |
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415 | } |
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416 | // step 3: reduce tails by linear reductions as well |
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417 | if (redtail <> 0) |
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418 | { |
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419 | dbprint(ppl,"// finished reducing leading monomials"); |
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420 | dbprint(ppl-1,string(J)); |
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421 | if (remembercoeffs <> 0) { dbprint(ppl-1,"// used reductions:" + string(M)); } |
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422 | int k; |
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423 | for (i=sZeros+1; i<=sI; i++) |
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424 | { |
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425 | lm = lmJ[i]; |
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426 | for (j=i+1; j<=sI; j++) |
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427 | { |
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428 | for (k=2; k<=size(J[j]); k++) // run over all terms in J[j] |
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429 | { |
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430 | if (ordJ[i] == ord(J[j][k])) |
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431 | { |
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432 | if (lm == normalize(J[j][k])) |
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433 | { |
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434 | c = leadcoef(J[j][k])/leadcoef(J[i]); |
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435 | dbprint(ppl-1,"// reducing " + string(J[j])); |
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436 | dbprint(ppl-1,"// with " + string(J[i])); |
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437 | J[j] = J[j] - c*J[i]; |
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438 | dbprint(ppl-1,"// to " + string(J[j])); |
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439 | if (remembercoeffs <> 0) { M[j] = M[j] - c * M[i]; } |
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440 | } |
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441 | } |
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442 | } |
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443 | } |
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444 | } |
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445 | } |
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446 | if (sI == sZeros) // if I=0,0,...,0, we now have one too much by construction due to sort |
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447 | { |
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448 | J = J[1..sZeros]; |
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449 | } |
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450 | if (remembercoeffs <> 0) { return(list(J,M)); } |
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451 | else { return(J); } |
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452 | } |
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453 | example |
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454 | { |
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455 | "EXAMPLE:"; echo = 2; |
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456 | ring r = 0,(x,y),dp; |
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457 | ideal I = 3,x+9,y4+5x,2y4+7x+2; |
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458 | linReduceIdeal(I); // reduces tails |
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459 | linReduceIdeal(I,0,0); // no reductions of tails |
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460 | list l = linReduceIdeal(I,1); // reduces tails and shows reductions used |
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461 | l; |
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462 | module M = I; |
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463 | matrix(l[1]) - M*l[2]; |
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464 | } |
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465 | |
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466 | proc linReduce(poly f, ideal I, list #) |
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467 | "USAGE: linReduce(f, I [,s,t,u]); f a poly, I an ideal, s,t,u optional ints |
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468 | RETURN: poly or list, linear reductum (over field) of f by elements from I |
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469 | PURPOSE: reduce a polynomial only by linear reductions (no monomial multiplications) |
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470 | NOTE: If s<>0, a list consisting of the reduced polynomial and the coefficient |
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471 | @* vector of the used reductions is returned, otherwise (and by default) |
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472 | @* only reduced polynomial is returned. |
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473 | @* If t<>0 (and by default) all monomials are reduced (if possible), |
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474 | @* otherwise, only leading monomials are reduced. |
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475 | @* If u<>0 (and by default), the ideal is linearly pre-reduced, i.e. |
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476 | @* instead of the given ideal, the output of @code{linReduceIdeal} is used. |
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477 | @* If u is set to 0 and the given ideal does not equal the output of |
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478 | @* @code{linReduceIdeal}, the result might not be as expected. |
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479 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
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480 | @* if printlevel>=2, all the debug messages will be printed. |
---|
481 | EXAMPLE: example linReduce; shows examples |
---|
482 | " |
---|
483 | { |
---|
484 | int ppl = printlevel - voice + 2; |
---|
485 | int remembercoeffs = 0; // default |
---|
486 | int redtail = 1; // default |
---|
487 | int prepareideal = 1; // default |
---|
488 | if (size(#)>0) |
---|
489 | { |
---|
490 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
491 | { |
---|
492 | remembercoeffs = #[1]; |
---|
493 | } |
---|
494 | if (size(#)>1) |
---|
495 | { |
---|
496 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
497 | { |
---|
498 | redtail = #[2]; |
---|
499 | } |
---|
500 | if (size(#)>2) |
---|
501 | { |
---|
502 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
503 | { |
---|
504 | prepareideal = #[3]; |
---|
505 | } |
---|
506 | } |
---|
507 | } |
---|
508 | } |
---|
509 | int i,j,k; |
---|
510 | int sI = ncols(I); |
---|
511 | // pre-reduce I: |
---|
512 | module M; |
---|
513 | if (prepareideal <> 0) |
---|
514 | { |
---|
515 | if (remembercoeffs <> 0) |
---|
516 | { |
---|
517 | list sortedI = linReduceIdeal(I,1,redtail); |
---|
518 | I = sortedI[1]; |
---|
519 | M = sortedI[2]; |
---|
520 | dbprint(ppl-1,"// prepared ideal:" +string(I)); |
---|
521 | dbprint(ppl-1,"// with operations:" +string(M)); |
---|
522 | } |
---|
523 | else { I = linReduceIdeal(I,0,redtail); } |
---|
524 | } |
---|
525 | else |
---|
526 | { |
---|
527 | if (remembercoeffs <> 0) |
---|
528 | { |
---|
529 | for (i=1; i<=sI; i++) { M[i] = gen(i); } |
---|
530 | } |
---|
531 | } |
---|
532 | ideal lmI,lcI,ordI; |
---|
533 | for (i=1; i<=sI; i++) |
---|
534 | { |
---|
535 | lmI[i] = leadmonom(I[i]); |
---|
536 | lcI[i] = leadcoef(I[i]); |
---|
537 | ordI[i] = ord(lmI[i]); |
---|
538 | } |
---|
539 | vector v; |
---|
540 | poly c; |
---|
541 | // === reduce leading monomials === |
---|
542 | poly lm = leadmonom(f); |
---|
543 | int ordf = ord(lm); |
---|
544 | for (i=sI; i>=1; i--) // I is sorted: smallest lm's on top |
---|
545 | { |
---|
546 | if (lm == 0) { break; } |
---|
547 | else |
---|
548 | { |
---|
549 | if (ordf == ordI[i]) |
---|
550 | { |
---|
551 | if (lm == lmI[i]) |
---|
552 | { |
---|
553 | c = leadcoef(f)/lcI[i]; |
---|
554 | f = f - c*I[i]; |
---|
555 | lm = leadmonom(f); |
---|
556 | ordf = ord(lm); |
---|
557 | if (remembercoeffs <> 0) { v = v - c * M[i]; } |
---|
558 | } |
---|
559 | } |
---|
560 | } |
---|
561 | } |
---|
562 | // === reduce tails as well === |
---|
563 | if (redtail <> 0) |
---|
564 | { |
---|
565 | dbprint(ppl,"// finished reducing leading monomials"); |
---|
566 | dbprint(ppl-1,string(f)); |
---|
567 | if (remembercoeffs <> 0) { dbprint(ppl-1,"// used reductions:" + string(v)); } |
---|
568 | for (i=1; i<=sI; i++) |
---|
569 | { |
---|
570 | dbprint(ppl,"// testing ideal entry "+string(i)); |
---|
571 | for (j=1; j<=size(f); j++) |
---|
572 | { |
---|
573 | if (ord(f[j]) == ordI[i]) |
---|
574 | { |
---|
575 | if (normalize(f[j]) == lmI[i]) |
---|
576 | { |
---|
577 | c = leadcoef(f[j])/lcI[i]; |
---|
578 | f = f - c*I[i]; |
---|
579 | dbprint(ppl-1,"// reducing with " + string(I[i])); |
---|
580 | dbprint(ppl-1,"// to " + string(f)); |
---|
581 | if (remembercoeffs <> 0) { v = v - c * M[i]; } |
---|
582 | } |
---|
583 | } |
---|
584 | } |
---|
585 | } |
---|
586 | } |
---|
587 | if (remembercoeffs <> 0) |
---|
588 | { |
---|
589 | list l = f,v; |
---|
590 | return(l); |
---|
591 | } |
---|
592 | else { return(f); } |
---|
593 | } |
---|
594 | example |
---|
595 | { |
---|
596 | "EXAMPLE:"; echo = 2; |
---|
597 | ring r = 0,(x,y),dp; |
---|
598 | ideal I = 1,y,xy; |
---|
599 | poly f = 5xy+7y+3; |
---|
600 | poly g = 7x+5y+3; |
---|
601 | linReduce(g,I); // reduces tails |
---|
602 | linReduce(g,I,0,0); // no reductions of tails |
---|
603 | linReduce(f,I,1); // reduces tails and shows reductions used |
---|
604 | f = x3+y2+x2+y+x; |
---|
605 | I = x3-y3, y3-x2,x3-y2,x2-y,y2-x; |
---|
606 | list l = linReduce(f,I,1); |
---|
607 | l; |
---|
608 | module M = I; |
---|
609 | f - (l[1]-(M*l[2])[1,1]); |
---|
610 | } |
---|
611 | |
---|
612 | proc linSyzSolve (ideal I, list #) |
---|
613 | "USAGE: linSyzSolve(I[,s]); I an ideal, s an optional int |
---|
614 | RETURN: vector, coefficient vector of linear combination of 0 in elements of I |
---|
615 | PURPOSE: compute a linear dependency between the elements of an ideal |
---|
616 | @* if such one exists |
---|
617 | NOTE: If s<>0, @code{std} is used for Groebner basis computations, |
---|
618 | @* otherwise, @code{slimgb} is used. |
---|
619 | @* By default, @code{slimgb} is used in char 0 and @code{std} in char >0. |
---|
620 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
621 | @* if printlevel>=2, all the debug messages will be printed. |
---|
622 | EXAMPLE: example linSyzSolve; shows examples |
---|
623 | " |
---|
624 | { |
---|
625 | int whichengine = 0; // default |
---|
626 | int enginespecified = 0; // default |
---|
627 | if (size(#)>0) |
---|
628 | { |
---|
629 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
630 | { |
---|
631 | whichengine = int( #[1]); |
---|
632 | enginespecified = 1; |
---|
633 | } |
---|
634 | } |
---|
635 | int ppl = printlevel - voice +2; |
---|
636 | int sI = ncols(I); |
---|
637 | // check if we are done |
---|
638 | if (I[sI]==0) |
---|
639 | { |
---|
640 | vector v = gen(sI); |
---|
641 | } |
---|
642 | else |
---|
643 | { |
---|
644 | // ------- 1. introduce undefined coeffs ------------------ |
---|
645 | def save = basering; |
---|
646 | list RL = ringlist(save); |
---|
647 | int nv = nvars(save); |
---|
648 | int np = npars(save); |
---|
649 | int p = char(save); |
---|
650 | string cs = "(" + charstr(save) + ")"; |
---|
651 | if (enginespecified == 0) |
---|
652 | { |
---|
653 | if (p <> 0) |
---|
654 | { |
---|
655 | whichengine = 1; |
---|
656 | } |
---|
657 | } |
---|
658 | int i; |
---|
659 | list Lvar; |
---|
660 | for (i=1; i<=sI; i++) |
---|
661 | { |
---|
662 | Lvar[i] = safeVarName("@a(" + string(i) + ")"); |
---|
663 | } |
---|
664 | list L@A = RL[1..4]; |
---|
665 | L@A[2] = Lvar; |
---|
666 | L@A[3] = list(list("lp",intvec(1:sI)),list("C",intvec(0))); |
---|
667 | def @A = ring(L@A); |
---|
668 | L@A[2] = list(safeVarName("@z")); |
---|
669 | L@A[3][1] = list("dp",intvec(1)); |
---|
670 | if (size(L@A[1])>1) |
---|
671 | { |
---|
672 | L@A[1][2] = L@A[1][2] + Lvar; |
---|
673 | L@A[1][3][size(L@A[1][3])+1] = list("lp",intvec(1:sI)); |
---|
674 | } |
---|
675 | else |
---|
676 | { |
---|
677 | L@A[1] = list(p,Lvar,list(list("lp",intvec(1:sI))),ideal(0)); |
---|
678 | } |
---|
679 | def @aA = ring(L@A); |
---|
680 | def @B = save + @aA; |
---|
681 | setring @B; |
---|
682 | ideal I = imap(save,I); |
---|
683 | // ------- 2. form the linear system for the undef coeffs --- |
---|
684 | poly W; ideal QQ; |
---|
685 | for (i=1; i<=sI; i++) |
---|
686 | { |
---|
687 | W = W + par(np+i)*I[i]; |
---|
688 | } |
---|
689 | while (W!=0) |
---|
690 | { |
---|
691 | QQ = QQ,leadcoef(W); |
---|
692 | W = W - lead(W); |
---|
693 | } |
---|
694 | // QQ consists of polynomial expressions in @a(i) of type number |
---|
695 | setring @A; |
---|
696 | ideal QQ = imap(@B,QQ); |
---|
697 | // ------- 3. this QQ is a polynomial ideal, so "solve" the system ----- |
---|
698 | dbprint(ppl, "// linSyzSolve: starting Groebner basis computation with engine:", whichengine); |
---|
699 | QQ = engine(QQ,whichengine); |
---|
700 | dbprint(ppl, "// QQ after engine:", QQ); |
---|
701 | if (dim(QQ) == -1) |
---|
702 | { |
---|
703 | dbprint(ppl+1, "// no solutions by linSyzSolve"); |
---|
704 | // output zeroes |
---|
705 | setring save; |
---|
706 | kill @A,@aA,@B; |
---|
707 | return(v); |
---|
708 | } |
---|
709 | // ------- 4. in order to get the numeric values ------- |
---|
710 | matrix AA = matrix(maxideal(1)); |
---|
711 | module MQQ = std(module(QQ)); |
---|
712 | AA = NF(AA,MQQ); // todo: we still receive NF warnings |
---|
713 | dbprint(ppl, "// AA after NF:",AA); |
---|
714 | // "AA after NF:"; print(AA); |
---|
715 | for(i=1; i<=sI; i++) |
---|
716 | { |
---|
717 | AA = subst(AA,var(i),1); |
---|
718 | } |
---|
719 | dbprint(ppl, "// AA after subst:",AA); |
---|
720 | // "AA after subst: "; print(AA); |
---|
721 | vector v = (module(transpose(AA)))[1]; |
---|
722 | setring save; |
---|
723 | vector v = imap(@A,v); |
---|
724 | kill @A,@aA,@B; |
---|
725 | } |
---|
726 | return(v); |
---|
727 | } |
---|
728 | example |
---|
729 | { |
---|
730 | "EXAMPLE:"; echo = 2; |
---|
731 | ring r = 0,x,dp; |
---|
732 | ideal I = x,2x; |
---|
733 | linSyzSolve(I); |
---|
734 | ideal J = x,x2; |
---|
735 | linSyzSolve(J); |
---|
736 | } |
---|
737 | |
---|
738 | proc pIntersect (poly s, ideal I, list #) |
---|
739 | "USAGE: pIntersect(f, I [,s]); f a poly, I an ideal, s an optional int |
---|
740 | RETURN: vector, coefficient vector of the monic polynomial |
---|
741 | PURPOSE: compute the intersection of ideal I with the subalgebra K[f] |
---|
742 | ASSUME: I is given as Groebner basis, basering is not a qring. |
---|
743 | NOTE: If the intersection is zero, this proc might not terminate. |
---|
744 | @* If s>0 is given, it is searched for the generator of the intersection |
---|
745 | @* only up to degree s. Otherwise (and by default), no bound is assumed. |
---|
746 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
747 | @* if printlevel>=2, all the debug messages will be printed. |
---|
748 | EXAMPLE: example pIntersect; shows examples |
---|
749 | " |
---|
750 | { |
---|
751 | def save = basering; |
---|
752 | int n = nvars(save); |
---|
753 | list RL = ringlist(save); |
---|
754 | int i,j,k; |
---|
755 | if (RL[4]<>0) |
---|
756 | { |
---|
757 | ERROR ("basering must not be a qring"); |
---|
758 | } |
---|
759 | // assume I is given in Groebner basis |
---|
760 | if (attrib(I,"isSB") <> 1) |
---|
761 | { |
---|
762 | print("// WARNING: The input has no SB attribute!"); |
---|
763 | print("// Treating it as if it were a Groebner basis and proceeding..."); |
---|
764 | attrib(I,"isSB",1); // set attribute for suppressing NF messages |
---|
765 | } |
---|
766 | int bound = 0; // default |
---|
767 | if (size(#)>0) |
---|
768 | { |
---|
769 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
770 | { |
---|
771 | bound = #[1]; |
---|
772 | } |
---|
773 | } |
---|
774 | int ppl = printlevel-voice+2; |
---|
775 | // ---case 1: I = basering--- |
---|
776 | if (size(I) == 1) |
---|
777 | { |
---|
778 | if (simplify(I,2)[1] == 1) |
---|
779 | { |
---|
780 | return(gen(1)); // = 1 |
---|
781 | } |
---|
782 | } |
---|
783 | // ---case 2: intersection is zero--- |
---|
784 | intvec degs = leadexp(s); |
---|
785 | intvec possdegbounds; |
---|
786 | list degI; |
---|
787 | i = 1; |
---|
788 | while (i <= ncols(I)) |
---|
789 | { |
---|
790 | if (i == ncols(I)+1) { break; } |
---|
791 | degI[i] = leadexp(I[i]); |
---|
792 | for (j=1; j<=n; j++) |
---|
793 | { |
---|
794 | if (degs[j] == 0) |
---|
795 | { |
---|
796 | if (degI[i][j] <> 0) { break; } |
---|
797 | } |
---|
798 | if (j == n) |
---|
799 | { |
---|
800 | k++; |
---|
801 | possdegbounds[k] = Max(degI[i]); |
---|
802 | } |
---|
803 | } |
---|
804 | i++; |
---|
805 | } |
---|
806 | int degbound = Min(possdegbounds); |
---|
807 | if (bound>0 && bound<degbound) // given bound is too small |
---|
808 | { |
---|
809 | print("// Try a bound of at least " + string(degbound)); |
---|
810 | return(vector(0)); |
---|
811 | } |
---|
812 | dbprint(ppl,"// lower bound for the degree of the insection is " +string(degbound)); |
---|
813 | if (degbound == 0) // lm(s) does not appear in lm(I) |
---|
814 | { |
---|
815 | print("// Intersection is zero"); |
---|
816 | return(vector(0)); |
---|
817 | } |
---|
818 | // ---case 3: intersection is non-trivial--- |
---|
819 | ideal redNI = 1; |
---|
820 | vector v; |
---|
821 | list l,ll; |
---|
822 | l[1] = vector(0); |
---|
823 | poly toNF,tobracket,newNF,rednewNF,oldNF,secNF; |
---|
824 | i = 1; |
---|
825 | while (1) |
---|
826 | { |
---|
827 | if (bound>0 && i>bound) { return(vector(0)); } |
---|
828 | dbprint(ppl,"// Testing degree "+string(i)); |
---|
829 | if (i>1) |
---|
830 | { |
---|
831 | oldNF = newNF; |
---|
832 | tobracket = s^(i-1) - oldNF; |
---|
833 | if (tobracket==0) // todo bug in bracket? |
---|
834 | { |
---|
835 | toNF = 0; |
---|
836 | } |
---|
837 | else |
---|
838 | { |
---|
839 | toNF = bracket(tobracket,secNF); |
---|
840 | } |
---|
841 | newNF = NF(toNF+oldNF*secNF,I); // = NF(s^i,I) |
---|
842 | } |
---|
843 | else |
---|
844 | { |
---|
845 | newNF = NF(s,I); |
---|
846 | secNF = newNF; |
---|
847 | } |
---|
848 | ll = linReduce(newNF,redNI,1); |
---|
849 | rednewNF = ll[1]; |
---|
850 | l[i+1] = ll[2]; |
---|
851 | dbprint(ppl-1,"// newNF is: "+string(newNF)); |
---|
852 | dbprint(ppl-1,"// rednewNF is: "+string(rednewNF)); |
---|
853 | if (rednewNF != 0) // no linear dependency |
---|
854 | { |
---|
855 | redNI[i+1] = rednewNF; |
---|
856 | i++; |
---|
857 | } |
---|
858 | else // there is a linear dependency, hence we are done |
---|
859 | { |
---|
860 | dbprint(ppl,"// degree of the generator of the intersection is: "+string(i)); |
---|
861 | break; |
---|
862 | } |
---|
863 | } |
---|
864 | dbprint(ppl-1,"// used linear reductions:", l); |
---|
865 | // we obtain the coefficients of the generator by the used reductions: |
---|
866 | list Lvar; |
---|
867 | for (j=1; j<=i+1; j++) |
---|
868 | { |
---|
869 | Lvar[j] = safeVarName("a("+string(j)+")"); |
---|
870 | } |
---|
871 | list Lord = list("dp",intvec(1:(i+1))),list("C",intvec(0)); |
---|
872 | list L@R = RL[1..4]; |
---|
873 | L@R[2] = Lvar; L@R[3] = Lord; |
---|
874 | def @R = ring(L@R); setring @R; |
---|
875 | list l = imap(save,l); |
---|
876 | ideal C; |
---|
877 | for (j=1;j<=i+1;j++) |
---|
878 | { |
---|
879 | C[j] = 0; |
---|
880 | for (k=1;k<=j;k++) |
---|
881 | { |
---|
882 | C[j] = C[j]+l[j][k]*var(k); |
---|
883 | } |
---|
884 | } |
---|
885 | for (j=i;j>=1;j--) |
---|
886 | { |
---|
887 | C[i+1]= subst(C[i+1],var(j),var(j)+C[j]); |
---|
888 | } |
---|
889 | matrix m = coeffs(C[i+1],maxideal(1)); |
---|
890 | vector v = gen(i+1); |
---|
891 | for (j=1;j<=i+1;j++) |
---|
892 | { |
---|
893 | v = v + m[j,1]*gen(j); |
---|
894 | } |
---|
895 | setring save; |
---|
896 | v = imap(@R,v); |
---|
897 | kill @R; |
---|
898 | return(v); |
---|
899 | } |
---|
900 | example |
---|
901 | { |
---|
902 | "EXAMPLE:"; echo = 2; |
---|
903 | ring r = 0,(x,y),dp; |
---|
904 | poly f = x^2+y^3+x*y^2; |
---|
905 | def D = initialMalgrange(f); |
---|
906 | setring D; |
---|
907 | inF; |
---|
908 | pIntersect(t*Dt,inF); |
---|
909 | pIntersect(t*Dt,inF,1); |
---|
910 | } |
---|
911 | |
---|
912 | proc pIntersectSyz (poly s, ideal I, list #) |
---|
913 | "USAGE: pIntersectSyz(f, I [,p,s,t]); f poly, I ideal, p,t optial ints, p prime |
---|
914 | RETURN: vector, coefficient vector of the monic polynomial |
---|
915 | PURPOSE: compute the intersection of an ideal I with the subalgebra K[f] |
---|
916 | ASSUME: I is given as Groebner basis. |
---|
917 | NOTE: If the intersection is zero, this procedure might not terminate. |
---|
918 | @* If p>0 is given, this proc computes the generator of the intersection in |
---|
919 | @* char p first and then only searches for a generator of the obtained |
---|
920 | @* degree in the basering. Otherwise, it searches for all degrees by |
---|
921 | @* computing syzygies. |
---|
922 | @* If s<>0, @code{std} is used for Groebner basis computations in char 0, |
---|
923 | @* otherwise, and by default, @code{slimgb} is used. |
---|
924 | @* If t<>0 and by default, @code{std} is used for Groebner basis |
---|
925 | @* computations in char >0, otherwise, @code{slimgb} is used. |
---|
926 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
927 | @* if printlevel>=2, all the debug messages will be printed. |
---|
928 | EXAMPLE: example pIntersectSyz; shows examples |
---|
929 | " |
---|
930 | { |
---|
931 | // assume I is given as Groebner basis |
---|
932 | if (attrib(I,"isSB") <> 1) |
---|
933 | { |
---|
934 | print("// WARNING: The input has no SB attribute!"); |
---|
935 | print("// Treating it as if it were a Groebner basis and proceeding..."); |
---|
936 | attrib(I,"isSB",1); // set attribute for suppressing NF messages |
---|
937 | } |
---|
938 | int ppl = printlevel-voice+2; |
---|
939 | int whichengine = 0; // default |
---|
940 | int modengine = 1; // default |
---|
941 | int solveincharp = 0; // default |
---|
942 | def save = basering; |
---|
943 | int n = nvars(save); |
---|
944 | if (size(#)>0) |
---|
945 | { |
---|
946 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
947 | { |
---|
948 | solveincharp = int(#[1]); |
---|
949 | } |
---|
950 | if (size(#)>1) |
---|
951 | { |
---|
952 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
953 | { |
---|
954 | whichengine = int(#[2]); |
---|
955 | } |
---|
956 | if (size(#)>2) |
---|
957 | { |
---|
958 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
959 | { |
---|
960 | modengine = int(#[3]); |
---|
961 | } |
---|
962 | } |
---|
963 | } |
---|
964 | } |
---|
965 | int i,j; |
---|
966 | vector v; |
---|
967 | poly tobracket,toNF,newNF,p; |
---|
968 | ideal NI = 1; |
---|
969 | newNF = NF(s,I); |
---|
970 | NI[2] = newNF; |
---|
971 | list RL = ringlist(save); |
---|
972 | if (solveincharp) |
---|
973 | { |
---|
974 | int psolveincharp = prime(solveincharp); |
---|
975 | if (solveincharp <> psolveincharp) |
---|
976 | { |
---|
977 | print("// " + string(solveincharp) + " is invalid characteristic of ground field."); |
---|
978 | print("// " + string(psolveincharp) + " is used."); |
---|
979 | solveincharp = psolveincharp; |
---|
980 | kill psolveincharp; |
---|
981 | } |
---|
982 | list RLp = RL[1..4]; |
---|
983 | if (typeof(RL[1]) == "int") { RLp[1] = solveincharp; } |
---|
984 | else { RLp[1][1] = solveincharp; } // parameters |
---|
985 | def @Rp = ring(RLp); |
---|
986 | setring @Rp; |
---|
987 | number c; |
---|
988 | setring save; |
---|
989 | int shortSave = short; // workaround for maps Q(a_i) -> Z/p(a_i) |
---|
990 | short = 0; |
---|
991 | string str; |
---|
992 | int badprime; |
---|
993 | i=1; |
---|
994 | while (badprime == 0 && i<=size(s)) // detect bad primes |
---|
995 | { |
---|
996 | str = string(denominator(leadcoef(s[i]))); |
---|
997 | str = "c = " + str + ";"; |
---|
998 | setring @Rp; |
---|
999 | execute(str); |
---|
1000 | if (c == 0) { badprime = 1; } |
---|
1001 | setring save; |
---|
1002 | i++; |
---|
1003 | } |
---|
1004 | str = "poly s = " + string(s) + ";"; |
---|
1005 | if (size(RL) > 4) // basering is NC-algebra |
---|
1006 | { |
---|
1007 | string RL5 = "@C = " + string(RL[5]) + ";"; |
---|
1008 | string RL6 = "@D = " + string(RL[6]) + ";"; |
---|
1009 | setring @Rp; |
---|
1010 | matrix @C[n][n]; matrix @D[n][n]; |
---|
1011 | execute(RL5); execute(RL6); |
---|
1012 | def Rp = nc_algebra(@C,@D); |
---|
1013 | } |
---|
1014 | else { def Rp = @Rp; } |
---|
1015 | setring Rp; |
---|
1016 | kill @Rp; |
---|
1017 | dbprint(ppl-1,"// solving in ring ", Rp); |
---|
1018 | execute(str); |
---|
1019 | vector v; |
---|
1020 | number c; |
---|
1021 | ideal NI = 1; |
---|
1022 | setring save; |
---|
1023 | i=1; |
---|
1024 | while (badprime == 0 && i<=size(I)) // detect bad primes |
---|
1025 | { |
---|
1026 | str = string(leadcoef(cleardenom(I[i]))); |
---|
1027 | str = "c = " + str + ";"; |
---|
1028 | setring Rp; |
---|
1029 | execute(str); |
---|
1030 | if (c == 0) { badprime = 1; } |
---|
1031 | setring save; |
---|
1032 | i++; |
---|
1033 | } |
---|
1034 | if (badprime == 1) |
---|
1035 | { |
---|
1036 | print("// WARNING: bad prime"); |
---|
1037 | short = shortSave; |
---|
1038 | return(vector(0)); |
---|
1039 | } |
---|
1040 | } |
---|
1041 | i = 1; |
---|
1042 | dbprint(ppl,"// pIntersectSyz starts..."); |
---|
1043 | dbprint(ppl-1,"// with ideal I=", I); |
---|
1044 | while (1) |
---|
1045 | { |
---|
1046 | dbprint(ppl,"// testing degree: "+string(i)); |
---|
1047 | if (i>1) |
---|
1048 | { |
---|
1049 | tobracket = s^(i-1)-NI[i]; |
---|
1050 | if (tobracket!=0) |
---|
1051 | { |
---|
1052 | toNF = bracket(tobracket,NI[2]) + NI[i]*NI[2]; |
---|
1053 | } |
---|
1054 | else |
---|
1055 | { |
---|
1056 | toNF = NI[i]*NI[2]; |
---|
1057 | } |
---|
1058 | newNF = NF(toNF,I); |
---|
1059 | NI[i+1] = newNF; |
---|
1060 | } |
---|
1061 | // look for a solution |
---|
1062 | dbprint(ppl-1,"// linSyzSolve starts with: "+string(matrix(NI))); |
---|
1063 | if (solveincharp) // modular method |
---|
1064 | { |
---|
1065 | for (j=1; j<=size(newNF); j++) |
---|
1066 | { |
---|
1067 | str = string(denominator(leadcoef(s[i]))); |
---|
1068 | str = "c = " + str + ";"; |
---|
1069 | setring Rp; |
---|
1070 | execute(str); |
---|
1071 | if (c == 0) |
---|
1072 | { |
---|
1073 | print("// WARNING: bad prime"); |
---|
1074 | setring save; |
---|
1075 | short = shortSave; |
---|
1076 | return(vector(0)); |
---|
1077 | } |
---|
1078 | setring save; |
---|
1079 | } |
---|
1080 | str = "NI[" + string(i) + "+1] = " + string(newNF) + ";"; |
---|
1081 | setring Rp; |
---|
1082 | execute(str); // NI[i+1] = [newNF]_{solveincharp} |
---|
1083 | v = linSyzSolve(NI,modengine); |
---|
1084 | if (v!=0) // there is a modular solution |
---|
1085 | { |
---|
1086 | dbprint(ppl,"// got solution in char "+string(solveincharp)+" of degree "+string(i)); |
---|
1087 | setring save; |
---|
1088 | v = linSyzSolve(NI,whichengine); |
---|
1089 | if (v==0) |
---|
1090 | { |
---|
1091 | break; |
---|
1092 | } |
---|
1093 | } |
---|
1094 | else // no modular solution |
---|
1095 | { |
---|
1096 | setring save; |
---|
1097 | v = 0; |
---|
1098 | } |
---|
1099 | } |
---|
1100 | else // non-modular method |
---|
1101 | { |
---|
1102 | v = linSyzSolve(NI,whichengine); |
---|
1103 | } |
---|
1104 | matrix MM[1][nrows(v)] = matrix(v); |
---|
1105 | dbprint(ppl-1,"// linSyzSolve ready with: "+string(MM)); |
---|
1106 | kill MM; |
---|
1107 | // "linSyzSolve ready with"; print(v); |
---|
1108 | if (v!=0) |
---|
1109 | { |
---|
1110 | // a solution: |
---|
1111 | //check for the reality of the solution |
---|
1112 | p = 0; |
---|
1113 | for (j=1; j<=i+1; j++) |
---|
1114 | { |
---|
1115 | p = p + v[j]*NI[j]; |
---|
1116 | } |
---|
1117 | if (p!=0) |
---|
1118 | { |
---|
1119 | dbprint(ppl,"// linSyzSolve: bad solution!"); |
---|
1120 | } |
---|
1121 | else |
---|
1122 | { |
---|
1123 | dbprint(ppl,"// linSyzSolve: got solution!"); |
---|
1124 | // "got solution!"; |
---|
1125 | break; |
---|
1126 | } |
---|
1127 | } |
---|
1128 | // no solution: |
---|
1129 | i++; |
---|
1130 | } |
---|
1131 | dbprint(ppl,"// pIntersectSyz finished"); |
---|
1132 | if (solveincharp) { short = shortSave; } |
---|
1133 | return(v); |
---|
1134 | } |
---|
1135 | example |
---|
1136 | { |
---|
1137 | "EXAMPLE:"; echo = 2; |
---|
1138 | ring r = 0,(x,y),dp; |
---|
1139 | poly f = x^2+y^3+x*y^2; |
---|
1140 | def D = initialMalgrange(f); |
---|
1141 | setring D; |
---|
1142 | inF; |
---|
1143 | poly s = t*Dt; |
---|
1144 | pIntersectSyz(s,inF); |
---|
1145 | int p = prime(20000); |
---|
1146 | pIntersectSyz(s,inF,p,0,0); |
---|
1147 | } |
---|
1148 | |
---|
1149 | proc vec2poly (list #) |
---|
1150 | "USAGE: vec2poly(v [,i]); v a vector or an intvec, i an optional int |
---|
1151 | RETURN: poly, an univariate polynomial in i-th variable with coefficients given by v |
---|
1152 | PURPOSE: constructs an univariate polynomial in K[var(i)] with given coefficients, |
---|
1153 | @* such that the coefficient at var(i)^{j-1} is v[j]. |
---|
1154 | NOTE: The optional argument i must be positive, by default i is 1. |
---|
1155 | EXAMPLE: example vec2poly; shows examples |
---|
1156 | " |
---|
1157 | { |
---|
1158 | def save = basering; |
---|
1159 | int i,ringvar; |
---|
1160 | ringvar = 1; // default |
---|
1161 | if (size(#) > 0) |
---|
1162 | { |
---|
1163 | if (typeof(#[1])=="vector" || typeof(#[1])=="intvec") |
---|
1164 | { |
---|
1165 | def v = #[1]; |
---|
1166 | } |
---|
1167 | else |
---|
1168 | { |
---|
1169 | ERROR("wrong input: expected vector/intvec expression"); |
---|
1170 | } |
---|
1171 | if (size(#) > 1) |
---|
1172 | { |
---|
1173 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1174 | { |
---|
1175 | ringvar = int(#[2]); |
---|
1176 | } |
---|
1177 | } |
---|
1178 | } |
---|
1179 | if (ringvar > nvars(save)) |
---|
1180 | { |
---|
1181 | ERROR("var out of range"); |
---|
1182 | } |
---|
1183 | poly p; |
---|
1184 | for (i=1; i<=nrows(v); i++) |
---|
1185 | { |
---|
1186 | p = p + v[i]*(var(ringvar))^(i-1); |
---|
1187 | } |
---|
1188 | return(p); |
---|
1189 | } |
---|
1190 | example |
---|
1191 | { |
---|
1192 | "EXAMPLE:"; echo = 2; |
---|
1193 | ring r = 0,(x,y),dp; |
---|
1194 | vector v = gen(1) + 3*gen(3) + 22/9*gen(4); |
---|
1195 | intvec iv = 3,2,1; |
---|
1196 | vec2poly(v,2); |
---|
1197 | vec2poly(iv); |
---|
1198 | } |
---|
1199 | |
---|
1200 | /* |
---|
1201 | // // listofroots and addRoots aren't needed anymore due to some modifications |
---|
1202 | // |
---|
1203 | // static proc listofroots (list #) |
---|
1204 | // { |
---|
1205 | // def save = basering; |
---|
1206 | // int n = nvars(save); |
---|
1207 | // int i; |
---|
1208 | // poly p; |
---|
1209 | // if (typeof(#[1])=="vector") |
---|
1210 | // { |
---|
1211 | // vector b = #[1]; |
---|
1212 | // for (i=1; i<=nrows(b); i++) |
---|
1213 | // { |
---|
1214 | // p = p + b[i]*(var(1))^(i-1); |
---|
1215 | // } |
---|
1216 | // } |
---|
1217 | // else |
---|
1218 | // { |
---|
1219 | // p = #[1]; |
---|
1220 | // } |
---|
1221 | // int substitution = int(#[2]); |
---|
1222 | // string s = safeVarName("s"); |
---|
1223 | // list RL = ringlist(save); RL = RL[1..4]; |
---|
1224 | // RL[2] = list(s); RL[3] = list(list("dp",intvec(1)),list("C",0)); |
---|
1225 | // def S = ring(RL); setring S; |
---|
1226 | // ideal J; |
---|
1227 | // for (i=1; i<=n; i++) |
---|
1228 | // { |
---|
1229 | // J[i] = var(1); |
---|
1230 | // } |
---|
1231 | // map @m = save,J; |
---|
1232 | // poly p = @m(p); |
---|
1233 | // if (substitution == 1) |
---|
1234 | // { |
---|
1235 | // p = subst(p,var(1),-var(1)-1); |
---|
1236 | // } |
---|
1237 | // // the rest of this proc is nicked from bernsteinBM from dmod.lib |
---|
1238 | // list P = factorize(p);//with constants and multiplicities |
---|
1239 | // ideal bs; intvec m; //the BS polynomial is monic, so we are not interested in constants |
---|
1240 | // for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
1241 | // { |
---|
1242 | // bs[i-1] = P[1][i]; |
---|
1243 | // m[i-1] = P[2][i]; |
---|
1244 | // } |
---|
1245 | // bs = normalize(bs); |
---|
1246 | // bs = -subst(bs,var(1),0); |
---|
1247 | // setring save; |
---|
1248 | // ideal bs = imap(S,bs); |
---|
1249 | // kill S; |
---|
1250 | // list BS = bs,m; |
---|
1251 | // return(BS); |
---|
1252 | // } |
---|
1253 | // |
---|
1254 | // |
---|
1255 | // static proc addRoot(number q, list L) |
---|
1256 | // { |
---|
1257 | // // add root to list in bFactor format |
---|
1258 | // int i,qInL; |
---|
1259 | // ideal I = L[1]; |
---|
1260 | // intvec v = L[2]; |
---|
1261 | // list LL; |
---|
1262 | // if (v == 0) |
---|
1263 | // { |
---|
1264 | // I = poly(q); |
---|
1265 | // v = 1; |
---|
1266 | // LL = I,v; |
---|
1267 | // } |
---|
1268 | // else |
---|
1269 | // { |
---|
1270 | // for (i=1; i<=ncols(I); i++) |
---|
1271 | // { |
---|
1272 | // if (I[i] == q) |
---|
1273 | // { |
---|
1274 | // qInL = i; |
---|
1275 | // break; |
---|
1276 | // } |
---|
1277 | // } |
---|
1278 | // if (qInL) |
---|
1279 | // { |
---|
1280 | // v[qInL] = v[qInL] + 1; |
---|
1281 | // } |
---|
1282 | // else |
---|
1283 | // { |
---|
1284 | // I = q,I; |
---|
1285 | // v = 1,v; |
---|
1286 | // } |
---|
1287 | // } |
---|
1288 | // LL = I,v; |
---|
1289 | // if (size(L) == 3) // irreducible factor |
---|
1290 | // { |
---|
1291 | // if (L[3] <> "0" && L[3] <> "1") |
---|
1292 | // { |
---|
1293 | // LL = LL + list(L[3]); |
---|
1294 | // } |
---|
1295 | // } |
---|
1296 | // return(LL); |
---|
1297 | // } |
---|
1298 | */ |
---|
1299 | |
---|
1300 | static proc bfctengine (poly f, int inorann, int whichengine, int addPD, int stdsum, int methodord, int methodpIntersect, int pIntersectchar, int modengine, intvec u0) |
---|
1301 | { |
---|
1302 | int printlevelsave = printlevel; |
---|
1303 | printlevel = printlevel + 1; |
---|
1304 | int ppl = printlevel - voice + 2; |
---|
1305 | int i; |
---|
1306 | def save = basering; |
---|
1307 | int n = nvars(save); |
---|
1308 | if (isCommutative() == 0) { ERROR("basering must be commutative"); } |
---|
1309 | if (char(save) <> 0) { ERROR("characteristic of basering has to be 0"); } |
---|
1310 | list L = ringlist(save); |
---|
1311 | int qr; |
---|
1312 | if (L[4] <> 0) // qring |
---|
1313 | { |
---|
1314 | print("// basering is qring:"); |
---|
1315 | print("// discarding the quotient and proceeding..."); |
---|
1316 | L[4] = ideal(0); |
---|
1317 | qr = 1; |
---|
1318 | def save2 = ring(L); setring save2; |
---|
1319 | poly f = imap(save,f); |
---|
1320 | } |
---|
1321 | if (inorann == 0) // bfct using initial ideal |
---|
1322 | { |
---|
1323 | // list L = ringlist(basering); |
---|
1324 | intvec iv = valvars(f)[1]; // heuristacally better ordering for initialMalgrange |
---|
1325 | list varL = L[2]; |
---|
1326 | varL = varL[iv]; |
---|
1327 | L[2] = varL; |
---|
1328 | if (u0 <> 0) |
---|
1329 | { |
---|
1330 | u0 = u0[iv]; |
---|
1331 | } |
---|
1332 | def newr = ring(L); |
---|
1333 | kill varL,iv,L; |
---|
1334 | setring newr; |
---|
1335 | poly f = imap(save,f); |
---|
1336 | dbprint(ppl,"// starting computation of the initial ideal of the Malgrange ideal..."); |
---|
1337 | def D = initialMalgrange(f,whichengine,methodord,u0); |
---|
1338 | dbprint(ppl,"// ...done"); |
---|
1339 | setring D; |
---|
1340 | ideal J = inF; |
---|
1341 | kill inF; |
---|
1342 | poly s = var(1)*var(n+2); |
---|
1343 | } |
---|
1344 | else // bfct using Ann(f^s) |
---|
1345 | { |
---|
1346 | dbprint(ppl,"// starting computation of the s-parametric annihilator..."); |
---|
1347 | def D = SannfsBFCT(f,addPD,whichengine,stdsum); |
---|
1348 | dbprint(ppl,"// ...done"); |
---|
1349 | setring D; |
---|
1350 | ideal J = LD; |
---|
1351 | kill LD; |
---|
1352 | poly s = var(1); |
---|
1353 | } |
---|
1354 | vector b; |
---|
1355 | dbprint(ppl,"// starting to intersect with subalgebra..."); |
---|
1356 | // try it modular |
---|
1357 | if (methodpIntersect <> 0) // pIntersectSyz |
---|
1358 | { |
---|
1359 | if (pIntersectchar == 0) // pIntersectSyz::modular |
---|
1360 | { |
---|
1361 | list L = ringlist(D); |
---|
1362 | int lb = 10000; int ub = 536870909; // bounds for random primes |
---|
1363 | list usedprimes; |
---|
1364 | int sJ = size(J); |
---|
1365 | int sLJq; |
---|
1366 | ideal LJ; |
---|
1367 | for (i=1; i<=sJ; i++) |
---|
1368 | { |
---|
1369 | LJ[i] = leadcoef(cleardenom(J[i])); |
---|
1370 | } |
---|
1371 | int short_save = short; // workaround for map Q(a_i) -> Z/q(a_i) |
---|
1372 | short = 0; |
---|
1373 | string strLJq = "ideal LJq = " + string(LJ) + ";"; |
---|
1374 | int nD = nvars(D); |
---|
1375 | string L5 = "matrix @C[nD][nD]; @C = " + string(L[5]) + ";"; |
---|
1376 | string L6 = "matrix @D[nD][nD]; @D = " + string(L[6]) + ";"; |
---|
1377 | L = L[1..4]; |
---|
1378 | i = 1; |
---|
1379 | while (b == 0) |
---|
1380 | { |
---|
1381 | dbprint(ppl,"// number of run in the loop: "+string(i)); |
---|
1382 | int q = prime(random(lb,ub)); |
---|
1383 | if (findFirst(usedprimes,q)==0) // if q has not been used already |
---|
1384 | { |
---|
1385 | usedprimes = usedprimes,q; |
---|
1386 | dbprint(ppl,"// using prime: "+string(q)); |
---|
1387 | if (typeof(L[1]) == "int") { L[1] = q; } |
---|
1388 | else { L[1][1] = q; } // parameters |
---|
1389 | def @Rq = ring(L); setring @Rq; |
---|
1390 | execute(L5); execute(L6); |
---|
1391 | def Rq = nc_algebra(@C,@D); // def Rq = nc_algebra(1,@D); |
---|
1392 | setring Rq; kill @Rq; |
---|
1393 | execute(strLJq); |
---|
1394 | sLJq = size(LJq); |
---|
1395 | setring D; kill Rq; |
---|
1396 | if (sLJq <> sJ ) // detect unlucky prime |
---|
1397 | { |
---|
1398 | dbprint(ppl,"// " +string(q) + " is unlucky"); |
---|
1399 | b = 0; |
---|
1400 | } |
---|
1401 | else |
---|
1402 | { |
---|
1403 | b = pIntersectSyz(s,J,q,whichengine,modengine); |
---|
1404 | } |
---|
1405 | } |
---|
1406 | i++; |
---|
1407 | } |
---|
1408 | short = short_save; |
---|
1409 | } |
---|
1410 | else // pIntersectSyz::non-modular |
---|
1411 | { |
---|
1412 | b = pIntersectSyz(s,J,0,whichengine); |
---|
1413 | } |
---|
1414 | } |
---|
1415 | else // pIntersect: linReduce |
---|
1416 | { |
---|
1417 | b = pIntersect(s,J); |
---|
1418 | } |
---|
1419 | dbprint(ppl,"// ...done"); // with the intersection |
---|
1420 | poly pb = vec2poly(b); |
---|
1421 | if (inorann == 0) |
---|
1422 | { |
---|
1423 | pb = subst(pb,var(1),-var(1)-1); |
---|
1424 | } |
---|
1425 | else // bfctAnn |
---|
1426 | { |
---|
1427 | if (addPD) |
---|
1428 | { |
---|
1429 | pb = pb*(var(1)+1); |
---|
1430 | } |
---|
1431 | } |
---|
1432 | list l = bFactor(pb); |
---|
1433 | setring save; |
---|
1434 | list l = imap(D,l); |
---|
1435 | printlevel = printlevelsave; |
---|
1436 | return(l); |
---|
1437 | } |
---|
1438 | |
---|
1439 | proc bfct (poly f, list #) |
---|
1440 | "USAGE: bfct(f [,s,t,v]); f a poly, s,t optional ints, v an optional intvec |
---|
1441 | RETURN: list of ideal and intvec |
---|
1442 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) |
---|
1443 | @* for the hypersurface defined by f. |
---|
1444 | ASSUME: The basering is commutative and of characteristic 0. |
---|
1445 | BACKGROUND: In this proc, the initial Malgrange ideal is computed according to |
---|
1446 | @* the algorithm by Masayuki Noro and then a system of linear equations is |
---|
1447 | @* solved by linear reductions. |
---|
1448 | NOTE: In the output list, the ideal contains all the roots |
---|
1449 | @* and the intvec their multiplicities. |
---|
1450 | @* If s<>0, @code{std} is used for GB computations, |
---|
1451 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1452 | @* If t<>0, a matrix ordering is used for Groebner basis computations, |
---|
1453 | @* otherwise, and by default, a block ordering is used. |
---|
1454 | @* If v is a positive weight vector, v is used for homogenization |
---|
1455 | @* computations, otherwise and by default, no weights are used. |
---|
1456 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1457 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1458 | EXAMPLE: example bfct; shows examples |
---|
1459 | " |
---|
1460 | { |
---|
1461 | int ppl = printlevel - voice +2; |
---|
1462 | int i; |
---|
1463 | int n = nvars(basering); |
---|
1464 | // in # we have two switches: |
---|
1465 | // one for the engine used for Groebner basis computations, |
---|
1466 | // one for M() ordering or its realization |
---|
1467 | // in # can also be the optional weight vector |
---|
1468 | int whichengine = 0; // default |
---|
1469 | int methodord = 0; // default |
---|
1470 | intvec u0 = 0; // default |
---|
1471 | if (size(#)>0) |
---|
1472 | { |
---|
1473 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1474 | { |
---|
1475 | whichengine = int(#[1]); |
---|
1476 | } |
---|
1477 | if (size(#)>1) |
---|
1478 | { |
---|
1479 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1480 | { |
---|
1481 | methodord = int(#[2]); |
---|
1482 | } |
---|
1483 | if (size(#)>2) |
---|
1484 | { |
---|
1485 | if (typeof(#[3])=="intvec" && size(#[3])==n && allPositive(#[3])==1) |
---|
1486 | { |
---|
1487 | u0 = #[3]; |
---|
1488 | } |
---|
1489 | } |
---|
1490 | } |
---|
1491 | } |
---|
1492 | list b = bfctengine(f,0,whichengine,0,0,methodord,0,0,0,u0); |
---|
1493 | return(b); |
---|
1494 | } |
---|
1495 | example |
---|
1496 | { |
---|
1497 | "EXAMPLE:"; echo = 2; |
---|
1498 | ring r = 0,(x,y),dp; |
---|
1499 | poly f = x^2+y^3+x*y^2; |
---|
1500 | bfct(f); |
---|
1501 | intvec v = 3,2; |
---|
1502 | bfct(f,1,0,v); |
---|
1503 | } |
---|
1504 | |
---|
1505 | proc bfctSyz (poly f, list #) |
---|
1506 | "USAGE: bfctSyz(f [,r,s,t,u,v]); f poly, r,s,t,u optional ints, v opt. intvec |
---|
1507 | RETURN: list of ideal and intvec |
---|
1508 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) |
---|
1509 | @* for the hypersurface defined by f |
---|
1510 | ASSUME: The basering is commutative and of characteristic 0. |
---|
1511 | BACKGROUND: In this proc, the initial Malgrange ideal is computed according to |
---|
1512 | @* the algorithm by Masayuki Noro and then a system of linear equations is |
---|
1513 | @* solved by computing syzygies. |
---|
1514 | NOTE: In the output list, the ideal contains all the roots and the intvec |
---|
1515 | @* their multiplicities. |
---|
1516 | @* If r<>0, @code{std} is used for GB computations in characteristic 0, |
---|
1517 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1518 | @* If s<>0, a matrix ordering is used for GB computations, otherwise, |
---|
1519 | @* and by default, a block ordering is used. |
---|
1520 | @* If t<>0, the computation of the intersection is solely performed over |
---|
1521 | @* charasteristic 0, otherwise and by default, a modular method is used. |
---|
1522 | @* If u<>0 and by default, @code{std} is used for GB computations in |
---|
1523 | @* characteristic >0, otherwise, @code{slimgb} is used. |
---|
1524 | @* If v is a positive weight vector, v is used for homogenization |
---|
1525 | @* computations, otherwise and by default, no weights are used. |
---|
1526 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1527 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1528 | EXAMPLE: example bfctSyz; shows examples |
---|
1529 | " |
---|
1530 | { |
---|
1531 | int ppl = printlevel - voice +2; |
---|
1532 | int i; |
---|
1533 | // in # we have four switches: |
---|
1534 | // one for the engine used for Groebner basis computations in char 0, |
---|
1535 | // one for M() ordering or its realization |
---|
1536 | // one for a modular method when computing the intersection |
---|
1537 | // and one for the engine used for Groebner basis computations in char >0 |
---|
1538 | // in # can also be the optional weight vector |
---|
1539 | int n = nvars(basering); |
---|
1540 | int whichengine = 0; // default |
---|
1541 | int methodord = 0; // default |
---|
1542 | int pIntersectchar = 0; // default |
---|
1543 | int modengine = 1; // default |
---|
1544 | intvec u0 = 0; // default |
---|
1545 | if (size(#)>0) |
---|
1546 | { |
---|
1547 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1548 | { |
---|
1549 | whichengine = int(#[1]); |
---|
1550 | } |
---|
1551 | if (size(#)>1) |
---|
1552 | { |
---|
1553 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1554 | { |
---|
1555 | methodord = int(#[2]); |
---|
1556 | } |
---|
1557 | if (size(#)>2) |
---|
1558 | { |
---|
1559 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
1560 | { |
---|
1561 | pIntersectchar = int(#[3]); |
---|
1562 | } |
---|
1563 | if (size(#)>3) |
---|
1564 | { |
---|
1565 | if (typeof(#[4])=="int" || typeof(#[4])=="number") |
---|
1566 | { |
---|
1567 | modengine = int(#[4]); |
---|
1568 | } |
---|
1569 | if (size(#)>4) |
---|
1570 | { |
---|
1571 | if (typeof(#[5])=="intvec" && size(#[5])==n && allPositive(#[5])==1) |
---|
1572 | { |
---|
1573 | u0 = #[5]; |
---|
1574 | } |
---|
1575 | } |
---|
1576 | } |
---|
1577 | } |
---|
1578 | } |
---|
1579 | } |
---|
1580 | list b = bfctengine(f,0,whichengine,0,0,methodord,1,pIntersectchar,modengine,u0); |
---|
1581 | return(b); |
---|
1582 | } |
---|
1583 | example |
---|
1584 | { |
---|
1585 | "EXAMPLE:"; echo = 2; |
---|
1586 | ring r = 0,(x,y),dp; |
---|
1587 | poly f = x^2+y^3+x*y^2; |
---|
1588 | bfctSyz(f); |
---|
1589 | intvec v = 3,2; |
---|
1590 | bfctSyz(f,0,1,1,0,v); |
---|
1591 | } |
---|
1592 | |
---|
1593 | proc bfctIdeal (ideal I, intvec w, list #) |
---|
1594 | "USAGE: bfctIdeal(I,w[,s,t]); I an ideal, w an intvec, s,t optional ints |
---|
1595 | RETURN: list of ideal and intvec |
---|
1596 | PURPOSE: computes the roots of the global b-function of I w.r.t. the weight (-w,w). |
---|
1597 | ASSUME: The basering is the n-th Weyl algebra in characteristic 0 and for all |
---|
1598 | @* 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 holds, i.e. the |
---|
1599 | @* sequence of variables is given by x(1),...,x(n),D(1),...,D(n), |
---|
1600 | @* where D(i) is the differential operator belonging to x(i). |
---|
1601 | @* Further we assume that I is holonomic. |
---|
1602 | BACKGROUND: In this proc, the initial ideal of I is computed according to the |
---|
1603 | @* algorithm by Masayuki Noro and then a system of linear equations is |
---|
1604 | @* solved by linear reductions. |
---|
1605 | NOTE: In the output list, say L, |
---|
1606 | @* - L[1] of type ideal contains all the rational roots of a b-function, |
---|
1607 | @* - L[2] of type intvec contains the multiplicities of above roots, |
---|
1608 | @* - optional L[3] of type string is the part of b-function without rational roots. |
---|
1609 | @* Note, that a b-function of degree 0 is encoded via L[1][1]=0, L[2]=0 and |
---|
1610 | @* L[3] is 1 (for nonzero constant) or 0 (for zero b-function). |
---|
1611 | @* If s<>0, @code{std} is used for GB computations in characteristic 0, |
---|
1612 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1613 | @* If t<>0, a matrix ordering is used for GB computations, otherwise, |
---|
1614 | @* and by default, a block ordering is used. |
---|
1615 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1616 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1617 | EXAMPLE: example bfctIdeal; shows examples |
---|
1618 | " |
---|
1619 | { |
---|
1620 | int ppl = printlevel - voice +2; |
---|
1621 | int i; |
---|
1622 | def save = basering; |
---|
1623 | int n = nvars(save)/2; |
---|
1624 | int whichengine = 0; // default |
---|
1625 | int methodord = 0; // default |
---|
1626 | if (size(#)>0) |
---|
1627 | { |
---|
1628 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1629 | { |
---|
1630 | whichengine = int(#[1]); |
---|
1631 | } |
---|
1632 | if (size(#)>1) |
---|
1633 | { |
---|
1634 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1635 | { |
---|
1636 | methodord = int(#[2]); |
---|
1637 | } |
---|
1638 | } |
---|
1639 | } |
---|
1640 | if (isWeyl()==0) { ERROR("basering is not a Weyl algebra"); } |
---|
1641 | for (i=1; i<=n; i++) |
---|
1642 | { |
---|
1643 | if (bracket(var(i+n),var(i))<>1) |
---|
1644 | { |
---|
1645 | ERROR(string(var(i+n))+" is not a differential operator for " +string(var(i))); |
---|
1646 | } |
---|
1647 | } |
---|
1648 | int isH = isHolonomic(I); |
---|
1649 | if (isH<>1) |
---|
1650 | { |
---|
1651 | print("WARNING: given ideal is not holonomic"); |
---|
1652 | print("... setting bound for degree of b-function to 10 and proceeding"); |
---|
1653 | isH = 10; |
---|
1654 | } |
---|
1655 | else { isH = 0; } // no degree bound for pIntersect |
---|
1656 | ideal J = initialIdealW(I,-w,w,whichengine,methodord); |
---|
1657 | poly s; |
---|
1658 | for (i=1; i<=n; i++) |
---|
1659 | { |
---|
1660 | s = s + w[i]*var(i)*var(n+i); |
---|
1661 | } |
---|
1662 | vector b = pIntersect(s,J,isH); |
---|
1663 | list RL = ringlist(save); RL = RL[1..4]; |
---|
1664 | RL[2] = list(safeVarName("s")); |
---|
1665 | RL[3] = list(list("dp",intvec(1)),list("C",intvec(0))); |
---|
1666 | def @S = ring(RL); setring @S; |
---|
1667 | vector b = imap(save,b); |
---|
1668 | poly bs = vec2poly(b); |
---|
1669 | list l = bFactor(bs); |
---|
1670 | setring save; |
---|
1671 | list l = imap(@S,l); |
---|
1672 | return(l); |
---|
1673 | } |
---|
1674 | example |
---|
1675 | { |
---|
1676 | "EXAMPLE:"; echo = 2; |
---|
1677 | ring @D = 0,(x,y,Dx,Dy),dp; |
---|
1678 | def D = Weyl(); |
---|
1679 | setring D; |
---|
1680 | ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6; I = std(I); |
---|
1681 | intvec w1 = 0,1; |
---|
1682 | intvec w2 = 2,3; |
---|
1683 | bfctIdeal(I,w1); |
---|
1684 | bfctIdeal(I,w2,0,1); |
---|
1685 | ideal J = I[size(I)]; // J is not holonomic by construction |
---|
1686 | bfctIdeal(J,w1); // b-function of D/J w.r.t. w1 is non-zero |
---|
1687 | bfctIdeal(J,w2); // b-function of D/J w.r.t. w2 is zero |
---|
1688 | } |
---|
1689 | |
---|
1690 | proc bfctOneGB (poly f,list #) |
---|
1691 | "USAGE: bfctOneGB(f [,s,t]); f a poly, s,t optional ints |
---|
1692 | RETURN: list of ideal and intvec |
---|
1693 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) for the |
---|
1694 | @* hypersurface defined by f, using only one GB computation |
---|
1695 | ASSUME: The basering is commutative and of characteristic 0. |
---|
1696 | BACKGROUND: In this proc, the initial Malgrange ideal is computed based on the |
---|
1697 | @* algorithm by Masayuki Noro and combined with an elimination ordering. |
---|
1698 | NOTE: In the output list, the ideal contains all the roots and the intvec |
---|
1699 | @* their multiplicities. |
---|
1700 | @* If s<>0, @code{std} is used for the GB computation, otherwise, |
---|
1701 | @* and by default, @code{slimgb} is used. |
---|
1702 | @* If t<>0, a matrix ordering is used for GB computations, |
---|
1703 | @* otherwise, and by default, a block ordering is used. |
---|
1704 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1705 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1706 | EXAMPLE: example bfctOneGB; shows examples |
---|
1707 | " |
---|
1708 | { |
---|
1709 | int ppl = printlevel - voice +2; |
---|
1710 | if (!isCommutative()) { ERROR("Basering must be commutative"); } |
---|
1711 | def save = basering; |
---|
1712 | int n = nvars(save); |
---|
1713 | if (char(save) <> 0) |
---|
1714 | { |
---|
1715 | ERROR("characteristic of basering has to be 0"); |
---|
1716 | } |
---|
1717 | list L = ringlist(save); |
---|
1718 | int qr; |
---|
1719 | if (L[4] <> 0) // qring? |
---|
1720 | { |
---|
1721 | print("// basering is qring:"); |
---|
1722 | print("// discarding the quotient and proceeding..."); |
---|
1723 | L[4] = ideal(0); |
---|
1724 | qr = 1; |
---|
1725 | def save2 = ring(L); |
---|
1726 | setring save2; |
---|
1727 | poly f = imap(save,f); |
---|
1728 | } |
---|
1729 | int N = 2*n+4; |
---|
1730 | int i; |
---|
1731 | int whichengine = 0; // default |
---|
1732 | int methodord = 0; // default |
---|
1733 | if (size(#)>0) |
---|
1734 | { |
---|
1735 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1736 | { |
---|
1737 | whichengine = int(#[1]); |
---|
1738 | } |
---|
1739 | if (size(#)>1) |
---|
1740 | { |
---|
1741 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1742 | { |
---|
1743 | methodord = int(#[2]); |
---|
1744 | } |
---|
1745 | } |
---|
1746 | } |
---|
1747 | // creating the homogenized extended Weyl algebra |
---|
1748 | // create names for vars |
---|
1749 | list Lvar; |
---|
1750 | Lvar[1] = safeVarName("t"); |
---|
1751 | Lvar[2] = safeVarName("s"); |
---|
1752 | Lvar[n+3] = safeVarName("D"+Lvar[1]); |
---|
1753 | Lvar[N] = safeVarName("h"); |
---|
1754 | for (i=1; i<=n; i++) |
---|
1755 | { |
---|
1756 | Lvar[i+2] = string(var(i)); |
---|
1757 | Lvar[i+n+3] = safeVarName("D" + string(var(i))); |
---|
1758 | } |
---|
1759 | // create ordering |
---|
1760 | intvec uv = -1; uv[n+3] = 1; uv[N] = 0; |
---|
1761 | intvec @a = 1:N; @a[2] = 2; |
---|
1762 | intvec @a2 = @a; @a2[2] = 0; @a2[2*n+4] = 0; |
---|
1763 | list Lord; |
---|
1764 | Lord[1] = list("a",@a); Lord[2] = list("a",@a2); |
---|
1765 | if (methodord == 0) // default: block ordering |
---|
1766 | { |
---|
1767 | //ring @Dh = 0,(t,s,x(n..1),Dt,D(n..1),h),(a(@a),a(@a2),a(uv),dp(N-1),lp(1)); |
---|
1768 | Lord[3] = list("a",uv); |
---|
1769 | Lord[4] = list("dp",intvec(1:(N-1))); |
---|
1770 | Lord[5] = list("lp",intvec(1)); |
---|
1771 | Lord[6] = list("C",intvec(0)); |
---|
1772 | } |
---|
1773 | else // M() ordering |
---|
1774 | { |
---|
1775 | intmat @Ord[N][N]; |
---|
1776 | @Ord[1,1..N] = uv; @Ord[2,1..N] = 1:(N-1); |
---|
1777 | for (i=1; i<=N-2; i++) { @Ord[2+i,N - i] = -1; } |
---|
1778 | dbprint(ppl,"// weights for ordering: "+string(transpose(@a))); |
---|
1779 | dbprint(ppl,"// the ordering matrix:",@Ord); |
---|
1780 | //ring @Dh = 0,(t,s,x(n..1),Dt,D(n..1),h),(a(@a),a(@a2),M(@Ord)); |
---|
1781 | Lord[3] = list("M",intvec(@Ord)); |
---|
1782 | Lord[4] = list("C",intvec(0)); |
---|
1783 | } |
---|
1784 | // create commutative ring |
---|
1785 | list L@Dh = ringlist(basering); |
---|
1786 | L@Dh = L@Dh[1..4]; // if basering is commutative nc_algebra |
---|
1787 | L@Dh[2] = Lvar; L@Dh[3] = Lord; |
---|
1788 | def @Dh = ring(L@Dh); setring @Dh; |
---|
1789 | dbprint(ppl,"// the ring @Dh:",@Dh); |
---|
1790 | // create non-commutative relations |
---|
1791 | matrix @relD[N][N]; |
---|
1792 | @relD[1,2] = var(1)*var(N)^2; // s*t = t*s + t*h^2 |
---|
1793 | @relD[2,n+3] = var(n+3)*var(N)^2; // Dt*s = s*Dt+Dt*h^2 |
---|
1794 | @relD[1,n+3] = var(N)^2; |
---|
1795 | for (i=1; i<=n; i++) |
---|
1796 | { |
---|
1797 | @relD[i+2,n+3+i] = var(N)^2; |
---|
1798 | } |
---|
1799 | dbprint(ppl,"// nc relations:",@relD); |
---|
1800 | def Dh = nc_algebra(1,@relD); |
---|
1801 | setring Dh; kill @Dh; |
---|
1802 | dbprint(ppl,"// computing in ring",Dh); |
---|
1803 | poly f = imap(save,f); |
---|
1804 | f = homog(f,h); |
---|
1805 | // create the Malgrange ideal |
---|
1806 | ideal I = var(1) - f, var(1)*var(n+3) - var(2); |
---|
1807 | for (i=1; i<=n; i++) |
---|
1808 | { |
---|
1809 | I[3+i] = var(i+n+3)+diff(f,var(i+2))*var(n+3); |
---|
1810 | } |
---|
1811 | dbprint(ppl-1, "// the Malgrange ideal: " +string(I)); |
---|
1812 | // the hard part: Groebner basis computation |
---|
1813 | dbprint(ppl, "// starting Groebner basis computation with engine: "+string(whichengine)); |
---|
1814 | I = engine(I, whichengine); |
---|
1815 | dbprint(ppl, "// finished Groebner basis computation"); |
---|
1816 | I = subst(I,h,1); // dehomogenization |
---|
1817 | dbprint(ppl-1,string(I)); |
---|
1818 | // 3.3 the initial form |
---|
1819 | I = inForm(I,uv); |
---|
1820 | dbprint(ppl, "// the initial ideal:", string(matrix(I))); |
---|
1821 | // read off the solution |
---|
1822 | intvec tonselect = 1; |
---|
1823 | for (i=3; i<=2*n+4; i++) { tonselect = tonselect,i; } |
---|
1824 | I = nselect(I,tonselect); |
---|
1825 | dbprint(ppl, "// generators containing only s:", string(matrix(I))); |
---|
1826 | I = engine(I, whichengine); // is now a principal ideal; |
---|
1827 | if (qr == 1) { setring save2; } |
---|
1828 | else { setring save; } |
---|
1829 | ideal J; J[2] = var(1); |
---|
1830 | map @m = Dh,J; |
---|
1831 | ideal I = @m(I); |
---|
1832 | poly p = I[1]; |
---|
1833 | p = subst(p,var(1),-var(1)-1); |
---|
1834 | list l = bFactor(p); |
---|
1835 | if (qr == 1) |
---|
1836 | { |
---|
1837 | setring save; |
---|
1838 | list l = imap(save2,l); |
---|
1839 | } |
---|
1840 | return(l); |
---|
1841 | } |
---|
1842 | example |
---|
1843 | { |
---|
1844 | "EXAMPLE:"; echo = 2; |
---|
1845 | ring r = 0,(x,y),dp; |
---|
1846 | poly f = x^2+y^3+x*y^2; |
---|
1847 | bfctOneGB(f); |
---|
1848 | bfctOneGB(f,1,1); |
---|
1849 | } |
---|
1850 | |
---|
1851 | |
---|
1852 | |
---|
1853 | proc bfctAnn (poly f, list #) |
---|
1854 | "USAGE: bfctAnn(f [,a,b,c]); f a poly, a, b, c optional ints |
---|
1855 | RETURN: list of ideal and intvec |
---|
1856 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) for the |
---|
1857 | @* hypersurface defined by f. |
---|
1858 | ASSUME: The basering is commutative and of characteristic 0. |
---|
1859 | BACKGROUND: In this proc, Ann(f^s) is computed and then a system of linear |
---|
1860 | @* equations is solved by linear reductions. |
---|
1861 | NOTE: In the output list, the ideal contains all the roots and the intvec |
---|
1862 | @* their multiplicities. |
---|
1863 | @* If a<>0, only f is appended to Ann(f^s), otherwise, and by default, |
---|
1864 | @* f and all its partial derivatives are appended. |
---|
1865 | @* If b<>0, @code{std} is used for GB computations, otherwise, and by |
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1866 | @* default, @code{slimgb} is used. |
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1867 | @* If c<>0, @code{std} is used for Groebner basis computations of ideals |
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1868 | @* <I+J> when I is already a Groebner basis of <I>. |
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1869 | @* Otherwise, and by default the engine determined by the switch b is used. |
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1870 | @* Note that in the case c<>0, the choice for b will be overwritten only |
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1871 | @* for the types of ideals mentioned above. |
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1872 | @* This means that if b<>0, specifying c has no effect. |
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1873 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
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1874 | @* if printlevel>=2, all the debug messages will be printed. |
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1875 | EXAMPLE: example bfctAnn; shows examples |
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1876 | " |
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1877 | { |
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1878 | def save = basering; |
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1879 | int ppl = printlevel - voice + 2; |
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1880 | int addPD = 1; // default |
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1881 | int whichengine = 0; // default |
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1882 | int stdsum = 0; // default |
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1883 | if (size(#)>0) |
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1884 | { |
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1885 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
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1886 | { |
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1887 | addPD = int(#[1]); |
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1888 | } |
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1889 | if (size(#)>1) |
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1890 | { |
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1891 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
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1892 | { |
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1893 | whichengine = int(#[2]); |
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1894 | } |
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1895 | if (size(#)>2) |
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1896 | { |
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1897 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
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1898 | { |
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1899 | stdsum = int(#[3]); |
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1900 | } |
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1901 | } |
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1902 | } |
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1903 | } |
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1904 | list b = bfctengine(f,1,whichengine,addPD,stdsum,0,0,0,0,0); |
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1905 | return(b); |
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1906 | } |
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1907 | example |
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1908 | { |
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1909 | "EXAMPLE:"; echo = 2; |
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1910 | ring r = 0,(x,y),dp; |
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1911 | poly f = x^2+y^3+x*y^2; |
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1912 | bfctAnn(f); |
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1913 | def R = reiffen(4,5); setring R; |
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1914 | RC; // the Reiffen curve in 4,5 |
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1915 | bfctAnn(RC,0,1); |
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1916 | } |
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1917 | |
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1918 | /* |
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1919 | static proc hardexamples () |
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1920 | { |
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1921 | // some hard examples |
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1922 | ring r1 = 0,(x,y,z,w),dp; |
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1923 | // ab34 |
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1924 | poly ab34 = (z3+w4)*(3z2x+4w3y); |
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1925 | bfct(ab34); |
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1926 | // ha3 |
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1927 | poly ha3 = xyzw*(x+y)*(x+z)*(x+w)*(y+z+w); |
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1928 | bfct(ha3); |
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1929 | // ha4 |
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1930 | poly ha4 = xyzw*(x+y)*(x+z)*(x+w)*(y+z)*(y+w); |
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1931 | bfct(ha4); |
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1932 | // chal4: reiffen(4,5)*reiffen(5,4) |
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1933 | ring r2 = 0,(x,y),dp; |
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1934 | poly chal4 = (x4+xy4+y5)*(x5+x4y+y4); |
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1935 | bfct(chal4); |
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1936 | // (xy+z)*reiffen(4,5) |
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1937 | ring r3 = 0,(x,y,z),dp; |
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1938 | poly xyzreiffen45 = (xy+z)*(y4+yz4+z5); |
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1939 | bfct(xyzreiffen45); |
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1940 | |
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1941 | // sparse ideal as suggested by Alex; gives 1 as std |
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1942 | ideal I1 = 28191*y^2+14628*x*Dy, 24865*x^2+24072*x*Dx+17756*Dy^2; |
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1943 | std(I1); |
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1944 | } |
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1945 | */ |
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