1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: gaussman.lib,v 1.56 2001-11-05 13:58:38 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 | sppairs(t); spectral pairs of t |
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20 | sppnf(a,w[,m][,v|V]); normalize spectral pairs |
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21 | vfilt(t); V-filtration of t on H''/H' |
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22 | vwfilt(t); weight refined V-filtration of t on H''/H' |
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23 | saito(t); matrix A0+A1*s of t on H'' |
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24 | endvfilt(V); endomorphism V-filtration |
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25 | spprint(sp); print spectrum sp |
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26 | sppprint(spp); print spectral pairs spp |
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27 | spadd(sp1,sp2); sum of spectra sp1 and sp2 |
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28 | spsub(sp1,sp2); difference of spectra sp1 and sp2 |
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29 | spmul(sp,k); product of spectrum sp and integer k |
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30 | spmul(sp,k); linear combination of spectra sp with coeffs k |
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31 | spissemicont(sp[,opt]); test spectrum sp for semicontinuity |
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32 | spsemicont(sp0,sp[,opt]); semicontinuity of spectra sp0 and sp |
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33 | spmilnor(sp); milnor number of spectrum sp |
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34 | spgeomgenus(sp); geometrical genus of spectrum sp |
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35 | spgamma(sp); gamma invariant of spectrum sp |
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36 | |
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37 | SEE ALSO: mondromy_lib, spectrum_lib |
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38 | |
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39 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
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40 | monodromy; spectrum; spectral pairs; |
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41 | mixed Hodge structure; V-filtration; weight filtration |
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42 | "; |
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43 | |
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44 | LIB "linalg.lib"; |
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45 | |
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46 | /////////////////////////////////////////////////////////////////////////////// |
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47 | |
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48 | static proc stdtrans(ideal I) |
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49 | { |
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50 | def R=basering; |
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51 | |
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52 | string os=ordstr(R); |
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53 | int j=find(os,",C"); |
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54 | if(j==0) |
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55 | { |
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56 | j=find(os,"C,"); |
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57 | } |
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58 | if(j==0) |
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59 | { |
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60 | j=find(os,",c"); |
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61 | } |
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62 | if(j==0) |
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63 | { |
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64 | j=find(os,"c,"); |
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65 | } |
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66 | if(j>0) |
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67 | { |
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68 | os[j..j+1]=" "; |
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69 | } |
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70 | |
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71 | execute("ring S="+charstr(R)+",(gmspoly,"+varstr(R)+"),(c,dp,"+os+");"); |
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72 | |
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73 | ideal I=homog(imap(R,I),gmspoly); |
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74 | module M=transpose(transpose(I)+freemodule(ncols(I))); |
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75 | M=std(M); |
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76 | |
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77 | setring(R); |
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78 | execute("map h=S,1,"+varstr(R)+";"); |
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79 | module M=h(M); |
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80 | |
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81 | for(int i=ncols(M);i>=1;i--) |
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82 | { |
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83 | for(j=ncols(M);j>=1;j--) |
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84 | { |
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85 | if(M[i][1]==0) |
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86 | { |
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87 | M[i]=0; |
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88 | } |
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89 | if(i!=j&&M[j][1]!=0) |
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90 | { |
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91 | if(lead(M[i][1])/lead(M[j][1])!=0) |
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92 | { |
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93 | M[i]=0; |
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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 | |
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99 | M=transpose(simplify(M,2)); |
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100 | I=M[1]; |
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101 | attrib(I,"isSB",1); |
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102 | M=M[2..ncols(M)]; |
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103 | module U=transpose(M); |
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104 | |
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105 | return(list(I,U)); |
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106 | } |
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107 | /////////////////////////////////////////////////////////////////////////////// |
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108 | |
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109 | proc gmsring(poly t,string s) |
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110 | "USAGE: gmsring(t,s); poly t, string s; |
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111 | ASSUME: basering with characteristic 0 and local degree ordering, |
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112 | t with isolated citical point 0 |
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113 | RETURN: |
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114 | @format |
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115 | ring G: C{{s}}*basering, |
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116 | poly gmspoly: image of t |
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117 | ideal gmsjacob: image of Jacobian ideal |
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118 | ideal gmsstd: image of standard basis of Jacobian ideal |
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119 | matrix gmsmatrix: matrix(gmsjacob)*gmsmatrix=matrix(gmsstd) |
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120 | ideal gmsbasis: image of monomial vector space basis of Jacobian algebra |
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121 | int gmsmaxweight: maximal weight of variables of basering |
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122 | @end format |
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123 | NOTE: do not modify gms variables if you want to use gms procedures |
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124 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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125 | EXAMPLE: example gms; shows examples |
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126 | " |
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127 | { |
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128 | def R=basering; |
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129 | if(charstr(R)!="0") |
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130 | { |
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131 | ERROR("characteristic 0 expected"); |
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132 | } |
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133 | for(int i=nvars(R);i>=1;i--) |
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134 | { |
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135 | if(var(i)>1) |
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136 | { |
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137 | ERROR("local ordering expected"); |
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138 | } |
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139 | } |
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140 | |
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141 | ideal dt=jacob(t); |
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142 | list l=stdtrans(dt); |
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143 | ideal g=l[1]; |
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144 | if(vdim(g)<=0) |
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145 | { |
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146 | if(vdim(g)==0) |
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147 | { |
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148 | ERROR("singularity at 0 expected"); |
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149 | } |
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150 | else |
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151 | { |
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152 | ERROR("isolated citical point 0 expected"); |
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153 | } |
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154 | } |
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155 | matrix a=l[2]; |
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156 | ideal m=kbase(g); |
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157 | |
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158 | int gmsmaxweight; |
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159 | for(i=nvars(R);i>=1;i--) |
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160 | { |
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161 | if(deg(var(i))>gmsmaxweight) |
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162 | { |
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163 | gmsmaxweight=deg(var(i)); |
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164 | } |
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165 | } |
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166 | |
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167 | string os=ordstr(R); |
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168 | int j=find(os,",C"); |
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169 | if(j==0) |
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170 | { |
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171 | j=find(os,"C,"); |
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172 | } |
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173 | if(j==0) |
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174 | { |
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175 | j=find(os,",c"); |
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176 | } |
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177 | if(j==0) |
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178 | { |
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179 | j=find(os,"c,"); |
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180 | } |
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181 | if(j>0) |
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182 | { |
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183 | os[j..j+1]=" "; |
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184 | } |
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185 | |
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186 | execute("ring G="+string(charstr(R))+",("+s+","+varstr(R)+"),(ws("+ |
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187 | string(deg(highcorner(g))+2*gmsmaxweight)+"),"+os+",c);"); |
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188 | |
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189 | poly gmspoly=imap(R,t); |
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190 | ideal gmsjacob=imap(R,dt); |
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191 | ideal gmsstd=imap(R,g); |
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192 | matrix gmsmatrix=imap(R,a); |
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193 | ideal gmsbasis=imap(R,m); |
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194 | |
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195 | attrib(gmsstd,"isSB",1); |
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196 | export gmspoly,gmsjacob,gmsstd,gmsmatrix,gmsbasis,gmsmaxweight; |
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197 | |
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198 | return(G); |
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199 | } |
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200 | example |
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201 | { "EXAMPLE:"; echo=2; |
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202 | ring R=0,(x,y),ds; |
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203 | poly t=x5+x2y2+y5; |
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204 | def G=gmsring(t,"s"); |
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205 | setring(G); |
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206 | gmspoly; |
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207 | print(gmsjacob); |
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208 | print(gmsstd); |
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209 | print(gmsmatrix); |
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210 | print(gmsbasis); |
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211 | gmsmaxweight; |
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212 | } |
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213 | /////////////////////////////////////////////////////////////////////////////// |
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214 | |
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215 | proc gmsnf(ideal p,int K,list #) |
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216 | "USAGE: gmsnf(p,K[,Kmax]); poly p, int K[, int Kmax]; |
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217 | ASSUME: basering constructed by gmsring, K<=Kmax |
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218 | RETURN: |
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219 | @format |
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220 | list l: |
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221 | ideal l[1]: projection of p to H''=C{{s}}*gmsbasis mod s^{K+1} |
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222 | ideal l[2]: p=l[1]+l[2] mod s^(Kmax+1) |
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223 | @end format |
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224 | NOTE: by setting p=l[2] the computation can be continued up to order |
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225 | at most Kmax, by default Kmax=infinity |
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226 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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227 | EXAMPLE: example gmsnf; shows examples |
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228 | " |
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229 | { |
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230 | int Kmax=-1; |
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231 | if(size(#)>0) |
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232 | { |
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233 | if(typeof(#[1])=="int") |
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234 | { |
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235 | Kmax=#[1]; |
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236 | if(K>Kmax) |
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237 | { |
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238 | Kmax=K; |
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239 | } |
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240 | } |
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241 | } |
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242 | |
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243 | intvec v=1; |
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244 | v[nvars(basering)]=0; |
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245 | |
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246 | int k; |
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247 | if(Kmax>=0) |
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248 | { |
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249 | p=jet(jet(p,K,v),(Kmax+1)*deg(var(1))-2*gmsmaxweight); |
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250 | } |
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251 | |
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252 | ideal r,q; |
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253 | r[ncols(p)]=0; |
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254 | q[ncols(p)]=0; |
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255 | |
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256 | poly s; |
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257 | int i,j; |
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258 | for(k=ncols(p);k>=1;k--) |
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259 | { |
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260 | while(p[k]!=0&°(lead(p[k]),v)<=K) |
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261 | { |
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262 | i=1; |
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263 | s=lead(p[k])/lead(gmsstd[i]); |
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264 | while(i<ncols(gmsstd)&&s==0) |
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265 | { |
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266 | i++; |
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267 | s=lead(p[k])/lead(gmsstd[i]); |
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268 | } |
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269 | if(s!=0) |
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270 | { |
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271 | p[k]=p[k]-s*gmsstd[i]; |
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272 | for(j=1;j<=nrows(gmsmatrix);j++) |
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273 | { |
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274 | if(Kmax>=0) |
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275 | { |
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276 | p[k]=p[k]+ |
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277 | jet(jet(diff(s*gmsmatrix[j,i],var(j+1))*var(1),Kmax,v), |
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278 | (Kmax+1)*deg(var(1))-2*gmsmaxweight); |
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279 | } |
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280 | else |
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281 | { |
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282 | p[k]=p[k]+diff(s*gmsmatrix[j,i],var(j+1))*var(1); |
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283 | } |
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284 | } |
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285 | } |
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286 | else |
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287 | { |
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288 | r[k]=r[k]+lead(p[k]); |
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289 | p[k]=p[k]-lead(p[k]); |
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290 | } |
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291 | while(deg(lead(p[k]))>(K+1)*deg(var(1))-2*gmsmaxweight&& |
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292 | deg(lead(p[k]),v)<=K) |
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293 | { |
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294 | q[k]=q[k]+lead(p[k]); |
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295 | p[k]=p[k]-lead(p[k]); |
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296 | } |
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297 | } |
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298 | q[k]=q[k]+p[k]; |
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299 | } |
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300 | |
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301 | return(list(r,q)); |
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302 | } |
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303 | example |
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304 | { "EXAMPLE:"; echo=2; |
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305 | ring R=0,(x,y),ds; |
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306 | poly t=x5+x2y2+y5; |
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307 | def G=gmsring(t,"s"); |
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308 | setring(G); |
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309 | list l0=gmsnf(gmspoly,0); |
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310 | print(l0[1]); |
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311 | list l1=gmsnf(gmspoly,1); |
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312 | print(l1[1]); |
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313 | list l=gmsnf(l0[2],1); |
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314 | print(l[1]); |
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315 | } |
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316 | /////////////////////////////////////////////////////////////////////////////// |
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317 | |
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318 | proc gmscoeffs(ideal p,int K,list #) |
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319 | "USAGE: gmscoeffs(p,K[,Kmax]); poly p, int K[, int Kmax]; |
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320 | ASSUME: basering constructed by gmsring, K<=Kmax |
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321 | RETURN: |
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322 | @format |
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323 | list l: |
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324 | matrix l[1]: projection of p to H''=C{{s}}*gmsbasis=C{{s}}^mu mod s^(K+1) |
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325 | ideal l[2]: p=matrix(gmsbasis)*l[1]+l[2] mod s^(Kmax+1) |
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326 | @end format |
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327 | NOTE: by setting p=l[2] the computation can be continued up to order |
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328 | at most Kmax, by default Kmax=infinity |
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329 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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330 | EXAMPLE: example gmscoeffs; shows examples |
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331 | " |
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332 | { |
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333 | list l=gmsnf(p,K,#); |
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334 | ideal r,q=l[1..2]; |
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335 | poly v=1; |
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336 | for(int i=2;i<=nvars(basering);i++) |
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337 | { |
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338 | v=v*var(i); |
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339 | } |
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340 | matrix C=coeffs(r,gmsbasis,v); |
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341 | return(C,q); |
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342 | } |
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343 | example |
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344 | { "EXAMPLE:"; echo=2; |
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345 | ring R=0,(x,y),ds; |
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346 | poly t=x5+x2y2+y5; |
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347 | def G=gmsring(t,"s"); |
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348 | setring(G); |
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349 | list l0=gmscoeffs(gmspoly,0); |
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350 | print(l0[1]); |
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351 | list l1=gmscoeffs(gmspoly,1); |
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352 | print(l1[1]); |
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353 | list l=gmscoeffs(l0[2],1); |
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354 | print(l[1]); |
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355 | } |
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356 | /////////////////////////////////////////////////////////////////////////////// |
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357 | |
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358 | static proc min(ideal e) |
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359 | { |
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360 | int i; |
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361 | number m=number(e[1]); |
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362 | for(i=2;i<=ncols(e);i++) |
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363 | { |
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364 | if(number(e[i])<m) |
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365 | { |
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366 | m=number(e[i]); |
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367 | } |
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368 | } |
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369 | return(m); |
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370 | } |
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371 | /////////////////////////////////////////////////////////////////////////////// |
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372 | |
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373 | static proc max(ideal e) |
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374 | { |
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375 | int i; |
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376 | number m=number(e[1]); |
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377 | for(i=2;i<=ncols(e);i++) |
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378 | { |
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379 | if(number(e[i])>m) |
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380 | { |
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381 | m=number(e[i]); |
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382 | } |
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383 | } |
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384 | return(m); |
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385 | } |
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386 | /////////////////////////////////////////////////////////////////////////////// |
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387 | |
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388 | static proc saturate(int K0) |
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389 | { |
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390 | int mu=ncols(gmsbasis); |
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391 | ideal r=gmspoly*gmsbasis; |
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392 | matrix A0[mu][mu],C; |
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393 | module H0; |
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394 | module H,H1=freemodule(mu),freemodule(mu); |
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395 | int k=-1; |
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396 | list l; |
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397 | |
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398 | while(size(reduce(H,std(H0*s)))>0) |
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399 | { |
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400 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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401 | k++; |
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402 | dbprint(printlevel-voice+2,"// k="+string(k)); |
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403 | l=gmscoeffs(r,k,mu+K0); |
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404 | C,r=l[1..2]; |
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405 | A0=A0+C; |
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406 | |
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407 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
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408 | H0=H; |
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409 | H1=jet(module(A0*H1+s^2*diff(matrix(H1),s)),k); |
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410 | H=H*s+H1; |
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411 | } |
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412 | |
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413 | A0=A0-k*s; |
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414 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
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415 | H=std(H0); |
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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,k); |
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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 k0,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 | dbprint(printlevel-voice+2,"// k="+string(K+k0+1)); |
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429 | list l=gmscoeffs(r,K+k0+1,K0+k0+1); |
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430 | matrix C; |
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431 | C,r=l[1..2]; |
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432 | A0=A0+C; |
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433 | |
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434 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
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435 | l=division(H*s,A0*H+s^2*diff(matrix(H),s)); |
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436 | matrix A=jet(l[1],l[2],K); |
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437 | |
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438 | return(A,A0,r); |
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439 | } |
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440 | /////////////////////////////////////////////////////////////////////////////// |
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441 | |
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442 | static proc eigenvals(matrix A0,ideal r,module H,int k0,int K0) |
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443 | { |
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444 | dbprint(printlevel-voice+2, |
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445 | "// compute eigenvalues e with multiplicities m of A"); |
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446 | matrix A; |
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447 | A,A0,r=tmatrix(A0,r,H,k0,0,K0); |
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448 | list l=eigenvalues(A); |
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449 | def e,m=l[1..2]; |
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450 | dbprint(printlevel-voice+2,"// e="+string(e)); |
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451 | dbprint(printlevel-voice+2,"// m="+string(m)); |
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452 | |
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453 | return(e,m,A0,r,int(max(e)-min(e))); |
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454 | } |
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455 | /////////////////////////////////////////////////////////////////////////////// |
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456 | |
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457 | static proc transform(matrix A,matrix A0,ideal r,module H,module H0,ideal e,intvec m,int k0,int k1,int K,int K0) |
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458 | { |
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459 | int mu=ncols(gmsbasis); |
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460 | |
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461 | dbprint(printlevel-voice+2,"// compute minimum e0 and maximum e1 of e"); |
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462 | number e0,e1=min(e),max(e); |
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463 | dbprint(printlevel-voice+2,"// e0="+string(e0)); |
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464 | dbprint(printlevel-voice+2,"// e1="+string(e1)); |
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465 | A,A0,r=tmatrix(A0,r,H,k0,K+k1,K0+k1); |
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466 | module U0=s^k0*freemodule(mu); |
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467 | |
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468 | if(e1>=e0+1) |
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469 | { |
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470 | int i,j,i0,j0,i1,j1; |
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471 | module U,V; |
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472 | list l; |
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473 | |
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474 | while(e1>=e0+1) |
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475 | { |
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476 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
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477 | U=0; |
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478 | for(i=1;i<=ncols(e);i++) |
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479 | { |
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480 | U=U+syz(power(jet(A,0)-e[i],m[i])); |
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481 | } |
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482 | V=inverse(U); |
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483 | A=V*A*U; |
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484 | H0=V*H0; |
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485 | U0=U0*U; |
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486 | |
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487 | dbprint(printlevel-voice+2,"// transform to reduce e1 by 1"); |
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488 | for(i0,i=1,1;i0<=ncols(e);i0++) |
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489 | { |
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490 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
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491 | { |
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492 | for(j0,j=1,1;j0<=ncols(e);j0++) |
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493 | { |
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494 | for(j1=1;j1<=m[j0];j1,j=j1+1,j+1) |
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495 | { |
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496 | if(number(e[i0])<e0+1&&number(e[j0])>=e0+1) |
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497 | { |
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498 | A[i,j]=A[i,j]/s; |
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499 | } |
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500 | if(number(e[i0])>=e0+1&&number(e[j0])<e0+1) |
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501 | { |
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502 | A[i,j]=A[i,j]*s; |
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503 | } |
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504 | } |
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505 | } |
---|
506 | } |
---|
507 | } |
---|
508 | |
---|
509 | H0=transpose(H0); |
---|
510 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
511 | { |
---|
512 | if(number(e[i0])>=e0+1) |
---|
513 | { |
---|
514 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
515 | { |
---|
516 | A[i,i]=A[i,i]-1; |
---|
517 | H0[i]=H0[i]*s; |
---|
518 | U0[i]=U0[i]/s; |
---|
519 | } |
---|
520 | e[i0]=e[i0]-1; |
---|
521 | } |
---|
522 | else |
---|
523 | { |
---|
524 | i=i+m[i0]; |
---|
525 | } |
---|
526 | } |
---|
527 | H0=transpose(H0); |
---|
528 | |
---|
529 | l=spnf(e,m); |
---|
530 | e,m=l[1..2]; |
---|
531 | |
---|
532 | e1=e1-1; |
---|
533 | dbprint(printlevel-voice+2,"// e1="+string(e1)); |
---|
534 | } |
---|
535 | |
---|
536 | A=jet(A,K); |
---|
537 | } |
---|
538 | |
---|
539 | return(A,A0,r,H0,U0,e,m); |
---|
540 | } |
---|
541 | /////////////////////////////////////////////////////////////////////////////// |
---|
542 | |
---|
543 | proc monodromy(poly t,list #) |
---|
544 | "USAGE: monodromy(t); poly t |
---|
545 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
546 | t with isolated citical point 0 |
---|
547 | RETURN: list l: Jordan data jordan(M) of a monodromy matrix exp(-2*pi*i*M) |
---|
548 | SEE ALSO: mondromy_lib |
---|
549 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; monodromy |
---|
550 | EXAMPLE: example monodromy; shows examples |
---|
551 | " |
---|
552 | { |
---|
553 | def R=basering; |
---|
554 | int n=nvars(R)-1; |
---|
555 | def G=gmsring(t,"s"); |
---|
556 | setring(G); |
---|
557 | |
---|
558 | matrix A; |
---|
559 | module U0; |
---|
560 | ideal e; |
---|
561 | intvec m; |
---|
562 | int k1; |
---|
563 | |
---|
564 | def A0,r,H,H0,k0=saturate(n); |
---|
565 | e,m,A0,r,k1=eigenvals(A0,r,H,k0,n); |
---|
566 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,k1,0,0); |
---|
567 | |
---|
568 | setring(R); |
---|
569 | return(jordan(imap(G,A),imap(G,e),m)); |
---|
570 | } |
---|
571 | example |
---|
572 | { "EXAMPLE:"; echo=2; |
---|
573 | ring R=0,(x,y),ds; |
---|
574 | poly t=x5+x2y2+y5; |
---|
575 | monodromy(t); |
---|
576 | } |
---|
577 | /////////////////////////////////////////////////////////////////////////////// |
---|
578 | |
---|
579 | proc spectrum(poly t) |
---|
580 | "USAGE: spectrum(t); poly t |
---|
581 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
582 | t with isolated citical point 0 |
---|
583 | RETURN: |
---|
584 | @format |
---|
585 | list sp: spectrum of t |
---|
586 | ideal sp[1]: V-filtration indices in increasing order |
---|
587 | intvec sp[2]: weight filtration indices in decreasing order |
---|
588 | intvec sp[3]: |
---|
589 | int sp[3][i]: multiplicity of spectral number sp[1][i] |
---|
590 | @end format |
---|
591 | SEE ALSO: spectrum_lib |
---|
592 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
593 | mixed Hodge structure; V-filtration; spectrum |
---|
594 | EXAMPLE: example spectrum; shows examples |
---|
595 | " |
---|
596 | { |
---|
597 | list l=vwfilt(t); |
---|
598 | return(spnf(l[1],l[3])); |
---|
599 | } |
---|
600 | example |
---|
601 | { "EXAMPLE:"; echo=2; |
---|
602 | ring R=0,(x,y),ds; |
---|
603 | poly t=x5+x2y2+y5; |
---|
604 | spprint(spectrum(t)); |
---|
605 | } |
---|
606 | /////////////////////////////////////////////////////////////////////////////// |
---|
607 | |
---|
608 | proc sppairs(poly t) |
---|
609 | "USAGE: sppairs(t); poly t |
---|
610 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
611 | t with isolated citical point 0 |
---|
612 | RETURN: |
---|
613 | @format |
---|
614 | list spp: spectrum of t |
---|
615 | ideal spp[1],intvec spp[2]: spectral pairs in in-/decreasing lex. order |
---|
616 | intvec spp[3]: |
---|
617 | int spp[3][i]: multiplicity of spectral pair (spp[1][i],spp[2][i]) |
---|
618 | @end format |
---|
619 | SEE ALSO: spectrum_lib |
---|
620 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
621 | mixed Hodge structure; V-filtration; weight filtration; |
---|
622 | spectrum; spectral pairs |
---|
623 | EXAMPLE: example sppairs; shows examples |
---|
624 | " |
---|
625 | { |
---|
626 | list l=vwfilt(t); |
---|
627 | return(list(l[1],l[2],l[3])); |
---|
628 | } |
---|
629 | example |
---|
630 | { "EXAMPLE:"; echo=2; |
---|
631 | ring R=0,(x,y),ds; |
---|
632 | poly t=x5+x2y2+y5; |
---|
633 | sppprint(sppairs(t)); |
---|
634 | } |
---|
635 | /////////////////////////////////////////////////////////////////////////////// |
---|
636 | |
---|
637 | proc sppnf(ideal a,intvec w,list #) |
---|
638 | "USAGE: sppnf(a,w[,m][,v|V]); ideal a, intvec w, intvec m, module v, list V |
---|
639 | RETURN: |
---|
640 | @format |
---|
641 | list spp: normalized spectral pairs (a,w,m[,V]) |
---|
642 | ideal spp[1]: numbers in a in increasing order |
---|
643 | intvec spp[2]: integers in w in decreasing order |
---|
644 | intvec spp[3]: |
---|
645 | int spp[3][i]: multiplicity of pair (spp[1][i],spp[2][i]) in (a,w) |
---|
646 | list spp[4]: |
---|
647 | module spp[4][i]: module associated to pair (spp[1][i],spp[2][i]) |
---|
648 | @end format |
---|
649 | EXAMPLE: example sppnorm; shows examples |
---|
650 | " |
---|
651 | { |
---|
652 | int n=ncols(a); |
---|
653 | intvec m; |
---|
654 | module v; |
---|
655 | list V; |
---|
656 | int i,j; |
---|
657 | while(i<size(#)) |
---|
658 | { |
---|
659 | i++; |
---|
660 | if(typeof(#[i])=="intvec") |
---|
661 | { |
---|
662 | m=#[i]; |
---|
663 | } |
---|
664 | if(typeof(#[i])=="module") |
---|
665 | { |
---|
666 | v=#[i]; |
---|
667 | for(j=n;j>=1;j--) |
---|
668 | { |
---|
669 | V[j]=module(v[j]); |
---|
670 | } |
---|
671 | } |
---|
672 | if(typeof(#[i])=="list") |
---|
673 | { |
---|
674 | V=#[i]; |
---|
675 | } |
---|
676 | } |
---|
677 | if(m==0) |
---|
678 | { |
---|
679 | for(i=n;i>=1;i--) |
---|
680 | { |
---|
681 | m[i]=1; |
---|
682 | } |
---|
683 | } |
---|
684 | |
---|
685 | int k; |
---|
686 | ideal a0; |
---|
687 | intvec w0,m0; |
---|
688 | list V0; |
---|
689 | number a1; |
---|
690 | int w1,m1; |
---|
691 | for(i=n;i>=1;i--) |
---|
692 | { |
---|
693 | if(m[i]!=0) |
---|
694 | { |
---|
695 | for(j=i-1;j>=1;j--) |
---|
696 | { |
---|
697 | if(m[j]!=0) |
---|
698 | { |
---|
699 | if(number(a[i])>number(a[j])|| |
---|
700 | (number(a[i])==number(a[j])&&w[i]<w[j])) |
---|
701 | { |
---|
702 | a1=number(a[i]); |
---|
703 | a[i]=a[j]; |
---|
704 | a[j]=a1; |
---|
705 | w1=w[i]; |
---|
706 | w[i]=w[j]; |
---|
707 | w[j]=w1; |
---|
708 | m1=m[i]; |
---|
709 | m[i]=m[j]; |
---|
710 | m[j]=m1; |
---|
711 | if(size(V)>0) |
---|
712 | { |
---|
713 | v=V[i]; |
---|
714 | V[i]=V[j]; |
---|
715 | V[j]=v; |
---|
716 | } |
---|
717 | } |
---|
718 | if(number(a[i])==number(a[j])&&w[i]==w[j]) |
---|
719 | { |
---|
720 | m[i]=m[i]+m[j]; |
---|
721 | m[j]=0; |
---|
722 | if(size(V)>0) |
---|
723 | { |
---|
724 | V[i]=V[i]+V[j]; |
---|
725 | } |
---|
726 | } |
---|
727 | } |
---|
728 | } |
---|
729 | k++; |
---|
730 | a0[k]=a[i]; |
---|
731 | w0[k]=w[i]; |
---|
732 | m0[k]=m[i]; |
---|
733 | if(size(V)>0) |
---|
734 | { |
---|
735 | V0[k]=V[i]; |
---|
736 | } |
---|
737 | } |
---|
738 | } |
---|
739 | |
---|
740 | if(size(V0)>0) |
---|
741 | { |
---|
742 | n=size(V0); |
---|
743 | module U=std(V0[n]); |
---|
744 | for(i=n-1;i>=1;i--) |
---|
745 | { |
---|
746 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
747 | if(i>=2) |
---|
748 | { |
---|
749 | U=std(U+V0[i]); |
---|
750 | } |
---|
751 | } |
---|
752 | } |
---|
753 | |
---|
754 | list l; |
---|
755 | if(k>0) |
---|
756 | { |
---|
757 | l=a0,w0,m0; |
---|
758 | if(size(V0)>0) |
---|
759 | { |
---|
760 | l[4]=V0; |
---|
761 | } |
---|
762 | } |
---|
763 | return(l); |
---|
764 | } |
---|
765 | example |
---|
766 | { "EXAMPLE:"; echo=2; |
---|
767 | } |
---|
768 | /////////////////////////////////////////////////////////////////////////////// |
---|
769 | |
---|
770 | proc vfilt(poly t) |
---|
771 | "USAGE: vfilt(t); poly t |
---|
772 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
773 | t with isolated citical point 0 |
---|
774 | RETURN:t |
---|
775 | @format |
---|
776 | list v: V-filtration on H''/H' |
---|
777 | ideal v[1]: spectral numbers in increasing order |
---|
778 | intvec v[2]: |
---|
779 | int v[2][i]: multiplicity of spectral number v[1][i] |
---|
780 | list v[4]: |
---|
781 | module v[4][i]: vector space basis of v[1][i]-th graded part |
---|
782 | in terms of v[4] |
---|
783 | ideal v[4]: monomial vector space basis of H''/H' |
---|
784 | ideal v[5]: standard basis of Jacobian ideal |
---|
785 | @end format |
---|
786 | SEE ALSO: spectrum_lib |
---|
787 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
788 | mixed Hodge structure; V-filtration; spectrum |
---|
789 | EXAMPLE: example vfilt; shows examples |
---|
790 | " |
---|
791 | { |
---|
792 | list l=vwfilt(t); |
---|
793 | return(spnf(l[1],l[3],l[4])+list(l[5],l[6])); |
---|
794 | } |
---|
795 | example |
---|
796 | { "EXAMPLE:"; echo=2; |
---|
797 | ring R=0,(x,y),ds; |
---|
798 | poly t=x5+x2y2+y5; |
---|
799 | vfilt(t); |
---|
800 | } |
---|
801 | /////////////////////////////////////////////////////////////////////////////// |
---|
802 | |
---|
803 | proc vwfilt(poly t) |
---|
804 | "USAGE: vwfilt(t); poly t |
---|
805 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
806 | t with isolated citical point 0 |
---|
807 | RETURN: |
---|
808 | @format |
---|
809 | list vw: weight refined V-filtration on H''/H' |
---|
810 | ideal vw[1]: spectral numbers in increasing order |
---|
811 | intvec vw[2]: weights in decreasing order |
---|
812 | intvec vw[3]: |
---|
813 | int vw[3][i]: multiplicity of spectral pair (vw[1][i],vw[2][i]) |
---|
814 | list vw[4]: |
---|
815 | module vw[4][i]: vector space basis of (vw[1][i],vw[2][i])-th graded part |
---|
816 | in terms of vw[5] |
---|
817 | ideal vw[5]: monomial vector space basis of H''/H' |
---|
818 | ideal vw[6]: standard basis of Jacobian ideal |
---|
819 | @end format |
---|
820 | SEE ALSO: spectrum_lib |
---|
821 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
822 | mixed Hodge structure; V-filtration; weight filtration; |
---|
823 | spectrum; spectral pairs |
---|
824 | EXAMPLE: example vwfilt; shows examples |
---|
825 | " |
---|
826 | { |
---|
827 | def R=basering; |
---|
828 | int n=nvars(R)-1; |
---|
829 | def G=gmsring(t,"s"); |
---|
830 | setring(G); |
---|
831 | |
---|
832 | int mu=ncols(gmsbasis); |
---|
833 | matrix A; |
---|
834 | module U0; |
---|
835 | ideal e; |
---|
836 | intvec m; |
---|
837 | int k1; |
---|
838 | |
---|
839 | def A0,r,H,H0,k0=saturate(n); |
---|
840 | e,m,A0,r,k1=eigenvals(A0,r,H,k0,n); |
---|
841 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,k1,0,0); |
---|
842 | |
---|
843 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
844 | list l=jordanbasis(A,e,m); |
---|
845 | def U,v=l[1..2]; |
---|
846 | kill l; |
---|
847 | vector u0; |
---|
848 | int v0; |
---|
849 | int i,j,k,l; |
---|
850 | for(k,l=1,1;l<=ncols(e);k,l=k+m[l],l+1) |
---|
851 | { |
---|
852 | for(i=k+m[l]-1;i>=k+1;i--) |
---|
853 | { |
---|
854 | for(j=i-1;j>=k;j--) |
---|
855 | { |
---|
856 | if(v[i]>v[j]) |
---|
857 | { |
---|
858 | v0=v[i];v[i]=v[j];v[j]=v0; |
---|
859 | u0=U[i];U[i]=U[j];U[j]=u0; |
---|
860 | } |
---|
861 | } |
---|
862 | } |
---|
863 | } |
---|
864 | |
---|
865 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
866 | matrix V=inverse(U); |
---|
867 | A=V*A*U; |
---|
868 | dbprint(printlevel-voice+2,"// compute normal form of H''"); |
---|
869 | H0=std(V*H0); |
---|
870 | U0=U0*U; |
---|
871 | |
---|
872 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
873 | ideal a; |
---|
874 | intvec w; |
---|
875 | for(i=1;i<=mu;i++) |
---|
876 | { |
---|
877 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
878 | a[i]=A[j,j]+ord(H0[i])/deg(s)-1; |
---|
879 | w[i]=v[j]+n; |
---|
880 | } |
---|
881 | kill v; |
---|
882 | module v=simplify(jet(H*U0*H0,2*k0)/s^(2*k0),1); |
---|
883 | |
---|
884 | setring(R); |
---|
885 | ideal g=imap(G,gmsstd); |
---|
886 | attrib(g,"isSB",1); |
---|
887 | return(sppnf(imap(G,a),w,imap(G,v))+list(imap(G,gmsbasis),g)); |
---|
888 | } |
---|
889 | example |
---|
890 | { "EXAMPLE:"; echo=2; |
---|
891 | ring R=0,(x,y),ds; |
---|
892 | poly t=x5+x2y2+y5; |
---|
893 | vwfilt(t); |
---|
894 | } |
---|
895 | /////////////////////////////////////////////////////////////////////////////// |
---|
896 | |
---|
897 | proc saito(poly t,list #) |
---|
898 | "USAGE: saito(t); poly t |
---|
899 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
900 | t with isolated citical point 0 |
---|
901 | RETURN: list A: matrix A[1]+A[2]*s of t on H'' |
---|
902 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
903 | mixed Hodge structure; opposite Hodge filtration; V-filtration; |
---|
904 | EXAMPLE: example saito; shows examples |
---|
905 | " |
---|
906 | { |
---|
907 | def R=basering; |
---|
908 | int n=nvars(R)-1; |
---|
909 | def G=gmsring(t,"s"); |
---|
910 | setring(G); |
---|
911 | |
---|
912 | int mu=ncols(gmsbasis); |
---|
913 | matrix A; |
---|
914 | module U0; |
---|
915 | ideal e; |
---|
916 | intvec m; |
---|
917 | int k1; |
---|
918 | |
---|
919 | def A0,r,H,H0,k0=saturate(2*n+mu-1); |
---|
920 | e,m,A0,r,k1=eigenvals(A0,r,H,k0,n); |
---|
921 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,k1,k0+k1,k0+k1); |
---|
922 | |
---|
923 | dbprint(printlevel-voice+2,"// transform to Jordan basis"); |
---|
924 | module U=jordanbasis(A,e,m)[1]; |
---|
925 | matrix V=inverse(U); |
---|
926 | A=V*A*U; |
---|
927 | H=V*H0; |
---|
928 | |
---|
929 | dbprint(printlevel-voice+2,"// compute splitting of V-filtration"); |
---|
930 | int i,j,k; |
---|
931 | U=freemodule(mu); |
---|
932 | V=matrix(0,mu,mu); |
---|
933 | matrix v[mu^2][1]; |
---|
934 | A0=commutator(jet(A,0)); |
---|
935 | for(k=1;k<=k0+k1;k++) |
---|
936 | { |
---|
937 | for(j=0;j<k;j++) |
---|
938 | { |
---|
939 | V=matrix(V)-(jet(A,k-j)/s^(k-j))*(jet(U,j)/s^j); |
---|
940 | } |
---|
941 | v=V[1..mu,1..mu]; |
---|
942 | v=inverse(A0+k)*v; |
---|
943 | V=v[1..mu^2,1]; |
---|
944 | U=matrix(U)+s^k*V; |
---|
945 | } |
---|
946 | |
---|
947 | dbprint(printlevel-voice+2,"// transform to V-splitting basis"); |
---|
948 | A=jet(A,0); |
---|
949 | H=jet(division(U,H)[1],k0+k1); |
---|
950 | H=std(H); |
---|
951 | |
---|
952 | dbprint(printlevel-voice+2,"// compute V-leading terms of H''"); |
---|
953 | int i0,j0; |
---|
954 | module H1=H; |
---|
955 | for(k=ncols(H1);k>=1;k--) |
---|
956 | { |
---|
957 | i0=leadexp(H1[k])[nvars(basering)+1]; |
---|
958 | j0=ord(H1[k])/deg(s); |
---|
959 | H0[k]=lead(H1[k]); |
---|
960 | H1[k]=H1[k]-lead(H1[k]); |
---|
961 | if(H1[k]!=0) |
---|
962 | { |
---|
963 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
964 | j=ord(H1[k])/deg(s); |
---|
965 | while(A[i,i]+j==A[i0,i0]+j0) |
---|
966 | { |
---|
967 | H0[k]=H0[k]+lead(H1[k]); |
---|
968 | H1[k]=H1[k]-lead(H1[k]); |
---|
969 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
970 | j=ord(H1[k])/deg(s); |
---|
971 | } |
---|
972 | } |
---|
973 | } |
---|
974 | H0=simplify(H0,1); |
---|
975 | |
---|
976 | dbprint(printlevel-voice+2,"// compute N"); |
---|
977 | matrix N=A; |
---|
978 | for(i=1;i<=ncols(N);i++) |
---|
979 | { |
---|
980 | N[i,i]=0; |
---|
981 | } |
---|
982 | |
---|
983 | dbprint(printlevel-voice+2,"// compute splitting of Hodge filtration"); |
---|
984 | U=0; |
---|
985 | module U1; |
---|
986 | module C; |
---|
987 | list F,I; |
---|
988 | module F0,I0; |
---|
989 | for(i0,j0=1,1;i0<=ncols(e);i0++) |
---|
990 | { |
---|
991 | C=matrix(0,mu,1); |
---|
992 | for(j=m[i0];j>=1;j,j0=j-1,j0+1) |
---|
993 | { |
---|
994 | C=C+gen(j0); |
---|
995 | } |
---|
996 | F0=intersect(C,H0); |
---|
997 | F=list(); |
---|
998 | j=0; |
---|
999 | while(size(F0)>0) |
---|
1000 | { |
---|
1001 | j++; |
---|
1002 | F[j]=matrix(0,mu,1); |
---|
1003 | if(size(jet(F0,0))>0) |
---|
1004 | { |
---|
1005 | for(i=ncols(F0);i>=1;i--) |
---|
1006 | { |
---|
1007 | if(ord(F0[i])==0) |
---|
1008 | { |
---|
1009 | F[j]=F[j]+F0[i]; |
---|
1010 | } |
---|
1011 | } |
---|
1012 | } |
---|
1013 | for(i=ncols(F0);i>=1;i--) |
---|
1014 | { |
---|
1015 | F0[i]=F0[i]/s; |
---|
1016 | } |
---|
1017 | } |
---|
1018 | |
---|
1019 | I=list(); |
---|
1020 | I0=module(); |
---|
1021 | U0=std(0); |
---|
1022 | for(i=size(F);i>=1;i--) |
---|
1023 | { |
---|
1024 | I[i]=module(); |
---|
1025 | } |
---|
1026 | for(i=1;i<=size(F);i++) |
---|
1027 | { |
---|
1028 | I0=reduce(F[i],U0); |
---|
1029 | j=i; |
---|
1030 | while(size(I0)>0) |
---|
1031 | { |
---|
1032 | U0=std(U0+I0); |
---|
1033 | I[j]=I[j]+I0; |
---|
1034 | I0=reduce(N*I0,U0); |
---|
1035 | j++; |
---|
1036 | } |
---|
1037 | } |
---|
1038 | |
---|
1039 | for(i=1;i<=size(I);i++) |
---|
1040 | { |
---|
1041 | U=U+I[i]; |
---|
1042 | } |
---|
1043 | } |
---|
1044 | |
---|
1045 | dbprint(printlevel-voice+2,"// transform to Hodge splitting basis"); |
---|
1046 | V=inverse(U); |
---|
1047 | A=V*A*U; |
---|
1048 | H=V*H; |
---|
1049 | |
---|
1050 | dbprint(printlevel-voice+2,"// compute reduced standard basis of H''"); |
---|
1051 | ring S=0,s,ds; |
---|
1052 | module H=imap(G,H); |
---|
1053 | degBound=k0+k1+1; |
---|
1054 | option("redSB"); |
---|
1055 | H=std(H); |
---|
1056 | degBound=0; |
---|
1057 | H=simplify(jet(H,k0+k1),1); |
---|
1058 | setring(G); |
---|
1059 | H=imap(S,H); |
---|
1060 | |
---|
1061 | dbprint(printlevel-voice+2,"// compute matrix A0+sA1 of t"); |
---|
1062 | list l=division(H,s*A*H+s^2*diff(matrix(H),s)); |
---|
1063 | A=jet(l[1],l[2],k0+k1+1); |
---|
1064 | A0=jet(A,0); |
---|
1065 | A=jet(A,1)/s; |
---|
1066 | |
---|
1067 | setring(R); |
---|
1068 | return(list(imap(G,A0),imap(G,A))); |
---|
1069 | } |
---|
1070 | example |
---|
1071 | { "EXAMPLE:"; echo=2; |
---|
1072 | ring R=0,(x,y),ds; |
---|
1073 | poly t=x5+x2y2+y5; |
---|
1074 | list A=saito(t); |
---|
1075 | print(A[1]); |
---|
1076 | print(A[2]); |
---|
1077 | } |
---|
1078 | /////////////////////////////////////////////////////////////////////////////// |
---|
1079 | |
---|
1080 | proc endvfilt(list V) |
---|
1081 | "USAGE: endvwfilt(V); list V |
---|
1082 | ASSUME: V computed by vfilt |
---|
1083 | RETURN: |
---|
1084 | @format |
---|
1085 | list EV: endomorphism V-filtration on the Jacobian algebra |
---|
1086 | ideal EV[1]: spectral numbers in increasing order |
---|
1087 | intvec EV[2]: |
---|
1088 | int EV[2][i]: multiplicity of spectral pair (EV[1][i],EV[2][i]) |
---|
1089 | list EV[3]: |
---|
1090 | module EV[3][i]: vector space basis of the (EV[1][i],EV[2][i])-th |
---|
1091 | graded part in terms of EV[4] |
---|
1092 | ideal EV[4]: monomial vector space basis |
---|
1093 | ideal EV[5]: standard basis of Jacobian ideal |
---|
1094 | @end format |
---|
1095 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1096 | mixed Hodge structure; V-filtration; endomorphism filtration |
---|
1097 | EXAMPLE: example endvwfilt; shows examples |
---|
1098 | " |
---|
1099 | { |
---|
1100 | def a,d,v,m,g=V[1..5]; |
---|
1101 | int mu=ncols(m); |
---|
1102 | |
---|
1103 | module V0=v[1]; |
---|
1104 | for(int i=2;i<=size(v);i++) |
---|
1105 | { |
---|
1106 | V0=V0,v[i]; |
---|
1107 | } |
---|
1108 | |
---|
1109 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
1110 | list M; |
---|
1111 | module U=freemodule(ncols(m)); |
---|
1112 | for(i=ncols(m);i>=1;i--) |
---|
1113 | { |
---|
1114 | M[i]=division(V0,coeffs(reduce(m[i]*m,U,g),m)*V0)[1]; |
---|
1115 | } |
---|
1116 | |
---|
1117 | int j,k,i0,j0,i1,j1; |
---|
1118 | number b0=number(a[1]-a[ncols(a)]); |
---|
1119 | number b1,b2; |
---|
1120 | matrix M0; |
---|
1121 | module L; |
---|
1122 | list v0=freemodule(ncols(m)); |
---|
1123 | ideal a0=b0; |
---|
1124 | |
---|
1125 | while(b0<number(a[ncols(a)]-a[1])) |
---|
1126 | { |
---|
1127 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
1128 | b1=number(a[ncols(a)]-a[1]); |
---|
1129 | for(j=ncols(a);j>=1;j--) |
---|
1130 | { |
---|
1131 | for(i=ncols(a);i>=1;i--) |
---|
1132 | { |
---|
1133 | b2=number(a[i]-a[j]); |
---|
1134 | if(b2>b0&&b2<b1) |
---|
1135 | { |
---|
1136 | b1=b2; |
---|
1137 | } |
---|
1138 | else |
---|
1139 | { |
---|
1140 | if(b2<=b0) |
---|
1141 | { |
---|
1142 | i=0; |
---|
1143 | } |
---|
1144 | } |
---|
1145 | } |
---|
1146 | } |
---|
1147 | b0=b1; |
---|
1148 | |
---|
1149 | list l=ideal(); |
---|
1150 | for(k=ncols(m);k>=2;k--) |
---|
1151 | { |
---|
1152 | l=l+list(ideal()); |
---|
1153 | } |
---|
1154 | |
---|
1155 | dbprint(printlevel-voice+2,"// collect conditions for EV["+string(b0)+"]"); |
---|
1156 | j=ncols(a); |
---|
1157 | j0=mu; |
---|
1158 | while(j>=1) |
---|
1159 | { |
---|
1160 | i0=1; |
---|
1161 | i=1; |
---|
1162 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
1163 | { |
---|
1164 | i0=i0+d[i]; |
---|
1165 | i++; |
---|
1166 | } |
---|
1167 | if(a[i]<a[j]+b0) |
---|
1168 | { |
---|
1169 | i0=i0+d[i]; |
---|
1170 | i++; |
---|
1171 | } |
---|
1172 | for(k=1;k<=ncols(m);k++) |
---|
1173 | { |
---|
1174 | M0=M[k]; |
---|
1175 | if(i0>1) |
---|
1176 | { |
---|
1177 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
1178 | } |
---|
1179 | } |
---|
1180 | j0=j0-d[j]; |
---|
1181 | j--; |
---|
1182 | } |
---|
1183 | |
---|
1184 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
1185 | L=transpose(module(l[1])); |
---|
1186 | for(k=2;k<=ncols(m);k++) |
---|
1187 | { |
---|
1188 | L=L,transpose(module(l[k])); |
---|
1189 | } |
---|
1190 | |
---|
1191 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
1192 | v0=v0+list(syz(L)); |
---|
1193 | a0=a0,b0; |
---|
1194 | } |
---|
1195 | |
---|
1196 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1197 | option(redSB); |
---|
1198 | for(i=1;i<size(v0);i++) |
---|
1199 | { |
---|
1200 | v0[i+1]=std(v0[i+1]); |
---|
1201 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
1202 | } |
---|
1203 | |
---|
1204 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
1205 | i=1; |
---|
1206 | while(size(v0[i])==0) |
---|
1207 | { |
---|
1208 | i++; |
---|
1209 | } |
---|
1210 | list v1=v0[i]; |
---|
1211 | intvec d1=size(v0[i]); |
---|
1212 | ideal a1=a0[i]; |
---|
1213 | i++; |
---|
1214 | while(i<=size(v0)) |
---|
1215 | { |
---|
1216 | if(size(v0[i])>0) |
---|
1217 | { |
---|
1218 | v1=v1+list(v0[i]); |
---|
1219 | d1=d1,size(v0[i]); |
---|
1220 | a1=a1,a0[i]; |
---|
1221 | } |
---|
1222 | i++; |
---|
1223 | } |
---|
1224 | return(list(a1,d1,v1,m,g)); |
---|
1225 | } |
---|
1226 | example |
---|
1227 | { "EXAMPLE:"; echo=2; |
---|
1228 | ring R=0,(x,y),ds; |
---|
1229 | poly t=x5+x2y2+y5; |
---|
1230 | endvfilt(vfilt(t)); |
---|
1231 | } |
---|
1232 | /////////////////////////////////////////////////////////////////////////////// |
---|
1233 | |
---|
1234 | proc spprint(list sp) |
---|
1235 | "USAGE: spprint(sp); list sp |
---|
1236 | RETURN: string: spectrum sp |
---|
1237 | EXAMPLE: example spprint; shows examples |
---|
1238 | " |
---|
1239 | { |
---|
1240 | string s; |
---|
1241 | for(int i=1;i<size(sp[2]);i++) |
---|
1242 | { |
---|
1243 | s=s+"("+string(sp[1][i])+","+string(sp[2][i])+"),"; |
---|
1244 | } |
---|
1245 | s=s+"("+string(sp[1][i])+","+string(sp[2][i])+")"; |
---|
1246 | return(s); |
---|
1247 | } |
---|
1248 | example |
---|
1249 | { "EXAMPLE:"; echo=2; |
---|
1250 | ring R=0,(x,y),ds; |
---|
1251 | 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)); |
---|
1252 | spprint(sp); |
---|
1253 | } |
---|
1254 | /////////////////////////////////////////////////////////////////////////////// |
---|
1255 | |
---|
1256 | proc sppprint(list spp) |
---|
1257 | "USAGE: sppprint(spp); list spp |
---|
1258 | RETURN: string: spectral pairs spp |
---|
1259 | EXAMPLE: example sppprint; shows examples |
---|
1260 | " |
---|
1261 | { |
---|
1262 | string s; |
---|
1263 | for(int i=1;i<size(spp[3]);i++) |
---|
1264 | { |
---|
1265 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+"),"; |
---|
1266 | } |
---|
1267 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+")"; |
---|
1268 | return(s); |
---|
1269 | } |
---|
1270 | example |
---|
1271 | { "EXAMPLE:"; echo=2; |
---|
1272 | ring R=0,(x,y),ds; |
---|
1273 | 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)); |
---|
1274 | sppprint(spp); |
---|
1275 | } |
---|
1276 | /////////////////////////////////////////////////////////////////////////////// |
---|
1277 | |
---|
1278 | proc spadd(list sp1,list sp2) |
---|
1279 | "USAGE: spadd(sp1,sp2); list sp1,sp2 |
---|
1280 | RETURN: list: sum of spectra sp1 and sp2 |
---|
1281 | EXAMPLE: example spadd; shows examples |
---|
1282 | " |
---|
1283 | { |
---|
1284 | ideal s; |
---|
1285 | intvec m; |
---|
1286 | int i,i1,i2=1,1,1; |
---|
1287 | while(i1<=size(sp1[2])||i2<=size(sp2[2])) |
---|
1288 | { |
---|
1289 | if(i1<=size(sp1[2])) |
---|
1290 | { |
---|
1291 | if(i2<=size(sp2[2])) |
---|
1292 | { |
---|
1293 | if(number(sp1[1][i1])<number(sp2[1][i2])) |
---|
1294 | { |
---|
1295 | s[i]=sp1[1][i1]; |
---|
1296 | m[i]=sp1[2][i1]; |
---|
1297 | i++; |
---|
1298 | i1++; |
---|
1299 | } |
---|
1300 | else |
---|
1301 | { |
---|
1302 | if(number(sp1[1][i1])>number(sp2[1][i2])) |
---|
1303 | { |
---|
1304 | s[i]=sp2[1][i2]; |
---|
1305 | m[i]=sp2[2][i2]; |
---|
1306 | i++; |
---|
1307 | i2++; |
---|
1308 | } |
---|
1309 | else |
---|
1310 | { |
---|
1311 | if(sp1[2][i1]+sp2[2][i2]!=0) |
---|
1312 | { |
---|
1313 | s[i]=sp1[1][i1]; |
---|
1314 | m[i]=sp1[2][i1]+sp2[2][i2]; |
---|
1315 | i++; |
---|
1316 | } |
---|
1317 | i1++; |
---|
1318 | i2++; |
---|
1319 | } |
---|
1320 | } |
---|
1321 | } |
---|
1322 | else |
---|
1323 | { |
---|
1324 | s[i]=sp1[1][i1]; |
---|
1325 | m[i]=sp1[2][i1]; |
---|
1326 | i++; |
---|
1327 | i1++; |
---|
1328 | } |
---|
1329 | } |
---|
1330 | else |
---|
1331 | { |
---|
1332 | s[i]=sp2[1][i2]; |
---|
1333 | m[i]=sp2[2][i2]; |
---|
1334 | i++; |
---|
1335 | i2++; |
---|
1336 | } |
---|
1337 | } |
---|
1338 | return(list(s,m)); |
---|
1339 | } |
---|
1340 | example |
---|
1341 | { "EXAMPLE:"; echo=2; |
---|
1342 | ring R=0,(x,y),ds; |
---|
1343 | 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)); |
---|
1344 | spprint(sp1); |
---|
1345 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1346 | spprint(sp2); |
---|
1347 | spprint(spadd(sp1,sp2)); |
---|
1348 | } |
---|
1349 | /////////////////////////////////////////////////////////////////////////////// |
---|
1350 | |
---|
1351 | proc spsub(list sp1,list sp2) |
---|
1352 | "USAGE: spsub(sp1,sp2); list sp1,sp2 |
---|
1353 | RETURN: list: difference of spectra sp1 and sp2 |
---|
1354 | EXAMPLE: example spsub; shows examples |
---|
1355 | " |
---|
1356 | { |
---|
1357 | return(spadd(sp1,spmul(sp2,-1))); |
---|
1358 | } |
---|
1359 | example |
---|
1360 | { "EXAMPLE:"; echo=2; |
---|
1361 | ring R=0,(x,y),ds; |
---|
1362 | 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)); |
---|
1363 | spprint(sp1); |
---|
1364 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1365 | spprint(sp2); |
---|
1366 | spprint(spsub(sp1,sp2)); |
---|
1367 | } |
---|
1368 | /////////////////////////////////////////////////////////////////////////////// |
---|
1369 | |
---|
1370 | proc spmul(list #) |
---|
1371 | "USAGE: |
---|
1372 | @format |
---|
1373 | 1) spmul(sp,k); list sp, int k |
---|
1374 | 2) spmul(sp,k); list sp, intvec k |
---|
1375 | @end format |
---|
1376 | RETURN: |
---|
1377 | @format |
---|
1378 | 1) list: product of spectrum sp and integer k |
---|
1379 | 2) list: linear combination of spectra sp with coefficients k |
---|
1380 | @end format |
---|
1381 | EXAMPLE: example spmul; shows examples |
---|
1382 | " |
---|
1383 | { |
---|
1384 | if(size(#)==2) |
---|
1385 | { |
---|
1386 | if(typeof(#[1])=="list") |
---|
1387 | { |
---|
1388 | if(typeof(#[2])=="int") |
---|
1389 | { |
---|
1390 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1391 | } |
---|
1392 | if(typeof(#[2])=="intvec") |
---|
1393 | { |
---|
1394 | list sp0=list(ideal(),intvec(0)); |
---|
1395 | for(int i=size(#[2]);i>=1;i--) |
---|
1396 | { |
---|
1397 | sp0=spadd(sp0,spmul(#[1][i],#[2][i])); |
---|
1398 | } |
---|
1399 | return(sp0); |
---|
1400 | } |
---|
1401 | } |
---|
1402 | } |
---|
1403 | return(list(ideal(),intvec(0))); |
---|
1404 | } |
---|
1405 | example |
---|
1406 | { "EXAMPLE:"; echo=2; |
---|
1407 | ring R=0,(x,y),ds; |
---|
1408 | 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)); |
---|
1409 | spprint(sp); |
---|
1410 | spprint(spmul(sp,2)); |
---|
1411 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1412 | spprint(sp1); |
---|
1413 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1414 | spprint(sp2); |
---|
1415 | spprint(spmul(list(sp1,sp2),intvec(1,2))); |
---|
1416 | } |
---|
1417 | /////////////////////////////////////////////////////////////////////////////// |
---|
1418 | |
---|
1419 | proc spissemicont(list sp,list #) |
---|
1420 | "USAGE: spissemicont(sp[,opt]); list sp, int opt |
---|
1421 | RETURN: |
---|
1422 | @format |
---|
1423 | int k= |
---|
1424 | if opt=0: |
---|
1425 | 1, if sum of spectrum sp over all intervals [a,a+1) is positive |
---|
1426 | 0, if sum of spectrum sp over some interval [a,a+1) is negative |
---|
1427 | if opt=1: |
---|
1428 | 1, if sum of spectrum sp over all intervals [a,a+1) and (a,a+1) is positive |
---|
1429 | 0, if sum of spectrum sp over some interval [a,a+1) or (a,a+1) is negative |
---|
1430 | default: opt=0 |
---|
1431 | @end format |
---|
1432 | EXAMPLE: example spissemicont; shows examples |
---|
1433 | " |
---|
1434 | { |
---|
1435 | int opt=0; |
---|
1436 | if(size(#)>0) |
---|
1437 | { |
---|
1438 | if(typeof(#[1])=="int") |
---|
1439 | { |
---|
1440 | opt=1; |
---|
1441 | } |
---|
1442 | } |
---|
1443 | int i,j,k=1,1,0; |
---|
1444 | while(j<=size(sp[2])) |
---|
1445 | { |
---|
1446 | while(j+1<=size(sp[2])&&sp[1][j]<sp[1][i]+1) |
---|
1447 | { |
---|
1448 | k=k+sp[2][j]; |
---|
1449 | j++; |
---|
1450 | } |
---|
1451 | if(j==size(sp[2])&&sp[1][j]<sp[1][i]+1) |
---|
1452 | { |
---|
1453 | k=k+sp[2][j]; |
---|
1454 | j++; |
---|
1455 | } |
---|
1456 | if(k<0) |
---|
1457 | { |
---|
1458 | return(0); |
---|
1459 | } |
---|
1460 | k=k-sp[2][i]; |
---|
1461 | if(k<0&&opt==1) |
---|
1462 | { |
---|
1463 | return(0); |
---|
1464 | } |
---|
1465 | i++; |
---|
1466 | } |
---|
1467 | return(1); |
---|
1468 | } |
---|
1469 | example |
---|
1470 | { "EXAMPLE:"; echo=2; |
---|
1471 | ring R=0,(x,y),ds; |
---|
1472 | 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)); |
---|
1473 | spprint(sp1); |
---|
1474 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1475 | spprint(sp2); |
---|
1476 | spissemicont(spsub(sp1,spmul(sp2,5))); |
---|
1477 | spissemicont(spsub(sp1,spmul(sp2,5)),1); |
---|
1478 | spissemicont(spsub(sp1,spmul(sp2,6))); |
---|
1479 | } |
---|
1480 | /////////////////////////////////////////////////////////////////////////////// |
---|
1481 | |
---|
1482 | proc spsemicont(list sp0,list sp,list #) |
---|
1483 | "USAGE: spsemicont(sp,k[,opt]); list sp0, list sp, int opt |
---|
1484 | RETURN: list of intvecs l: |
---|
1485 | spissemicont(sub(sp0,spmul(sp,k)),opt)==1 iff k<=l[i] for some i |
---|
1486 | NOTE: if the spectra sp occur with multiplicities k in a deformation |
---|
1487 | of the [quasihomogeneous] spectrum sp0 then |
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1488 | spissemicont(sub(sp0,spmul(sp,k))[,1])==1 |
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1489 | EXAMPLE: example spsemicont; shows examples |
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1490 | " |
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1491 | { |
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1492 | list l,l0; |
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1493 | int i,j,k; |
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1494 | while(spissemicont(sp0,#)) |
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1495 | { |
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1496 | if(size(sp)>1) |
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1497 | { |
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1498 | l0=spsemicont(sp0,list(sp[1..size(sp)-1])); |
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1499 | for(i=1;i<=size(l0);i++) |
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1500 | { |
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1501 | if(size(l)>0) |
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1502 | { |
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1503 | j=1; |
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1504 | while(j<size(l)&&l[j]!=l0[i]) |
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1505 | { |
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1506 | j++; |
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1507 | } |
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1508 | if(l[j]==l0[i]) |
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1509 | { |
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1510 | l[j][size(sp)]=k; |
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1511 | } |
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1512 | else |
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1513 | { |
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1514 | l0[i][size(sp)]=k; |
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1515 | l=l+list(l0[i]); |
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1516 | } |
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1517 | } |
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1518 | else |
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1519 | { |
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1520 | l=l0; |
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1521 | } |
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1522 | } |
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1523 | } |
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1524 | sp0=spsub(sp0,sp[size(sp)]); |
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1525 | k++; |
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1526 | } |
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1527 | if(size(sp)>1) |
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1528 | { |
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1529 | return(l); |
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1530 | } |
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1531 | else |
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1532 | { |
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1533 | return(list(intvec(k-1))); |
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1534 | } |
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1535 | } |
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1536 | example |
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1537 | { "EXAMPLE:"; echo=2; |
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1538 | ring R=0,(x,y),ds; |
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1539 | 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)); |
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1540 | spprint(sp0); |
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1541 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
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1542 | spprint(sp1); |
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1543 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
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1544 | spprint(sp2); |
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1545 | list sp=sp1,sp2; |
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1546 | list l=spsemicont(sp0,sp); |
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1547 | l; |
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1548 | spissemicont(spsub(sp0,spmul(sp,l[1]))); |
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1549 | spissemicont(spsub(sp0,spmul(sp,l[1]-1))); |
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1550 | spissemicont(spsub(sp0,spmul(sp,l[1]+1))); |
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1551 | } |
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1552 | /////////////////////////////////////////////////////////////////////////////// |
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1553 | |
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1554 | proc spmilnor(list sp) |
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1555 | "USAGE: spmilnor(sp); list sp |
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1556 | RETURN: int: Milnor number of spectrum sp |
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1557 | EXAMPLE: example spmilnor; shows examples |
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1558 | " |
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1559 | { |
---|
1560 | return(sum(sp[2])); |
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1561 | } |
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1562 | example |
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1563 | { "EXAMPLE:"; echo=2; |
---|
1564 | ring R=0,(x,y),ds; |
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1565 | 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)); |
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1566 | spprint(sp); |
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1567 | spmilnor(sp); |
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1568 | } |
---|
1569 | /////////////////////////////////////////////////////////////////////////////// |
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1570 | |
---|
1571 | proc spgeomgenus(list sp) |
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1572 | "USAGE: spgeomgenus(sp); list sp |
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1573 | RETURN: int: geometrical genus of spectrum sp |
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1574 | EXAMPLE: example spgeomgenus; shows examples |
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1575 | " |
---|
1576 | { |
---|
1577 | int g=0; |
---|
1578 | int i=1; |
---|
1579 | while(i+1<=size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1580 | { |
---|
1581 | g=g+sp[2][i]; |
---|
1582 | i++; |
---|
1583 | } |
---|
1584 | if(i==size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1585 | { |
---|
1586 | g=g+sp[2][i]; |
---|
1587 | } |
---|
1588 | return(g); |
---|
1589 | } |
---|
1590 | example |
---|
1591 | { "EXAMPLE:"; echo=2; |
---|
1592 | ring R=0,(x,y),ds; |
---|
1593 | 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)); |
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1594 | spprint(sp); |
---|
1595 | spgeomgenus(sp); |
---|
1596 | } |
---|
1597 | /////////////////////////////////////////////////////////////////////////////// |
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1598 | |
---|
1599 | proc spgamma(list sp) |
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1600 | "USAGE: spgamma(sp); list sp |
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1601 | RETURN: number: gamma invariant of spectrum sp |
---|
1602 | EXAMPLE: example spgamma; shows examples |
---|
1603 | " |
---|
1604 | { |
---|
1605 | int i,j; |
---|
1606 | number g=0; |
---|
1607 | for(i=1;i<=ncols(sp[1]);i++) |
---|
1608 | { |
---|
1609 | for(j=1;j<=sp[2][i];j++) |
---|
1610 | { |
---|
1611 | g=g+(number(sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1612 | } |
---|
1613 | } |
---|
1614 | g=-g/4+sum(sp[2])*number(sp[1][ncols(sp[1])]-sp[1][1])/48; |
---|
1615 | return(g); |
---|
1616 | } |
---|
1617 | example |
---|
1618 | { "EXAMPLE:"; echo=2; |
---|
1619 | ring R=0,(x,y),ds; |
---|
1620 | 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)); |
---|
1621 | spprint(sp); |
---|
1622 | spgamma(sp); |
---|
1623 | } |
---|
1624 | /////////////////////////////////////////////////////////////////////////////// |
---|