1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: gaussman.lib,v 1.54 2001-08-31 18:18:08 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 | gmsring(t,s); Brieskorn lattice in Gauss-Manin system of t |
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15 | gmsnf(p,K[,Kmax]); Gauss-Manin system normal form |
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16 | gmscoeffs(p,K[,Kmax]); Gauss-Manin system basis representation |
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17 | monodromy(t); Jordan data of monodromy of t |
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18 | spectrum(t); spectrum of t |
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19 | sppnormalize(a,w[,m]); normalize spectral pairs |
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20 | sppairs(t); spectral pairs of t |
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21 | saito(t); matrix A0+A1*s of t on H'' |
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22 | vfilt(t[,opt]); V-filtration on H''/H' or spectrum of t |
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23 | endfilt(t,V); endomorphism filtration of V-filtration V |
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24 | spprint(Sp); print spectrum Sp |
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25 | sppprint(Spp); print spectral pairs Spp |
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26 | spadd(Sp1,Sp2); sum of spectra Sp1 and Sp2 |
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27 | spsub(Sp1,Sp2); difference of spectra Sp1 and Sp2 |
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28 | spmul(Sp,k); product of spectrum Sp and integer k |
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29 | spmul(Sp,k); linear combination of spectra Sp with coeffs k |
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30 | spissemicont(Sp[,opt]); test spectrum Sp for semicontinuity |
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31 | spsemicont(Sp0,Sp[,opt]); semicontinuity of spectra Sp0 and Sp |
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32 | spmilnor(Sp); milnor number of spectrum Sp |
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33 | spgeomgenus(Sp); geometrical genus of spectrum Sp |
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34 | spgamma(Sp); gamma invariant of spectrum Sp |
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35 | |
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36 | SEE ALSO: mondromy_lib, spectrum_lib |
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37 | |
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38 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
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39 | monodromy; spectrum; spectral pairs; |
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40 | Hodge filtration; V-filtration; weight filtration |
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41 | "; |
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42 | |
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43 | LIB "linalg.lib"; |
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44 | |
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45 | /////////////////////////////////////////////////////////////////////////////// |
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46 | |
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47 | static proc stdtrans(ideal I) |
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48 | { |
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49 | def R=basering; |
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50 | |
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51 | string os=ordstr(R); |
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52 | int j=find(os,",C"); |
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53 | if(j==0) |
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54 | { |
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55 | j=find(os,"C,"); |
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56 | } |
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57 | if(j==0) |
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58 | { |
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59 | j=find(os,",c"); |
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60 | } |
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61 | if(j==0) |
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62 | { |
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63 | j=find(os,"c,"); |
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64 | } |
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65 | if(j>0) |
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66 | { |
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67 | os[j..j+1]=" "; |
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68 | } |
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69 | |
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70 | execute("ring S="+charstr(R)+",(gmspoly,"+varstr(R)+"),(c,dp,"+os+");"); |
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71 | |
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72 | ideal I=homog(imap(R,I),gmspoly); |
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73 | module M=transpose(transpose(I)+freemodule(ncols(I))); |
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74 | M=std(M); |
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75 | |
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76 | setring(R); |
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77 | execute("map h=S,1,"+varstr(R)+";"); |
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78 | module M=h(M); |
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79 | |
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80 | for(int i=ncols(M);i>=1;i--) |
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81 | { |
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82 | for(j=ncols(M);j>=1;j--) |
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83 | { |
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84 | if(M[i][1]==0) |
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85 | { |
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86 | M[i]=0; |
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87 | } |
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88 | if(i!=j&&M[j][1]!=0) |
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89 | { |
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90 | if(lead(M[i][1])/lead(M[j][1])!=0) |
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91 | { |
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92 | M[i]=0; |
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93 | } |
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94 | } |
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95 | } |
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96 | } |
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97 | |
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98 | M=transpose(simplify(M,2)); |
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99 | I=M[1]; |
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100 | attrib(I,"isSB",1); |
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101 | M=M[2..ncols(M)]; |
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102 | module U=transpose(M); |
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103 | |
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104 | return(list(I,U)); |
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105 | } |
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106 | /////////////////////////////////////////////////////////////////////////////// |
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107 | |
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108 | proc gmsring(poly t,string s) |
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109 | "USAGE: gmsring(t,s); poly t, string s; |
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110 | ASSUME: basering with characteristic 0 and local degree ordering, |
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111 | t with isolated citical point 0 |
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112 | RETURN: |
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113 | @format |
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114 | ring G: C{{s}}*basering, |
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115 | poly gmspoly: image of t |
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116 | ideal gmsjacob: image of Jacobian ideal |
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117 | ideal gmsstd: image of standard basis of Jacobian ideal |
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118 | matrix gmsmatrix: matrix(gmsjacob)*gmsmatrix=matrix(gmsstd) |
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119 | ideal gmsbasis: image of monomial vector space basis of Jacobian algebra |
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120 | int gmsmaxweight: maximal weight of variables of basering |
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121 | @end format |
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122 | NOTE: do not modify gms variables if you want to use gms procedures |
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123 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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124 | EXAMPLE: example gms; shows examples |
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125 | " |
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126 | { |
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127 | def R=basering; |
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128 | if(charstr(R)!="0") |
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129 | { |
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130 | ERROR("characteristic 0 expected"); |
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131 | } |
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132 | for(int i=nvars(R);i>=1;i--) |
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133 | { |
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134 | if(var(i)>1) |
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135 | { |
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136 | ERROR("local ordering expected"); |
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137 | } |
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138 | } |
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139 | |
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140 | ideal dt=jacob(t); |
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141 | list l=stdtrans(dt); |
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142 | ideal g=l[1]; |
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143 | if(vdim(g)<=0) |
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144 | { |
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145 | if(vdim(g)==0) |
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146 | { |
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147 | ERROR("singularity at 0 expected"); |
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148 | } |
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149 | else |
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150 | { |
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151 | ERROR("isolated citical point 0 expected"); |
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152 | } |
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153 | } |
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154 | matrix a=l[2]; |
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155 | ideal m=kbase(g); |
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156 | |
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157 | int gmsmaxweight; |
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158 | for(i=nvars(R);i>=1;i--) |
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159 | { |
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160 | if(deg(var(i))>gmsmaxweight) |
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161 | { |
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162 | gmsmaxweight=deg(var(i)); |
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163 | } |
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164 | } |
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165 | |
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166 | string os=ordstr(R); |
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167 | int j=find(os,",C"); |
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168 | if(j==0) |
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169 | { |
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170 | j=find(os,"C,"); |
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171 | } |
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172 | if(j==0) |
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173 | { |
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174 | j=find(os,",c"); |
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175 | } |
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176 | if(j==0) |
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177 | { |
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178 | j=find(os,"c,"); |
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179 | } |
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180 | if(j>0) |
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181 | { |
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182 | os[j..j+1]=" "; |
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183 | } |
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184 | |
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185 | execute("ring G="+string(charstr(R))+",("+s+","+varstr(R)+"),(ws("+ |
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186 | string(deg(highcorner(g))+2*gmsmaxweight)+"),"+os+",c);"); |
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187 | |
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188 | poly gmspoly=imap(R,t); |
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189 | ideal gmsjacob=imap(R,dt); |
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190 | ideal gmsstd=imap(R,g); |
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191 | matrix gmsmatrix=imap(R,a); |
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192 | ideal gmsbasis=imap(R,m); |
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193 | |
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194 | attrib(gmsstd,"isSB",1); |
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195 | export gmspoly,gmsjacob,gmsstd,gmsmatrix,gmsbasis,gmsmaxweight; |
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196 | |
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197 | return(G); |
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198 | } |
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199 | example |
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200 | { "EXAMPLE:"; echo=2; |
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201 | ring R=0,(x,y),ds; |
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202 | poly t=x5+x2y2+y5; |
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203 | def G=gmsring(t,"s"); |
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204 | setring(G); |
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205 | gmspoly; |
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206 | print(gmsjacob); |
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207 | print(gmsstd); |
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208 | print(gmsmatrix); |
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209 | print(gmsbasis); |
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210 | gmsmaxweight; |
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211 | } |
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212 | /////////////////////////////////////////////////////////////////////////////// |
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213 | |
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214 | proc gmsnf(ideal p,int K,list #) |
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215 | "USAGE: gmsnf(p,K[,Kmax]); poly p, int K[, int Kmax]; |
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216 | ASSUME: basering constructed by gmsring, K<=Kmax |
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217 | RETURN: |
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218 | @format |
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219 | list l: |
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220 | ideal l[1]: projection of p to H''=C{{s}}*gmsbasis mod s^{K+1} |
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221 | ideal l[2]: p=l[1]+l[2] mod s^(Kmax+1) |
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222 | @end format |
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223 | NOTE: by setting p=l[2] the computation can be continued up to order |
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224 | at most Kmax, by default Kmax=infinity |
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225 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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226 | EXAMPLE: example gmsnf; shows examples |
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227 | " |
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228 | { |
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229 | int Kmax=-1; |
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230 | if(size(#)>0) |
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231 | { |
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232 | if(typeof(#[1])=="int") |
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233 | { |
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234 | Kmax=#[1]; |
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235 | if(K>Kmax) |
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236 | { |
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237 | Kmax=K; |
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238 | } |
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239 | } |
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240 | } |
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241 | |
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242 | intvec v=1; |
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243 | v[nvars(basering)]=0; |
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244 | |
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245 | int k; |
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246 | if(Kmax>=0) |
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247 | { |
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248 | p=jet(jet(p,K,v),(Kmax+1)*deg(var(1))-2*gmsmaxweight); |
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249 | } |
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250 | |
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251 | ideal r,q; |
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252 | r[ncols(p)]=0; |
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253 | q[ncols(p)]=0; |
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254 | |
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255 | poly s; |
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256 | int i,j; |
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257 | for(k=ncols(p);k>=1;k--) |
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258 | { |
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259 | while(p[k]!=0&°(lead(p[k]),v)<=K) |
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260 | { |
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261 | i=1; |
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262 | s=lead(p[k])/lead(gmsstd[i]); |
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263 | while(i<ncols(gmsstd)&&s==0) |
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264 | { |
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265 | i++; |
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266 | s=lead(p[k])/lead(gmsstd[i]); |
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267 | } |
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268 | if(s!=0) |
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269 | { |
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270 | p[k]=p[k]-s*gmsstd[i]; |
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271 | for(j=1;j<=nrows(gmsmatrix);j++) |
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272 | { |
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273 | if(Kmax>=0) |
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274 | { |
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275 | p[k]=p[k]+ |
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276 | jet(jet(diff(s*gmsmatrix[j,i],var(j+1))*var(1),Kmax,v), |
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277 | (Kmax+1)*deg(var(1))-2*gmsmaxweight); |
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278 | } |
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279 | else |
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280 | { |
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281 | p[k]=p[k]+diff(s*gmsmatrix[j,i],var(j+1))*var(1); |
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282 | } |
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283 | } |
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284 | } |
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285 | else |
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286 | { |
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287 | r[k]=r[k]+lead(p[k]); |
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288 | p[k]=p[k]-lead(p[k]); |
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289 | } |
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290 | while(deg(lead(p[k]))>(K+1)*deg(var(1))-2*gmsmaxweight&& |
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291 | deg(lead(p[k]),v)<=K) |
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292 | { |
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293 | q[k]=q[k]+lead(p[k]); |
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294 | p[k]=p[k]-lead(p[k]); |
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295 | } |
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296 | } |
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297 | q[k]=q[k]+p[k]; |
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298 | } |
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299 | |
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300 | return(list(r,q)); |
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301 | } |
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302 | example |
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303 | { "EXAMPLE:"; echo=2; |
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304 | ring R=0,(x,y),ds; |
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305 | poly t=x5+x2y2+y5; |
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306 | def G=gmsring(t,"s"); |
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307 | setring(G); |
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308 | list l0=gmsnf(gmspoly,0); |
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309 | print(l0[1]); |
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310 | list l1=gmsnf(gmspoly,1); |
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311 | print(l1[1]); |
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312 | list l=gmsnf(l0[2],1); |
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313 | print(l[1]); |
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314 | } |
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315 | /////////////////////////////////////////////////////////////////////////////// |
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316 | |
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317 | proc gmscoeffs(ideal p,int K,list #) |
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318 | "USAGE: gmscoeffs(p,K[,Kmax]); poly p, int K[, int Kmax]; |
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319 | ASSUME: basering constructed by gmsring, K<=Kmax |
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320 | RETURN: |
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321 | @format |
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322 | list l: |
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323 | matrix l[1]: projection of p to H''=C{{s}}*gmsbasis=C{{s}}^mu mod s^(K+1) |
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324 | ideal l[2]: p=matrix(gmsbasis)*l[1]+l[2] mod s^(Kmax+1) |
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325 | @end format |
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326 | NOTE: by setting p=l[2] the computation can be continued up to order |
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327 | at most Kmax, by default Kmax=infinity |
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328 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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329 | EXAMPLE: example gmscoeffs; shows examples |
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330 | " |
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331 | { |
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332 | list l=gmsnf(p,K,#); |
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333 | ideal r,q=l[1..2]; |
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334 | poly v=1; |
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335 | for(int i=2;i<=nvars(basering);i++) |
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336 | { |
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337 | v=v*var(i); |
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338 | } |
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339 | matrix C=coeffs(r,gmsbasis,v); |
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340 | return(C,q); |
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341 | } |
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342 | example |
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343 | { "EXAMPLE:"; echo=2; |
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344 | ring R=0,(x,y),ds; |
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345 | poly t=x5+x2y2+y5; |
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346 | def G=gmsring(t,"s"); |
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347 | setring(G); |
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348 | list l0=gmscoeffs(gmspoly,0); |
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349 | print(l0[1]); |
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350 | list l1=gmscoeffs(gmspoly,1); |
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351 | print(l1[1]); |
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352 | list l=gmscoeffs(l0[2],1); |
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353 | print(l[1]); |
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354 | } |
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355 | /////////////////////////////////////////////////////////////////////////////// |
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356 | |
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357 | static proc min(ideal e) |
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358 | { |
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359 | int i; |
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360 | number m=number(e[1]); |
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361 | for(i=2;i<=ncols(e);i++) |
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362 | { |
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363 | if(number(e[i])<m) |
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364 | { |
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365 | m=number(e[i]); |
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366 | } |
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367 | } |
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368 | return(m); |
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369 | } |
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370 | /////////////////////////////////////////////////////////////////////////////// |
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371 | |
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372 | static proc max(ideal e) |
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373 | { |
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374 | int i; |
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375 | number m=number(e[1]); |
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376 | for(i=2;i<=ncols(e);i++) |
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377 | { |
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378 | if(number(e[i])>m) |
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379 | { |
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380 | m=number(e[i]); |
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381 | } |
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382 | } |
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383 | return(m); |
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384 | } |
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385 | /////////////////////////////////////////////////////////////////////////////// |
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386 | |
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387 | static proc saturate(int K0) |
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388 | { |
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389 | int mu=ncols(gmsbasis); |
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390 | ideal r=gmspoly*gmsbasis; |
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391 | matrix A0[mu][mu],C; |
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392 | module H0; |
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393 | module H,H1=freemodule(mu),freemodule(mu); |
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394 | int k=-1; |
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395 | list l; |
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396 | |
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397 | while(size(reduce(H,std(H0*s)))>0) |
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398 | { |
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399 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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400 | k++; |
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401 | dbprint(printlevel-voice+2,"// k="+string(k)); |
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402 | l=gmscoeffs(r,k,mu+K0); |
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403 | C,r=l[1..2]; |
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404 | A0=A0+C; |
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405 | |
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406 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
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407 | H0=H; |
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408 | H1=jet(module(A0*H1+s^2*diff(matrix(H1),s)),k); |
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409 | H=H*s+H1; |
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410 | } |
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411 | |
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412 | A0=A0-k*s; |
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413 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
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414 | H=std(H0); |
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415 | int d0=maxdeg1(H); |
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416 | |
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417 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
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418 | l=division(H,freemodule(mu)*s^k); |
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419 | H0=l[1]; |
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420 | |
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421 | return(A0,r,H,H0); |
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422 | } |
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423 | /////////////////////////////////////////////////////////////////////////////// |
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424 | |
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425 | static proc tmatrix(matrix A0,ideal r,module H,int K,int K0) |
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426 | { |
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427 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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428 | int d0=maxdeg1(H); |
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429 | dbprint(printlevel-voice+2,"// k="+string(K+d0+1)); |
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430 | list l=gmscoeffs(r,K+d0+1,K0+d0+1); |
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431 | matrix C; |
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432 | C,r=l[1..2]; |
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433 | A0=A0+C; |
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434 | |
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435 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
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436 | l=division(H*s,A0*H+s^2*diff(matrix(H),s)); |
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437 | matrix A=jet(l[1],l[2],K); |
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438 | |
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439 | return(A,A0,r); |
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440 | } |
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441 | /////////////////////////////////////////////////////////////////////////////// |
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442 | |
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443 | static proc eigenvals(matrix A0,ideal r,module H,int K0) |
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444 | { |
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445 | dbprint(printlevel-voice+2, |
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446 | "// compute eigenvalues e with multiplicities m of A"); |
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447 | matrix A; |
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448 | A,A0,r=tmatrix(A0,r,H,0,K0); |
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449 | list l=eigenvalues(A); |
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450 | def e,m=l[1..2]; |
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451 | dbprint(printlevel-voice+2,"// e="+string(e)); |
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452 | dbprint(printlevel-voice+2,"// m="+string(m)); |
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453 | |
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454 | return(e,m,A0,r); |
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455 | } |
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456 | /////////////////////////////////////////////////////////////////////////////// |
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457 | |
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458 | static proc transform(ideal e,intvec m,matrix A,matrix A0,ideal r,module H,module H0,int K,int K0) |
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459 | { |
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460 | dbprint(printlevel-voice+2,"// compute minimum e0 and maximum e1 of e"); |
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461 | number e0,e1=min(e),max(e); |
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462 | dbprint(printlevel-voice+2,"// e0="+string(e0)); |
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463 | dbprint(printlevel-voice+2,"// e1="+string(e1)); |
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464 | int d1=int(e1-e0); |
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465 | A,A0,r=tmatrix(A0,r,H,K+d1,K0+d1); |
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466 | |
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467 | if(e1>=e0+1) |
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468 | { |
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469 | int i,j,i0,j0,i1,j1; |
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470 | module U,V; |
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471 | list l; |
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472 | |
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473 | while(e1>=e0+1) |
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474 | { |
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475 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
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476 | U=0; |
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477 | for(i=1;i<=ncols(e);i++) |
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478 | { |
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479 | U=U+syz(power(jet(A,0)-e[i],m[i])); |
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480 | } |
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481 | V=inverse(U); |
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482 | A=V*A*U; |
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483 | H0=V*H0; |
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484 | |
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485 | dbprint(printlevel-voice+2,"// transform to reduce e1 by 1"); |
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486 | for(i0,i=1,1;i0<=ncols(e);i0++) |
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487 | { |
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488 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
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489 | { |
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490 | for(j0,j=1,1;j0<=ncols(e);j0++) |
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491 | { |
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492 | for(j1=1;j1<=m[j0];j1,j=j1+1,j+1) |
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493 | { |
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494 | if(number(e[i0])<e0+1&&number(e[j0])>=e0+1) |
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495 | { |
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496 | A[i,j]=A[i,j]/s; |
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497 | } |
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498 | if(number(e[i0])>=e0+1&&number(e[j0])<e0+1) |
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499 | { |
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500 | A[i,j]=A[i,j]*s; |
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501 | } |
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502 | } |
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503 | } |
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504 | } |
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505 | } |
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506 | |
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507 | H0=transpose(H0); |
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508 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
509 | { |
---|
510 | if(number(e[i0])>=e0+1) |
---|
511 | { |
---|
512 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
513 | { |
---|
514 | A[i,i]=A[i,i]-1; |
---|
515 | H0[i]=H0[i]*s; |
---|
516 | } |
---|
517 | e[i0]=e[i0]-1; |
---|
518 | } |
---|
519 | else |
---|
520 | { |
---|
521 | i=i+m[i0]; |
---|
522 | } |
---|
523 | } |
---|
524 | H0=transpose(H0); |
---|
525 | |
---|
526 | l=spnormalize(e,m); |
---|
527 | e,m=l[1..2]; |
---|
528 | |
---|
529 | e1=e1-1; |
---|
530 | dbprint(printlevel-voice+2,"// e1="+string(e1)); |
---|
531 | } |
---|
532 | |
---|
533 | A=jet(A,K); |
---|
534 | } |
---|
535 | |
---|
536 | return(e,m,A,A0,r,H0); |
---|
537 | } |
---|
538 | /////////////////////////////////////////////////////////////////////////////// |
---|
539 | |
---|
540 | proc monodromy(poly t,list #) |
---|
541 | "USAGE: monodromy(t); poly t |
---|
542 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
543 | t with isolated citical point 0 |
---|
544 | RETURN: list l: Jordan data jordan(M) of a monodromy matrix exp(-2*pi*i*M) |
---|
545 | SEE ALSO: mondromy_lib |
---|
546 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; monodromy |
---|
547 | EXAMPLE: example monodromy; shows examples |
---|
548 | " |
---|
549 | { |
---|
550 | def R=basering; |
---|
551 | int n=nvars(R)-1; |
---|
552 | def G=gmsring(t,"s"); |
---|
553 | setring(G); |
---|
554 | |
---|
555 | int mu=ncols(gmsbasis); |
---|
556 | matrix A; |
---|
557 | ideal e; |
---|
558 | intvec m; |
---|
559 | |
---|
560 | def A0,r,H,H0=saturate(n); |
---|
561 | e,m,A0,r=eigenvals(A0,r,H,n); |
---|
562 | e,m,A,A0,r,H0=transform(e,m,A,A0,r,H,H0,0,0); |
---|
563 | |
---|
564 | setring(R); |
---|
565 | matrix A=imap(G,A); |
---|
566 | ideal e=imap(G,e); |
---|
567 | |
---|
568 | return(jordan(A)); |
---|
569 | } |
---|
570 | example |
---|
571 | { "EXAMPLE:"; echo=2; |
---|
572 | ring R=0,(x,y),ds; |
---|
573 | poly f=x5+x2y2+y5; |
---|
574 | print(monodromy(f)); |
---|
575 | } |
---|
576 | /////////////////////////////////////////////////////////////////////////////// |
---|
577 | |
---|
578 | proc spectrum(poly t) |
---|
579 | "USAGE: spectrum(t); poly t |
---|
580 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
581 | t with isolated citical point 0 |
---|
582 | RETURN: |
---|
583 | @format |
---|
584 | list Sp: spectrum of t |
---|
585 | ideal Sp[1]: spectral numbers in increasing order |
---|
586 | intvec Sp[2]: |
---|
587 | int Sp[2][i]: multiplicity of spectral number Sp[1][i] |
---|
588 | @end format |
---|
589 | SEE ALSO: spectrum_lib |
---|
590 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; spectrum |
---|
591 | EXAMPLE: example spnumbers; shows examples |
---|
592 | " |
---|
593 | { |
---|
594 | list l=sppairs(t); |
---|
595 | return(spnormalize(l[1],l[3])); |
---|
596 | } |
---|
597 | example |
---|
598 | { "EXAMPLE:"; echo=2; |
---|
599 | ring R=0,(x,y),ds; |
---|
600 | poly t=x5+x2y2+y5; |
---|
601 | spprint(spectrum(t)); |
---|
602 | } |
---|
603 | /////////////////////////////////////////////////////////////////////////////// |
---|
604 | |
---|
605 | static proc sppappend(list l,number a,int w,int m) |
---|
606 | { |
---|
607 | if(size(l)==0) |
---|
608 | { |
---|
609 | l=list(ideal(a),intvec(w),intvec(m)); |
---|
610 | } |
---|
611 | else |
---|
612 | { |
---|
613 | int n=ncols(l[1]); |
---|
614 | l[1][n+1]=a; |
---|
615 | l[2][n+1]=w; |
---|
616 | l[3][n+1]=m; |
---|
617 | } |
---|
618 | return(l); |
---|
619 | } |
---|
620 | /////////////////////////////////////////////////////////////////////////////// |
---|
621 | |
---|
622 | proc sppnormalize(ideal a,intvec w,list #) |
---|
623 | "USAGE: sppnormalize(a,w[,m]); |
---|
624 | RETURN: |
---|
625 | @format |
---|
626 | list Spp: normalized spectral pairs (a,w,m) |
---|
627 | ideal Spp[1]: numbers in a in increasing order |
---|
628 | intvec Spp[2]: integers in w in decreasing order |
---|
629 | intvec Spp[3]: |
---|
630 | int Spp[3][i]: multiplicity of pair (Spp[1][i],Spp[2][i]) in a,w |
---|
631 | @end format |
---|
632 | EXAMPLE: example sppnormalize; shows examples |
---|
633 | " |
---|
634 | { |
---|
635 | intvec m; |
---|
636 | int i,j; |
---|
637 | if(size(#)==0) |
---|
638 | { |
---|
639 | for(i=ncols(a);i>=1;i--) |
---|
640 | { |
---|
641 | m[i]=1; |
---|
642 | } |
---|
643 | } |
---|
644 | else |
---|
645 | { |
---|
646 | m=#[1]; |
---|
647 | } |
---|
648 | |
---|
649 | list l; |
---|
650 | number a0; |
---|
651 | int w0,m0; |
---|
652 | for(i=ncols(a);i>=1;i--) |
---|
653 | { |
---|
654 | if(m[i]!=0) |
---|
655 | { |
---|
656 | for(j=i-1;j>=1;j--) |
---|
657 | { |
---|
658 | if(m[j]!=0) |
---|
659 | { |
---|
660 | if(number(a[i])>number(a[j])|| |
---|
661 | (number(a[i])==number(a[j])&&w[i]<w[j])) |
---|
662 | { |
---|
663 | a0=number(a[i]); |
---|
664 | a[i]=a[j]; |
---|
665 | a[j]=a0; |
---|
666 | w0=w[i]; |
---|
667 | w[i]=w[j]; |
---|
668 | w[j]=w0; |
---|
669 | m0=m[i]; |
---|
670 | m[i]=m[j]; |
---|
671 | m[j]=m0; |
---|
672 | } |
---|
673 | if(number(a[i])==number(a[j])&&w[i]==w[j]) |
---|
674 | { |
---|
675 | m[i]=m[i]+m[j]; |
---|
676 | m[j]=0; |
---|
677 | } |
---|
678 | } |
---|
679 | } |
---|
680 | l=sppappend(l,number(a[i]),w[i],m[i]); |
---|
681 | } |
---|
682 | } |
---|
683 | |
---|
684 | return(l); |
---|
685 | } |
---|
686 | example |
---|
687 | { "EXAMPLE:"; echo=2; |
---|
688 | } |
---|
689 | /////////////////////////////////////////////////////////////////////////////// |
---|
690 | |
---|
691 | proc sppairs(poly t) |
---|
692 | "USAGE: sppairs(t); poly t |
---|
693 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
694 | t with isolated citical point 0 |
---|
695 | RETURN: |
---|
696 | @format |
---|
697 | list Spp: |
---|
698 | ideal Spp[1]: spectral numbers in increasing order |
---|
699 | intvec Spp[2]: weight filtration indices in decreasing order |
---|
700 | intvec Spp[3]: |
---|
701 | int Spp[3][i]: multiplicity of spectral pair (Spp[1][i],Spp[2][i]) |
---|
702 | @end format |
---|
703 | SEE ALSO: spectrum_lib |
---|
704 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
705 | spectrum; spectral pairs |
---|
706 | EXAMPLE: example sppairs; shows examples |
---|
707 | " |
---|
708 | { |
---|
709 | def R=basering; |
---|
710 | int n=nvars(R)-1; |
---|
711 | def G=gmsring(t,"s"); |
---|
712 | setring(G); |
---|
713 | |
---|
714 | int mu=ncols(gmsbasis); |
---|
715 | matrix A; |
---|
716 | ideal e; |
---|
717 | intvec m; |
---|
718 | |
---|
719 | def A0,r,H,H0=saturate(n); |
---|
720 | e,m,A0,r=eigenvals(A0,r,H,n); |
---|
721 | e,m,A,A0,r,H0=transform(e,m,A,A0,r,H,H0,0,0); |
---|
722 | |
---|
723 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
724 | list l=jordanbasis(A,e,m); |
---|
725 | def U,v=l[1..2]; |
---|
726 | vector u0; |
---|
727 | int v0; |
---|
728 | int i,j,k,k0; |
---|
729 | for(k,k0=1,1;k0<=ncols(e);k,k0=k+m[k0],k0+1) |
---|
730 | { |
---|
731 | for(i=k+m[k0]-1;i>=k+1;i--) |
---|
732 | { |
---|
733 | for(j=i-1;j>=k;j--) |
---|
734 | { |
---|
735 | if(v[i]>v[j]) |
---|
736 | { |
---|
737 | v0=v[i];v[i]=v[j];v[j]=v0; |
---|
738 | u0=U[i];U[i]=U[j];U[j]=u0; |
---|
739 | } |
---|
740 | } |
---|
741 | } |
---|
742 | } |
---|
743 | |
---|
744 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
745 | matrix V=inverse(U); |
---|
746 | A=V*A*U; |
---|
747 | dbprint(printlevel-voice+2,"// compute normal form of H''"); |
---|
748 | H0=std(V*H0); |
---|
749 | |
---|
750 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
751 | ideal a; |
---|
752 | intvec w; |
---|
753 | for(i=1;i<=mu;i++) |
---|
754 | { |
---|
755 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
756 | a[i]=A[j,j]+deg(lead(H0[i]))/deg(s)-1; |
---|
757 | w[i]=v[j]+n; |
---|
758 | } |
---|
759 | |
---|
760 | setring(R); |
---|
761 | |
---|
762 | return(sppnormalize(imap(G,a),w)); |
---|
763 | } |
---|
764 | example |
---|
765 | { "EXAMPLE:"; echo=2; |
---|
766 | ring R=0,(x,y),ds; |
---|
767 | poly t=x5+x2y2+y5; |
---|
768 | sppprint(sppairs(t)); |
---|
769 | } |
---|
770 | /////////////////////////////////////////////////////////////////////////////// |
---|
771 | |
---|
772 | static proc grmat(matrix A,int k) |
---|
773 | { |
---|
774 | int i,j; |
---|
775 | for(i=1;i<=ncols(A);i++) |
---|
776 | { |
---|
777 | for(j=1;j<=nrows(A);j++) |
---|
778 | { |
---|
779 | A[i,j]=jet(A[i,j]/var(1)^k,0); |
---|
780 | } |
---|
781 | } |
---|
782 | return(A); |
---|
783 | } |
---|
784 | /////////////////////////////////////////////////////////////////////////////// |
---|
785 | |
---|
786 | proc saito(poly t,list #) |
---|
787 | "USAGE: saito(t); poly t |
---|
788 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
789 | t with isolated citical point 0 |
---|
790 | RETURN: list A: matrix A[1]+A[2]*s of t on H'' |
---|
791 | SEE ALSO: spectrum_lib |
---|
792 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
793 | monodromy; spectrum; spectral pairs |
---|
794 | EXAMPLE: example saito; shows examples |
---|
795 | " |
---|
796 | { |
---|
797 | def R=basering; |
---|
798 | int n=nvars(R)-1; |
---|
799 | def G=gmsring(t,"s"); |
---|
800 | setring(G); |
---|
801 | |
---|
802 | int mu=ncols(gmsbasis); |
---|
803 | matrix A; |
---|
804 | ideal e; |
---|
805 | intvec m; |
---|
806 | |
---|
807 | def A0,r,H,H0=saturate(2*n+mu+1); |
---|
808 | e,m,A0,r=eigenvals(A0,r,H,n+1); |
---|
809 | int d0,d1=maxdeg1(H),int(max(e)-min(e)); |
---|
810 | e,m,A,A0,r,H0=transform(e,m,A,A0,r,H,H0,d0+d1+1,d0+d1+1); |
---|
811 | d0=maxdeg1(H0); |
---|
812 | |
---|
813 | dbprint(printlevel-voice+2,"// transform to Jordan basis"); |
---|
814 | module U=jordanbasis(A,e,m)[1]; |
---|
815 | module V=inverse(U); |
---|
816 | A=V*A*U; |
---|
817 | H=V*H0; |
---|
818 | |
---|
819 | dbprint(printlevel-voice+2,"// compute splitting of V filtration"); |
---|
820 | int i,j,k; |
---|
821 | U=freemodule(mu); |
---|
822 | matrix U0[mu][mu]; |
---|
823 | matrix u0[mu^2][1]; |
---|
824 | A0=commutator(jet(A,0)); |
---|
825 | for(k=1;k<=d0+1;k++) |
---|
826 | { |
---|
827 | V=0; |
---|
828 | for(j=0;j<k;j++) |
---|
829 | { |
---|
830 | U0=matrix(U0)-grmat(A,k-j)*grmat(U,j); |
---|
831 | } |
---|
832 | u0=U0[1..mu,1..mu]; |
---|
833 | u0=inverse(A0+k)*u0; |
---|
834 | U0=u0[1..mu^2,1]; |
---|
835 | U=matrix(U)+s^k*U0; |
---|
836 | } |
---|
837 | |
---|
838 | dbprint(printlevel-voice+2,"// transform to splitting basis"); |
---|
839 | A=jet(A,0); |
---|
840 | H=std(jet(division(U,H)[1],d0)); |
---|
841 | |
---|
842 | dbprint(printlevel-voice+2,"// compute N"); |
---|
843 | matrix N=A; |
---|
844 | for(i=1;i<=ncols(N);i++) |
---|
845 | { |
---|
846 | N[i,i]=0; |
---|
847 | } |
---|
848 | |
---|
849 | dbprint(printlevel-voice+2,"// compute N splitting of Hodge filtration"); |
---|
850 | module F; |
---|
851 | matrix K[mu][1]; |
---|
852 | U=0; |
---|
853 | i=1; |
---|
854 | for(k=1;k<=ncols(e);k++) |
---|
855 | { |
---|
856 | j=0; |
---|
857 | K=matrix(0,mu,0); |
---|
858 | while(j<m[k]) |
---|
859 | { |
---|
860 | K=K+gen(i); |
---|
861 | i++; |
---|
862 | j++; |
---|
863 | } |
---|
864 | j=0; |
---|
865 | F=grmat(H,0); |
---|
866 | V=intersect(F,K); |
---|
867 | while(size(V)==0) |
---|
868 | { |
---|
869 | j++; |
---|
870 | V=V+grmat(H,j); |
---|
871 | V=intersect(F,K) |
---|
872 | } |
---|
873 | while(size(V)>0) |
---|
874 | { |
---|
875 | U=U+V; |
---|
876 | V=N*V; |
---|
877 | } |
---|
878 | } |
---|
879 | |
---|
880 | dbprint(printlevel-voice+2,"// transform to N splitting basis"); |
---|
881 | V=inverse(U); |
---|
882 | A=V*A*U; |
---|
883 | H=V*H*U; |
---|
884 | |
---|
885 | dbprint(printlevel-voice+2,"// compute matrix A0+sA1 of t"); |
---|
886 | // degBound=d0; |
---|
887 | // option("redSB"); |
---|
888 | // H=std(H); |
---|
889 | H=simplify(lead(std(H)),1); |
---|
890 | // degBound=0; |
---|
891 | list l=division(H,s*A*H+s^2*diff(matrix(H),s)); |
---|
892 | A=jet(l[1],l[2],d0+1); |
---|
893 | A0=grmat(A,0); |
---|
894 | A=grmat(A,1); |
---|
895 | |
---|
896 | setring(R); |
---|
897 | |
---|
898 | return(list(imap(G,A0),imap(G,A))); |
---|
899 | } |
---|
900 | example |
---|
901 | { "EXAMPLE:"; echo=2; |
---|
902 | ring R=0,(x,y),ds; |
---|
903 | poly t=x5+x2y2+y5; |
---|
904 | list A=saito(t); |
---|
905 | print(A[1]); |
---|
906 | print(A[2]); |
---|
907 | } |
---|
908 | /////////////////////////////////////////////////////////////////////////////// |
---|
909 | |
---|
910 | proc vfilt(poly t,list #) |
---|
911 | "USAGE: vfilt(t[,opt]); poly t, int opt |
---|
912 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
913 | t with isolated citical point 0 |
---|
914 | RETURN: |
---|
915 | @format |
---|
916 | list V: V-filtration of t on H''/H' |
---|
917 | intvec V[1]: spectral numbers in increasing order |
---|
918 | intvec V[2]: |
---|
919 | int V[2][i]: multiplicity of spectral number V[1][i]/V[2][i] |
---|
920 | if opt>=1: |
---|
921 | list V[4]: |
---|
922 | module V[3][i]: vector space basis of V[1][i]/V[2][i]-th graded part |
---|
923 | in terms of V[5] |
---|
924 | ideal V[4]: monomial vector space basis of H''/H' |
---|
925 | ideal V[5]: standard basis of Jacobian ideal |
---|
926 | default: opt=1 |
---|
927 | @end format |
---|
928 | NOTE: H' and H'' denote the Brieskorn lattices |
---|
929 | SEE ALSO: spectrum_lib |
---|
930 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
931 | Hodge filtration; V-filtration; spectrum |
---|
932 | EXAMPLE: example vfilt; shows examples |
---|
933 | " |
---|
934 | { |
---|
935 | int opt=1; |
---|
936 | if(size(#)>0) |
---|
937 | { |
---|
938 | if(typeof(#[1])=="int") |
---|
939 | { |
---|
940 | opt=#[1]; |
---|
941 | } |
---|
942 | } |
---|
943 | |
---|
944 | def R=basering; |
---|
945 | int n=nvars(R)-1; |
---|
946 | def G=gmsring(t,"s"); |
---|
947 | setring G; |
---|
948 | |
---|
949 | int mu=ncols(gmsbasis); |
---|
950 | ideal r=gmspoly*gmsbasis; |
---|
951 | list l; |
---|
952 | matrix A[mu][mu],C; |
---|
953 | module H,H1=freemodule(mu),freemodule(mu); |
---|
954 | module H0; |
---|
955 | int k=-1; |
---|
956 | int N=n+1; |
---|
957 | |
---|
958 | while(size(reduce(H,std(H0*s)))>0) |
---|
959 | { |
---|
960 | k++; |
---|
961 | dbprint(printlevel-voice+2,"// k="+string(k)); |
---|
962 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
963 | l=gmscoeffs(r,k); |
---|
964 | C,r=l[1..2]; |
---|
965 | A=A+C; |
---|
966 | |
---|
967 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
---|
968 | H0=H; |
---|
969 | H1=jet(module(A*H1+s^2*diff(matrix(H1),s)),k); |
---|
970 | H=H*s+H1; |
---|
971 | } |
---|
972 | A=A-k*s; |
---|
973 | |
---|
974 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
---|
975 | H=std(H0); |
---|
976 | int d0=maxdeg1(H); |
---|
977 | dbprint(printlevel-voice+2,"// k="+string(d0+N)); |
---|
978 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
979 | l=gmscoeffs(r,d0+N,d0+N); |
---|
980 | C,r=l[1..2]; |
---|
981 | A=A+C; |
---|
982 | |
---|
983 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
---|
984 | l=division(H,freemodule(mu)*s^k); |
---|
985 | H0=jet(l[1],l[2],N-1); |
---|
986 | |
---|
987 | dbprint(printlevel-voice+2,"// compute vector spaces"); |
---|
988 | poly p; |
---|
989 | int i0,j0,i1,j1; |
---|
990 | matrix V0[mu*N][mu*N]; |
---|
991 | matrix V1[mu*N][mu*(N-1)]; |
---|
992 | for(i0=mu;i0>=1;i0--) |
---|
993 | { |
---|
994 | for(i1=mu;i1>=1;i1--) |
---|
995 | { |
---|
996 | p=H0[i1,i0]; |
---|
997 | while(p!=0) |
---|
998 | { |
---|
999 | j1=leadexp(p)[1]; |
---|
1000 | for(j0=N-j1-1;j0>=0;j0--) |
---|
1001 | { |
---|
1002 | V0[i1+(j1+j0)*mu,i0+j0*mu]=V0[i1+(j1+j0)*mu,i0+j0*mu]+leadcoef(p); |
---|
1003 | if(j1+j0+1<N) |
---|
1004 | { |
---|
1005 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]= |
---|
1006 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]+leadcoef(p); |
---|
1007 | } |
---|
1008 | } |
---|
1009 | p=p-lead(p); |
---|
1010 | } |
---|
1011 | } |
---|
1012 | } |
---|
1013 | |
---|
1014 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
---|
1015 | l=division(H*s,A*H+s^2*diff(matrix(H),s)); |
---|
1016 | A=jet(l[1],l[2],N-1); |
---|
1017 | |
---|
1018 | dbprint(printlevel-voice+2,"// compute matrix M of A"); |
---|
1019 | matrix M[mu*N][mu*N]; |
---|
1020 | for(i0=mu;i0>=1;i0--) |
---|
1021 | { |
---|
1022 | for(i1=mu;i1>=1;i1--) |
---|
1023 | { |
---|
1024 | p=A[i1,i0]; |
---|
1025 | while(p!=0) |
---|
1026 | { |
---|
1027 | j1=leadexp(p)[1]; |
---|
1028 | for(j0=N-j1-1;j0>=0;j0--) |
---|
1029 | { |
---|
1030 | M[i1+(j0+j1)*mu,i0+j0*mu]=leadcoef(p); |
---|
1031 | } |
---|
1032 | p=p-lead(p); |
---|
1033 | } |
---|
1034 | } |
---|
1035 | } |
---|
1036 | for(i0=mu;i0>=1;i0--) |
---|
1037 | { |
---|
1038 | for(j0=N-1;j0>=0;j0--) |
---|
1039 | { |
---|
1040 | M[i0+j0*mu,i0+j0*mu]=M[i0+j0*mu,i0+j0*mu]+j0; |
---|
1041 | } |
---|
1042 | } |
---|
1043 | |
---|
1044 | dbprint(printlevel-voice+2,"// compute eigenvalues eA of A"); |
---|
1045 | ideal eA=eigenvalues(jet(A,0))[1]; |
---|
1046 | dbprint(printlevel-voice+2,"// eA="+string(eA)); |
---|
1047 | |
---|
1048 | dbprint(printlevel-voice+2,"// compute eigenvalues eM of M"); |
---|
1049 | ideal eM; |
---|
1050 | k=0; |
---|
1051 | intvec u; |
---|
1052 | for(int i=N;i>=1;i--) |
---|
1053 | { |
---|
1054 | u[i]=1; |
---|
1055 | } |
---|
1056 | i0=1; |
---|
1057 | while(u[N]<=ncols(eA)) |
---|
1058 | { |
---|
1059 | for(i,i1=i0+1,i0;i<=N;i++) |
---|
1060 | { |
---|
1061 | if(eA[u[i]]+i<eA[u[i1]]+i1) |
---|
1062 | { |
---|
1063 | i1=i; |
---|
1064 | } |
---|
1065 | } |
---|
1066 | k++; |
---|
1067 | eM[k]=eA[u[i1]]+i1-1; |
---|
1068 | u[i1]=u[i1]+1; |
---|
1069 | if(u[i1]>ncols(eA)) |
---|
1070 | { |
---|
1071 | i0=i1+1; |
---|
1072 | } |
---|
1073 | } |
---|
1074 | dbprint(printlevel-voice+2,"// eM="+string(eM)); |
---|
1075 | |
---|
1076 | dbprint(printlevel-voice+2,"// compute V-filtration on H''/sH''"); |
---|
1077 | ideal a; |
---|
1078 | k=0; |
---|
1079 | list V; |
---|
1080 | V[ncols(eM)+1]=interred(V1); |
---|
1081 | intvec d; |
---|
1082 | if(opt<=0) |
---|
1083 | { |
---|
1084 | for(i=ncols(eM);number(eM[i])-1>number(n-1)/2;i--) |
---|
1085 | { |
---|
1086 | dbprint(printlevel-voice+2,"// compute V["+string(i)+"]"); |
---|
1087 | V1=module(V1)+syz(power(M-eM[i],n+1)); |
---|
1088 | V[i]=interred(intersect(V1,V0)); |
---|
1089 | |
---|
1090 | if(size(V[i])>size(V[i+1])) |
---|
1091 | { |
---|
1092 | k++; |
---|
1093 | a[k]=eM[i]-1; |
---|
1094 | d[k]=size(V[i])-size(V[i+1]); |
---|
1095 | } |
---|
1096 | } |
---|
1097 | |
---|
1098 | dbprint(printlevel-voice+2,"// symmetry index found"); |
---|
1099 | int j=k; |
---|
1100 | |
---|
1101 | if(number(eM[i])-1==number(n-1)/2) |
---|
1102 | { |
---|
1103 | dbprint(printlevel-voice+2,"// compute V["+string(i)+"]"); |
---|
1104 | V1=module(V1)+syz(power(M-eM[i],n+1)); |
---|
1105 | V[i]=interred(intersect(V1,V0)); |
---|
1106 | |
---|
1107 | if(size(V[i])>size(V[i+1])) |
---|
1108 | { |
---|
1109 | k++; |
---|
1110 | a[k]=eM[i]-1; |
---|
1111 | d[k]=size(V[i])-size(V[i+1]); |
---|
1112 | } |
---|
1113 | } |
---|
1114 | |
---|
1115 | dbprint(printlevel-voice+2,"// apply symmetry"); |
---|
1116 | while(j>=1) |
---|
1117 | { |
---|
1118 | k++; |
---|
1119 | a[k]=a[j]; |
---|
1120 | a[j]=n-1-a[k]; |
---|
1121 | d[k]=d[j]; |
---|
1122 | j--; |
---|
1123 | } |
---|
1124 | |
---|
1125 | setring(R); |
---|
1126 | ideal a=imap(G,a); |
---|
1127 | return(list(a,d)); |
---|
1128 | } |
---|
1129 | else |
---|
1130 | { |
---|
1131 | list v; |
---|
1132 | int j=-1; |
---|
1133 | for(i=ncols(eM);i>=1;i--) |
---|
1134 | { |
---|
1135 | dbprint(printlevel-voice+2,"// compute V["+string(i)+"]"); |
---|
1136 | V1=module(V1)+syz(power(M-eM[i],n+1)); |
---|
1137 | V[i]=interred(intersect(V1,V0)); |
---|
1138 | |
---|
1139 | if(size(V[i])>size(V[i+1])) |
---|
1140 | { |
---|
1141 | if(number(eM[i]-1)>=number(n-1)/2) |
---|
1142 | { |
---|
1143 | k++; |
---|
1144 | a[k]=eM[i]-1; |
---|
1145 | v[k]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
1146 | } |
---|
1147 | else |
---|
1148 | { |
---|
1149 | if(j<0) |
---|
1150 | { |
---|
1151 | if(a[k]==number(n-1)/2) |
---|
1152 | { |
---|
1153 | j=k-1; |
---|
1154 | } |
---|
1155 | else |
---|
1156 | { |
---|
1157 | j=k; |
---|
1158 | } |
---|
1159 | } |
---|
1160 | k++; |
---|
1161 | a[k]=a[j]; |
---|
1162 | a[j]=eM[i]-1; |
---|
1163 | v[k]=v[j]; |
---|
1164 | v[j]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
1165 | j--; |
---|
1166 | } |
---|
1167 | } |
---|
1168 | } |
---|
1169 | |
---|
1170 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1171 | for(k=1;k<size(v);k++) |
---|
1172 | { |
---|
1173 | v[k]=interred(reduce(v[k],std(v[k+1]))); |
---|
1174 | d[k]=size(v[k]); |
---|
1175 | } |
---|
1176 | v[k]=interred(v[k]); |
---|
1177 | d[k]=size(v[k]); |
---|
1178 | |
---|
1179 | setring(R); |
---|
1180 | ideal a=imap(G,a); |
---|
1181 | list v=imap(G,v); |
---|
1182 | ideal m=imap(G,gmsbasis); |
---|
1183 | ideal g=imap(G,gmsstd); |
---|
1184 | attrib(g,"isSB",1); |
---|
1185 | return(list(a,d,v,m,g)); |
---|
1186 | } |
---|
1187 | } |
---|
1188 | example |
---|
1189 | { "EXAMPLE:"; echo=2; |
---|
1190 | ring R=0,(x,y),ds; |
---|
1191 | poly t=x5+x2y2+y5; |
---|
1192 | vfilt(t); |
---|
1193 | } |
---|
1194 | /////////////////////////////////////////////////////////////////////////////// |
---|
1195 | |
---|
1196 | proc endfilt(list V) |
---|
1197 | "USAGE: endfilt(V); list V |
---|
1198 | ASSUME: V computed by vfilt |
---|
1199 | RETURN: |
---|
1200 | @format |
---|
1201 | list V1: endomorphim filtration of V on the Jacobian algebra |
---|
1202 | ideal V1[1]: spectral numbers in increasing order |
---|
1203 | intvec V1[2]: |
---|
1204 | int V1[2][i]: multiplicity of spectral number V1[1][i] |
---|
1205 | list V1[3]: |
---|
1206 | module V1[3][i]: vector space basis of the V1[1][i]-th graded part |
---|
1207 | in terms of V1[4] |
---|
1208 | ideal V1[4]: monomial vector space basis |
---|
1209 | @end format |
---|
1210 | SEE ALSO: spectrum_lib |
---|
1211 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; spectrum; |
---|
1212 | Hodge filtration; V-filtration |
---|
1213 | EXAMPLE: example endfilt; shows examples |
---|
1214 | " |
---|
1215 | { |
---|
1216 | def a,d,v,m,g=V[1..5]; |
---|
1217 | int mu=ncols(m); |
---|
1218 | |
---|
1219 | module V0=v[1]; |
---|
1220 | for(int i=2;i<=size(v);i++) |
---|
1221 | { |
---|
1222 | V0=V0,v[i]; |
---|
1223 | } |
---|
1224 | |
---|
1225 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
1226 | list M; |
---|
1227 | module U=freemodule(ncols(m)); |
---|
1228 | for(i=ncols(m);i>=1;i--) |
---|
1229 | { |
---|
1230 | M[i]=division(V0,coeffs(reduce(m[i]*m,U,g),m)*V0)[1]; |
---|
1231 | } |
---|
1232 | |
---|
1233 | int j,k,i0,j0,i1,j1; |
---|
1234 | number b0=number(a[1]-a[ncols(a)]); |
---|
1235 | number b1,b2; |
---|
1236 | matrix M0; |
---|
1237 | module L; |
---|
1238 | list v0=freemodule(ncols(m)); |
---|
1239 | ideal a0=b0; |
---|
1240 | |
---|
1241 | while(b0<number(a[ncols(a)]-a[1])) |
---|
1242 | { |
---|
1243 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
1244 | b1=number(a[ncols(a)]-a[1]); |
---|
1245 | for(j=ncols(a);j>=1;j--) |
---|
1246 | { |
---|
1247 | for(i=ncols(a);i>=1;i--) |
---|
1248 | { |
---|
1249 | b2=number(a[i]-a[j]); |
---|
1250 | if(b2>b0&&b2<b1) |
---|
1251 | { |
---|
1252 | b1=b2; |
---|
1253 | } |
---|
1254 | else |
---|
1255 | { |
---|
1256 | if(b2<=b0) |
---|
1257 | { |
---|
1258 | i=0; |
---|
1259 | } |
---|
1260 | } |
---|
1261 | } |
---|
1262 | } |
---|
1263 | b0=b1; |
---|
1264 | |
---|
1265 | list l=ideal(); |
---|
1266 | for(k=ncols(m);k>=2;k--) |
---|
1267 | { |
---|
1268 | l=l+list(ideal()); |
---|
1269 | } |
---|
1270 | |
---|
1271 | dbprint(printlevel-voice+2,"// collect conditions for V1["+string(b0)+"]"); |
---|
1272 | j=ncols(a); |
---|
1273 | j0=mu; |
---|
1274 | while(j>=1) |
---|
1275 | { |
---|
1276 | i0=1; |
---|
1277 | i=1; |
---|
1278 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
1279 | { |
---|
1280 | i0=i0+d[i]; |
---|
1281 | i++; |
---|
1282 | } |
---|
1283 | if(a[i]<a[j]+b0) |
---|
1284 | { |
---|
1285 | i0=i0+d[i]; |
---|
1286 | i++; |
---|
1287 | } |
---|
1288 | for(k=1;k<=ncols(m);k++) |
---|
1289 | { |
---|
1290 | M0=M[k]; |
---|
1291 | if(i0>1) |
---|
1292 | { |
---|
1293 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
1294 | } |
---|
1295 | } |
---|
1296 | j0=j0-d[j]; |
---|
1297 | j--; |
---|
1298 | } |
---|
1299 | |
---|
1300 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
1301 | L=transpose(module(l[1])); |
---|
1302 | for(k=2;k<=ncols(m);k++) |
---|
1303 | { |
---|
1304 | L=L,transpose(module(l[k])); |
---|
1305 | } |
---|
1306 | |
---|
1307 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
1308 | v0=v0+list(syz(L)); |
---|
1309 | a0=a0,b0; |
---|
1310 | } |
---|
1311 | |
---|
1312 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1313 | option(redSB); |
---|
1314 | for(i=1;i<size(v0);i++) |
---|
1315 | { |
---|
1316 | v0[i+1]=std(v0[i+1]); |
---|
1317 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
1318 | } |
---|
1319 | |
---|
1320 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
1321 | i=1; |
---|
1322 | while(size(v0[i])==0) |
---|
1323 | { |
---|
1324 | i++; |
---|
1325 | } |
---|
1326 | list v1=v0[i]; |
---|
1327 | intvec d1=size(v0[i]); |
---|
1328 | ideal a1=a0[i]; |
---|
1329 | i++; |
---|
1330 | while(i<=size(v0)) |
---|
1331 | { |
---|
1332 | if(size(v0[i])>0) |
---|
1333 | { |
---|
1334 | v1=v1+list(v0[i]); |
---|
1335 | d1=d1,size(v0[i]); |
---|
1336 | a1=a1,a0[i]; |
---|
1337 | } |
---|
1338 | i++; |
---|
1339 | } |
---|
1340 | return(list(a1,d1,v1,m)); |
---|
1341 | } |
---|
1342 | example |
---|
1343 | { "EXAMPLE:"; echo=2; |
---|
1344 | ring R=0,(x,y),ds; |
---|
1345 | poly t=x5+x2y2+y5; |
---|
1346 | endfilt(vfilt(t)); |
---|
1347 | } |
---|
1348 | /////////////////////////////////////////////////////////////////////////////// |
---|
1349 | |
---|
1350 | proc spprint(list Sp) |
---|
1351 | "USAGE: spprint(Sp); list Sp |
---|
1352 | RETURN: string: spectrum Sp |
---|
1353 | EXAMPLE: example spprint; shows examples |
---|
1354 | " |
---|
1355 | { |
---|
1356 | string s; |
---|
1357 | for(int i=1;i<size(Sp[2]);i++) |
---|
1358 | { |
---|
1359 | s=s+"("+string(Sp[1][i])+","+string(Sp[2][i])+"),"; |
---|
1360 | } |
---|
1361 | s=s+"("+string(Sp[1][i])+","+string(Sp[2][i])+")"; |
---|
1362 | return(s); |
---|
1363 | } |
---|
1364 | example |
---|
1365 | { "EXAMPLE:"; echo=2; |
---|
1366 | ring R=0,(x,y),ds; |
---|
1367 | list Sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1368 | spprint(Sp); |
---|
1369 | } |
---|
1370 | /////////////////////////////////////////////////////////////////////////////// |
---|
1371 | |
---|
1372 | proc sppprint(list Spp) |
---|
1373 | "USAGE: sppprint(Sp); list Spp |
---|
1374 | RETURN: string: spectral pairs Spp |
---|
1375 | EXAMPLE: example sppprint; shows examples |
---|
1376 | " |
---|
1377 | { |
---|
1378 | string s; |
---|
1379 | for(int i=1;i<size(Spp[3]);i++) |
---|
1380 | { |
---|
1381 | s=s+"(("+string(Spp[1][i])+","+string(Spp[2][i])+"),"+string(Spp[3][i])+"),"; |
---|
1382 | } |
---|
1383 | s=s+"(("+string(Spp[1][i])+","+string(Spp[2][i])+"),"+string(Spp[3][i])+")"; |
---|
1384 | return(s); |
---|
1385 | } |
---|
1386 | example |
---|
1387 | { "EXAMPLE:"; echo=2; |
---|
1388 | ring R=0,(x,y),ds; |
---|
1389 | list Spp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(2,1,1,1,1,1,0),intvec(1,2,2,1,2,2,1)); |
---|
1390 | sppprint(Spp); |
---|
1391 | } |
---|
1392 | /////////////////////////////////////////////////////////////////////////////// |
---|
1393 | |
---|
1394 | proc spadd(list Sp1,list Sp2) |
---|
1395 | "USAGE: spadd(Sp1,Sp2); list Sp1,Sp2 |
---|
1396 | RETURN: list: sum of spectra Sp1 and Sp2 |
---|
1397 | EXAMPLE: example spadd; shows examples |
---|
1398 | " |
---|
1399 | { |
---|
1400 | ideal s; |
---|
1401 | intvec m; |
---|
1402 | int i,i1,i2=1,1,1; |
---|
1403 | while(i1<=size(Sp1[2])||i2<=size(Sp2[2])) |
---|
1404 | { |
---|
1405 | if(i1<=size(Sp1[2])) |
---|
1406 | { |
---|
1407 | if(i2<=size(Sp2[2])) |
---|
1408 | { |
---|
1409 | if(number(Sp1[1][i1])<number(Sp2[1][i2])) |
---|
1410 | { |
---|
1411 | s[i]=Sp1[1][i1]; |
---|
1412 | m[i]=Sp1[2][i1]; |
---|
1413 | i++; |
---|
1414 | i1++; |
---|
1415 | } |
---|
1416 | else |
---|
1417 | { |
---|
1418 | if(number(Sp1[1][i1])>number(Sp2[1][i2])) |
---|
1419 | { |
---|
1420 | s[i]=Sp2[1][i2]; |
---|
1421 | m[i]=Sp2[2][i2]; |
---|
1422 | i++; |
---|
1423 | i2++; |
---|
1424 | } |
---|
1425 | else |
---|
1426 | { |
---|
1427 | if(Sp1[2][i1]+Sp2[2][i2]!=0) |
---|
1428 | { |
---|
1429 | s[i]=Sp1[1][i1]; |
---|
1430 | m[i]=Sp1[2][i1]+Sp2[2][i2]; |
---|
1431 | i++; |
---|
1432 | } |
---|
1433 | i1++; |
---|
1434 | i2++; |
---|
1435 | } |
---|
1436 | } |
---|
1437 | } |
---|
1438 | else |
---|
1439 | { |
---|
1440 | s[i]=Sp1[1][i1]; |
---|
1441 | m[i]=Sp1[2][i1]; |
---|
1442 | i++; |
---|
1443 | i1++; |
---|
1444 | } |
---|
1445 | } |
---|
1446 | else |
---|
1447 | { |
---|
1448 | s[i]=Sp2[1][i2]; |
---|
1449 | m[i]=Sp2[2][i2]; |
---|
1450 | i++; |
---|
1451 | i2++; |
---|
1452 | } |
---|
1453 | } |
---|
1454 | return(list(s,m)); |
---|
1455 | } |
---|
1456 | example |
---|
1457 | { "EXAMPLE:"; echo=2; |
---|
1458 | ring R=0,(x,y),ds; |
---|
1459 | list Sp1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1460 | spprint(Sp1); |
---|
1461 | list Sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1462 | spprint(Sp2); |
---|
1463 | spprint(spadd(Sp1,Sp2)); |
---|
1464 | } |
---|
1465 | /////////////////////////////////////////////////////////////////////////////// |
---|
1466 | |
---|
1467 | proc spsub(list Sp1,list Sp2) |
---|
1468 | "USAGE: spsub(Sp1,Sp2); list Sp1,Sp2 |
---|
1469 | RETURN: list: difference of spectra Sp1 and Sp2 |
---|
1470 | EXAMPLE: example spsub; shows examples |
---|
1471 | " |
---|
1472 | { |
---|
1473 | return(spadd(Sp1,spmul(Sp2,-1))); |
---|
1474 | } |
---|
1475 | example |
---|
1476 | { "EXAMPLE:"; echo=2; |
---|
1477 | ring R=0,(x,y),ds; |
---|
1478 | list Sp1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1479 | spprint(Sp1); |
---|
1480 | list Sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1481 | spprint(Sp2); |
---|
1482 | spprint(spsub(Sp1,Sp2)); |
---|
1483 | } |
---|
1484 | /////////////////////////////////////////////////////////////////////////////// |
---|
1485 | |
---|
1486 | proc spmul(list #) |
---|
1487 | "USAGE: |
---|
1488 | @format |
---|
1489 | 1) spmul(Sp,k); list Sp, int k |
---|
1490 | 2) spmul(Sp,k); list Sp, intvec k |
---|
1491 | @end format |
---|
1492 | RETURN: |
---|
1493 | @format |
---|
1494 | 1) list: product of spectrum Sp and integer k |
---|
1495 | 2) list: linear combination of spectra Sp with coefficients k |
---|
1496 | @end format |
---|
1497 | EXAMPLE: example spmul; shows examples |
---|
1498 | " |
---|
1499 | { |
---|
1500 | if(size(#)==2) |
---|
1501 | { |
---|
1502 | if(typeof(#[1])=="list") |
---|
1503 | { |
---|
1504 | if(typeof(#[2])=="int") |
---|
1505 | { |
---|
1506 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1507 | } |
---|
1508 | if(typeof(#[2])=="intvec") |
---|
1509 | { |
---|
1510 | list Sp0=list(ideal(),intvec(0)); |
---|
1511 | for(int i=size(#[2]);i>=1;i--) |
---|
1512 | { |
---|
1513 | Sp0=spadd(Sp0,spmul(#[1][i],#[2][i])); |
---|
1514 | } |
---|
1515 | return(Sp0); |
---|
1516 | } |
---|
1517 | } |
---|
1518 | } |
---|
1519 | return(list(ideal(),intvec(0))); |
---|
1520 | } |
---|
1521 | example |
---|
1522 | { "EXAMPLE:"; echo=2; |
---|
1523 | ring R=0,(x,y),ds; |
---|
1524 | list Sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1525 | spprint(Sp); |
---|
1526 | spprint(spmul(Sp,2)); |
---|
1527 | list Sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1528 | spprint(Sp1); |
---|
1529 | list Sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1530 | spprint(Sp2); |
---|
1531 | spprint(spmul(list(Sp1,Sp2),intvec(1,2))); |
---|
1532 | } |
---|
1533 | /////////////////////////////////////////////////////////////////////////////// |
---|
1534 | |
---|
1535 | proc spissemicont(list Sp,list #) |
---|
1536 | "USAGE: spissemicont(Sp[,opt]); list Sp, int opt |
---|
1537 | RETURN: |
---|
1538 | @format |
---|
1539 | int k= |
---|
1540 | if opt=0: |
---|
1541 | 1, if sum of spectrum Sp over all intervals [a,a+1) is positive |
---|
1542 | 0, if sum of spectrum Sp over some interval [a,a+1) is negative |
---|
1543 | if opt=1: |
---|
1544 | 1, if sum of spectrum Sp over all intervals [a,a+1) and (a,a+1) is positive |
---|
1545 | 0, if sum of spectrum Sp over some interval [a,a+1) or (a,a+1) is negative |
---|
1546 | default: opt=0 |
---|
1547 | @end format |
---|
1548 | EXAMPLE: example spissemicont; shows examples |
---|
1549 | " |
---|
1550 | { |
---|
1551 | int opt=0; |
---|
1552 | if(size(#)>0) |
---|
1553 | { |
---|
1554 | if(typeof(#[1])=="int") |
---|
1555 | { |
---|
1556 | opt=1; |
---|
1557 | } |
---|
1558 | } |
---|
1559 | int i,j,k=1,1,0; |
---|
1560 | while(j<=size(Sp[2])) |
---|
1561 | { |
---|
1562 | while(j+1<=size(Sp[2])&&Sp[1][j]<Sp[1][i]+1) |
---|
1563 | { |
---|
1564 | k=k+Sp[2][j]; |
---|
1565 | j++; |
---|
1566 | } |
---|
1567 | if(j==size(Sp[2])&&Sp[1][j]<Sp[1][i]+1) |
---|
1568 | { |
---|
1569 | k=k+Sp[2][j]; |
---|
1570 | j++; |
---|
1571 | } |
---|
1572 | if(k<0) |
---|
1573 | { |
---|
1574 | return(0); |
---|
1575 | } |
---|
1576 | k=k-Sp[2][i]; |
---|
1577 | if(k<0&&opt==1) |
---|
1578 | { |
---|
1579 | return(0); |
---|
1580 | } |
---|
1581 | i++; |
---|
1582 | } |
---|
1583 | return(1); |
---|
1584 | } |
---|
1585 | example |
---|
1586 | { "EXAMPLE:"; echo=2; |
---|
1587 | ring R=0,(x,y),ds; |
---|
1588 | list Sp1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1589 | spprint(Sp1); |
---|
1590 | list Sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1591 | spprint(Sp2); |
---|
1592 | spissemicont(spsub(Sp1,spmul(Sp2,5))); |
---|
1593 | spissemicont(spsub(Sp1,spmul(Sp2,5)),1); |
---|
1594 | spissemicont(spsub(Sp1,spmul(Sp2,6))); |
---|
1595 | } |
---|
1596 | /////////////////////////////////////////////////////////////////////////////// |
---|
1597 | |
---|
1598 | proc spsemicont(list Sp0,list Sp,list #) |
---|
1599 | "USAGE: spsemicont(Sp,k[,opt]); list Sp0, list Sp, int opt |
---|
1600 | RETURN: list of intvecs l: |
---|
1601 | spissemicont(sub(Sp0,spmul(Sp,k)),opt)==1 iff k<=l[i] for some i |
---|
1602 | NOTE: if the spectra Sp occur with multiplicities k in a deformation |
---|
1603 | of the [quasihomogeneous] spectrum Sp0 then |
---|
1604 | spissemicont(sub(Sp0,spmul(Sp,k))[,1])==1 |
---|
1605 | EXAMPLE: example spsemicont; shows examples |
---|
1606 | " |
---|
1607 | { |
---|
1608 | list l,l0; |
---|
1609 | int i,j,k; |
---|
1610 | while(spissemicont(Sp0,#)) |
---|
1611 | { |
---|
1612 | if(size(Sp)>1) |
---|
1613 | { |
---|
1614 | l0=spsemicont(Sp0,list(Sp[1..size(Sp)-1])); |
---|
1615 | for(i=1;i<=size(l0);i++) |
---|
1616 | { |
---|
1617 | if(size(l)>0) |
---|
1618 | { |
---|
1619 | j=1; |
---|
1620 | while(j<size(l)&&l[j]!=l0[i]) |
---|
1621 | { |
---|
1622 | j++; |
---|
1623 | } |
---|
1624 | if(l[j]==l0[i]) |
---|
1625 | { |
---|
1626 | l[j][size(Sp)]=k; |
---|
1627 | } |
---|
1628 | else |
---|
1629 | { |
---|
1630 | l0[i][size(Sp)]=k; |
---|
1631 | l=l+list(l0[i]); |
---|
1632 | } |
---|
1633 | } |
---|
1634 | else |
---|
1635 | { |
---|
1636 | l=l0; |
---|
1637 | } |
---|
1638 | } |
---|
1639 | } |
---|
1640 | Sp0=spsub(Sp0,Sp[size(Sp)]); |
---|
1641 | k++; |
---|
1642 | } |
---|
1643 | if(size(Sp)>1) |
---|
1644 | { |
---|
1645 | return(l); |
---|
1646 | } |
---|
1647 | else |
---|
1648 | { |
---|
1649 | return(list(intvec(k-1))); |
---|
1650 | } |
---|
1651 | } |
---|
1652 | example |
---|
1653 | { "EXAMPLE:"; echo=2; |
---|
1654 | ring R=0,(x,y),ds; |
---|
1655 | list Sp0=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1656 | spprint(Sp0); |
---|
1657 | list Sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1658 | spprint(Sp1); |
---|
1659 | list Sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1660 | spprint(Sp2); |
---|
1661 | list Sp=Sp1,Sp2; |
---|
1662 | list l=spsemicont(Sp0,Sp); |
---|
1663 | l; |
---|
1664 | spissemicont(spsub(Sp0,spmul(Sp,l[1]))); |
---|
1665 | spissemicont(spsub(Sp0,spmul(Sp,l[1]-1))); |
---|
1666 | spissemicont(spsub(Sp0,spmul(Sp,l[1]+1))); |
---|
1667 | } |
---|
1668 | /////////////////////////////////////////////////////////////////////////////// |
---|
1669 | |
---|
1670 | proc spmilnor(list Sp) |
---|
1671 | "USAGE: spmilnor(Sp); list Sp |
---|
1672 | RETURN: int: Milnor number of spectrum Sp |
---|
1673 | EXAMPLE: example spmilnor; shows examples |
---|
1674 | " |
---|
1675 | { |
---|
1676 | return(sum(Sp[2])); |
---|
1677 | } |
---|
1678 | example |
---|
1679 | { "EXAMPLE:"; echo=2; |
---|
1680 | ring R=0,(x,y),ds; |
---|
1681 | list Sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1682 | spprint(Sp); |
---|
1683 | spmilnor(Sp); |
---|
1684 | } |
---|
1685 | /////////////////////////////////////////////////////////////////////////////// |
---|
1686 | |
---|
1687 | proc spgeomgenus(list Sp) |
---|
1688 | "USAGE: spgeomgenus(Sp); list Sp |
---|
1689 | RETURN: int: geometrical genus of spectrum Sp |
---|
1690 | EXAMPLE: example spgeomgenus; shows examples |
---|
1691 | " |
---|
1692 | { |
---|
1693 | int g=0; |
---|
1694 | int i=1; |
---|
1695 | while(i+1<=size(Sp[2])&&number(Sp[1][i])<=number(0)) |
---|
1696 | { |
---|
1697 | g=g+Sp[2][i]; |
---|
1698 | i++; |
---|
1699 | } |
---|
1700 | if(i==size(Sp[2])&&number(Sp[1][i])<=number(0)) |
---|
1701 | { |
---|
1702 | g=g+Sp[2][i]; |
---|
1703 | } |
---|
1704 | return(g); |
---|
1705 | } |
---|
1706 | example |
---|
1707 | { "EXAMPLE:"; echo=2; |
---|
1708 | ring R=0,(x,y),ds; |
---|
1709 | list Sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1710 | spprint(Sp); |
---|
1711 | spgeomgenus(Sp); |
---|
1712 | } |
---|
1713 | /////////////////////////////////////////////////////////////////////////////// |
---|
1714 | |
---|
1715 | proc spgamma(list Sp) |
---|
1716 | "USAGE: spgamma(Sp); list Sp |
---|
1717 | RETURN: number: gamma invariant of spectrum Sp |
---|
1718 | EXAMPLE: example spgamma; shows examples |
---|
1719 | " |
---|
1720 | { |
---|
1721 | int i,j; |
---|
1722 | number g=0; |
---|
1723 | for(i=1;i<=ncols(Sp[1]);i++) |
---|
1724 | { |
---|
1725 | for(j=1;j<=Sp[2][i];j++) |
---|
1726 | { |
---|
1727 | g=g+(number(Sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1728 | } |
---|
1729 | } |
---|
1730 | g=-g/4+sum(Sp[2])*number(Sp[1][ncols(Sp[1])]-Sp[1][1])/48; |
---|
1731 | return(g); |
---|
1732 | } |
---|
1733 | example |
---|
1734 | { "EXAMPLE:"; echo=2; |
---|
1735 | ring R=0,(x,y),ds; |
---|
1736 | list Sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1737 | spprint(Sp); |
---|
1738 | spgamma(Sp); |
---|
1739 | } |
---|
1740 | /////////////////////////////////////////////////////////////////////////////// |
---|