1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: gaussman.lib,v 1.36 2001-03-20 18:19:48 mschulze Exp $"; |
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3 | category="Singularities"; |
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4 | |
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5 | info=" |
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6 | LIBRARY: gaussman.lib Gauss-Manin Connection of a Singularity |
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7 | |
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8 | AUTHOR: Mathias Schulze, email: mschulze@mathematik.uni-kl.de |
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9 | |
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10 | OVERVIEW: A library to compute invariants related to the Gauss-Manin connection |
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11 | of a an isolated hypersurface singularity |
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12 | |
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13 | PROCEDURES: |
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14 | monodromy(f[,opt]); monodromy matrix, spectrum of monodromy of f |
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15 | vfiltration(f[,opt]); V-filtration on H''/H', singularity spectrum of f |
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16 | spectrum(f); singularity spectrum of f |
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17 | endfilt(poly f,list V); endomorphism filtration of filtration V |
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18 | spprint(list S); print spectrum S |
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19 | spadd(list S1,list S2); sum of spectra S1 and S2 |
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20 | spsub(list S1,list S2); difference of spectra S1 and S2 |
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21 | spmul(list S,int k); product of spectrum S and integer k |
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22 | spmul(list S,intvec k); linear combination of spectra S with coefficients k |
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23 | spissemicont(list S[,opt]); test spectrum S for semicontinuity |
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24 | spsemicont(list S0,list S[,opt]); relative semicontinuity of spectra S0 and S |
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25 | spmilnor(list S); milnor number of spectrum S |
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26 | spgeomgenus(list S); geometrical genus of spectrum S |
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27 | spgamma(list S); gamma invariant of spectrum S |
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28 | |
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29 | SEE ALSO: mondromy_lib, spectrum_lib |
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30 | |
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31 | KEYWORDS: singularities; Gauss-Manin connection; monodromy; spectrum; |
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32 | Brieskorn lattice; Hodge filtration; V-filtration |
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33 | "; |
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34 | |
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35 | LIB "linalg.lib"; |
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36 | |
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37 | /////////////////////////////////////////////////////////////////////////////// |
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38 | |
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39 | static proc maxintdif(ideal e) |
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40 | { |
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41 | dbprint(printlevel-voice+2,"//gaussman::maxintdif"); |
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42 | int i,j,id; |
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43 | int mid=0; |
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44 | for(i=ncols(e);i>=1;i--) |
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45 | { |
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46 | for(j=i-1;j>=1;j--) |
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47 | { |
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48 | id=int(e[i]-e[j]); |
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49 | if(id<0) |
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50 | { |
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51 | id=-id; |
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52 | } |
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53 | if(id>mid) |
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54 | { |
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55 | mid=id; |
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56 | } |
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57 | } |
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58 | } |
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59 | return(mid); |
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60 | } |
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61 | /////////////////////////////////////////////////////////////////////////////// |
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62 | |
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63 | static proc maxorddif(matrix H) |
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64 | { |
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65 | dbprint(printlevel-voice+2,"//gaussman::maxorddif"); |
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66 | int i,j,d; |
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67 | int d0,d1=-1,-1; |
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68 | for(i=nrows(H);i>=1;i--) |
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69 | { |
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70 | for(j=ncols(H);j>=1;j--) |
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71 | { |
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72 | d=ord(H[i,j]); |
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73 | if(d>=0) |
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74 | { |
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75 | if(d0<0||d<d0) |
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76 | { |
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77 | d0=d; |
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78 | } |
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79 | if(d1<0||d>d1) |
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80 | { |
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81 | d1=d; |
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82 | } |
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83 | } |
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84 | } |
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85 | } |
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86 | return(d1-d0); |
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87 | } |
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88 | /////////////////////////////////////////////////////////////////////////////// |
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89 | |
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90 | proc monodromy(poly f,list #) |
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91 | "USAGE: monodromy(f[,opt]); poly f, int opt |
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92 | ASSUME: basering has characteristic 0 and local ordering, |
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93 | f has isolated singularity at 0 |
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94 | RETURN: |
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95 | @format |
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96 | if opt==0: |
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97 | matrix M: exp(-2*pi*i*M) is a monodromy matrix of f |
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98 | if opt==1: |
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99 | ideal e: exp(-2*pi*i*e) are the eigenvalues of the monodromy of f |
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100 | default: opt=1 |
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101 | @end format |
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102 | SEE ALSO: mondromy_lib |
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103 | KEYWORDS: singularities; Gauss-Manin connection; monodromy |
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104 | EXAMPLE: example monodromy; shows an example |
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105 | " |
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106 | { |
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107 | if(charstr(basering)!="0") |
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108 | { |
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109 | ERROR("characteristic 0 expected"); |
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110 | } |
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111 | int n=nvars(basering)-1; |
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112 | for(int i=n+1;i>=1;i--) |
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113 | { |
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114 | if(var(i)>1) |
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115 | { |
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116 | ERROR("local ordering expected"); |
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117 | } |
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118 | } |
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119 | ideal J=jacob(f); |
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120 | ideal sJ=std(J); |
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121 | if(vdim(sJ)<=0) |
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122 | { |
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123 | if(vdim(sJ)==0) |
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124 | { |
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125 | ERROR("singularity at 0 expected"); |
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126 | } |
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127 | else |
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128 | { |
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129 | ERROR("isolated singularity at 0 expected"); |
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130 | } |
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131 | } |
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132 | int opt=1; |
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133 | if(size(#)>0) |
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134 | { |
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135 | if(typeof(#[1])=="int") |
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136 | { |
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137 | opt=#[1]; |
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138 | } |
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139 | } |
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140 | |
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141 | ideal m=kbase(sJ); |
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142 | int mu,modm=ncols(m),maxorddif(m); |
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143 | |
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144 | ideal w=f*m; |
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145 | matrix U=freemodule(mu); |
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146 | matrix A0[mu][mu],A,C,D; |
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147 | list l; |
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148 | module H,dH=freemodule(mu),freemodule(mu); |
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149 | module H0; |
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150 | int sdH=1; |
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151 | int k=-1; |
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152 | int j,K,N,mide; |
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153 | |
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154 | while(k<K||sdH>0) |
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155 | { |
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156 | k++; |
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157 | dbprint(printlevel-voice+2,"//gaussman::monodromy: k="+string(k)); |
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158 | |
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159 | dbprint(printlevel-voice+2,"//gaussman::monodromy: compute C"); |
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160 | C=coeffs(reduce(w,U,sJ),m); |
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161 | A0=A0+C*var(1)^k; |
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162 | |
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163 | if(sdH>0) |
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164 | { |
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165 | H0=H; |
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166 | dbprint(printlevel-voice+2,"//gaussman::monodromy: compute dH"); |
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167 | dH=jet(module(A0*dH+var(1)^2*diff(matrix(dH),var(1))),k); |
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168 | H=H*var(1)+dH; |
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169 | |
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170 | dbprint(printlevel-voice+2,"//gaussman::monodromy: test dH==0"); |
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171 | sdH=size(reduce(H,std(H0*var(1)))); |
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172 | if(sdH>0) |
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173 | { |
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174 | A0=A0-var(1); |
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175 | } |
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176 | else |
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177 | { |
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178 | dbprint(printlevel-voice+2, |
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179 | "//gaussman::monodromy: compute basis of saturation"); |
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180 | H=minbase(H0); |
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181 | int modH=maxorddif(H); |
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182 | K=modH+1; |
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183 | } |
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184 | } |
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185 | |
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186 | if(k==K&&sdH==0) |
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187 | { |
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188 | N=k-modH; |
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189 | dbprint(printlevel-voice+2, |
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190 | "//gaussman::monodromy: compute A on saturation"); |
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191 | l=division(H*var(1),A0*H+var(1)^2*diff(matrix(H),var(1))); |
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192 | A=jet(l[1],l[2],N-1); |
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193 | if(mide==0) |
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194 | { |
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195 | dbprint(printlevel-voice+2, |
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196 | "//gaussman::monodromy: compute eigenvalues e and"+ |
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197 | "multiplicities b of A"); |
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198 | l=system("eigenval",jet(A,0)); |
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199 | ideal e=l[1]; |
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200 | intvec b=l[2]; |
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201 | dbprint(printlevel-voice+2,"//gaussman::monodromy: e="+string(e)); |
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202 | dbprint(printlevel-voice+2,"//gaussman::monodromy: b="+string(b)); |
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203 | if(opt==1) |
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204 | { |
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205 | mide=maxintdif(e); |
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206 | K=K+mide; |
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207 | } |
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208 | } |
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209 | } |
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210 | |
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211 | if(k<K||sdH>0) |
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212 | { |
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213 | dbprint(printlevel-voice+2,"//gaussman::monodromy: divide by J"); |
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214 | l=division(J,ideal(matrix(w)-matrix(m)*C*U)); |
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215 | D=l[1]; |
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216 | |
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217 | dbprint(printlevel-voice+2,"//gaussman::monodromy: compute w/U"); |
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218 | for(j=mu;j>=1;j--) |
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219 | { |
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220 | if(l[2][j,j]!=0) |
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221 | { |
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222 | dbprint(printlevel-voice+2, |
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223 | "//gaussman::monodromy: compute U["+string(j)+"]"); |
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224 | U[j,j]=U[j,j]*l[2][j,j]; |
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225 | } |
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226 | dbprint(printlevel-voice+2, |
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227 | "//gaussman::monodromy: compute w["+string(j)+"]"); |
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228 | w[j]=0; |
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229 | for(i=n+1;i>=1;i--) |
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230 | { |
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231 | w[j]=w[j]+U[j,j]*diff(D[i,j],var(i))-diff(U[j,j],var(i))*D[i,j]; |
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232 | } |
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233 | } |
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234 | U=U*U; |
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235 | } |
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236 | } |
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237 | |
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238 | if(mide>0) |
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239 | { |
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240 | intvec ide; |
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241 | ide[mu]=0; |
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242 | module dU; |
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243 | matrix A0e; |
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244 | for(i=ncols(e);i>=1;i--) |
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245 | { |
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246 | for(j=i-1;j>=1;j--) |
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247 | { |
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248 | k=int(e[j]-e[i]); |
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249 | if(k>ide[i]) |
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250 | { |
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251 | ide[i]=k; |
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252 | } |
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253 | if(-k>ide[j]) |
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254 | { |
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255 | ide[j]=-k; |
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256 | } |
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257 | } |
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258 | } |
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259 | for(j,k=ncols(e),mu;j>=1;j--) |
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260 | { |
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261 | for(i=b[j];i>=1;i--) |
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262 | { |
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263 | ide[k]=ide[j]; |
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264 | k--; |
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265 | } |
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266 | } |
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267 | } |
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268 | while(mide>0) |
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269 | { |
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270 | dbprint(printlevel-voice+2,"//gaussman::monodromy: mide="+string(mide)); |
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271 | |
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272 | U=0; |
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273 | A0=jet(A,0); |
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274 | for(i=ncols(e);i>=1;i--) |
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275 | { |
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276 | A0e=freemodule(mu); |
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277 | for(j=n;j>=0;j--) // Potenzen von Matrizen? |
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278 | { |
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279 | A0e=A0e*(A0-e[i]); |
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280 | } |
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281 | dU=syz(A0e); |
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282 | U=dU+U; |
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283 | } |
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284 | A=division(U,A*U)[1]; |
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285 | |
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286 | for(i=mu;i>=1;i--) |
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287 | { |
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288 | for(j=mu;j>=1;j--) |
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289 | { |
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290 | if(ide[i]==0&&ide[j]>0) |
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291 | { |
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292 | A[i,j]=A[i,j]*var(1); |
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293 | } |
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294 | else |
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295 | { |
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296 | if(ide[i]>0&&ide[j]==0) |
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297 | { |
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298 | A[i,j]=A[i,j]/var(1); |
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299 | } |
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300 | } |
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301 | } |
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302 | } |
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303 | for(i=mu;i>=1;i--) |
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304 | { |
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305 | if(ide[i]>0) |
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306 | { |
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307 | A[i,i]=A[i,i]+1; |
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308 | e[i]=e[i]+1; |
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309 | ide[i]=ide[i]-1; |
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310 | } |
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311 | } |
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312 | mide--; |
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313 | } |
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314 | |
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315 | if(opt==1) |
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316 | { |
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317 | return(jet(A,0)); |
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318 | } |
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319 | else |
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320 | { |
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321 | return(e); |
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322 | } |
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323 | } |
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324 | example |
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325 | { "EXAMPLE:"; echo=2; |
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326 | ring R=0,(x,y),ds; |
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327 | poly f=x5+x2y2+y5; |
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328 | print(monodromy(f)); |
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329 | } |
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330 | /////////////////////////////////////////////////////////////////////////////// |
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331 | |
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332 | proc vfiltration(poly f,list #) |
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333 | "USAGE: vfiltration(f[,opt]); poly f, int opt |
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334 | ASSUME: basering has characteristic 0 and local ordering, |
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335 | f has isolated singularity at 0 |
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336 | RETURN: |
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337 | @format |
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338 | list V: V-filtration of f on H''/H' |
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339 | if opt==0 or opt==1: |
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340 | ideal V[1]: spectral numbers in increasing order |
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341 | intvec V[2]: |
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342 | int V[2][i]: multiplicity of spectral number V[1][i] |
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343 | if opt==1: |
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344 | list V[3]: |
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345 | module V[3][i]: vector space basis of V[1][i]-th graded part |
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346 | in terms of V[4] |
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347 | ideal V[4]: monomial vector space basis |
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348 | default: opt=1 |
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349 | @end format |
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350 | NOTE: H' and H'' denote the Brieskorn lattices |
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351 | SEE ALSO: spectrum_lib |
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352 | KEYWORDS: singularities; Gauss-Manin connection; |
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353 | Brieskorn lattice; Hodge filtration; V-filtration; spectrum |
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354 | EXAMPLE: example vfiltration; shows an example |
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355 | " |
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356 | { |
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357 | if(charstr(basering)!="0") |
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358 | { |
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359 | ERROR("characteristic 0 expected"); |
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360 | } |
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361 | int n=nvars(basering)-1; |
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362 | for(int i=n+1;i>=1;i--) |
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363 | { |
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364 | if(var(i)>1) |
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365 | { |
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366 | ERROR("local ordering expected"); |
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367 | } |
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368 | } |
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369 | ideal J=jacob(f); |
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370 | ideal sJ=std(J); |
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371 | if(vdim(sJ)<=0) |
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372 | { |
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373 | if(vdim(sJ)==0) |
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374 | { |
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375 | ERROR("singularity at 0 expected"); |
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376 | } |
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377 | else |
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378 | { |
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379 | ERROR("isolated singularity at 0 expected"); |
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380 | } |
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381 | } |
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382 | int opt=1; |
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383 | if(size(#)>0) |
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384 | { |
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385 | if(typeof(#[1])=="int") |
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386 | { |
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387 | opt=#[1]; |
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388 | } |
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389 | } |
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390 | |
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391 | ideal m=kbase(sJ); |
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392 | int mu,modm=ncols(m),maxorddif(m); |
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393 | |
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394 | ideal w=f*m; |
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395 | matrix U=freemodule(mu); |
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396 | matrix A[mu][mu],C,D; |
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397 | list l; |
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398 | module H,dH=freemodule(mu),freemodule(mu); |
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399 | module H0; |
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400 | int sdH=1; |
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401 | int k=-1; |
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402 | int j,K; |
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403 | |
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404 | while(k<K||sdH>0) |
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405 | { |
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406 | k++; |
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407 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: k="+string(k)); |
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408 | |
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409 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute C"); |
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410 | C=coeffs(reduce(w,U,sJ),m); |
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411 | A=A+C*var(1)^k; |
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412 | |
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413 | if(sdH>0) |
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414 | { |
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415 | H0=H; |
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416 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute dH"); |
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417 | dH=jet(module(A*dH+var(1)^2*diff(matrix(dH),var(1))),k); |
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418 | H=H*var(1)+dH; |
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419 | |
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420 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: test dH==0"); |
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421 | sdH=size(reduce(H,std(H0*var(1)))); |
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422 | if(sdH>0) |
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423 | { |
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424 | A=A-var(1); |
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425 | } |
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426 | else |
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427 | { |
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428 | dbprint(printlevel-voice+2, |
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429 | "//gaussman::vfiltration: compute basis of saturation"); |
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430 | H=minbase(H0); |
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431 | int modH=maxorddif(H); |
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432 | K=modH+n+1; |
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433 | H0=freemodule(mu)*var(1)^k; |
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434 | } |
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435 | } |
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436 | |
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437 | if(k<K||sdH>0) |
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438 | { |
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439 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: divide by J"); |
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440 | l=division(J,ideal(matrix(w)-matrix(m)*C*U)); |
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441 | D=l[1]; |
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442 | |
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443 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute w/U"); |
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444 | for(j=mu;j>=1;j--) |
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445 | { |
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446 | if(l[2][j,j]!=0) |
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447 | { |
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448 | dbprint(printlevel-voice+2, |
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449 | "//gaussman::vfiltration: compute U["+string(j)+"]"); |
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450 | U[j,j]=U[j,j]*l[2][j,j]; |
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451 | } |
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452 | dbprint(printlevel-voice+2, |
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453 | "//gaussman::vfiltration: compute w["+string(j)+"]"); |
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454 | w[j]=0; |
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455 | for(i=n+1;i>=1;i--) |
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456 | { |
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457 | w[j]=w[j]+U[j,j]*diff(D[i,j],var(i))-diff(U[j,j],var(i))*D[i,j]; |
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458 | } |
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459 | } |
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460 | U=U*U; |
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461 | } |
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462 | } |
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463 | int N=k-modH; |
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464 | |
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465 | dbprint(printlevel-voice+2, |
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466 | "//gaussman::vfiltration: transform H0 to saturation"); |
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467 | l=division(H,H0); |
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468 | H0=jet(l[1],l[2],N-1); |
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469 | |
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470 | dbprint(printlevel-voice+2, |
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471 | "//gaussman::vfiltration: compute H0 as vector space V0"); |
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472 | dbprint(printlevel-voice+2, |
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473 | "//gaussman::vfiltration: compute H1 as vector space V1"); |
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474 | poly p; |
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475 | int i0,j0,i1,j1; |
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476 | matrix V0[mu*N][mu*N]; |
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477 | matrix V1[mu*N][mu*(N-1)]; |
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478 | for(i0=mu;i0>=1;i0--) |
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479 | { |
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480 | for(i1=mu;i1>=1;i1--) |
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481 | { |
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482 | p=H0[i1,i0]; |
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483 | while(p!=0) |
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484 | { |
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485 | j1=leadexp(p)[1]; |
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486 | for(j0=N-j1-1;j0>=0;j0--) |
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487 | { |
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488 | V0[i1+(j1+j0)*mu,i0+j0*mu]=V0[i1+(j1+j0)*mu,i0+j0*mu]+leadcoef(p); |
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489 | if(j1+j0+1<N) |
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490 | { |
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491 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]= |
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492 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]+leadcoef(p); |
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493 | } |
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494 | } |
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495 | p=p-lead(p); |
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496 | } |
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497 | } |
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498 | } |
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499 | |
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500 | dbprint(printlevel-voice+2, |
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501 | "//gaussman::vfiltration: compute A on saturation"); |
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502 | l=division(H*var(1),A*H+var(1)^2*diff(matrix(H),var(1))); |
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503 | A=jet(l[1],l[2],N-1); |
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504 | |
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505 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute matrix M of A"); |
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506 | matrix M[mu*N][mu*N]; |
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507 | for(i0=mu;i0>=1;i0--) |
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508 | { |
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509 | for(i1=mu;i1>=1;i1--) |
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510 | { |
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511 | p=A[i1,i0]; |
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512 | while(p!=0) |
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513 | { |
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514 | j1=leadexp(p)[1]; |
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515 | for(j0=N-j1-1;j0>=0;j0--) |
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516 | { |
---|
517 | M[i1+(j0+j1)*mu,i0+j0*mu]=leadcoef(p); |
---|
518 | } |
---|
519 | p=p-lead(p); |
---|
520 | } |
---|
521 | } |
---|
522 | } |
---|
523 | for(i0=mu;i0>=1;i0--) |
---|
524 | { |
---|
525 | for(j0=N-1;j0>=0;j0--) |
---|
526 | { |
---|
527 | M[i0+j0*mu,i0+j0*mu]=M[i0+j0*mu,i0+j0*mu]+j0; |
---|
528 | } |
---|
529 | } |
---|
530 | |
---|
531 | dbprint(printlevel-voice+2, |
---|
532 | "//gaussman::vfiltration: compute eigenvalues eA of A"); |
---|
533 | ideal eA=system("eigenval",jet(A,0))[1]; |
---|
534 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: eA="+string(eA)); |
---|
535 | |
---|
536 | dbprint(printlevel-voice+2, |
---|
537 | "//gaussman::vfiltration: compute eigenvalues eM of M"); |
---|
538 | ideal eM; |
---|
539 | k=0; |
---|
540 | intvec u; |
---|
541 | for(i=N;i>=1;i--) |
---|
542 | { |
---|
543 | u[i]=1; |
---|
544 | } |
---|
545 | i0=1; |
---|
546 | while(u[N]<=ncols(eA)) |
---|
547 | { |
---|
548 | for(i,i1=i0+1,i0;i<=N;i++) |
---|
549 | { |
---|
550 | if(eA[u[i]]+i<eA[u[i1]]+i1) |
---|
551 | { |
---|
552 | i1=i; |
---|
553 | } |
---|
554 | } |
---|
555 | k++; |
---|
556 | eM[k]=eA[u[i1]]+i1-1; |
---|
557 | u[i1]=u[i1]+1; |
---|
558 | if(u[i1]>ncols(eA)) |
---|
559 | { |
---|
560 | i0=i1+1; |
---|
561 | } |
---|
562 | } |
---|
563 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: eM="+string(eM)); |
---|
564 | |
---|
565 | dbprint(printlevel-voice+2, |
---|
566 | "//gaussman::vfiltration: compute V-filtration on H0/H1"); |
---|
567 | ideal a; |
---|
568 | k=0; |
---|
569 | list V; |
---|
570 | matrix Me; |
---|
571 | V[ncols(eM)+1]=std(V1); |
---|
572 | intvec d; |
---|
573 | if(opt==0) |
---|
574 | { |
---|
575 | for(i=ncols(eM);number(eM[i])-1>number(n-1)/2;i--) |
---|
576 | { |
---|
577 | Me=freemodule(mu*N); |
---|
578 | for(i0=n;i0>=0;i0--) // Potenzen von Matrizen? |
---|
579 | { |
---|
580 | Me=Me*(M-eM[i]); |
---|
581 | } |
---|
582 | |
---|
583 | dbprint(printlevel-voice+2, |
---|
584 | "//gaussman::vfiltration: compute V["+string(i)+"]"); |
---|
585 | V1=module(V1)+syz(Me); |
---|
586 | V[i]=std(intersect(V1,V0)); |
---|
587 | |
---|
588 | if(size(V[i])>size(V[i+1])) |
---|
589 | { |
---|
590 | k++; |
---|
591 | a[k]=eM[i]-1; |
---|
592 | d[k]=size(V[i])-size(V[i+1]); |
---|
593 | } |
---|
594 | } |
---|
595 | |
---|
596 | dbprint(printlevel-voice+2, |
---|
597 | "//gaussman::vfiltration: symmetry index found"); |
---|
598 | j=k; |
---|
599 | |
---|
600 | if(number(eM[i])-1==number(n-1)/2) |
---|
601 | { |
---|
602 | Me=freemodule(mu*N); |
---|
603 | for(i0=n;i0>=0;i0--) // Potenzen von Matrizen? |
---|
604 | { |
---|
605 | Me=Me*(M-eM[i]); |
---|
606 | } |
---|
607 | |
---|
608 | dbprint(printlevel-voice+2, |
---|
609 | "//gaussman::vfiltration: compute V["+string(i)+"]"); |
---|
610 | V1=module(V1)+syz(Me); |
---|
611 | V[i]=std(intersect(V1,V0)); |
---|
612 | |
---|
613 | if(size(V[i])>size(V[i+1])) |
---|
614 | { |
---|
615 | k++; |
---|
616 | a[k]=eM[i]-1; |
---|
617 | d[k]=size(V[i])-size(V[i+1]); |
---|
618 | } |
---|
619 | } |
---|
620 | |
---|
621 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: apply symmetry"); |
---|
622 | while(j>=1) |
---|
623 | { |
---|
624 | k++; |
---|
625 | a[k]=a[j]; |
---|
626 | a[j]=n-1-a[k]; |
---|
627 | d[k]=d[j]; |
---|
628 | j--; |
---|
629 | } |
---|
630 | |
---|
631 | return(list(a,d)); |
---|
632 | } |
---|
633 | else |
---|
634 | { |
---|
635 | list v; |
---|
636 | j=-1; |
---|
637 | for(i=ncols(eM);i>=1;i--) |
---|
638 | { |
---|
639 | Me=freemodule(mu*N); |
---|
640 | for(i0=n;i0>=0;i0--) // Potenzen von Matrizen? |
---|
641 | { |
---|
642 | Me=Me*(M-eM[i]); |
---|
643 | } |
---|
644 | |
---|
645 | dbprint(printlevel-voice+2, |
---|
646 | "//gaussman::vfiltration: compute V["+string(i)+"]"); |
---|
647 | V1=module(V1)+syz(Me); |
---|
648 | V[i]=std(intersect(V1,V0)); |
---|
649 | |
---|
650 | if(size(V[i])>size(V[i+1])) |
---|
651 | { |
---|
652 | if(number(eM[i]-1)>=number(n-1)/2) |
---|
653 | { |
---|
654 | k++; |
---|
655 | a[k]=eM[i]-1; |
---|
656 | dbprint(printlevel-voice+2, |
---|
657 | "//gaussman::vfiltration: transform to V0"); |
---|
658 | v[k]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
659 | } |
---|
660 | else |
---|
661 | { |
---|
662 | if(j<0) |
---|
663 | { |
---|
664 | if(a[k]==number(n-1)/2) |
---|
665 | { |
---|
666 | j=k-1; |
---|
667 | } |
---|
668 | else |
---|
669 | { |
---|
670 | j=k; |
---|
671 | } |
---|
672 | } |
---|
673 | k++; |
---|
674 | a[k]=a[j]; |
---|
675 | a[j]=eM[i]-1; |
---|
676 | v[k]=v[j]; |
---|
677 | dbprint(printlevel-voice+2, |
---|
678 | "//gaussman::vfiltration: transform to V0"); |
---|
679 | v[j]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
680 | j--; |
---|
681 | } |
---|
682 | } |
---|
683 | } |
---|
684 | |
---|
685 | dbprint(printlevel-voice+2, |
---|
686 | "//gaussman::vfiltration: compute graded parts"); |
---|
687 | option(redSB); |
---|
688 | for(k=1;k<size(v);k++) |
---|
689 | { |
---|
690 | v[k]=std(reduce(v[k],std(v[k+1]))); |
---|
691 | d[k]=size(v[k]); |
---|
692 | } |
---|
693 | v[k]=std(v[k]); |
---|
694 | d[k]=size(v[k]); |
---|
695 | |
---|
696 | return(list(a,d,v,m)); |
---|
697 | } |
---|
698 | } |
---|
699 | example |
---|
700 | { "EXAMPLE:"; echo=2; |
---|
701 | ring R=0,(x,y),ds; |
---|
702 | poly f=x5+x2y2+y5; |
---|
703 | vfiltration(f); |
---|
704 | } |
---|
705 | /////////////////////////////////////////////////////////////////////////////// |
---|
706 | |
---|
707 | proc spectrum(poly f) |
---|
708 | "USAGE: spectrum(f); poly f |
---|
709 | ASSUME: basering has characteristic 0 and local ordering, |
---|
710 | f has isolated singularity at 0 |
---|
711 | RETURN: |
---|
712 | @format |
---|
713 | list S: singularity spectrum of f |
---|
714 | ideal S[1]: spectral numbers in increasing order |
---|
715 | intvec S[2]: |
---|
716 | int S[2][i]: multiplicity of spectral number S[1][i] |
---|
717 | @end format |
---|
718 | SEE ALSO: spectrum_lib |
---|
719 | KEYWORDS: singularities; Gauss-Manin connection; spectrum |
---|
720 | EXAMPLE: example spectrum; shows an example |
---|
721 | " |
---|
722 | { |
---|
723 | return(vfiltration(f,0)); |
---|
724 | } |
---|
725 | example |
---|
726 | { "EXAMPLE:"; echo=2; |
---|
727 | ring R=0,(x,y),ds; |
---|
728 | poly f=x5+x2y2+y5; |
---|
729 | spprint(spectrum(f)); |
---|
730 | } |
---|
731 | /////////////////////////////////////////////////////////////////////////////// |
---|
732 | |
---|
733 | proc endfilt(poly f,list V) |
---|
734 | "USAGE: endfilt(f,V); poly f, list V |
---|
735 | ASSUME: basering has characteristic 0 and local ordering, |
---|
736 | f has isolated singularity at 0 |
---|
737 | RETURN: |
---|
738 | @format |
---|
739 | list V1: endomorphim filtration of V on the Jacobian algebra of f |
---|
740 | ideal V1[1]: spectral numbers in increasing order |
---|
741 | intvec V1[2]: |
---|
742 | int V1[2][i]: multiplicity of spectral number V1[1][i] |
---|
743 | list V1[3]: |
---|
744 | module V1[3][i]: vector space basis of the V1[1][i]-th graded part |
---|
745 | in terms of V1[4] |
---|
746 | ideal V1[4]: monomial vector space basis |
---|
747 | @end format |
---|
748 | SEE ALSO: spectrum_lib |
---|
749 | KEYWORDS: singularities; Gauss-Manin connection; spectrum; |
---|
750 | Brieskorn lattice; Hodge filtration; V-filtration |
---|
751 | EXAMPLE: example endfilt; shows an example |
---|
752 | " |
---|
753 | { |
---|
754 | if(charstr(basering)!="0") |
---|
755 | { |
---|
756 | ERROR("characteristic 0 expected"); |
---|
757 | } |
---|
758 | int n=nvars(basering)-1; |
---|
759 | for(int i=n+1;i>=1;i--) |
---|
760 | { |
---|
761 | if(var(i)>1) |
---|
762 | { |
---|
763 | ERROR("local ordering expected"); |
---|
764 | } |
---|
765 | } |
---|
766 | ideal sJ=std(jacob(f)); |
---|
767 | if(vdim(sJ)<=0) |
---|
768 | { |
---|
769 | if(vdim(sJ)==0) |
---|
770 | { |
---|
771 | ERROR("singularity at 0 expected"); |
---|
772 | } |
---|
773 | else |
---|
774 | { |
---|
775 | ERROR("isolated singularity at 0 expected"); |
---|
776 | } |
---|
777 | } |
---|
778 | |
---|
779 | def a,d,v,m=V[1..4]; |
---|
780 | int mu=ncols(m); |
---|
781 | |
---|
782 | module V0=v[1]; |
---|
783 | for(i=2;i<=size(v);i++) |
---|
784 | { |
---|
785 | V0=V0,v[i]; |
---|
786 | } |
---|
787 | |
---|
788 | dbprint(printlevel-voice+2, |
---|
789 | "//gaussman::endfilt: compute multiplication in Jacobian algebra"); |
---|
790 | list M; |
---|
791 | matrix U=freemodule(ncols(m)); |
---|
792 | for(i=ncols(m);i>=1;i--) |
---|
793 | { |
---|
794 | M[i]=lift(V0,coeffs(reduce(m[i]*m,U,sJ),m)*V0); |
---|
795 | } |
---|
796 | |
---|
797 | int j,k,i0,j0,i1,j1; |
---|
798 | number b0=number(a[1]-a[ncols(a)]); |
---|
799 | number b1,b2; |
---|
800 | matrix M0; |
---|
801 | module L; |
---|
802 | list v0=freemodule(ncols(m)); |
---|
803 | ideal a0=b0; |
---|
804 | |
---|
805 | while(b0<number(a[ncols(a)]-a[1])) |
---|
806 | { |
---|
807 | dbprint(printlevel-voice+2, |
---|
808 | "//gaussman::endfilt: find next possible index"); |
---|
809 | b1=number(a[ncols(a)]-a[1]); |
---|
810 | for(j=ncols(a);j>=1;j--) |
---|
811 | { |
---|
812 | for(i=ncols(a);i>=1;i--) |
---|
813 | { |
---|
814 | b2=number(a[i]-a[j]); |
---|
815 | if(b2>b0&&b2<b1) |
---|
816 | { |
---|
817 | b1=b2; |
---|
818 | } |
---|
819 | else |
---|
820 | { |
---|
821 | if(b2<=b0) |
---|
822 | { |
---|
823 | i=0; |
---|
824 | } |
---|
825 | } |
---|
826 | } |
---|
827 | } |
---|
828 | b0=b1; |
---|
829 | |
---|
830 | list l=ideal(); |
---|
831 | for(k=ncols(m);k>=2;k--) |
---|
832 | { |
---|
833 | l=l+list(ideal()); |
---|
834 | } |
---|
835 | |
---|
836 | dbprint(printlevel-voice+2, |
---|
837 | "//gaussman::endfilt: collect conditions for V1["+string(b0)+"]"); |
---|
838 | j=ncols(a); |
---|
839 | j0=mu; |
---|
840 | while(j>=1) |
---|
841 | { |
---|
842 | i0=1; |
---|
843 | i=1; |
---|
844 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
845 | { |
---|
846 | i0=i0+d[i]; |
---|
847 | i++; |
---|
848 | } |
---|
849 | if(a[i]<a[j]+b0) |
---|
850 | { |
---|
851 | i0=i0+d[i]; |
---|
852 | i++; |
---|
853 | } |
---|
854 | for(k=1;k<=ncols(m);k++) |
---|
855 | { |
---|
856 | M0=M[k]; |
---|
857 | if(i0>1) |
---|
858 | { |
---|
859 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
860 | } |
---|
861 | } |
---|
862 | j0=j0-d[j]; |
---|
863 | j--; |
---|
864 | } |
---|
865 | |
---|
866 | dbprint(printlevel-voice+2, |
---|
867 | "//gaussman::endfilt: compose condition matrix"); |
---|
868 | L=transpose(module(l[1])); |
---|
869 | for(k=2;k<=ncols(m);k++) |
---|
870 | { |
---|
871 | L=L,transpose(module(l[k])); |
---|
872 | } |
---|
873 | |
---|
874 | dbprint(printlevel-voice+2, |
---|
875 | "//gaussman::endfilt: compute kernel of condition matrix"); |
---|
876 | v0=v0+list(syz(L)); |
---|
877 | a0=a0,b0; |
---|
878 | } |
---|
879 | |
---|
880 | dbprint(printlevel-voice+2,"//gaussman::endfilt: compute graded parts"); |
---|
881 | option(redSB); |
---|
882 | for(i=1;i<size(v0);i++) |
---|
883 | { |
---|
884 | v0[i+1]=std(v0[i+1]); |
---|
885 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
886 | } |
---|
887 | |
---|
888 | dbprint(printlevel-voice+2, |
---|
889 | "//gaussman::endfilt: remove trivial graded parts"); |
---|
890 | i=1; |
---|
891 | while(size(v0[i])==0) |
---|
892 | { |
---|
893 | i++; |
---|
894 | } |
---|
895 | list v1=v0[i]; |
---|
896 | intvec d1=size(v0[i]); |
---|
897 | ideal a1=a0[i]; |
---|
898 | i++; |
---|
899 | while(i<=size(v0)) |
---|
900 | { |
---|
901 | if(size(v0[i])>0) |
---|
902 | { |
---|
903 | v1=v1+list(v0[i]); |
---|
904 | d1=d1,size(v0[i]); |
---|
905 | a1=a1,a0[i]; |
---|
906 | } |
---|
907 | i++; |
---|
908 | } |
---|
909 | return(list(a1,d1,v1,m)); |
---|
910 | } |
---|
911 | example |
---|
912 | { "EXAMPLE:"; echo=2; |
---|
913 | ring R=0,(x,y),ds; |
---|
914 | poly f=x5+x2y2+y5; |
---|
915 | endfilt(f,vfiltration(f)); |
---|
916 | } |
---|
917 | /////////////////////////////////////////////////////////////////////////////// |
---|
918 | |
---|
919 | proc spprint(list S) |
---|
920 | "USAGE: spprint(S); list S |
---|
921 | RETURN: string: spectrum S |
---|
922 | EXAMPLE: example spprint; shows an example |
---|
923 | " |
---|
924 | { |
---|
925 | string s; |
---|
926 | for(int i=1;i<size(S[2]);i++) |
---|
927 | { |
---|
928 | s=s+"("+string(S[1][i])+","+string(S[2][i])+"),"; |
---|
929 | } |
---|
930 | s=s+"("+string(S[1][i])+","+string(S[2][i])+")"; |
---|
931 | return(s); |
---|
932 | } |
---|
933 | example |
---|
934 | { "EXAMPLE:"; echo=2; |
---|
935 | ring R=0,(x,y),ds; |
---|
936 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
937 | spprint(S); |
---|
938 | } |
---|
939 | /////////////////////////////////////////////////////////////////////////////// |
---|
940 | |
---|
941 | proc spadd(list S1,list S2) |
---|
942 | "USAGE: spadd(S1,S2); list S1,S2 |
---|
943 | RETURN: list: sum of spectra S1 and S2 |
---|
944 | EXAMPLE: example spadd; shows an example |
---|
945 | " |
---|
946 | { |
---|
947 | ideal s; |
---|
948 | intvec m; |
---|
949 | int i,i1,i2=1,1,1; |
---|
950 | while(i1<=size(S1[2])||i2<=size(S2[2])) |
---|
951 | { |
---|
952 | if(i1<=size(S1[2])) |
---|
953 | { |
---|
954 | if(i2<=size(S2[2])) |
---|
955 | { |
---|
956 | if(number(S1[1][i1])<number(S2[1][i2])) |
---|
957 | { |
---|
958 | s[i]=S1[1][i1]; |
---|
959 | m[i]=S1[2][i1]; |
---|
960 | i++; |
---|
961 | i1++; |
---|
962 | } |
---|
963 | else |
---|
964 | { |
---|
965 | if(number(S1[1][i1])>number(S2[1][i2])) |
---|
966 | { |
---|
967 | s[i]=S2[1][i2]; |
---|
968 | m[i]=S2[2][i2]; |
---|
969 | i++; |
---|
970 | i2++; |
---|
971 | } |
---|
972 | else |
---|
973 | { |
---|
974 | if(S1[2][i1]+S2[2][i2]!=0) |
---|
975 | { |
---|
976 | s[i]=S1[1][i1]; |
---|
977 | m[i]=S1[2][i1]+S2[2][i2]; |
---|
978 | i++; |
---|
979 | } |
---|
980 | i1++; |
---|
981 | i2++; |
---|
982 | } |
---|
983 | } |
---|
984 | } |
---|
985 | else |
---|
986 | { |
---|
987 | s[i]=S1[1][i1]; |
---|
988 | m[i]=S1[2][i1]; |
---|
989 | i++; |
---|
990 | i1++; |
---|
991 | } |
---|
992 | } |
---|
993 | else |
---|
994 | { |
---|
995 | s[i]=S2[1][i2]; |
---|
996 | m[i]=S2[2][i2]; |
---|
997 | i++; |
---|
998 | i2++; |
---|
999 | } |
---|
1000 | } |
---|
1001 | return(list(s,m)); |
---|
1002 | } |
---|
1003 | example |
---|
1004 | { "EXAMPLE:"; echo=2; |
---|
1005 | ring R=0,(x,y),ds; |
---|
1006 | list S1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1007 | spprint(S1); |
---|
1008 | list S2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1009 | spprint(S2); |
---|
1010 | spprint(spadd(S1,S2)); |
---|
1011 | } |
---|
1012 | /////////////////////////////////////////////////////////////////////////////// |
---|
1013 | |
---|
1014 | proc spsub(list S1,list S2) |
---|
1015 | "USAGE: spsub(S1,S2); list S1,S2 |
---|
1016 | RETURN: list: difference of spectra S1 and S2 |
---|
1017 | EXAMPLE: example spsub; shows an example |
---|
1018 | " |
---|
1019 | { |
---|
1020 | return(spadd(S1,spmul(S2,-1))); |
---|
1021 | } |
---|
1022 | example |
---|
1023 | { "EXAMPLE:"; echo=2; |
---|
1024 | ring R=0,(x,y),ds; |
---|
1025 | list S1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1026 | spprint(S1); |
---|
1027 | list S2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1028 | spprint(S2); |
---|
1029 | spprint(spsub(S1,S2)); |
---|
1030 | } |
---|
1031 | /////////////////////////////////////////////////////////////////////////////// |
---|
1032 | |
---|
1033 | proc spmul(list #) |
---|
1034 | "USAGE: |
---|
1035 | @format |
---|
1036 | 1) spmul(S,k); list S, int k |
---|
1037 | 2) spmul(S,k); list S, intvec k |
---|
1038 | @end format |
---|
1039 | RETURN: |
---|
1040 | @format |
---|
1041 | 1) list: product of spectrum S and integer k |
---|
1042 | 2) list: linear combination of spectra S with coefficients k |
---|
1043 | @end format |
---|
1044 | EXAMPLE: example spmul; shows an example |
---|
1045 | " |
---|
1046 | { |
---|
1047 | if(size(#)==2) |
---|
1048 | { |
---|
1049 | if(typeof(#[1])=="list") |
---|
1050 | { |
---|
1051 | if(typeof(#[2])=="int") |
---|
1052 | { |
---|
1053 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1054 | } |
---|
1055 | if(typeof(#[2])=="intvec") |
---|
1056 | { |
---|
1057 | list S0=list(ideal(),intvec(0)); |
---|
1058 | for(int i=size(#[2]);i>=1;i--) |
---|
1059 | { |
---|
1060 | S0=spadd(S0,spmul(#[1][i],#[2][i])); |
---|
1061 | } |
---|
1062 | return(S0); |
---|
1063 | } |
---|
1064 | } |
---|
1065 | } |
---|
1066 | return(list(ideal(),intvec(0))); |
---|
1067 | } |
---|
1068 | example |
---|
1069 | { "EXAMPLE:"; echo=2; |
---|
1070 | ring R=0,(x,y),ds; |
---|
1071 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1072 | spprint(S); |
---|
1073 | spprint(spmul(S,2)); |
---|
1074 | list S1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1075 | spprint(S1); |
---|
1076 | list S2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1077 | spprint(S2); |
---|
1078 | spprint(spmul(list(S1,S2),intvec(1,2))); |
---|
1079 | } |
---|
1080 | /////////////////////////////////////////////////////////////////////////////// |
---|
1081 | |
---|
1082 | proc spissemicont(list S,list #) |
---|
1083 | "USAGE: spissemicont(S[,opt]); list S, int opt |
---|
1084 | RETURN: |
---|
1085 | @format |
---|
1086 | int k= |
---|
1087 | if opt==0: |
---|
1088 | 1, if sum of spectrum S over all intervals [a,a+1) is positive |
---|
1089 | 0, if sum of spectrum S over some interval [a,a+1) is negative |
---|
1090 | if opt==1: |
---|
1091 | 1, if sum of spectrum S over all intervals [a,a+1) and (a,a+1) is positive |
---|
1092 | 0, if sum of spectrum S over some interval [a,a+1) or (a,a+1) is negative |
---|
1093 | default: opt=0 |
---|
1094 | @end format |
---|
1095 | EXAMPLE: example spissemicont; shows an example |
---|
1096 | " |
---|
1097 | { |
---|
1098 | int opt=0; |
---|
1099 | if(size(#)>0) |
---|
1100 | { |
---|
1101 | if(typeof(#[1])=="int") |
---|
1102 | { |
---|
1103 | opt=1; |
---|
1104 | } |
---|
1105 | } |
---|
1106 | int i,j,k=1,1,0; |
---|
1107 | while(j<=size(S[2])) |
---|
1108 | { |
---|
1109 | while(j+1<=size(S[2])&&S[1][j]<S[1][i]+1) |
---|
1110 | { |
---|
1111 | k=k+S[2][j]; |
---|
1112 | j++; |
---|
1113 | } |
---|
1114 | if(j==size(S[2])&&S[1][j]<S[1][i]+1) |
---|
1115 | { |
---|
1116 | k=k+S[2][j]; |
---|
1117 | j++; |
---|
1118 | } |
---|
1119 | if(k<0) |
---|
1120 | { |
---|
1121 | return(0); |
---|
1122 | } |
---|
1123 | k=k-S[2][i]; |
---|
1124 | if(k<0&&opt==1) |
---|
1125 | { |
---|
1126 | return(0); |
---|
1127 | } |
---|
1128 | i++; |
---|
1129 | } |
---|
1130 | return(1); |
---|
1131 | } |
---|
1132 | example |
---|
1133 | { "EXAMPLE:"; echo=2; |
---|
1134 | ring R=0,(x,y),ds; |
---|
1135 | list S1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1136 | spprint(S1); |
---|
1137 | list S2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1138 | spprint(S2); |
---|
1139 | spissemicont(spsub(S1,spmul(S2,5))); |
---|
1140 | spissemicont(spsub(S1,spmul(S2,5)),1); |
---|
1141 | spissemicont(spsub(S1,spmul(S2,6))); |
---|
1142 | } |
---|
1143 | /////////////////////////////////////////////////////////////////////////////// |
---|
1144 | |
---|
1145 | proc spsemicont(list S0,list S,list #) |
---|
1146 | "USAGE: spsemicont(S,k[,opt]); list S0, list S, int opt |
---|
1147 | RETURN: list of intvecs l: |
---|
1148 | spissemicont(sub(S0,spmul(S,k)),opt)==1 iff k<=l[i] for some i |
---|
1149 | NOTE: if the spectra S occur with multiplicities k in a deformation |
---|
1150 | of the [quasihomogeneous] spectrum S0 then |
---|
1151 | spissemicont(sub(S0,spmul(S,k))[,1])==1 |
---|
1152 | EXAMPLE: example spsemicont; shows an example |
---|
1153 | " |
---|
1154 | { |
---|
1155 | list l,l0; |
---|
1156 | int i,k; |
---|
1157 | while(spissemicont(S0,#)) |
---|
1158 | { |
---|
1159 | if(size(S)>1) |
---|
1160 | { |
---|
1161 | l0=spsemicont(S0,list(S[1..size(S)-1])); |
---|
1162 | for(i=size(l0);i>=1;i--) |
---|
1163 | { |
---|
1164 | l0[i][size(S)]=k; |
---|
1165 | } |
---|
1166 | l=l+l0; |
---|
1167 | } |
---|
1168 | S0=spsub(S0,S[size(S)]); |
---|
1169 | k++; |
---|
1170 | } |
---|
1171 | if(size(S)>1) |
---|
1172 | { |
---|
1173 | return(l); |
---|
1174 | } |
---|
1175 | else |
---|
1176 | { |
---|
1177 | return(list(intvec(k-1))); |
---|
1178 | } |
---|
1179 | } |
---|
1180 | example |
---|
1181 | { "EXAMPLE:"; echo=2; |
---|
1182 | ring R=0,(x,y),ds; |
---|
1183 | list S0=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1184 | spprint(S0); |
---|
1185 | list S1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1186 | spprint(S1); |
---|
1187 | list S2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1188 | spprint(S2); |
---|
1189 | list S=S1,S2; |
---|
1190 | list l=spsemicont(S0,S); |
---|
1191 | l; |
---|
1192 | spissemicont(spsub(S0,spmul(S,l[1]))); |
---|
1193 | spissemicont(spsub(S0,spmul(S,l[1]-1))); |
---|
1194 | spissemicont(spsub(S0,spmul(S,l[1]+1))); |
---|
1195 | } |
---|
1196 | /////////////////////////////////////////////////////////////////////////////// |
---|
1197 | |
---|
1198 | proc spmilnor(list S) |
---|
1199 | "USAGE: spmilnor(S); list S |
---|
1200 | RETURN: int: Milnor number of spectrum S |
---|
1201 | EXAMPLE: example spmilnor; shows an example |
---|
1202 | " |
---|
1203 | { |
---|
1204 | return(sum(S[2])); |
---|
1205 | } |
---|
1206 | example |
---|
1207 | { "EXAMPLE:"; echo=2; |
---|
1208 | ring R=0,(x,y),ds; |
---|
1209 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1210 | spprint(S); |
---|
1211 | spmilnor(S); |
---|
1212 | } |
---|
1213 | /////////////////////////////////////////////////////////////////////////////// |
---|
1214 | |
---|
1215 | proc spgeomgenus(list S) |
---|
1216 | "USAGE: spgeomgenus(S); list S |
---|
1217 | RETURN: int: geometrical genus of spectrum S |
---|
1218 | EXAMPLE: example spgeomgenus; shows an example |
---|
1219 | " |
---|
1220 | { |
---|
1221 | int g=0; |
---|
1222 | int i=1; |
---|
1223 | while(i+1<=size(S[2])&&number(S[1][i])<=number(0)) |
---|
1224 | { |
---|
1225 | g=g+S[2][i]; |
---|
1226 | i++; |
---|
1227 | } |
---|
1228 | if(i==size(S[2])&&number(S[1][i])<=number(0)) |
---|
1229 | { |
---|
1230 | g=g+S[2][i]; |
---|
1231 | } |
---|
1232 | return(g); |
---|
1233 | } |
---|
1234 | example |
---|
1235 | { "EXAMPLE:"; echo=2; |
---|
1236 | ring R=0,(x,y),ds; |
---|
1237 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1238 | spprint(S); |
---|
1239 | spgeomgenus(S); |
---|
1240 | } |
---|
1241 | /////////////////////////////////////////////////////////////////////////////// |
---|
1242 | |
---|
1243 | proc spgamma(list S) |
---|
1244 | "USAGE: spgamma(S); list S |
---|
1245 | RETURN: number: gamma invariant of spectrum S |
---|
1246 | EXAMPLE: example spgamma; shows an example |
---|
1247 | " |
---|
1248 | { |
---|
1249 | int i,j; |
---|
1250 | number g=0; |
---|
1251 | for(i=1;i<=ncols(S[1]);i++) |
---|
1252 | { |
---|
1253 | for(j=1;j<=S[2][i];j++) |
---|
1254 | { |
---|
1255 | g=g+(number(S[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1256 | } |
---|
1257 | } |
---|
1258 | g=-g/4+sum(S[2])*number(S[1][ncols(S[1])]-S[1][1])/48; |
---|
1259 | return(g); |
---|
1260 | } |
---|
1261 | example |
---|
1262 | { "EXAMPLE:"; echo=2; |
---|
1263 | ring R=0,(x,y),ds; |
---|
1264 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1265 | spprint(S); |
---|
1266 | spgamma(S); |
---|
1267 | } |
---|
1268 | /////////////////////////////////////////////////////////////////////////////// |
---|