1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: gaussman.lib,v 1.50 2001-08-13 11:40:58 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[,opt]); Jordan data or eigenvalues of monodromy of t |
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18 | spectrum(t); spectrum of t |
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19 | sppairs(t[,opt]); spectral pairs or spectrum of t |
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20 | vfilt(t[,opt]); V-filtration on H''/H' or spectrum of t |
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21 | endfilt(t,V); endomorphism filtration of V-filtration V |
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22 | spgen(a); generate spectrum defined by a |
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23 | sppgen(a,w); generate spectral pairs defined by a and w |
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24 | spprint(list Sp); print spectrum or spectral pairs Sp |
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25 | spadd(list Sp1,list Sp2); sum of spectra Sp1 and Sp2 |
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26 | spsub(list Sp1,list Sp2); difference of spectra Sp1 and Sp2 |
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27 | spmul(list Sp,int k); product of spectrum Sp and integer k |
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28 | spmul(list Sp,intvec k); linear combination of spectra Sp with coeffs k |
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29 | spissemicont(list Sp[,opt]); test spectrum Sp for semicontinuity |
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30 | spsemicont(list Sp0,list Sp[,opt]); semicontinuity of spectra Sp0 and Sp |
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31 | spmilnor(list Sp); milnor number of spectrum Sp |
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32 | spgeomgenus(list Sp); geometrical genus of spectrum Sp |
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33 | spgamma(list Sp); gamma invariant of spectrum Sp |
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34 | |
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35 | SEE ALSO: mondromy_lib, spectrum_lib |
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36 | |
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37 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
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38 | monodromy; spectrum; spectral pairs; |
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39 | Hodge filtration; V-filtration; weight filtration |
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40 | "; |
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41 | |
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42 | LIB "linalg.lib"; |
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43 | |
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44 | /////////////////////////////////////////////////////////////////////////////// |
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45 | |
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46 | static proc stdtrans(ideal I) |
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47 | { |
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48 | def R=basering; |
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49 | |
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50 | string os=ordstr(R); |
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51 | int j=find(os,",C"); |
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52 | if(j==0) |
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53 | { |
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54 | j=find(os,"C,"); |
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55 | } |
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56 | if(j==0) |
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57 | { |
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58 | j=find(os,",c"); |
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59 | } |
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60 | if(j==0) |
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61 | { |
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62 | j=find(os,"c,"); |
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63 | } |
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64 | if(j>0) |
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65 | { |
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66 | os[j..j+1]=" "; |
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67 | } |
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68 | |
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69 | execute("ring S="+charstr(R)+",(gmspoly,"+varstr(R)+"),(c,dp,"+os+");"); |
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70 | |
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71 | ideal I=homog(imap(R,I),gmspoly); |
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72 | module M=transpose(transpose(I)+freemodule(ncols(I))); |
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73 | M=std(M); |
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74 | |
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75 | setring(R); |
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76 | execute("map h=S,1,"+varstr(R)+";"); |
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77 | module M=h(M); |
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78 | |
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79 | for(int i=ncols(M);i>=1;i--) |
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80 | { |
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81 | for(j=ncols(M);j>=1;j--) |
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82 | { |
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83 | if(M[i][1]==0) |
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84 | { |
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85 | M[i]=0; |
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86 | } |
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87 | if(i!=j&&M[j][1]!=0) |
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88 | { |
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89 | if(lead(M[i][1])/lead(M[j][1])!=0) |
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90 | { |
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91 | M[i]=0; |
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92 | } |
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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 | M=transpose(simplify(M,2)); |
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98 | I=M[1]; |
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99 | attrib(I,"isSB",1); |
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100 | M=M[2..ncols(M)]; |
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101 | module U=transpose(M); |
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102 | |
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103 | return(list(I,U)); |
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104 | } |
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105 | /////////////////////////////////////////////////////////////////////////////// |
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106 | |
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107 | proc gmsring(poly t,string s) |
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108 | "USAGE: gmsring(t,s); poly t, string s; |
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109 | ASSUME: basering with characteristic 0 and local degree ordering, |
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110 | t with isolated singularity at 0 |
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111 | RETURN: |
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112 | @format |
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113 | ring G: C{{s}}*basering, |
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114 | poly gmspoly: image of t |
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115 | ideal gmsjacob: image of Jacobian ideal |
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116 | ideal gmsstd: image of standard basis of Jacobian ideal |
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117 | matrix gmsmatrix: matrix(gmsjacob)*gmsmatrix=matrix(gmsstd) |
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118 | ideal gmsbasis: image of monomial vector space basis of Jacobian algebra |
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119 | int gmsmaxweight: maximal weight of variables of basering |
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120 | @end format |
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121 | NOTE: do not modify gms variables if you want to use gms procedures |
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122 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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123 | EXAMPLE: example gms; shows examples |
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124 | " |
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125 | { |
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126 | def R=basering; |
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127 | if(charstr(R)!="0") |
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128 | { |
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129 | ERROR("characteristic 0 expected"); |
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130 | } |
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131 | for(int i=nvars(R);i>=1;i--) |
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132 | { |
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133 | if(var(i)>1) |
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134 | { |
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135 | ERROR("local ordering expected"); |
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136 | } |
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137 | } |
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138 | |
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139 | ideal dt=jacob(t); |
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140 | list l=stdtrans(dt); |
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141 | ideal g=l[1]; |
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142 | if(vdim(g)<=0) |
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143 | { |
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144 | if(vdim(g)==0) |
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145 | { |
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146 | ERROR("singularity at 0 expected"); |
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147 | } |
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148 | else |
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149 | { |
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150 | ERROR("isolated singularity at 0 expected"); |
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151 | } |
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152 | } |
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153 | matrix a=l[2]; |
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154 | ideal m=kbase(g); |
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155 | |
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156 | int gmsmaxweight; |
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157 | for(i=nvars(R);i>=1;i--) |
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158 | { |
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159 | if(deg(var(i))>gmsmaxweight) |
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160 | { |
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161 | gmsmaxweight=deg(var(i)); |
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162 | } |
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163 | } |
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164 | |
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165 | string os=ordstr(R); |
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166 | int j=find(os,",C"); |
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167 | if(j==0) |
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168 | { |
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169 | j=find(os,"C,"); |
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170 | } |
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171 | if(j==0) |
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172 | { |
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173 | j=find(os,",c"); |
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174 | } |
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175 | if(j==0) |
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176 | { |
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177 | j=find(os,"c,"); |
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178 | } |
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179 | if(j>0) |
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180 | { |
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181 | os[j..j+1]=" "; |
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182 | } |
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183 | |
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184 | execute("ring G="+string(charstr(R))+",("+s+","+varstr(R)+"),(ws("+ |
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185 | string(deg(highcorner(g))+2*gmsmaxweight)+"),"+os+",c);"); |
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186 | |
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187 | poly gmspoly=imap(R,t); |
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188 | ideal gmsjacob=imap(R,dt); |
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189 | ideal gmsstd=imap(R,g); |
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190 | matrix gmsmatrix=imap(R,a); |
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191 | ideal gmsbasis=imap(R,m); |
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192 | |
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193 | attrib(gmsstd,"isSB",1); |
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194 | export gmspoly,gmsjacob,gmsstd,gmsmatrix,gmsbasis,gmsmaxweight; |
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195 | |
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196 | return(G); |
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197 | } |
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198 | example |
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199 | { "EXAMPLE:"; echo=2; |
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200 | ring R=0,(x,y),ds; |
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201 | poly t=x5+x2y2+y5; |
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202 | def G=gmsring(t,"s"); |
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203 | setring(G); |
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204 | gmspoly; |
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205 | print(gmsjacob); |
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206 | print(gmsstd); |
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207 | print(gmsmatrix); |
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208 | print(gmsbasis); |
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209 | gmsmaxweight; |
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210 | } |
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211 | /////////////////////////////////////////////////////////////////////////////// |
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212 | |
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213 | proc gmsnf(ideal p,int K,list #) |
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214 | "USAGE: gmsnf(p,K[,Kmax]); poly p, int K[, int Kmax]; |
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215 | ASSUME: basering constructed by gmsring, K<=Kmax |
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216 | RETURN: |
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217 | @format |
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218 | list l: |
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219 | ideal l[1]: projection of p to H''=C{{s}}*gmsbasis mod s^{K+1} |
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220 | ideal l[2]: p=l[1]+l[2] mod s^(Kmax+1) |
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221 | @end format |
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222 | NOTE: by setting p=l[2] the computation can be continued up to order |
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223 | at most Kmax, by default Kmax=infinity |
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224 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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225 | EXAMPLE: example gmsnf; shows examples |
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226 | " |
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227 | { |
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228 | int Kmax=-1; |
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229 | if(size(#)>0) |
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230 | { |
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231 | if(typeof(#[1])=="int") |
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232 | { |
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233 | Kmax=#[1]; |
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234 | if(K>Kmax) |
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235 | { |
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236 | Kmax=K; |
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237 | } |
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238 | } |
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239 | } |
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240 | |
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241 | intvec v=1; |
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242 | v[nvars(basering)]=0; |
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243 | |
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244 | int k; |
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245 | if(Kmax>=0) |
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246 | { |
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247 | p=jet(jet(p,K,v),(Kmax+1)*deg(var(1))-2*gmsmaxweight); |
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248 | } |
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249 | |
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250 | ideal r,q; |
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251 | r[ncols(p)]=0; |
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252 | q[ncols(p)]=0; |
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253 | |
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254 | poly s; |
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255 | int i,j; |
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256 | for(k=ncols(p);k>=1;k--) |
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257 | { |
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258 | while(p[k]!=0&°(lead(p[k]),v)<=K) |
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259 | { |
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260 | i=1; |
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261 | s=lead(p[k])/lead(gmsstd[i]); |
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262 | while(i<ncols(gmsstd)&&s==0) |
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263 | { |
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264 | i++; |
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265 | s=lead(p[k])/lead(gmsstd[i]); |
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266 | } |
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267 | if(s!=0) |
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268 | { |
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269 | p[k]=p[k]-s*gmsstd[i]; |
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270 | for(j=1;j<=nrows(gmsmatrix);j++) |
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271 | { |
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272 | if(Kmax>=0) |
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273 | { |
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274 | p[k]=p[k]+ |
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275 | jet(jet(diff(s*gmsmatrix[j,i],var(j+1))*var(1),Kmax,v), |
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276 | (Kmax+1)*deg(var(1))-2*gmsmaxweight); |
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277 | } |
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278 | else |
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279 | { |
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280 | p[k]=p[k]+diff(s*gmsmatrix[j,i],var(j+1))*var(1); |
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281 | } |
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282 | } |
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283 | } |
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284 | else |
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285 | { |
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286 | r[k]=r[k]+lead(p[k]); |
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287 | p[k]=p[k]-lead(p[k]); |
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288 | } |
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289 | while(deg(lead(p[k]))>(K+1)*deg(var(1))-2*gmsmaxweight&& |
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290 | deg(lead(p[k]),v)<=K) |
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291 | { |
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292 | q[k]=q[k]+lead(p[k]); |
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293 | p[k]=p[k]-lead(p[k]); |
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294 | } |
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295 | } |
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296 | q[k]=q[k]+p[k]; |
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297 | } |
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298 | |
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299 | return(list(r,q)); |
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300 | } |
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301 | example |
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302 | { "EXAMPLE:"; echo=2; |
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303 | ring R=0,(x,y),ds; |
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304 | poly t=x5+x2y2+y5; |
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305 | def G=gmsring(t,"s"); |
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306 | setring(G); |
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307 | list l0=gmsnf(gmspoly,0); |
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308 | print(l0[1]); |
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309 | list l1=gmsnf(gmspoly,1); |
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310 | print(l1[1]); |
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311 | list l=gmsnf(l0[2],1); |
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312 | print(l[1]); |
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313 | } |
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314 | /////////////////////////////////////////////////////////////////////////////// |
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315 | |
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316 | proc gmscoeffs(ideal p,int K,list #) |
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317 | "USAGE: gmscoeffs(p,K[,Kmax]); poly p, int K[, int Kmax]; |
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318 | ASSUME: basering constructed by gmsring, K<=Kmax |
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319 | RETURN: |
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320 | @format |
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321 | list l: |
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322 | matrix l[1]: projection of p to H''=C{{s}}*gmsbasis=C{{s}}^mu mod s^(K+1) |
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323 | ideal l[2]: p=matrix(gmsbasis)*l[1]+l[2] mod s^(Kmax+1) |
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324 | @end format |
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325 | NOTE: by setting p=l[2] the computation can be continued up to order |
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326 | at most Kmax, by default Kmax=infinity |
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327 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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328 | EXAMPLE: example gmscoeffs; shows examples |
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329 | " |
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330 | { |
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331 | list l=gmsnf(p,K,#); |
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332 | ideal r,q=l[1..2]; |
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333 | poly v=1; |
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334 | for(int i=2;i<=nvars(basering);i++) |
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335 | { |
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336 | v=v*var(i); |
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337 | } |
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338 | matrix C=coeffs(r,gmsbasis,v); |
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339 | return(C,q); |
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340 | } |
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341 | example |
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342 | { "EXAMPLE:"; echo=2; |
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343 | ring R=0,(x,y),ds; |
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344 | poly t=x5+x2y2+y5; |
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345 | def G=gmsring(t,"s"); |
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346 | setring(G); |
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347 | list l0=gmscoeffs(gmspoly,0); |
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348 | print(l0[1]); |
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349 | list l1=gmscoeffs(gmspoly,1); |
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350 | print(l1[1]); |
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351 | list l=gmscoeffs(l0[2],1); |
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352 | print(l[1]); |
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353 | } |
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354 | /////////////////////////////////////////////////////////////////////////////// |
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355 | |
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356 | static proc maxintdif(ideal e) |
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357 | { |
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358 | int i,j,d; |
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359 | int d0=0; |
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360 | for(i=ncols(e);i>=1;i--) |
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361 | { |
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362 | for(j=i-1;j>=1;j--) |
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363 | { |
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364 | d=int(e[i]-e[j]); |
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365 | if(d<0) |
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366 | { |
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367 | d=-d; |
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368 | } |
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369 | if(d>d0) |
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370 | { |
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371 | d0=d; |
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372 | } |
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373 | } |
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374 | } |
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375 | return(d0); |
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376 | } |
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377 | /////////////////////////////////////////////////////////////////////////////// |
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378 | |
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379 | proc monodromy(poly t,list #) |
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380 | "USAGE: monodromy(t[,opt]); poly t, int opt |
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381 | ASSUME: basering with characteristic 0 and local degree ordering, |
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382 | t with isolated singularity at 0 |
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383 | RETURN: |
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384 | @format |
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385 | if opt<=0: |
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386 | list l: |
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387 | ideal l[1]: exp(-2*pi*i*l[1]) are the eigenvalues of the monodromy |
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388 | if opt>=1: |
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389 | list l: Jordan data jordan(M) of a monodromy matrix exp(-2*pi*i*M) |
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390 | default: opt=1 |
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391 | @end format |
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392 | SEE ALSO: mondromy_lib |
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393 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; monodromy |
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394 | EXAMPLE: example monodromy; shows examples |
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395 | " |
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396 | { |
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397 | int opt=1; |
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398 | if(size(#)>0) |
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399 | { |
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400 | if(typeof(#[1])=="int") |
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401 | { |
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402 | opt=#[1]; |
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403 | } |
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404 | } |
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405 | |
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406 | def R=basering; |
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407 | int n=nvars(R)-1; |
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408 | def G=gmsring(t,"s"); |
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409 | setring G; |
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410 | |
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411 | int mu=ncols(gmsbasis); |
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412 | ideal r=gmspoly*gmsbasis; |
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413 | list l; |
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414 | matrix A0[mu][mu],C; |
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415 | module H,H1=freemodule(mu),freemodule(mu); |
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416 | module H0; |
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417 | int k=-1; |
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418 | while(size(reduce(H,std(H0*s)))>0) |
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419 | { |
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420 | k++; |
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421 | dbprint(printlevel-voice+2,"// k="+string(k)); |
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422 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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423 | if(opt<=0) |
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424 | { |
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425 | l=gmscoeffs(r,k,mu); |
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426 | } |
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427 | else |
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428 | { |
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429 | l=gmscoeffs(r,k,mu+n); |
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430 | } |
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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,"// compute saturation of H''"); |
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435 | H0=H; |
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436 | H1=jet(module(A0*H1+s^2*diff(matrix(H1),s)),k); |
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437 | H=H*s+H1; |
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438 | } |
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439 | A0=A0-k*s; |
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440 | |
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441 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
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442 | H=std(H0); |
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443 | int d0=maxdeg1(H); |
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444 | dbprint(printlevel-voice+2,"// k="+string(d0+1)); |
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445 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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446 | if(opt<=0) |
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447 | { |
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448 | l=gmscoeffs(r,d0+1,d0+1); |
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449 | } |
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450 | else |
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451 | { |
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452 | l=gmscoeffs(r,d0+1,d0+n+1); |
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453 | } |
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454 | C,r=l[1..2]; |
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455 | A0=A0+C; |
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456 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
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457 | l=division(H*s,A0*H+s^2*diff(matrix(H),s)); |
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458 | matrix A=jet(l[1],l[2],0); |
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459 | |
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460 | dbprint(printlevel-voice+2, |
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461 | "// compute eigenvalues e with multiplicities m of A"); |
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462 | l=eigenval(jet(A,0)); |
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463 | def e,m=l[1..2]; |
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464 | dbprint(printlevel-voice+2,"// e="+string(e)); |
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465 | dbprint(printlevel-voice+2,"// m="+string(m)); |
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466 | |
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467 | if(opt<=0) |
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468 | { |
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469 | setring(R); |
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470 | ideal e=imap(G,e); |
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471 | return(list(e)); |
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472 | } |
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473 | |
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474 | dbprint(printlevel-voice+2,"// compute maximal integer difference d1 of e"); |
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475 | int d1=maxintdif(e); |
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476 | dbprint(printlevel-voice+2,"// d1="+string(d1)); |
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477 | |
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478 | module U; |
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479 | if(d1>0) |
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480 | { |
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481 | dbprint(printlevel-voice+2,"// k="+string(d0+d1+1)); |
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482 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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483 | l=gmscoeffs(r,d0+d1+1,d0+d1+1); |
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484 | C,r=l[1..2]; |
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485 | A0=A0+C; |
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486 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
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487 | l=division(H*s,A0*H+s^2*diff(matrix(H),s)); |
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488 | A=jet(l[1],l[2],d1); |
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489 | |
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490 | intvec d; |
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491 | d[mu]=0; |
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492 | int i,j; |
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493 | for(i=ncols(e);i>=1;i--) |
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494 | { |
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495 | for(j=i-1;j>=1;j--) |
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496 | { |
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497 | k=int(e[j]-e[i]); |
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498 | if(k>d[j]) |
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499 | { |
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500 | d[j]=k; |
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501 | } |
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502 | if(-k>d[i]) |
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503 | { |
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504 | d[i]=-k; |
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505 | } |
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506 | } |
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507 | } |
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508 | for(j,k=ncols(e),mu;j>=1;j--) |
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509 | { |
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510 | for(i=m[j];i>=1;i--) |
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511 | { |
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512 | d[k]=d[j]; |
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513 | k--; |
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514 | } |
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515 | } |
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516 | |
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517 | while(d1>0) |
---|
518 | { |
---|
519 | dbprint(printlevel-voice+2,"// transform basis to reduce d1 by 1"); |
---|
520 | |
---|
521 | A0=jet(A,0); |
---|
522 | U=0; |
---|
523 | for(i=ncols(e);i>=1;i--) |
---|
524 | { |
---|
525 | U=syz(power(A0-e[i],n+1))+U; |
---|
526 | } |
---|
527 | A=lift(U,A*U); |
---|
528 | |
---|
529 | for(i=mu;i>=1;i--) |
---|
530 | { |
---|
531 | for(j=mu;j>=1;j--) |
---|
532 | { |
---|
533 | if(d[i]==0&&d[j]>0) |
---|
534 | { |
---|
535 | A[i,j]=A[i,j]/s; |
---|
536 | } |
---|
537 | else |
---|
538 | { |
---|
539 | if(d[i]>0&&d[j]==0) |
---|
540 | { |
---|
541 | A[i,j]=A[i,j]*s; |
---|
542 | } |
---|
543 | } |
---|
544 | } |
---|
545 | } |
---|
546 | for(i=mu;i>=1;i--) |
---|
547 | { |
---|
548 | if(d[i]>0) |
---|
549 | { |
---|
550 | A[i,i]=A[i,i]-1; |
---|
551 | d[i]=d[i]-1; |
---|
552 | } |
---|
553 | } |
---|
554 | |
---|
555 | d1--; |
---|
556 | dbprint(printlevel-voice+2,"// d1="+string(d1)); |
---|
557 | } |
---|
558 | |
---|
559 | A=jet(A,0); |
---|
560 | } |
---|
561 | |
---|
562 | setring(R); |
---|
563 | matrix A=imap(G,A); |
---|
564 | return(jordan(A)); |
---|
565 | } |
---|
566 | example |
---|
567 | { "EXAMPLE:"; echo=2; |
---|
568 | ring R=0,(x,y),ds; |
---|
569 | poly f=x5+x2y2+y5; |
---|
570 | print(monodromy(f)); |
---|
571 | } |
---|
572 | /////////////////////////////////////////////////////////////////////////////// |
---|
573 | |
---|
574 | proc spectrum(poly t) |
---|
575 | "USAGE: spectrum(t); poly t |
---|
576 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
577 | t with isolated singularity at 0 |
---|
578 | RETURN: |
---|
579 | @format |
---|
580 | list Sp: spectrum of t |
---|
581 | ideal Sp[1]: spectral numbers in increasing order |
---|
582 | intvec Sp[2]: |
---|
583 | int Sp[2][i]: multiplicity of spectral number Sp[1][i] |
---|
584 | @end format |
---|
585 | SEE ALSO: spectrum_lib |
---|
586 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; spectrum |
---|
587 | EXAMPLE: example spnumbers; shows examples |
---|
588 | " |
---|
589 | { |
---|
590 | return(sppairs(t,0)); |
---|
591 | } |
---|
592 | example |
---|
593 | { "EXAMPLE:"; echo=2; |
---|
594 | ring R=0,(x,y),ds; |
---|
595 | poly t=x5+x2y2+y5; |
---|
596 | spprint(spectrum(t)); |
---|
597 | } |
---|
598 | /////////////////////////////////////////////////////////////////////////////// |
---|
599 | |
---|
600 | proc sppairs(poly t,list #) |
---|
601 | "USAGE: sppairs(t[,opt]); poly t, int opt |
---|
602 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
603 | t with isolated singularity at 0 |
---|
604 | RETURN: list l: |
---|
605 | @format |
---|
606 | ideal l[1]: spectral numbers in increasing order |
---|
607 | if opt<=0: |
---|
608 | intvec l[2]: |
---|
609 | int l[2][i]: multiplicity of spectral number l[1][i] |
---|
610 | if opt>=1: |
---|
611 | intvec l[2]: weight filtration indices in decreasing order |
---|
612 | intvec l[3]: |
---|
613 | int l[3][i]: multiplicity of spectral pair (l[1][i],l[2][i]) |
---|
614 | default: opt=1 |
---|
615 | @end format |
---|
616 | SEE ALSO: spectrum_lib |
---|
617 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
618 | spectrum; spectral pairs |
---|
619 | EXAMPLE: example sppairs; shows examples |
---|
620 | " |
---|
621 | { |
---|
622 | int opt=1; |
---|
623 | if(size(#)>0) |
---|
624 | { |
---|
625 | if(typeof(#[1])=="int") |
---|
626 | { |
---|
627 | opt=#[1]; |
---|
628 | } |
---|
629 | } |
---|
630 | |
---|
631 | def R=basering; |
---|
632 | int n=nvars(R)-1; |
---|
633 | def G=gmsring(t,"s"); |
---|
634 | setring(G); |
---|
635 | |
---|
636 | int mu=ncols(gmsbasis); |
---|
637 | ideal r=gmspoly*gmsbasis; |
---|
638 | list l; |
---|
639 | matrix A0[mu][mu],C; |
---|
640 | module H0; |
---|
641 | module H,H1=freemodule(mu),freemodule(mu); |
---|
642 | int k=-1; |
---|
643 | while(size(reduce(H,std(H0*s)))>0) |
---|
644 | { |
---|
645 | k++; |
---|
646 | dbprint(printlevel-voice+2,"// k="+string(k)); |
---|
647 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
648 | if(opt<=0) |
---|
649 | { |
---|
650 | l=gmscoeffs(r,k,mu); |
---|
651 | } |
---|
652 | else |
---|
653 | { |
---|
654 | l=gmscoeffs(r,k,mu+n); |
---|
655 | } |
---|
656 | C,r=l[1..2]; |
---|
657 | A0=A0+C; |
---|
658 | |
---|
659 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
---|
660 | H0=H; |
---|
661 | H1=jet(module(A0*H1+s^2*diff(matrix(H1),s)),k); |
---|
662 | H=H*s+H1; |
---|
663 | } |
---|
664 | A0=A0-k*s; |
---|
665 | |
---|
666 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
---|
667 | H=std(H0); |
---|
668 | int d0=maxdeg1(H); |
---|
669 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
---|
670 | l=division(H,freemodule(mu)*s^k); |
---|
671 | H0=l[1]; |
---|
672 | |
---|
673 | dbprint(printlevel-voice+2,"// k="+string(d0+1)); |
---|
674 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
675 | if(opt<=0) |
---|
676 | { |
---|
677 | l=gmscoeffs(r,d0+1,d0+1); |
---|
678 | } |
---|
679 | else |
---|
680 | { |
---|
681 | l=gmscoeffs(r,d0+1,d0+n+1); |
---|
682 | } |
---|
683 | C,r=l[1..2]; |
---|
684 | A0=A0+C; |
---|
685 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
---|
686 | l=division(H*s,A0*H+s^2*diff(matrix(H),s)); |
---|
687 | matrix A=jet(l[1],l[2],0); |
---|
688 | |
---|
689 | int i,j; |
---|
690 | module U,V; |
---|
691 | if(opt<=0) |
---|
692 | { |
---|
693 | dbprint(printlevel-voice+2,"// transform to Jordan basis"); |
---|
694 | U=jordanbasis(A)[1]; |
---|
695 | V=lift(U,freemodule(mu)); |
---|
696 | A=V*A*U; |
---|
697 | dbprint(printlevel-voice+2,"// compute normal form of H''"); |
---|
698 | H0=std(V*H0); |
---|
699 | |
---|
700 | dbprint(printlevel-voice+2,"// compute spectral numbers"); |
---|
701 | ideal a; |
---|
702 | for(i=1;i<=mu;i++) |
---|
703 | { |
---|
704 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
705 | a[i]=A[j,j]+deg(lead(H0[i]))/deg(s)-1; |
---|
706 | } |
---|
707 | |
---|
708 | setring(R); |
---|
709 | return(spgen(imap(G,a))); |
---|
710 | } |
---|
711 | |
---|
712 | dbprint(printlevel-voice+2, |
---|
713 | "// compute eigenvalues e with multiplicities m of A"); |
---|
714 | l=eigenval(A); |
---|
715 | def e,m=l[1..2]; |
---|
716 | dbprint(printlevel-voice+2,"// e="+string(e)); |
---|
717 | dbprint(printlevel-voice+2,"// m="+string(m)); |
---|
718 | dbprint(printlevel-voice+2,"// compute maximal integer difference d1 of e"); |
---|
719 | int d1=maxintdif(e); |
---|
720 | dbprint(printlevel-voice+2,"// d1="+string(d1)); |
---|
721 | |
---|
722 | if(d1>0) |
---|
723 | { |
---|
724 | dbprint(printlevel-voice+2,"// k="+string(d0+d1+1)); |
---|
725 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
726 | l=gmscoeffs(r,d0+d1+1,d0+d1+1); |
---|
727 | C,r=l[1..2]; |
---|
728 | A0=A0+C; |
---|
729 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
---|
730 | l=division(H*s,A0*H+s^2*diff(matrix(H),s)); |
---|
731 | A=jet(l[1],l[2],d1); |
---|
732 | |
---|
733 | intvec d; |
---|
734 | d[mu]=0; |
---|
735 | for(i=ncols(e);i>=1;i--) |
---|
736 | { |
---|
737 | for(j=i-1;j>=1;j--) |
---|
738 | { |
---|
739 | k=int(e[j]-e[i]); |
---|
740 | if(k>d[j]) |
---|
741 | { |
---|
742 | d[j]=k; |
---|
743 | } |
---|
744 | if(-k>d[i]) |
---|
745 | { |
---|
746 | d[i]=-k; |
---|
747 | } |
---|
748 | } |
---|
749 | } |
---|
750 | for(j,k=ncols(e),mu;j>=1;j--) |
---|
751 | { |
---|
752 | for(i=m[j];i>=1;i--) |
---|
753 | { |
---|
754 | d[k]=d[j]; |
---|
755 | k--; |
---|
756 | } |
---|
757 | } |
---|
758 | |
---|
759 | while(d1>0) |
---|
760 | { |
---|
761 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
---|
762 | A0=jet(A,0); |
---|
763 | U=0; |
---|
764 | for(i=ncols(e);i>=1;i--) |
---|
765 | { |
---|
766 | U=syz(power(A0-e[i],n+1))+U; |
---|
767 | } |
---|
768 | V=lift(U,freemodule(mu)); |
---|
769 | A=V*A*U; |
---|
770 | H0=V*H0; |
---|
771 | |
---|
772 | dbprint(printlevel-voice+2,"// transform to reduce d1 by 1"); |
---|
773 | for(i=mu;i>=1;i--) |
---|
774 | { |
---|
775 | for(j=mu;j>=1;j--) |
---|
776 | { |
---|
777 | if(d[i]==0&&d[j]>0) |
---|
778 | { |
---|
779 | A[i,j]=A[i,j]/s; |
---|
780 | } |
---|
781 | else |
---|
782 | { |
---|
783 | if(d[i]>0&&d[j]==0) |
---|
784 | { |
---|
785 | A[i,j]=A[i,j]*s; |
---|
786 | } |
---|
787 | } |
---|
788 | } |
---|
789 | } |
---|
790 | H0=transpose(H0); |
---|
791 | for(i=mu;i>=1;i--) |
---|
792 | { |
---|
793 | if(d[i]>0) |
---|
794 | { |
---|
795 | A[i,i]=A[i,i]-1; |
---|
796 | d[i]=d[i]-1; |
---|
797 | H0[i]=H0[i]*s; |
---|
798 | } |
---|
799 | } |
---|
800 | H0=transpose(H0); |
---|
801 | |
---|
802 | d1--; |
---|
803 | dbprint(printlevel-voice+2,"// d1="+string(d1)); |
---|
804 | } |
---|
805 | |
---|
806 | A=jet(A,0); |
---|
807 | } |
---|
808 | |
---|
809 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
810 | intvec w0; |
---|
811 | l=jordanbasis(A); |
---|
812 | U,w0=l[1..2]; |
---|
813 | V=lift(U,freemodule(mu)); |
---|
814 | A0=jet(V*A*U,0); |
---|
815 | vector u; |
---|
816 | i=1; |
---|
817 | while(i<ncols(A0)) |
---|
818 | { |
---|
819 | j=i+1; |
---|
820 | while(j<ncols(A0)&&A0[i,i]==A0[j,j]) |
---|
821 | { |
---|
822 | if(w0[i]<w0[j]) |
---|
823 | { |
---|
824 | k=w0[i]; |
---|
825 | w0[i]=w0[j]; |
---|
826 | w0[i]=k; |
---|
827 | u=U[i]; |
---|
828 | U[i]=U[j]; |
---|
829 | U[j]=u; |
---|
830 | } |
---|
831 | j++; |
---|
832 | } |
---|
833 | if(j==ncols(A0)&&A0[i,i]==A0[j,j]&&w0[i]<w0[j]) |
---|
834 | { |
---|
835 | k=w0[i]; |
---|
836 | w0[i]=w0[j]; |
---|
837 | w0[i]=k; |
---|
838 | u=U[i]; |
---|
839 | U[i]=U[j]; |
---|
840 | U[j]=u; |
---|
841 | } |
---|
842 | i=j; |
---|
843 | } |
---|
844 | |
---|
845 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
846 | V=lift(U,freemodule(mu)); |
---|
847 | A=V*A*U; |
---|
848 | dbprint(printlevel-voice+2,"// compute normal form of H''"); |
---|
849 | H0=std(V*H0); |
---|
850 | |
---|
851 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
852 | ideal a; |
---|
853 | intvec w; |
---|
854 | for(i=1;i<=mu;i++) |
---|
855 | { |
---|
856 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
857 | a[i]=A[j,j]+deg(lead(H0[i]))/deg(s)-1; |
---|
858 | w[i]=w0[j]+n; |
---|
859 | } |
---|
860 | setring(R); |
---|
861 | return(sppgen(imap(G,a),w)); |
---|
862 | } |
---|
863 | example |
---|
864 | { "EXAMPLE:"; echo=2; |
---|
865 | ring R=0,(x,y),ds; |
---|
866 | poly t=x5+x2y2+y5; |
---|
867 | spprint(sppairs(t)); |
---|
868 | } |
---|
869 | /////////////////////////////////////////////////////////////////////////////// |
---|
870 | |
---|
871 | proc vfilt(poly t,list #) |
---|
872 | "USAGE: vfilt(t[,opt]); poly t, int opt |
---|
873 | ASSUME: basering with characteristic 0 and local degree ordering, |
---|
874 | t with isolated singularity at 0 |
---|
875 | RETURN: |
---|
876 | @format |
---|
877 | list V: V-filtration of t on H''/H' |
---|
878 | intvec V[1]: spectral numbers in increasing order |
---|
879 | intvec V[2]: |
---|
880 | int V[2][i]: multiplicity of spectral number V[1][i]/V[2][i] |
---|
881 | if opt>=1: |
---|
882 | list V[4]: |
---|
883 | module V[3][i]: vector space basis of V[1][i]/V[2][i]-th graded part |
---|
884 | in terms of V[5] |
---|
885 | ideal V[4]: monomial vector space basis of H''/H' |
---|
886 | ideal V[5]: standard basis of Jacobian ideal |
---|
887 | default: opt=1 |
---|
888 | @end format |
---|
889 | NOTE: H' and H'' denote the Brieskorn lattices |
---|
890 | SEE ALSO: spectrum_lib |
---|
891 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
892 | Hodge filtration; V-filtration; spectrum |
---|
893 | EXAMPLE: example vfilt; shows examples |
---|
894 | " |
---|
895 | { |
---|
896 | int opt=1; |
---|
897 | if(size(#)>0) |
---|
898 | { |
---|
899 | if(typeof(#[1])=="int") |
---|
900 | { |
---|
901 | opt=#[1]; |
---|
902 | } |
---|
903 | } |
---|
904 | |
---|
905 | def R=basering; |
---|
906 | int n=nvars(R)-1; |
---|
907 | def G=gmsring(t,"s"); |
---|
908 | setring G; |
---|
909 | |
---|
910 | int mu=ncols(gmsbasis); |
---|
911 | ideal r=gmspoly*gmsbasis; |
---|
912 | list l; |
---|
913 | matrix A[mu][mu],C; |
---|
914 | module H,H1=freemodule(mu),freemodule(mu); |
---|
915 | module H0; |
---|
916 | int k=-1; |
---|
917 | int N=n+1; |
---|
918 | |
---|
919 | while(size(reduce(H,std(H0*s)))>0) |
---|
920 | { |
---|
921 | k++; |
---|
922 | dbprint(printlevel-voice+2,"// k="+string(k)); |
---|
923 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
924 | l=gmscoeffs(r,k); |
---|
925 | C,r=l[1..2]; |
---|
926 | A=A+C; |
---|
927 | |
---|
928 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
---|
929 | H0=H; |
---|
930 | H1=jet(module(A*H1+s^2*diff(matrix(H1),s)),k); |
---|
931 | H=H*s+H1; |
---|
932 | } |
---|
933 | A=A-k*s; |
---|
934 | |
---|
935 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
---|
936 | H=std(H0); |
---|
937 | int d0=maxdeg1(H); |
---|
938 | dbprint(printlevel-voice+2,"// k="+string(d0+N)); |
---|
939 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
940 | l=gmscoeffs(r,d0+N,d0+N); |
---|
941 | C,r=l[1..2]; |
---|
942 | A=A+C; |
---|
943 | |
---|
944 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
---|
945 | l=division(H,freemodule(mu)*s^k); |
---|
946 | H0=jet(l[1],l[2],N-1); |
---|
947 | |
---|
948 | dbprint(printlevel-voice+2,"// compute vector spaces"); |
---|
949 | poly p; |
---|
950 | int i0,j0,i1,j1; |
---|
951 | matrix V0[mu*N][mu*N]; |
---|
952 | matrix V1[mu*N][mu*(N-1)]; |
---|
953 | for(i0=mu;i0>=1;i0--) |
---|
954 | { |
---|
955 | for(i1=mu;i1>=1;i1--) |
---|
956 | { |
---|
957 | p=H0[i1,i0]; |
---|
958 | while(p!=0) |
---|
959 | { |
---|
960 | j1=leadexp(p)[1]; |
---|
961 | for(j0=N-j1-1;j0>=0;j0--) |
---|
962 | { |
---|
963 | V0[i1+(j1+j0)*mu,i0+j0*mu]=V0[i1+(j1+j0)*mu,i0+j0*mu]+leadcoef(p); |
---|
964 | if(j1+j0+1<N) |
---|
965 | { |
---|
966 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]= |
---|
967 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]+leadcoef(p); |
---|
968 | } |
---|
969 | } |
---|
970 | p=p-lead(p); |
---|
971 | } |
---|
972 | } |
---|
973 | } |
---|
974 | |
---|
975 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
---|
976 | l=division(H*s,A*H+s^2*diff(matrix(H),s)); |
---|
977 | A=jet(l[1],l[2],N-1); |
---|
978 | |
---|
979 | dbprint(printlevel-voice+2,"// compute matrix M of A"); |
---|
980 | matrix M[mu*N][mu*N]; |
---|
981 | for(i0=mu;i0>=1;i0--) |
---|
982 | { |
---|
983 | for(i1=mu;i1>=1;i1--) |
---|
984 | { |
---|
985 | p=A[i1,i0]; |
---|
986 | while(p!=0) |
---|
987 | { |
---|
988 | j1=leadexp(p)[1]; |
---|
989 | for(j0=N-j1-1;j0>=0;j0--) |
---|
990 | { |
---|
991 | M[i1+(j0+j1)*mu,i0+j0*mu]=leadcoef(p); |
---|
992 | } |
---|
993 | p=p-lead(p); |
---|
994 | } |
---|
995 | } |
---|
996 | } |
---|
997 | for(i0=mu;i0>=1;i0--) |
---|
998 | { |
---|
999 | for(j0=N-1;j0>=0;j0--) |
---|
1000 | { |
---|
1001 | M[i0+j0*mu,i0+j0*mu]=M[i0+j0*mu,i0+j0*mu]+j0; |
---|
1002 | } |
---|
1003 | } |
---|
1004 | |
---|
1005 | dbprint(printlevel-voice+2,"// compute eigenvalues eA of A"); |
---|
1006 | ideal eA=eigenval(jet(A,0))[1]; |
---|
1007 | dbprint(printlevel-voice+2,"// eA="+string(eA)); |
---|
1008 | |
---|
1009 | dbprint(printlevel-voice+2,"// compute eigenvalues eM of M"); |
---|
1010 | ideal eM; |
---|
1011 | k=0; |
---|
1012 | intvec u; |
---|
1013 | for(int i=N;i>=1;i--) |
---|
1014 | { |
---|
1015 | u[i]=1; |
---|
1016 | } |
---|
1017 | i0=1; |
---|
1018 | while(u[N]<=ncols(eA)) |
---|
1019 | { |
---|
1020 | for(i,i1=i0+1,i0;i<=N;i++) |
---|
1021 | { |
---|
1022 | if(eA[u[i]]+i<eA[u[i1]]+i1) |
---|
1023 | { |
---|
1024 | i1=i; |
---|
1025 | } |
---|
1026 | } |
---|
1027 | k++; |
---|
1028 | eM[k]=eA[u[i1]]+i1-1; |
---|
1029 | u[i1]=u[i1]+1; |
---|
1030 | if(u[i1]>ncols(eA)) |
---|
1031 | { |
---|
1032 | i0=i1+1; |
---|
1033 | } |
---|
1034 | } |
---|
1035 | dbprint(printlevel-voice+2,"// eM="+string(eM)); |
---|
1036 | |
---|
1037 | dbprint(printlevel-voice+2,"// compute V-filtration on H''/sH''"); |
---|
1038 | ideal a; |
---|
1039 | k=0; |
---|
1040 | list V; |
---|
1041 | V[ncols(eM)+1]=interred(V1); |
---|
1042 | intvec d; |
---|
1043 | if(opt<=0) |
---|
1044 | { |
---|
1045 | for(i=ncols(eM);number(eM[i])-1>number(n-1)/2;i--) |
---|
1046 | { |
---|
1047 | dbprint(printlevel-voice+2,"// compute V["+string(i)+"]"); |
---|
1048 | V1=module(V1)+syz(power(M-eM[i],n+1)); |
---|
1049 | V[i]=interred(intersect(V1,V0)); |
---|
1050 | |
---|
1051 | if(size(V[i])>size(V[i+1])) |
---|
1052 | { |
---|
1053 | k++; |
---|
1054 | a[k]=eM[i]-1; |
---|
1055 | d[k]=size(V[i])-size(V[i+1]); |
---|
1056 | } |
---|
1057 | } |
---|
1058 | |
---|
1059 | dbprint(printlevel-voice+2,"// symmetry index found"); |
---|
1060 | int j=k; |
---|
1061 | |
---|
1062 | if(number(eM[i])-1==number(n-1)/2) |
---|
1063 | { |
---|
1064 | dbprint(printlevel-voice+2,"// compute V["+string(i)+"]"); |
---|
1065 | V1=module(V1)+syz(power(M-eM[i],n+1)); |
---|
1066 | V[i]=interred(intersect(V1,V0)); |
---|
1067 | |
---|
1068 | if(size(V[i])>size(V[i+1])) |
---|
1069 | { |
---|
1070 | k++; |
---|
1071 | a[k]=eM[i]-1; |
---|
1072 | d[k]=size(V[i])-size(V[i+1]); |
---|
1073 | } |
---|
1074 | } |
---|
1075 | |
---|
1076 | dbprint(printlevel-voice+2,"// apply symmetry"); |
---|
1077 | while(j>=1) |
---|
1078 | { |
---|
1079 | k++; |
---|
1080 | a[k]=a[j]; |
---|
1081 | a[j]=n-1-a[k]; |
---|
1082 | d[k]=d[j]; |
---|
1083 | j--; |
---|
1084 | } |
---|
1085 | |
---|
1086 | setring(R); |
---|
1087 | ideal a=imap(G,a); |
---|
1088 | return(list(a,d)); |
---|
1089 | } |
---|
1090 | else |
---|
1091 | { |
---|
1092 | list v; |
---|
1093 | int j=-1; |
---|
1094 | for(i=ncols(eM);i>=1;i--) |
---|
1095 | { |
---|
1096 | dbprint(printlevel-voice+2,"// compute V["+string(i)+"]"); |
---|
1097 | V1=module(V1)+syz(power(M-eM[i],n+1)); |
---|
1098 | V[i]=interred(intersect(V1,V0)); |
---|
1099 | |
---|
1100 | if(size(V[i])>size(V[i+1])) |
---|
1101 | { |
---|
1102 | if(number(eM[i]-1)>=number(n-1)/2) |
---|
1103 | { |
---|
1104 | k++; |
---|
1105 | a[k]=eM[i]-1; |
---|
1106 | v[k]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
1107 | } |
---|
1108 | else |
---|
1109 | { |
---|
1110 | if(j<0) |
---|
1111 | { |
---|
1112 | if(a[k]==number(n-1)/2) |
---|
1113 | { |
---|
1114 | j=k-1; |
---|
1115 | } |
---|
1116 | else |
---|
1117 | { |
---|
1118 | j=k; |
---|
1119 | } |
---|
1120 | } |
---|
1121 | k++; |
---|
1122 | a[k]=a[j]; |
---|
1123 | a[j]=eM[i]-1; |
---|
1124 | v[k]=v[j]; |
---|
1125 | v[j]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
1126 | j--; |
---|
1127 | } |
---|
1128 | } |
---|
1129 | } |
---|
1130 | |
---|
1131 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1132 | for(k=1;k<size(v);k++) |
---|
1133 | { |
---|
1134 | v[k]=interred(reduce(v[k],std(v[k+1]))); |
---|
1135 | d[k]=size(v[k]); |
---|
1136 | } |
---|
1137 | v[k]=interred(v[k]); |
---|
1138 | d[k]=size(v[k]); |
---|
1139 | |
---|
1140 | setring(R); |
---|
1141 | ideal a=imap(G,a); |
---|
1142 | list v=imap(G,v); |
---|
1143 | ideal m=imap(G,gmsbasis); |
---|
1144 | ideal g=imap(G,gmsstd); |
---|
1145 | attrib(g,"isSB",1); |
---|
1146 | return(list(a,d,v,m,g)); |
---|
1147 | } |
---|
1148 | } |
---|
1149 | example |
---|
1150 | { "EXAMPLE:"; echo=2; |
---|
1151 | ring R=0,(x,y),ds; |
---|
1152 | poly t=x5+x2y2+y5; |
---|
1153 | vfilt(t); |
---|
1154 | } |
---|
1155 | /////////////////////////////////////////////////////////////////////////////// |
---|
1156 | |
---|
1157 | proc endfilt(list V) |
---|
1158 | "USAGE: endfilt(V); list V |
---|
1159 | ASSUME: V computed by vfilt |
---|
1160 | RETURN: |
---|
1161 | @format |
---|
1162 | list V1: endomorphim filtration of V on the Jacobian algebra |
---|
1163 | ideal V1[1]: spectral numbers in increasing order |
---|
1164 | intvec V1[2]: |
---|
1165 | int V1[2][i]: multiplicity of spectral number V1[1][i] |
---|
1166 | list V1[3]: |
---|
1167 | module V1[3][i]: vector space basis of the V1[1][i]-th graded part |
---|
1168 | in terms of V1[4] |
---|
1169 | ideal V1[4]: monomial vector space basis |
---|
1170 | @end format |
---|
1171 | SEE ALSO: spectrum_lib |
---|
1172 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; spectrum; |
---|
1173 | Hodge filtration; V-filtration |
---|
1174 | EXAMPLE: example endfilt; shows examples |
---|
1175 | " |
---|
1176 | { |
---|
1177 | def a,d,v,m,g=V[1..5]; |
---|
1178 | int mu=ncols(m); |
---|
1179 | |
---|
1180 | module V0=v[1]; |
---|
1181 | for(int i=2;i<=size(v);i++) |
---|
1182 | { |
---|
1183 | V0=V0,v[i]; |
---|
1184 | } |
---|
1185 | |
---|
1186 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
1187 | list M; |
---|
1188 | module U=freemodule(ncols(m)); |
---|
1189 | for(i=ncols(m);i>=1;i--) |
---|
1190 | { |
---|
1191 | M[i]=lift(V0,coeffs(reduce(m[i]*m,U,g),m)*V0); |
---|
1192 | } |
---|
1193 | |
---|
1194 | int j,k,i0,j0,i1,j1; |
---|
1195 | number b0=number(a[1]-a[ncols(a)]); |
---|
1196 | number b1,b2; |
---|
1197 | matrix M0; |
---|
1198 | module L; |
---|
1199 | list v0=freemodule(ncols(m)); |
---|
1200 | ideal a0=b0; |
---|
1201 | |
---|
1202 | while(b0<number(a[ncols(a)]-a[1])) |
---|
1203 | { |
---|
1204 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
1205 | b1=number(a[ncols(a)]-a[1]); |
---|
1206 | for(j=ncols(a);j>=1;j--) |
---|
1207 | { |
---|
1208 | for(i=ncols(a);i>=1;i--) |
---|
1209 | { |
---|
1210 | b2=number(a[i]-a[j]); |
---|
1211 | if(b2>b0&&b2<b1) |
---|
1212 | { |
---|
1213 | b1=b2; |
---|
1214 | } |
---|
1215 | else |
---|
1216 | { |
---|
1217 | if(b2<=b0) |
---|
1218 | { |
---|
1219 | i=0; |
---|
1220 | } |
---|
1221 | } |
---|
1222 | } |
---|
1223 | } |
---|
1224 | b0=b1; |
---|
1225 | |
---|
1226 | list l=ideal(); |
---|
1227 | for(k=ncols(m);k>=2;k--) |
---|
1228 | { |
---|
1229 | l=l+list(ideal()); |
---|
1230 | } |
---|
1231 | |
---|
1232 | dbprint(printlevel-voice+2,"// collect conditions for V1["+string(b0)+"]"); |
---|
1233 | j=ncols(a); |
---|
1234 | j0=mu; |
---|
1235 | while(j>=1) |
---|
1236 | { |
---|
1237 | i0=1; |
---|
1238 | i=1; |
---|
1239 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
1240 | { |
---|
1241 | i0=i0+d[i]; |
---|
1242 | i++; |
---|
1243 | } |
---|
1244 | if(a[i]<a[j]+b0) |
---|
1245 | { |
---|
1246 | i0=i0+d[i]; |
---|
1247 | i++; |
---|
1248 | } |
---|
1249 | for(k=1;k<=ncols(m);k++) |
---|
1250 | { |
---|
1251 | M0=M[k]; |
---|
1252 | if(i0>1) |
---|
1253 | { |
---|
1254 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
1255 | } |
---|
1256 | } |
---|
1257 | j0=j0-d[j]; |
---|
1258 | j--; |
---|
1259 | } |
---|
1260 | |
---|
1261 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
1262 | L=transpose(module(l[1])); |
---|
1263 | for(k=2;k<=ncols(m);k++) |
---|
1264 | { |
---|
1265 | L=L,transpose(module(l[k])); |
---|
1266 | } |
---|
1267 | |
---|
1268 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
1269 | v0=v0+list(syz(L)); |
---|
1270 | a0=a0,b0; |
---|
1271 | } |
---|
1272 | |
---|
1273 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1274 | option(redSB); |
---|
1275 | for(i=1;i<size(v0);i++) |
---|
1276 | { |
---|
1277 | v0[i+1]=std(v0[i+1]); |
---|
1278 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
1279 | } |
---|
1280 | |
---|
1281 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
1282 | i=1; |
---|
1283 | while(size(v0[i])==0) |
---|
1284 | { |
---|
1285 | i++; |
---|
1286 | } |
---|
1287 | list v1=v0[i]; |
---|
1288 | intvec d1=size(v0[i]); |
---|
1289 | ideal a1=a0[i]; |
---|
1290 | i++; |
---|
1291 | while(i<=size(v0)) |
---|
1292 | { |
---|
1293 | if(size(v0[i])>0) |
---|
1294 | { |
---|
1295 | v1=v1+list(v0[i]); |
---|
1296 | d1=d1,size(v0[i]); |
---|
1297 | a1=a1,a0[i]; |
---|
1298 | } |
---|
1299 | i++; |
---|
1300 | } |
---|
1301 | return(list(a1,d1,v1,m)); |
---|
1302 | } |
---|
1303 | example |
---|
1304 | { "EXAMPLE:"; echo=2; |
---|
1305 | ring R=0,(x,y),ds; |
---|
1306 | poly t=x5+x2y2+y5; |
---|
1307 | endfilt(vfilt(t)); |
---|
1308 | } |
---|
1309 | /////////////////////////////////////////////////////////////////////////////// |
---|
1310 | |
---|
1311 | proc spgen(ideal a) |
---|
1312 | "USAGE: spgen(a); ideal a |
---|
1313 | RETURN: |
---|
1314 | @format |
---|
1315 | list Sp: numbers in a with multiplicities |
---|
1316 | ideal Sp[1]: numbers in a in increasing order |
---|
1317 | intvec Sp[2]: |
---|
1318 | int Sp[2][i]: multiplicity of number Sp[1][i] in a |
---|
1319 | @end format |
---|
1320 | EXAMPLE: example spgen; shows examples |
---|
1321 | " |
---|
1322 | { |
---|
1323 | ideal a0=jet(a,0); |
---|
1324 | int i,j; |
---|
1325 | poly p; |
---|
1326 | for(i=1;i<=ncols(a0);i++) |
---|
1327 | { |
---|
1328 | for(j=i+1;j<=ncols(a0);j++) |
---|
1329 | { |
---|
1330 | if(number(a0[i])>number(a0[j])) |
---|
1331 | { |
---|
1332 | p=a0[i]; |
---|
1333 | a0[i]=a0[j]; |
---|
1334 | a0[j]=p; |
---|
1335 | } |
---|
1336 | } |
---|
1337 | } |
---|
1338 | j=1; |
---|
1339 | a=a0[1]; |
---|
1340 | intvec m=1; |
---|
1341 | for(i=2;i<=ncols(a0);i++) |
---|
1342 | { |
---|
1343 | if(a0[i]==a[j]) |
---|
1344 | { |
---|
1345 | m[j]=m[j]+1; |
---|
1346 | } |
---|
1347 | else |
---|
1348 | { |
---|
1349 | j++; |
---|
1350 | a[j]=a0[i]; |
---|
1351 | m[j]=1; |
---|
1352 | } |
---|
1353 | } |
---|
1354 | return(list(a,m)); |
---|
1355 | } |
---|
1356 | example |
---|
1357 | { "EXAMPLE:"; echo=2; |
---|
1358 | ring R=0,(x,y),ds; |
---|
1359 | ideal a=-1/2,-3/10,-3/10,-1/10,-1/10,0,1/10,1/10,3/10,3/10,1/2; |
---|
1360 | spprint(spgen(a)); |
---|
1361 | } |
---|
1362 | /////////////////////////////////////////////////////////////////////////////// |
---|
1363 | |
---|
1364 | proc sppgen(ideal a,intvec w) |
---|
1365 | "USAGE: sppgen(a,w); ideal a, intvec w |
---|
1366 | RETURN: |
---|
1367 | @format |
---|
1368 | list Spp: pairs in a and w with multiplicities |
---|
1369 | ideal Spp[1]: numbers in a in increasing order |
---|
1370 | intvec Spp[2]: integers in w in decreasing order |
---|
1371 | intvec Spp[3]: |
---|
1372 | int Spp[3][i]: multiplicity of pair (Spp[1][i],Spp[2][i]) in a,w |
---|
1373 | @end format |
---|
1374 | EXAMPLE: example sppgen; shows examples |
---|
1375 | " |
---|
1376 | { |
---|
1377 | ideal a0=jet(a,0); |
---|
1378 | intvec w0=w; |
---|
1379 | int i,j,k; |
---|
1380 | poly p; |
---|
1381 | for(i=1;i<=ncols(a0);i++) |
---|
1382 | { |
---|
1383 | for(j=i+1;j<=ncols(a0);j++) |
---|
1384 | { |
---|
1385 | if(number(a0[i])>number(a0[j])||a0[i]==a0[j]&&w0[i]>w0[j]) |
---|
1386 | { |
---|
1387 | p=a0[i]; |
---|
1388 | a0[i]=a0[j]; |
---|
1389 | a0[j]=p; |
---|
1390 | k=w0[i]; |
---|
1391 | w0[i]=w0[j]; |
---|
1392 | w0[j]=k; |
---|
1393 | } |
---|
1394 | } |
---|
1395 | } |
---|
1396 | j=1; |
---|
1397 | a=a0[1]; |
---|
1398 | w=w0[1]; |
---|
1399 | intvec m=1; |
---|
1400 | for(i=2;i<=ncols(a0);i++) |
---|
1401 | { |
---|
1402 | if(a0[i]==a[j]&&w0[i]==w[j]) |
---|
1403 | { |
---|
1404 | m[j]=m[j]+1; |
---|
1405 | } |
---|
1406 | else |
---|
1407 | { |
---|
1408 | j++; |
---|
1409 | a[j]=a0[i]; |
---|
1410 | w[j]=w0[i]; |
---|
1411 | m[j]=1; |
---|
1412 | } |
---|
1413 | } |
---|
1414 | return(list(a,w,m)); |
---|
1415 | } |
---|
1416 | example |
---|
1417 | { "EXAMPLE:"; echo=2; |
---|
1418 | ring R=0,(x,y),ds; |
---|
1419 | ideal a=-1/2,-3/10,-3/10,-1/10,-1/10,0,1/10,1/10,3/10,3/10,1/2; |
---|
1420 | intvec w=2,1,1,1,1,1,1,1,1,1,0; |
---|
1421 | spprint(sppgen(a,w)); |
---|
1422 | } |
---|
1423 | /////////////////////////////////////////////////////////////////////////////// |
---|
1424 | |
---|
1425 | proc spprint(list Sp) |
---|
1426 | "USAGE: spprint(Sp); list Sp |
---|
1427 | RETURN: string: spectrum or spectral pairs Sp |
---|
1428 | EXAMPLE: example spprint; shows examples |
---|
1429 | " |
---|
1430 | { |
---|
1431 | string s; |
---|
1432 | if(size(Sp)==2) |
---|
1433 | { |
---|
1434 | for(int i=1;i<size(Sp[2]);i++) |
---|
1435 | { |
---|
1436 | s=s+"("+string(Sp[1][i])+","+string(Sp[2][i])+"),"; |
---|
1437 | } |
---|
1438 | s=s+"("+string(Sp[1][i])+","+string(Sp[2][i])+")"; |
---|
1439 | } |
---|
1440 | else |
---|
1441 | { |
---|
1442 | for(int i=1;i<size(Sp[3]);i++) |
---|
1443 | { |
---|
1444 | s=s+"(("+string(Sp[1][i])+","+string(Sp[2][i])+"),"+string(Sp[3][i])+"),"; |
---|
1445 | } |
---|
1446 | s=s+"(("+string(Sp[1][i])+","+string(Sp[2][i])+"),"+string(Sp[3][i])+")"; |
---|
1447 | } |
---|
1448 | return(s); |
---|
1449 | } |
---|
1450 | example |
---|
1451 | { "EXAMPLE:"; echo=2; |
---|
1452 | ring R=0,(x,y),ds; |
---|
1453 | 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)); |
---|
1454 | spprint(Sp); |
---|
1455 | } |
---|
1456 | /////////////////////////////////////////////////////////////////////////////// |
---|
1457 | |
---|
1458 | proc spadd(list Sp1,list Sp2) |
---|
1459 | "USAGE: spadd(Sp1,Sp2); list Sp1,Sp2 |
---|
1460 | RETURN: list: sum of spectra Sp1 and Sp2 |
---|
1461 | EXAMPLE: example spadd; shows examples |
---|
1462 | " |
---|
1463 | { |
---|
1464 | ideal s; |
---|
1465 | intvec m; |
---|
1466 | int i,i1,i2=1,1,1; |
---|
1467 | while(i1<=size(Sp1[2])||i2<=size(Sp2[2])) |
---|
1468 | { |
---|
1469 | if(i1<=size(Sp1[2])) |
---|
1470 | { |
---|
1471 | if(i2<=size(Sp2[2])) |
---|
1472 | { |
---|
1473 | if(number(Sp1[1][i1])<number(Sp2[1][i2])) |
---|
1474 | { |
---|
1475 | s[i]=Sp1[1][i1]; |
---|
1476 | m[i]=Sp1[2][i1]; |
---|
1477 | i++; |
---|
1478 | i1++; |
---|
1479 | } |
---|
1480 | else |
---|
1481 | { |
---|
1482 | if(number(Sp1[1][i1])>number(Sp2[1][i2])) |
---|
1483 | { |
---|
1484 | s[i]=Sp2[1][i2]; |
---|
1485 | m[i]=Sp2[2][i2]; |
---|
1486 | i++; |
---|
1487 | i2++; |
---|
1488 | } |
---|
1489 | else |
---|
1490 | { |
---|
1491 | if(Sp1[2][i1]+Sp2[2][i2]!=0) |
---|
1492 | { |
---|
1493 | s[i]=Sp1[1][i1]; |
---|
1494 | m[i]=Sp1[2][i1]+Sp2[2][i2]; |
---|
1495 | i++; |
---|
1496 | } |
---|
1497 | i1++; |
---|
1498 | i2++; |
---|
1499 | } |
---|
1500 | } |
---|
1501 | } |
---|
1502 | else |
---|
1503 | { |
---|
1504 | s[i]=Sp1[1][i1]; |
---|
1505 | m[i]=Sp1[2][i1]; |
---|
1506 | i++; |
---|
1507 | i1++; |
---|
1508 | } |
---|
1509 | } |
---|
1510 | else |
---|
1511 | { |
---|
1512 | s[i]=Sp2[1][i2]; |
---|
1513 | m[i]=Sp2[2][i2]; |
---|
1514 | i++; |
---|
1515 | i2++; |
---|
1516 | } |
---|
1517 | } |
---|
1518 | return(list(s,m)); |
---|
1519 | } |
---|
1520 | example |
---|
1521 | { "EXAMPLE:"; echo=2; |
---|
1522 | ring R=0,(x,y),ds; |
---|
1523 | 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)); |
---|
1524 | spprint(Sp1); |
---|
1525 | list Sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1526 | spprint(Sp2); |
---|
1527 | spprint(spadd(Sp1,Sp2)); |
---|
1528 | } |
---|
1529 | /////////////////////////////////////////////////////////////////////////////// |
---|
1530 | |
---|
1531 | proc spsub(list Sp1,list Sp2) |
---|
1532 | "USAGE: spsub(Sp1,Sp2); list Sp1,Sp2 |
---|
1533 | RETURN: list: difference of spectra Sp1 and Sp2 |
---|
1534 | EXAMPLE: example spsub; shows examples |
---|
1535 | " |
---|
1536 | { |
---|
1537 | return(spadd(Sp1,spmul(Sp2,-1))); |
---|
1538 | } |
---|
1539 | example |
---|
1540 | { "EXAMPLE:"; echo=2; |
---|
1541 | ring R=0,(x,y),ds; |
---|
1542 | 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)); |
---|
1543 | spprint(Sp1); |
---|
1544 | list Sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1545 | spprint(Sp2); |
---|
1546 | spprint(spsub(Sp1,Sp2)); |
---|
1547 | } |
---|
1548 | /////////////////////////////////////////////////////////////////////////////// |
---|
1549 | |
---|
1550 | proc spmul(list #) |
---|
1551 | "USAGE: |
---|
1552 | @format |
---|
1553 | 1) spmul(Sp,k); list Sp, int k |
---|
1554 | 2) spmul(Sp,k); list Sp, intvec k |
---|
1555 | @end format |
---|
1556 | RETURN: |
---|
1557 | @format |
---|
1558 | 1) list: product of spectrum Sp and integer k |
---|
1559 | 2) list: linear combination of spectra Sp with coefficients k |
---|
1560 | @end format |
---|
1561 | EXAMPLE: example spmul; shows examples |
---|
1562 | " |
---|
1563 | { |
---|
1564 | if(size(#)==2) |
---|
1565 | { |
---|
1566 | if(typeof(#[1])=="list") |
---|
1567 | { |
---|
1568 | if(typeof(#[2])=="int") |
---|
1569 | { |
---|
1570 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1571 | } |
---|
1572 | if(typeof(#[2])=="intvec") |
---|
1573 | { |
---|
1574 | list Sp0=list(ideal(),intvec(0)); |
---|
1575 | for(int i=size(#[2]);i>=1;i--) |
---|
1576 | { |
---|
1577 | Sp0=spadd(Sp0,spmul(#[1][i],#[2][i])); |
---|
1578 | } |
---|
1579 | return(Sp0); |
---|
1580 | } |
---|
1581 | } |
---|
1582 | } |
---|
1583 | return(list(ideal(),intvec(0))); |
---|
1584 | } |
---|
1585 | example |
---|
1586 | { "EXAMPLE:"; echo=2; |
---|
1587 | ring R=0,(x,y),ds; |
---|
1588 | 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)); |
---|
1589 | spprint(Sp); |
---|
1590 | spprint(spmul(Sp,2)); |
---|
1591 | list Sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1592 | spprint(Sp1); |
---|
1593 | list Sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1594 | spprint(Sp2); |
---|
1595 | spprint(spmul(list(Sp1,Sp2),intvec(1,2))); |
---|
1596 | } |
---|
1597 | /////////////////////////////////////////////////////////////////////////////// |
---|
1598 | |
---|
1599 | proc spissemicont(list Sp,list #) |
---|
1600 | "USAGE: spissemicont(Sp[,opt]); list Sp, int opt |
---|
1601 | RETURN: |
---|
1602 | @format |
---|
1603 | int k= |
---|
1604 | if opt=0: |
---|
1605 | 1, if sum of spectrum Sp over all intervals [a,a+1) is positive |
---|
1606 | 0, if sum of spectrum Sp over some interval [a,a+1) is negative |
---|
1607 | if opt=1: |
---|
1608 | 1, if sum of spectrum Sp over all intervals [a,a+1) and (a,a+1) is positive |
---|
1609 | 0, if sum of spectrum Sp over some interval [a,a+1) or (a,a+1) is negative |
---|
1610 | default: opt=0 |
---|
1611 | @end format |
---|
1612 | EXAMPLE: example spissemicont; shows examples |
---|
1613 | " |
---|
1614 | { |
---|
1615 | int opt=0; |
---|
1616 | if(size(#)>0) |
---|
1617 | { |
---|
1618 | if(typeof(#[1])=="int") |
---|
1619 | { |
---|
1620 | opt=1; |
---|
1621 | } |
---|
1622 | } |
---|
1623 | int i,j,k=1,1,0; |
---|
1624 | while(j<=size(Sp[2])) |
---|
1625 | { |
---|
1626 | while(j+1<=size(Sp[2])&&Sp[1][j]<Sp[1][i]+1) |
---|
1627 | { |
---|
1628 | k=k+Sp[2][j]; |
---|
1629 | j++; |
---|
1630 | } |
---|
1631 | if(j==size(Sp[2])&&Sp[1][j]<Sp[1][i]+1) |
---|
1632 | { |
---|
1633 | k=k+Sp[2][j]; |
---|
1634 | j++; |
---|
1635 | } |
---|
1636 | if(k<0) |
---|
1637 | { |
---|
1638 | return(0); |
---|
1639 | } |
---|
1640 | k=k-Sp[2][i]; |
---|
1641 | if(k<0&&opt==1) |
---|
1642 | { |
---|
1643 | return(0); |
---|
1644 | } |
---|
1645 | i++; |
---|
1646 | } |
---|
1647 | return(1); |
---|
1648 | } |
---|
1649 | example |
---|
1650 | { "EXAMPLE:"; echo=2; |
---|
1651 | ring R=0,(x,y),ds; |
---|
1652 | 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)); |
---|
1653 | spprint(Sp1); |
---|
1654 | list Sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1655 | spprint(Sp2); |
---|
1656 | spissemicont(spsub(Sp1,spmul(Sp2,5))); |
---|
1657 | spissemicont(spsub(Sp1,spmul(Sp2,5)),1); |
---|
1658 | spissemicont(spsub(Sp1,spmul(Sp2,6))); |
---|
1659 | } |
---|
1660 | /////////////////////////////////////////////////////////////////////////////// |
---|
1661 | |
---|
1662 | proc spsemicont(list Sp0,list Sp,list #) |
---|
1663 | "USAGE: spsemicont(Sp,k[,opt]); list Sp0, list Sp, int opt |
---|
1664 | RETURN: list of intvecs l: |
---|
1665 | spissemicont(sub(Sp0,spmul(Sp,k)),opt)==1 iff k<=l[i] for some i |
---|
1666 | NOTE: if the spectra Sp occur with multiplicities k in a deformation |
---|
1667 | of the [quasihomogeneous] spectrum Sp0 then |
---|
1668 | spissemicont(sub(Sp0,spmul(Sp,k))[,1])==1 |
---|
1669 | EXAMPLE: example spsemicont; shows examples |
---|
1670 | " |
---|
1671 | { |
---|
1672 | list l,l0; |
---|
1673 | int i,j,k; |
---|
1674 | while(spissemicont(Sp0,#)) |
---|
1675 | { |
---|
1676 | if(size(Sp)>1) |
---|
1677 | { |
---|
1678 | l0=spsemicont(Sp0,list(Sp[1..size(Sp)-1])); |
---|
1679 | for(i=1;i<=size(l0);i++) |
---|
1680 | { |
---|
1681 | if(size(l)>0) |
---|
1682 | { |
---|
1683 | j=1; |
---|
1684 | while(j<size(l)&&l[j]!=l0[i]) |
---|
1685 | { |
---|
1686 | j++; |
---|
1687 | } |
---|
1688 | if(l[j]==l0[i]) |
---|
1689 | { |
---|
1690 | l[j][size(Sp)]=k; |
---|
1691 | } |
---|
1692 | else |
---|
1693 | { |
---|
1694 | l0[i][size(Sp)]=k; |
---|
1695 | l=l+list(l0[i]); |
---|
1696 | } |
---|
1697 | } |
---|
1698 | else |
---|
1699 | { |
---|
1700 | l=l0; |
---|
1701 | } |
---|
1702 | } |
---|
1703 | } |
---|
1704 | Sp0=spsub(Sp0,Sp[size(Sp)]); |
---|
1705 | k++; |
---|
1706 | } |
---|
1707 | if(size(Sp)>1) |
---|
1708 | { |
---|
1709 | return(l); |
---|
1710 | } |
---|
1711 | else |
---|
1712 | { |
---|
1713 | return(list(intvec(k-1))); |
---|
1714 | } |
---|
1715 | } |
---|
1716 | example |
---|
1717 | { "EXAMPLE:"; echo=2; |
---|
1718 | ring R=0,(x,y),ds; |
---|
1719 | 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)); |
---|
1720 | spprint(Sp0); |
---|
1721 | list Sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1722 | spprint(Sp1); |
---|
1723 | list Sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1724 | spprint(Sp2); |
---|
1725 | list Sp=Sp1,Sp2; |
---|
1726 | list l=spsemicont(Sp0,Sp); |
---|
1727 | l; |
---|
1728 | spissemicont(spsub(Sp0,spmul(Sp,l[1]))); |
---|
1729 | spissemicont(spsub(Sp0,spmul(Sp,l[1]-1))); |
---|
1730 | spissemicont(spsub(Sp0,spmul(Sp,l[1]+1))); |
---|
1731 | } |
---|
1732 | /////////////////////////////////////////////////////////////////////////////// |
---|
1733 | |
---|
1734 | proc spmilnor(list Sp) |
---|
1735 | "USAGE: spmilnor(Sp); list Sp |
---|
1736 | RETURN: int: Milnor number of spectrum Sp |
---|
1737 | EXAMPLE: example spmilnor; shows examples |
---|
1738 | " |
---|
1739 | { |
---|
1740 | return(sum(Sp[2])); |
---|
1741 | } |
---|
1742 | example |
---|
1743 | { "EXAMPLE:"; echo=2; |
---|
1744 | ring R=0,(x,y),ds; |
---|
1745 | 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)); |
---|
1746 | spprint(Sp); |
---|
1747 | spmilnor(Sp); |
---|
1748 | } |
---|
1749 | /////////////////////////////////////////////////////////////////////////////// |
---|
1750 | |
---|
1751 | proc spgeomgenus(list Sp) |
---|
1752 | "USAGE: spgeomgenus(Sp); list Sp |
---|
1753 | RETURN: int: geometrical genus of spectrum Sp |
---|
1754 | EXAMPLE: example spgeomgenus; shows examples |
---|
1755 | " |
---|
1756 | { |
---|
1757 | int g=0; |
---|
1758 | int i=1; |
---|
1759 | while(i+1<=size(Sp[2])&&number(Sp[1][i])<=number(0)) |
---|
1760 | { |
---|
1761 | g=g+Sp[2][i]; |
---|
1762 | i++; |
---|
1763 | } |
---|
1764 | if(i==size(Sp[2])&&number(Sp[1][i])<=number(0)) |
---|
1765 | { |
---|
1766 | g=g+Sp[2][i]; |
---|
1767 | } |
---|
1768 | return(g); |
---|
1769 | } |
---|
1770 | example |
---|
1771 | { "EXAMPLE:"; echo=2; |
---|
1772 | ring R=0,(x,y),ds; |
---|
1773 | 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)); |
---|
1774 | spprint(Sp); |
---|
1775 | spgeomgenus(Sp); |
---|
1776 | } |
---|
1777 | /////////////////////////////////////////////////////////////////////////////// |
---|
1778 | |
---|
1779 | proc spgamma(list Sp) |
---|
1780 | "USAGE: spgamma(Sp); list Sp |
---|
1781 | RETURN: number: gamma invariant of spectrum Sp |
---|
1782 | EXAMPLE: example spgamma; shows examples |
---|
1783 | " |
---|
1784 | { |
---|
1785 | int i,j; |
---|
1786 | number g=0; |
---|
1787 | for(i=1;i<=ncols(Sp[1]);i++) |
---|
1788 | { |
---|
1789 | for(j=1;j<=Sp[2][i];j++) |
---|
1790 | { |
---|
1791 | g=g+(number(Sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1792 | } |
---|
1793 | } |
---|
1794 | g=-g/4+sum(Sp[2])*number(Sp[1][ncols(Sp[1])]-Sp[1][1])/48; |
---|
1795 | return(g); |
---|
1796 | } |
---|
1797 | example |
---|
1798 | { "EXAMPLE:"; echo=2; |
---|
1799 | ring R=0,(x,y),ds; |
---|
1800 | 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)); |
---|
1801 | spprint(Sp); |
---|
1802 | spgamma(Sp); |
---|
1803 | } |
---|
1804 | /////////////////////////////////////////////////////////////////////////////// |
---|