1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: gaussman.lib,v 1.84 2002-07-16 14:35:09 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 Algorithmic Gauss-Manin Connection |
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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 Hodge-theoretic invariants |
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11 | of isolated hypersurface singularities |
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12 | |
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13 | PROCEDURES: |
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14 | gmsring(t,s); Gauss-Manin system of t with variable s |
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15 | gmsnf(p,K); Gauss-Manin normal form of p |
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16 | gmscoeffs(p,K); Gauss-Manin basis representation of p |
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17 | bernstein(t); roots of the Bernstein polynomial of t |
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18 | monodromy(t); Jordan data of complex monodromy of t |
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19 | spectrum(t); singularity spectrum of t |
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20 | sppairs(t); spectral pairs of t |
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21 | spnf(a[,m][,V]); spectrum normal form of (a,m,V) |
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22 | sppnf(a,w[,m][,V]); spectral pairs normal form of (a,w,m,V) |
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23 | vfilt(t); V-filtration of t on Brieskorn lattice |
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24 | vwfilt(t); weighted V-filtration of t on Brieskorn lattice |
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25 | tmatrix(t); C[[s]]-matrix of t on Brieskorn lattice |
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26 | endvfilt(V); endomorphism V-filtration on Jacobian algebra |
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27 | spprint(sp); print spectrum sp |
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28 | sppprint(spp); print spectral pairs spp |
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29 | spadd(sp1,sp2); sum of spectra sp1 and sp2 |
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30 | spsub(sp1,sp2); difference of spectra sp1 and sp2 |
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31 | spmul(sp0,k); linear combination of spectra sp |
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32 | spissemicont(sp[,opt]); semicontinuity test of spectrum sp |
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33 | spsemicont(sp0,sp[,opt]); semicontinuous combinations of spectra sp0 in sp |
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34 | spmilnor(sp); Milnor number of spectrum sp |
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35 | spgeomgenus(sp); geometrical genus of spectrum sp |
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36 | spgamma(sp); gamma invariant of spectrum sp |
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37 | |
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38 | SEE ALSO: mondromy_lib, spectrum_lib |
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39 | |
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40 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
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41 | monodromy; spectrum; spectral pairs; |
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42 | mixed Hodge structure; V-filtration; weight filtration |
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43 | "; |
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44 | |
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45 | LIB "linalg.lib"; |
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46 | |
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47 | /////////////////////////////////////////////////////////////////////////////// |
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48 | |
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49 | static proc stdtrans(ideal I) |
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50 | { |
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51 | def R=basering; |
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52 | |
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53 | string os=ordstr(R); |
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54 | int j=find(os,",C"); |
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55 | if(j==0) |
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56 | { |
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57 | j=find(os,"C,"); |
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58 | } |
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59 | if(j==0) |
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60 | { |
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61 | j=find(os,",c"); |
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62 | } |
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63 | if(j==0) |
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64 | { |
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65 | j=find(os,"c,"); |
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66 | } |
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67 | if(j>0) |
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68 | { |
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69 | os[j..j+1]=" "; |
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70 | } |
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71 | |
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72 | execute("ring S="+charstr(R)+",(gmspoly,"+varstr(R)+"),(c,dp,"+os+");"); |
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73 | |
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74 | ideal I=homog(imap(R,I),gmspoly); |
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75 | module M=transpose(transpose(I)+freemodule(ncols(I))); |
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76 | M=std(M); |
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77 | |
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78 | setring(R); |
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79 | execute("map h=S,1,"+varstr(R)+";"); |
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80 | module M=h(M); |
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81 | |
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82 | for(int i=ncols(M);i>=1;i--) |
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83 | { |
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84 | for(j=ncols(M);j>=1;j--) |
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85 | { |
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86 | if(M[i][1]==0) |
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87 | { |
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88 | M[i]=0; |
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89 | } |
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90 | if(i!=j&&M[j][1]!=0) |
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91 | { |
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92 | if(lead(M[i][1])/lead(M[j][1])!=0) |
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93 | { |
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94 | M[i]=0; |
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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 | |
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100 | M=transpose(simplify(M,2)); |
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101 | I=M[1]; |
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102 | attrib(I,"isSB",1); |
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103 | M=M[2..ncols(M)]; |
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104 | module U=transpose(M); |
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105 | |
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106 | return(list(I,U)); |
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107 | } |
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108 | /////////////////////////////////////////////////////////////////////////////// |
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109 | |
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110 | proc gmsring(poly t,string s) |
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111 | "USAGE: gmsring(t,s); poly t, string s |
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112 | ASSUME: characteristic 0; local degree ordering; |
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113 | isolated critical point 0 of t |
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114 | RETURN: |
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115 | @format |
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116 | ring G; Gauss-Manin system of t with variable s |
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117 | poly gmspoly=t; |
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118 | ideal gmsjacob; Jacobian ideal of t |
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119 | ideal gmsstd; standard basis of Jacobian ideal |
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120 | matrix gmsmatrix; matrix(gmsjacob)*gmsmatrix==matrix(gmsstd) |
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121 | ideal gmsbasis; monomial vector space basis of Jacobian algebra |
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122 | int gmsmaxdeg; maximal weight of variables |
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123 | @end format |
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124 | NOTE: gmsbasis is a C[[s]]-basis of H'' and [t,s]=s^2 |
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125 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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126 | EXAMPLE: example gmsring; shows examples |
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127 | " |
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128 | { |
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129 | def R=basering; |
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130 | if(charstr(R)!="0") |
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131 | { |
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132 | ERROR("characteristic 0 expected"); |
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133 | } |
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134 | for(int i=nvars(R);i>=1;i--) |
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135 | { |
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136 | if(var(i)>1) |
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137 | { |
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138 | ERROR("local ordering expected"); |
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139 | } |
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140 | } |
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141 | |
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142 | ideal dt=jacob(t); |
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143 | list l=stdtrans(dt); |
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144 | ideal g=l[1]; |
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145 | if(vdim(g)<=0) |
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146 | { |
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147 | if(vdim(g)==0) |
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148 | { |
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149 | ERROR("singularity at 0 expected"); |
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150 | } |
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151 | else |
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152 | { |
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153 | ERROR("isolated critical point 0 expected"); |
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154 | } |
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155 | } |
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156 | matrix B=l[2]; |
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157 | ideal m=kbase(g); |
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158 | |
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159 | int gmsmaxdeg; |
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160 | for(i=nvars(R);i>=1;i--) |
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161 | { |
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162 | if(deg(var(i))>gmsmaxdeg) |
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163 | { |
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164 | gmsmaxdeg=deg(var(i)); |
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165 | } |
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166 | } |
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167 | |
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168 | string os=ordstr(R); |
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169 | int j=find(os,",C"); |
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170 | if(j==0) |
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171 | { |
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172 | j=find(os,"C,"); |
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173 | } |
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174 | if(j==0) |
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175 | { |
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176 | j=find(os,",c"); |
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177 | } |
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178 | if(j==0) |
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179 | { |
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180 | j=find(os,"c,"); |
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181 | } |
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182 | if(j>0) |
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183 | { |
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184 | os[j..j+1]=" "; |
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185 | } |
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186 | |
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187 | execute("ring G="+string(charstr(R))+",("+s+","+varstr(R)+"),(ws("+ |
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188 | string(deg(highcorner(g))+2*gmsmaxdeg)+"),"+os+",c);"); |
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189 | |
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190 | poly gmspoly=imap(R,t); |
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191 | ideal gmsjacob=imap(R,dt); |
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192 | ideal gmsstd=imap(R,g); |
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193 | matrix gmsmatrix=imap(R,B); |
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194 | ideal gmsbasis=imap(R,m); |
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195 | |
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196 | attrib(gmsstd,"isSB",1); |
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197 | export gmspoly,gmsjacob,gmsstd,gmsmatrix,gmsbasis,gmsmaxdeg; |
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198 | |
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199 | return(G); |
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200 | } |
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201 | example |
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202 | { "EXAMPLE:"; echo=2; |
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203 | ring R=0,(x,y),ds; |
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204 | poly t=x5+x2y2+y5; |
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205 | def G=gmsring(t,"s"); |
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206 | setring(G); |
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207 | gmspoly; |
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208 | print(gmsjacob); |
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209 | print(gmsstd); |
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210 | print(gmsmatrix); |
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211 | print(gmsbasis); |
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212 | gmsmaxdeg; |
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213 | } |
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214 | /////////////////////////////////////////////////////////////////////////////// |
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215 | |
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216 | proc gmsnf(ideal p,int K) |
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217 | "USAGE: gmsnf(p,K); poly p, int K |
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218 | ASSUME: basering returned by gmsring |
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219 | RETURN: |
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220 | @format |
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221 | list nf; |
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222 | ideal nf[1]; projection of p to <gmsbasis>C[[s]] mod s^(K+1) |
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223 | ideal nf[2]; p==nf[1]+nf[2] |
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224 | @end format |
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225 | NOTE: computation can be continued by setting p=nf[2] |
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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 | if(system("with","gms")) |
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231 | { |
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232 | return(system("gmsnf",p,gmsstd,gmsmatrix,(K+1)*deg(var(1))-2*gmsmaxdeg,K)); |
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233 | } |
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234 | |
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235 | intvec v=1; |
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236 | v[nvars(basering)]=0; |
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237 | |
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238 | int k; |
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239 | ideal r,q; |
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240 | r[ncols(p)]=0; |
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241 | q[ncols(p)]=0; |
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242 | |
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243 | poly s; |
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244 | int i,j; |
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245 | for(k=ncols(p);k>=1;k--) |
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246 | { |
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247 | while(p[k]!=0&°(lead(p[k]),v)<=K) |
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248 | { |
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249 | i=1; |
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250 | s=lead(p[k])/lead(gmsstd[i]); |
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251 | while(i<ncols(gmsstd)&&s==0) |
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252 | { |
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253 | i++; |
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254 | s=lead(p[k])/lead(gmsstd[i]); |
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255 | } |
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256 | if(s!=0) |
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257 | { |
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258 | p[k]=p[k]-s*gmsstd[i]; |
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259 | for(j=1;j<=nrows(gmsmatrix);j++) |
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260 | { |
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261 | p[k]=p[k]+diff(s*gmsmatrix[j,i],var(j+1))*var(1); |
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262 | } |
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263 | } |
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264 | else |
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265 | { |
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266 | r[k]=r[k]+lead(p[k]); |
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267 | p[k]=p[k]-lead(p[k]); |
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268 | } |
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269 | while(deg(lead(p[k]))>(K+1)*deg(var(1))-2*gmsmaxdeg&& |
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270 | deg(lead(p[k]),v)<=K) |
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271 | { |
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272 | q[k]=q[k]+lead(p[k]); |
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273 | p[k]=p[k]-lead(p[k]); |
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274 | } |
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275 | } |
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276 | q[k]=q[k]+p[k]; |
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277 | } |
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278 | |
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279 | return(list(r,q)); |
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280 | } |
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281 | example |
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282 | { "EXAMPLE:"; echo=2; |
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283 | ring R=0,(x,y),ds; |
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284 | poly t=x5+x2y2+y5; |
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285 | def G=gmsring(t,"s"); |
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286 | setring(G); |
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287 | list l0=gmsnf(gmspoly,0); |
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288 | print(l0[1]); |
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289 | list l1=gmsnf(gmspoly,1); |
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290 | print(l1[1]); |
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291 | list l=gmsnf(l0[2],1); |
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292 | print(l[1]); |
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293 | } |
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294 | /////////////////////////////////////////////////////////////////////////////// |
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295 | |
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296 | proc gmscoeffs(ideal p,int K) |
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297 | "USAGE: gmscoeffs(p,K); poly p, int K |
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298 | ASSUME: basering constructed by gmsring |
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299 | RETURN: |
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300 | @format |
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301 | list l; |
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302 | matrix l[1]; C[[s]]-basis representation of p mod s^(K+1) |
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303 | ideal l[2]; p==matrix(gmsbasis)*l[1]+l[2] |
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304 | @end format |
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305 | NOTE: computation can be continued by setting p=l[2] |
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306 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice |
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307 | EXAMPLE: example gmscoeffs; shows examples |
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308 | " |
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309 | { |
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310 | list l=gmsnf(p,K); |
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311 | ideal r,q=l[1..2]; |
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312 | poly v=1; |
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313 | for(int i=2;i<=nvars(basering);i++) |
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314 | { |
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315 | v=v*var(i); |
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316 | } |
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317 | matrix C=coeffs(r,gmsbasis,v); |
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318 | return(C,q); |
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319 | } |
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320 | example |
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321 | { "EXAMPLE:"; echo=2; |
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322 | ring R=0,(x,y),ds; |
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323 | poly t=x5+x2y2+y5; |
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324 | def G=gmsring(t,"s"); |
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325 | setring(G); |
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326 | list l0=gmscoeffs(gmspoly,0); |
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327 | print(l0[1]); |
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328 | list l1=gmscoeffs(gmspoly,1); |
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329 | print(l1[1]); |
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330 | list l=gmscoeffs(l0[2],1); |
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331 | print(l[1]); |
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332 | } |
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333 | /////////////////////////////////////////////////////////////////////////////// |
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334 | |
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335 | static proc nmin(ideal e) |
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336 | { |
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337 | int i; |
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338 | number m=number(e[1]); |
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339 | for(i=2;i<=ncols(e);i++) |
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340 | { |
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341 | if(number(e[i])<m) |
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342 | { |
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343 | m=number(e[i]); |
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344 | } |
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345 | } |
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346 | return(m); |
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347 | } |
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348 | /////////////////////////////////////////////////////////////////////////////// |
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349 | |
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350 | static proc nmax(ideal e) |
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351 | { |
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352 | int i; |
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353 | number m=number(e[1]); |
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354 | for(i=2;i<=ncols(e);i++) |
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355 | { |
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356 | if(number(e[i])>m) |
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357 | { |
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358 | m=number(e[i]); |
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359 | } |
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360 | } |
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361 | return(m); |
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362 | } |
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363 | /////////////////////////////////////////////////////////////////////////////// |
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364 | |
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365 | static proc saturate() |
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366 | { |
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367 | int mu=ncols(gmsbasis); |
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368 | ideal r=gmspoly*gmsbasis; |
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369 | matrix A0[mu][mu],C; |
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370 | module H0; |
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371 | module H,H1=freemodule(mu),freemodule(mu); |
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372 | int k=-1; |
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373 | list l; |
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374 | |
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375 | while(size(reduce(H,std(H0*s)))>0) |
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376 | { |
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377 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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378 | k++; |
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379 | dbprint(printlevel-voice+2,"// k="+string(k)); |
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380 | l=gmscoeffs(r,k); |
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381 | C,r=l[1..2]; |
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382 | A0=A0+C; |
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383 | |
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384 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
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385 | H0=H; |
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386 | H1=jet(module(A0*H1+s^2*diff(matrix(H1),s)),k); |
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387 | H=H*s+H1; |
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388 | } |
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389 | |
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390 | A0=A0-k*s; |
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391 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
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392 | H=std(H0); |
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393 | |
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394 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
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395 | H0=division(freemodule(mu)*s^k,H,k*deg(s))[1]; |
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396 | |
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397 | return(A0,r,H,H0,k); |
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398 | } |
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399 | /////////////////////////////////////////////////////////////////////////////// |
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400 | |
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401 | static proc tjet(matrix A0,ideal r,module H,int k0,int K) |
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402 | { |
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403 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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404 | dbprint(printlevel-voice+2,"// k="+string(K+k0+1)); |
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405 | list l=gmscoeffs(r,K+k0+1); |
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406 | matrix C; |
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407 | C,r=l[1..2]; |
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408 | A0=A0+C; |
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409 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
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410 | matrix A=division(A0*H+s^2*diff(matrix(H),s),H,(K+1)*deg(s))[1]/s; |
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411 | |
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412 | return(A,A0,r); |
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413 | } |
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414 | /////////////////////////////////////////////////////////////////////////////// |
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415 | |
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416 | static proc eigenval(matrix A0,ideal r,module H,int k0) |
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417 | { |
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418 | dbprint(printlevel-voice+2, |
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419 | "// compute eigenvalues e with multiplicities m of A0"); |
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420 | matrix A; |
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421 | A,A0,r=tjet(A0,r,H,k0,0); |
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422 | list l=eigenvals(A); |
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423 | def e,m=l[1..2]; |
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424 | dbprint(printlevel-voice+2,"// e="+string(e)); |
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425 | dbprint(printlevel-voice+2,"// m="+string(m)); |
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426 | |
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427 | return(e,m,A0,r); |
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428 | } |
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429 | /////////////////////////////////////////////////////////////////////////////// |
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430 | |
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431 | static proc transform(matrix A,matrix A0,ideal r,module H,module H0,ideal e,intvec m,int k0,int K,int opt) |
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432 | { |
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433 | int mu=ncols(gmsbasis); |
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434 | |
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435 | number e0,e1=nmin(e),nmax(e); |
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436 | |
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437 | int i,j,k; |
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438 | int k1; |
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439 | intvec d; |
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440 | d[ncols(e)]=0; |
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441 | if(opt) |
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442 | { |
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443 | dbprint(printlevel-voice+2, |
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444 | "// compute maximal differences d of e"); |
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445 | for(i=1;i<=ncols(e);i++) |
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446 | { |
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447 | d[i]=int(e[i]-e0); |
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448 | } |
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449 | } |
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450 | else |
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451 | { |
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452 | dbprint(printlevel-voice+2, |
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453 | "// compute maximal integer differences d of e"); |
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454 | for(i=1;i<ncols(e);i++) |
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455 | { |
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456 | for(j=i+1;j<=ncols(e);j++) |
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457 | { |
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458 | k=int(e[i]-e[j]); |
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459 | if(number(e[i]-e[j])==k) |
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460 | { |
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461 | if(k>d[i]) |
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462 | { |
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463 | d[i]=k; |
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464 | } |
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465 | if(-k>d[j]) |
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466 | { |
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467 | d[j]=-k; |
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468 | } |
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469 | } |
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470 | } |
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471 | } |
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472 | } |
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473 | dbprint(printlevel-voice+2,"// d="+string(d)); |
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474 | |
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475 | for(i,k=1,0;i<=size(d);i++) |
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476 | { |
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477 | if(k<d[i]) |
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478 | { |
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479 | k=d[i]; |
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480 | } |
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481 | } |
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482 | |
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483 | A,A0,r=tjet(A0,r,H,k0,K+k1); |
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484 | module U0=s^k0*freemodule(mu); |
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485 | |
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486 | if(k>0) |
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487 | { |
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488 | int i0,j0,i1,j1; |
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489 | module U,V; |
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490 | list l; |
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491 | |
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492 | while(k>0) |
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493 | { |
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494 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
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495 | U=0; |
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496 | for(i=1;i<=ncols(e);i++) |
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497 | { |
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498 | U=U+syz(power(jet(A,0)-e[i],m[i])); |
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499 | } |
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500 | V=inverse(U); |
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501 | A=V*A*U; |
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502 | H0=V*H0; |
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503 | U0=U0*U; |
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504 | |
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505 | dbprint(printlevel-voice+2, |
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506 | "// transform to reduce d by 1"); |
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507 | for(i0,i=1,1;i0<=ncols(e);i0++) |
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508 | { |
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509 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
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510 | { |
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511 | for(j0,j=1,1;j0<=ncols(e);j0++) |
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512 | { |
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513 | for(j1=1;j1<=m[j0];j1,j=j1+1,j+1) |
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514 | { |
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515 | if(d[i0]==0&&d[j0]>0) |
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516 | { |
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517 | A[i,j]=A[i,j]/s; |
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518 | } |
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519 | if(d[i0]>0&&d[j0]==0) |
---|
520 | { |
---|
521 | A[i,j]=A[i,j]*s; |
---|
522 | } |
---|
523 | } |
---|
524 | } |
---|
525 | } |
---|
526 | } |
---|
527 | |
---|
528 | H0=transpose(H0); |
---|
529 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
530 | { |
---|
531 | if(d[i0]>0) |
---|
532 | { |
---|
533 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
534 | { |
---|
535 | A[i,i]=A[i,i]-1; |
---|
536 | H0[i]=H0[i]*s; |
---|
537 | U0[i]=U0[i]/s; |
---|
538 | } |
---|
539 | e[i0]=e[i0]-1; |
---|
540 | d[i0]=d[i0]-1; |
---|
541 | } |
---|
542 | else |
---|
543 | { |
---|
544 | i=i+m[i0]; |
---|
545 | } |
---|
546 | } |
---|
547 | H0=transpose(H0); |
---|
548 | |
---|
549 | l=sppnf(e,d,m); |
---|
550 | e,d,m=l[1..3]; |
---|
551 | |
---|
552 | k--; |
---|
553 | } |
---|
554 | |
---|
555 | A=jet(A,K); |
---|
556 | } |
---|
557 | |
---|
558 | return(A,A0,r,H0,U0,e,m); |
---|
559 | } |
---|
560 | /////////////////////////////////////////////////////////////////////////////// |
---|
561 | |
---|
562 | proc bernstein(poly t) |
---|
563 | "USAGE: bernstein(t); poly t |
---|
564 | ASSUME: characteristic 0; local degree ordering; |
---|
565 | isolated critical point 0 of t |
---|
566 | RETURN: ideal r; roots of the Bernstein polynomial b excluding the root -1 |
---|
567 | NOTE: the roots of b are negative rational numbers and -1 is a root of b |
---|
568 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
569 | Bernstein polynomial |
---|
570 | EXAMPLE: example bernstein; shows examples |
---|
571 | " |
---|
572 | { |
---|
573 | def R=basering; |
---|
574 | int n=nvars(R)-1; |
---|
575 | def G=gmsring(t,"s"); |
---|
576 | setring(G); |
---|
577 | |
---|
578 | matrix A; |
---|
579 | module U0; |
---|
580 | ideal e; |
---|
581 | intvec m; |
---|
582 | |
---|
583 | def A0,r,H,H0,k0=saturate(); |
---|
584 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
585 | |
---|
586 | ideal e0; |
---|
587 | for(int i,j=1,1;i<=ncols(e);i++) |
---|
588 | { |
---|
589 | if(e[i]!=1) |
---|
590 | { |
---|
591 | e0[j]=e[i]; |
---|
592 | j++; |
---|
593 | } |
---|
594 | } |
---|
595 | |
---|
596 | setring(R); |
---|
597 | ideal r=-imap(G,e0); |
---|
598 | kill G,gmsmaxdeg; |
---|
599 | |
---|
600 | return(r); |
---|
601 | } |
---|
602 | example |
---|
603 | { "EXAMPLE:"; echo=2; |
---|
604 | ring R=0,(x,y),ds; |
---|
605 | poly t=x5+x2y2+y5; |
---|
606 | bernstein(t); |
---|
607 | } |
---|
608 | /////////////////////////////////////////////////////////////////////////////// |
---|
609 | |
---|
610 | proc monodromy(poly t) |
---|
611 | "USAGE: monodromy(t); poly t |
---|
612 | ASSUME: characteristic 0; local degree ordering; |
---|
613 | isolated critical point 0 of t |
---|
614 | RETURN: |
---|
615 | @format |
---|
616 | list l; Jordan data jordan(M) of monodromy matrix exp(-2*pi*i*M) |
---|
617 | ideal l[1]; |
---|
618 | number l[1][i]; eigenvalue of i-th Jordan block of M |
---|
619 | intvec l[2]; |
---|
620 | int l[2][i]; size of i-th Jordan block of M |
---|
621 | intvec l[3]; |
---|
622 | int l[3][i]; multiplicity of i-th Jordan block of M |
---|
623 | @end format |
---|
624 | SEE ALSO: mondromy_lib, linalg_lib |
---|
625 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; monodromy |
---|
626 | EXAMPLE: example monodromy; shows examples |
---|
627 | " |
---|
628 | { |
---|
629 | def R=basering; |
---|
630 | int n=nvars(R)-1; |
---|
631 | def G=gmsring(t,"s"); |
---|
632 | setring(G); |
---|
633 | |
---|
634 | matrix A; |
---|
635 | module U0; |
---|
636 | ideal e; |
---|
637 | intvec m; |
---|
638 | |
---|
639 | def A0,r,H,H0,k0=saturate(); |
---|
640 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
641 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,0,0); |
---|
642 | |
---|
643 | list l=jordan(A,e,m); |
---|
644 | setring(R); |
---|
645 | list l=imap(G,l); |
---|
646 | kill G,gmsmaxdeg; |
---|
647 | |
---|
648 | return(l); |
---|
649 | } |
---|
650 | example |
---|
651 | { "EXAMPLE:"; echo=2; |
---|
652 | ring R=0,(x,y),ds; |
---|
653 | poly t=x5+x2y2+y5; |
---|
654 | monodromy(t); |
---|
655 | } |
---|
656 | /////////////////////////////////////////////////////////////////////////////// |
---|
657 | |
---|
658 | proc spectrum(poly t) |
---|
659 | "USAGE: spectrum(t); poly t |
---|
660 | ASSUME: characteristic 0; local degree ordering; |
---|
661 | isolated critical point 0 of t |
---|
662 | RETURN: |
---|
663 | @format |
---|
664 | list sp; singularity spectrum of t |
---|
665 | ideal sp[1]; |
---|
666 | number sp[1][i]; i-th spectral number |
---|
667 | intvec sp[2]; |
---|
668 | int sp[2][i]; multiplicity of i-th spectral number |
---|
669 | @end format |
---|
670 | SEE ALSO: spectrum_lib |
---|
671 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
672 | mixed Hodge structure; V-filtration; spectrum |
---|
673 | EXAMPLE: example spectrum; shows examples |
---|
674 | " |
---|
675 | { |
---|
676 | list l=vwfilt(t); |
---|
677 | return(spnf(l[1],l[3])); |
---|
678 | } |
---|
679 | example |
---|
680 | { "EXAMPLE:"; echo=2; |
---|
681 | ring R=0,(x,y),ds; |
---|
682 | poly t=x5+x2y2+y5; |
---|
683 | spprint(spectrum(t)); |
---|
684 | } |
---|
685 | /////////////////////////////////////////////////////////////////////////////// |
---|
686 | |
---|
687 | proc sppairs(poly t) |
---|
688 | "USAGE: sppairs(t); poly t |
---|
689 | ASSUME: characteristic 0; local degree ordering; |
---|
690 | isolated critical point 0 of t |
---|
691 | RETURN: |
---|
692 | @format |
---|
693 | list spp; spectral pairs of t |
---|
694 | ideal spp[1]; |
---|
695 | number spp[1][i]; V-filtration index of i-th spectral pair |
---|
696 | intvec spp[2]; |
---|
697 | int spp[2][i]; weight filtration index of i-th spectral pair |
---|
698 | intvec spp[3]; |
---|
699 | int spp[3][i]; multiplicity of i-th spectral pair |
---|
700 | @end format |
---|
701 | SEE ALSO: spectrum_lib |
---|
702 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
703 | mixed Hodge structure; V-filtration; weight filtration; |
---|
704 | spectrum; spectral pairs |
---|
705 | EXAMPLE: example sppairs; shows examples |
---|
706 | " |
---|
707 | { |
---|
708 | list l=vwfilt(t); |
---|
709 | return(list(l[1],l[2],l[3])); |
---|
710 | } |
---|
711 | example |
---|
712 | { "EXAMPLE:"; echo=2; |
---|
713 | ring R=0,(x,y),ds; |
---|
714 | poly t=x5+x2y2+y5; |
---|
715 | sppprint(sppairs(t)); |
---|
716 | } |
---|
717 | /////////////////////////////////////////////////////////////////////////////// |
---|
718 | |
---|
719 | proc spnf(ideal a,list #) |
---|
720 | "USAGE: spnf(a[,m][,V]); ideal a, intvec m, list V |
---|
721 | ASSUME: ncols(a)==size(m)==size(V); typeof(V[i])=="int" |
---|
722 | RETURN: order (a[i][,V[i]]) with multiplicity m[i] lexicographically |
---|
723 | EXAMPLE: example spnf; shows examples |
---|
724 | " |
---|
725 | { |
---|
726 | int n=ncols(a); |
---|
727 | intvec m; |
---|
728 | module v; |
---|
729 | list V; |
---|
730 | int i,j; |
---|
731 | while(i<size(#)) |
---|
732 | { |
---|
733 | i++; |
---|
734 | if(typeof(#[i])=="intvec") |
---|
735 | { |
---|
736 | m=#[i]; |
---|
737 | } |
---|
738 | if(typeof(#[i])=="module") |
---|
739 | { |
---|
740 | v=#[i]; |
---|
741 | for(j=n;j>=1;j--) |
---|
742 | { |
---|
743 | V[j]=module(v[j]); |
---|
744 | } |
---|
745 | } |
---|
746 | if(typeof(#[i])=="list") |
---|
747 | { |
---|
748 | V=#[i]; |
---|
749 | } |
---|
750 | } |
---|
751 | if(m==0) |
---|
752 | { |
---|
753 | for(i=n;i>=1;i--) |
---|
754 | { |
---|
755 | m[i]=1; |
---|
756 | } |
---|
757 | } |
---|
758 | |
---|
759 | int k; |
---|
760 | ideal a0; |
---|
761 | intvec m0; |
---|
762 | list V0; |
---|
763 | number a1; |
---|
764 | int m1; |
---|
765 | for(i=n;i>=1;i--) |
---|
766 | { |
---|
767 | if(m[i]!=0) |
---|
768 | { |
---|
769 | for(j=i-1;j>=1;j--) |
---|
770 | { |
---|
771 | if(m[j]!=0) |
---|
772 | { |
---|
773 | if(number(a[i])>number(a[j])) |
---|
774 | { |
---|
775 | a1=number(a[i]); |
---|
776 | a[i]=a[j]; |
---|
777 | a[j]=a1; |
---|
778 | m1=m[i]; |
---|
779 | m[i]=m[j]; |
---|
780 | m[j]=m1; |
---|
781 | if(size(V)>0) |
---|
782 | { |
---|
783 | v=V[i]; |
---|
784 | V[i]=V[j]; |
---|
785 | V[j]=v; |
---|
786 | } |
---|
787 | } |
---|
788 | if(number(a[i])==number(a[j])) |
---|
789 | { |
---|
790 | m[i]=m[i]+m[j]; |
---|
791 | m[j]=0; |
---|
792 | if(size(V)>0) |
---|
793 | { |
---|
794 | V[i]=V[i]+V[j]; |
---|
795 | } |
---|
796 | } |
---|
797 | } |
---|
798 | } |
---|
799 | k++; |
---|
800 | a0[k]=a[i]; |
---|
801 | m0[k]=m[i]; |
---|
802 | if(size(V)>0) |
---|
803 | { |
---|
804 | V0[k]=V[i]; |
---|
805 | } |
---|
806 | } |
---|
807 | } |
---|
808 | |
---|
809 | if(size(V0)>0) |
---|
810 | { |
---|
811 | n=size(V0); |
---|
812 | module U=std(V0[n]); |
---|
813 | for(i=n-1;i>=1;i--) |
---|
814 | { |
---|
815 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
816 | if(i>=2) |
---|
817 | { |
---|
818 | U=std(U+V0[i]); |
---|
819 | } |
---|
820 | } |
---|
821 | } |
---|
822 | |
---|
823 | list l; |
---|
824 | if(k>0) |
---|
825 | { |
---|
826 | l=a0,m0; |
---|
827 | if(size(V0)>0) |
---|
828 | { |
---|
829 | l[3]=V0; |
---|
830 | } |
---|
831 | } |
---|
832 | return(l); |
---|
833 | } |
---|
834 | example |
---|
835 | { "EXAMPLE:"; echo=2; |
---|
836 | } |
---|
837 | /////////////////////////////////////////////////////////////////////////////// |
---|
838 | |
---|
839 | proc sppnf(ideal a,intvec w,list #) |
---|
840 | "USAGE: sppnf(a,w[,m][,V]); ideal a, intvec w, intvec m, list V |
---|
841 | ASSUME: ncols(e)=size(w)=size(m)=size(V); typeof(V[i])=="module" |
---|
842 | RETURN: order (a[i][,w[i]][,V[i]]) with multiplicity m[i] lexicographically |
---|
843 | EXAMPLE: example sppnorm; shows examples |
---|
844 | " |
---|
845 | { |
---|
846 | int n=ncols(a); |
---|
847 | intvec m; |
---|
848 | module v; |
---|
849 | list V; |
---|
850 | int i,j; |
---|
851 | while(i<size(#)) |
---|
852 | { |
---|
853 | i++; |
---|
854 | if(typeof(#[i])=="intvec") |
---|
855 | { |
---|
856 | m=#[i]; |
---|
857 | } |
---|
858 | if(typeof(#[i])=="module") |
---|
859 | { |
---|
860 | v=#[i]; |
---|
861 | for(j=n;j>=1;j--) |
---|
862 | { |
---|
863 | V[j]=module(v[j]); |
---|
864 | } |
---|
865 | } |
---|
866 | if(typeof(#[i])=="list") |
---|
867 | { |
---|
868 | V=#[i]; |
---|
869 | } |
---|
870 | } |
---|
871 | if(m==0) |
---|
872 | { |
---|
873 | for(i=n;i>=1;i--) |
---|
874 | { |
---|
875 | m[i]=1; |
---|
876 | } |
---|
877 | } |
---|
878 | |
---|
879 | int k; |
---|
880 | ideal a0; |
---|
881 | intvec w0,m0; |
---|
882 | list V0; |
---|
883 | number a1; |
---|
884 | int w1,m1; |
---|
885 | for(i=n;i>=1;i--) |
---|
886 | { |
---|
887 | if(m[i]!=0) |
---|
888 | { |
---|
889 | for(j=i-1;j>=1;j--) |
---|
890 | { |
---|
891 | if(m[j]!=0) |
---|
892 | { |
---|
893 | if(number(a[i])>number(a[j])|| |
---|
894 | (number(a[i])==number(a[j])&&w[i]<w[j])) |
---|
895 | { |
---|
896 | a1=number(a[i]); |
---|
897 | a[i]=a[j]; |
---|
898 | a[j]=a1; |
---|
899 | w1=w[i]; |
---|
900 | w[i]=w[j]; |
---|
901 | w[j]=w1; |
---|
902 | m1=m[i]; |
---|
903 | m[i]=m[j]; |
---|
904 | m[j]=m1; |
---|
905 | if(size(V)>0) |
---|
906 | { |
---|
907 | v=V[i]; |
---|
908 | V[i]=V[j]; |
---|
909 | V[j]=v; |
---|
910 | } |
---|
911 | } |
---|
912 | if(number(a[i])==number(a[j])&&w[i]==w[j]) |
---|
913 | { |
---|
914 | m[i]=m[i]+m[j]; |
---|
915 | m[j]=0; |
---|
916 | if(size(V)>0) |
---|
917 | { |
---|
918 | V[i]=V[i]+V[j]; |
---|
919 | } |
---|
920 | } |
---|
921 | } |
---|
922 | } |
---|
923 | k++; |
---|
924 | a0[k]=a[i]; |
---|
925 | w0[k]=w[i]; |
---|
926 | m0[k]=m[i]; |
---|
927 | if(size(V)>0) |
---|
928 | { |
---|
929 | V0[k]=V[i]; |
---|
930 | } |
---|
931 | } |
---|
932 | } |
---|
933 | |
---|
934 | if(size(V0)>0) |
---|
935 | { |
---|
936 | n=size(V0); |
---|
937 | module U=std(V0[n]); |
---|
938 | for(i=n-1;i>=1;i--) |
---|
939 | { |
---|
940 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
941 | if(i>=2) |
---|
942 | { |
---|
943 | U=std(U+V0[i]); |
---|
944 | } |
---|
945 | } |
---|
946 | } |
---|
947 | |
---|
948 | list l; |
---|
949 | if(k>0) |
---|
950 | { |
---|
951 | l=a0,w0,m0; |
---|
952 | if(size(V0)>0) |
---|
953 | { |
---|
954 | l[4]=V0; |
---|
955 | } |
---|
956 | } |
---|
957 | return(l); |
---|
958 | } |
---|
959 | example |
---|
960 | { "EXAMPLE:"; echo=2; |
---|
961 | } |
---|
962 | /////////////////////////////////////////////////////////////////////////////// |
---|
963 | |
---|
964 | proc vfilt(poly t) |
---|
965 | "USAGE: vfilt(t); poly t |
---|
966 | ASSUME: characteristic 0; local degree ordering; |
---|
967 | isolated critical point 0 of t |
---|
968 | RETURN: |
---|
969 | @format |
---|
970 | list v; V-filtration on H''/s*H'' |
---|
971 | ideal v[1]; |
---|
972 | number v[1][i]; V-filtration index of i-th spectral number |
---|
973 | intvec v[2]; |
---|
974 | int v[2][i]; multiplicity of i-th spectral number |
---|
975 | list v[3]; |
---|
976 | module v[3][i]; vector space of i-th graded part in terms of v[4] |
---|
977 | ideal v[4]; monomial vector space basis of H''/s*H'' |
---|
978 | ideal v[5]; standard basis of Jacobian ideal |
---|
979 | @end format |
---|
980 | SEE ALSO: spectrum_lib |
---|
981 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
982 | mixed Hodge structure; V-filtration; spectrum |
---|
983 | EXAMPLE: example vfilt; shows examples |
---|
984 | " |
---|
985 | { |
---|
986 | list l=vwfilt(t); |
---|
987 | return(spnf(l[1],l[3],l[4])+list(l[5],l[6])); |
---|
988 | } |
---|
989 | example |
---|
990 | { "EXAMPLE:"; echo=2; |
---|
991 | ring R=0,(x,y),ds; |
---|
992 | poly t=x5+x2y2+y5; |
---|
993 | vfilt(t); |
---|
994 | } |
---|
995 | /////////////////////////////////////////////////////////////////////////////// |
---|
996 | |
---|
997 | proc vwfilt(poly t) |
---|
998 | "USAGE: vwfilt(t); poly t |
---|
999 | ASSUME: characteristic 0; local degree ordering; |
---|
1000 | isolated critical point 0 of t |
---|
1001 | RETURN: |
---|
1002 | @format |
---|
1003 | list vw; weighted V-filtration on H''/s*H'' |
---|
1004 | ideal vw[1]; |
---|
1005 | number vw[1][i]; V-filtration index of i-th spectral pair |
---|
1006 | intvec vw[2]; |
---|
1007 | int vw[2][i]; weight filtration index of i-th spectral pair |
---|
1008 | intvec vw[3]; |
---|
1009 | int vw[3][i]; multiplicity of i-th spectral pair |
---|
1010 | list vw[4]; |
---|
1011 | module vw[4][i]; vector space of i-th graded part in terms of vw[5] |
---|
1012 | ideal vw[5]; monomial vector space basis of H''/s*H'' |
---|
1013 | ideal vw[6]; standard basis of Jacobian ideal |
---|
1014 | @end format |
---|
1015 | SEE ALSO: spectrum_lib |
---|
1016 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1017 | mixed Hodge structure; V-filtration; weight filtration; |
---|
1018 | spectrum; spectral pairs |
---|
1019 | EXAMPLE: example vwfilt; shows examples |
---|
1020 | " |
---|
1021 | { |
---|
1022 | def R=basering; |
---|
1023 | int n=nvars(R)-1; |
---|
1024 | def G=gmsring(t,"s"); |
---|
1025 | setring(G); |
---|
1026 | |
---|
1027 | int mu=ncols(gmsbasis); |
---|
1028 | matrix A; |
---|
1029 | module U0; |
---|
1030 | ideal e; |
---|
1031 | intvec m; |
---|
1032 | |
---|
1033 | def A0,r,H,H0,k0=saturate(); |
---|
1034 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
1035 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,0,1); |
---|
1036 | |
---|
1037 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
1038 | list l=jordanbasis(A,e,m); |
---|
1039 | def U,v=l[1..2]; |
---|
1040 | kill l; |
---|
1041 | vector u0; |
---|
1042 | int v0; |
---|
1043 | int i,j,k,l; |
---|
1044 | for(k,l=1,1;l<=ncols(e);k,l=k+m[l],l+1) |
---|
1045 | { |
---|
1046 | for(i=k+m[l]-1;i>=k+1;i--) |
---|
1047 | { |
---|
1048 | for(j=i-1;j>=k;j--) |
---|
1049 | { |
---|
1050 | if(v[i]>v[j]) |
---|
1051 | { |
---|
1052 | v0=v[i];v[i]=v[j];v[j]=v0; |
---|
1053 | u0=U[i];U[i]=U[j];U[j]=u0; |
---|
1054 | } |
---|
1055 | } |
---|
1056 | } |
---|
1057 | } |
---|
1058 | |
---|
1059 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
1060 | matrix V=inverse(U); |
---|
1061 | A=V*A*U; |
---|
1062 | dbprint(printlevel-voice+2,"// compute normal form of H''"); |
---|
1063 | H0=std(V*H0); |
---|
1064 | U0=U0*U; |
---|
1065 | |
---|
1066 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
1067 | ideal a; |
---|
1068 | intvec w; |
---|
1069 | for(i=1;i<=mu;i++) |
---|
1070 | { |
---|
1071 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
1072 | a[i]=A[j,j]+ord(H0[i])/deg(s)-1; |
---|
1073 | w[i]=v[j]+n; |
---|
1074 | } |
---|
1075 | kill v; |
---|
1076 | module v=simplify(jet(H*U0*H0,2*k0)/s^(2*k0),1); |
---|
1077 | |
---|
1078 | kill l; |
---|
1079 | list l=sppnf(a,w,v)+list(gmsbasis,gmsstd); |
---|
1080 | setring(R); |
---|
1081 | list l=imap(G,l); |
---|
1082 | kill G,gmsmaxdeg; |
---|
1083 | attrib(l[5],"isSB",1); |
---|
1084 | |
---|
1085 | return(l); |
---|
1086 | } |
---|
1087 | example |
---|
1088 | { "EXAMPLE:"; echo=2; |
---|
1089 | ring R=0,(x,y),ds; |
---|
1090 | poly t=x5+x2y2+y5; |
---|
1091 | vwfilt(t); |
---|
1092 | } |
---|
1093 | /////////////////////////////////////////////////////////////////////////////// |
---|
1094 | |
---|
1095 | static proc commutator(matrix A) |
---|
1096 | { |
---|
1097 | int n=ncols(A); |
---|
1098 | int i,j,k; |
---|
1099 | matrix C[n^2][n^2]; |
---|
1100 | for(i=0;i<n;i++) |
---|
1101 | { |
---|
1102 | for(j=0;j<n;j++) |
---|
1103 | { |
---|
1104 | for(k=0;k<n;k++) |
---|
1105 | { |
---|
1106 | C[i*n+j+1,k*n+j+1]=C[i*n+j+1,k*n+j+1]+A[i+1,k+1]; |
---|
1107 | C[i*n+j+1,i*n+k+1]=C[i*n+j+1,i*n+k+1]-A[k+1,j+1]; |
---|
1108 | } |
---|
1109 | } |
---|
1110 | } |
---|
1111 | return(C); |
---|
1112 | } |
---|
1113 | |
---|
1114 | /////////////////////////////////////////////////////////////////////////////// |
---|
1115 | |
---|
1116 | proc tmatrix(poly t,list #) |
---|
1117 | "USAGE: tmatrix(t); poly t |
---|
1118 | ASSUME: characteristic 0; local degree ordering; |
---|
1119 | isolated critical point 0 of t |
---|
1120 | RETURN: |
---|
1121 | @format |
---|
1122 | list A; C[[s]]-matrix A[1]+s*A[2] of t on H'' |
---|
1123 | matrix A[1]; |
---|
1124 | matrix A[2]; |
---|
1125 | @end format |
---|
1126 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1127 | mixed Hodge structure; opposite Hodge filtration; V-filtration |
---|
1128 | EXAMPLE: example tmatrix; shows examples |
---|
1129 | " |
---|
1130 | { |
---|
1131 | def R=basering; |
---|
1132 | int n=nvars(R)-1; |
---|
1133 | def G=gmsring(t,"s"); |
---|
1134 | setring(G); |
---|
1135 | |
---|
1136 | int mu=ncols(gmsbasis); |
---|
1137 | matrix A; |
---|
1138 | module U0; |
---|
1139 | ideal e; |
---|
1140 | intvec m; |
---|
1141 | |
---|
1142 | def A0,r,H,H0,k0=saturate(); |
---|
1143 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
1144 | int k1=int(nmax(e)-nmin(e)); |
---|
1145 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,k0+k1,1); |
---|
1146 | |
---|
1147 | ring S=0,s,(ds,c); |
---|
1148 | matrix A=imap(G,A); |
---|
1149 | module H0=imap(G,H0); |
---|
1150 | ideal e=imap(G,e); |
---|
1151 | kill G,gmsmaxdeg; |
---|
1152 | |
---|
1153 | dbprint(printlevel-voice+2,"// transform to Jordan basis"); |
---|
1154 | module U=jordanbasis(A,e,m)[1]; |
---|
1155 | matrix V=inverse(U); |
---|
1156 | A=V*A*U; |
---|
1157 | module H=V*H0; |
---|
1158 | |
---|
1159 | dbprint(printlevel-voice+2,"// compute splitting of V-filtration"); |
---|
1160 | int i,j,k; |
---|
1161 | U=freemodule(mu); |
---|
1162 | V=matrix(0,mu,mu); |
---|
1163 | matrix v[mu^2][1]; |
---|
1164 | matrix A0=commutator(jet(A,0)); |
---|
1165 | for(k=1;k<=k0+k1;k++) |
---|
1166 | { |
---|
1167 | for(j=0;j<k;j++) |
---|
1168 | { |
---|
1169 | V=matrix(V)-(jet(A,k-j)/s^(k-j))*(jet(U,j)/s^j); |
---|
1170 | } |
---|
1171 | v=V[1..mu,1..mu]; |
---|
1172 | v=inverse(A0+k)*v; |
---|
1173 | V=v[1..mu^2,1]; |
---|
1174 | U=matrix(U)+s^k*V; |
---|
1175 | } |
---|
1176 | attrib(U,"isSB",1); |
---|
1177 | |
---|
1178 | dbprint(printlevel-voice+2,"// transform to V-splitting basis"); |
---|
1179 | A=jet(A,0); |
---|
1180 | H=std(division(H,U,(k0+k1)*deg(s))[1]); |
---|
1181 | |
---|
1182 | dbprint(printlevel-voice+2,"// compute V-leading terms of H''"); |
---|
1183 | int i0,j0; |
---|
1184 | module H1=H; |
---|
1185 | for(k=ncols(H1);k>=1;k--) |
---|
1186 | { |
---|
1187 | i0=leadexp(H1[k])[nvars(basering)+1]; |
---|
1188 | j0=ord(H1[k]);//deg(s); |
---|
1189 | H0[k]=lead(H1[k]); |
---|
1190 | H1[k]=H1[k]-lead(H1[k]); |
---|
1191 | if(H1[k]!=0) |
---|
1192 | { |
---|
1193 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
1194 | j=ord(H1[k]);//deg(s); |
---|
1195 | while(A[i,i]+j==A[i0,i0]+j0) |
---|
1196 | { |
---|
1197 | H0[k]=H0[k]+lead(H1[k]); |
---|
1198 | H1[k]=H1[k]-lead(H1[k]); |
---|
1199 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
1200 | j=ord(H1[k]);//deg(s); |
---|
1201 | } |
---|
1202 | } |
---|
1203 | } |
---|
1204 | H0=simplify(H0,1); |
---|
1205 | |
---|
1206 | dbprint(printlevel-voice+2,"// compute N"); |
---|
1207 | matrix N=A; |
---|
1208 | for(i=1;i<=ncols(N);i++) |
---|
1209 | { |
---|
1210 | N[i,i]=0; |
---|
1211 | } |
---|
1212 | |
---|
1213 | dbprint(printlevel-voice+2,"// compute splitting of Hodge filtration"); |
---|
1214 | U=0; |
---|
1215 | module U1; |
---|
1216 | module C; |
---|
1217 | list F,I; |
---|
1218 | module F0,I0,U0; |
---|
1219 | for(i0,j0=1,1;i0<=ncols(e);i0++) |
---|
1220 | { |
---|
1221 | C=matrix(0,mu,1); |
---|
1222 | for(j=m[i0];j>=1;j,j0=j-1,j0+1) |
---|
1223 | { |
---|
1224 | C=C+gen(j0); |
---|
1225 | } |
---|
1226 | F0=intersect(C,H0); |
---|
1227 | |
---|
1228 | F=list(); |
---|
1229 | j=0; |
---|
1230 | while(size(F0)>0) |
---|
1231 | { |
---|
1232 | j++; |
---|
1233 | F[j]=matrix(0,mu,1); |
---|
1234 | if(size(jet(F0,0))>0) |
---|
1235 | { |
---|
1236 | for(i=ncols(F0);i>=1;i--) |
---|
1237 | { |
---|
1238 | if(ord(F0[i])==0) |
---|
1239 | { |
---|
1240 | F[j]=F[j]+F0[i]; |
---|
1241 | } |
---|
1242 | } |
---|
1243 | } |
---|
1244 | for(i=ncols(F0);i>=1;i--) |
---|
1245 | { |
---|
1246 | F0[i]=F0[i]/s; |
---|
1247 | } |
---|
1248 | } |
---|
1249 | |
---|
1250 | I=list(); |
---|
1251 | I0=module(); |
---|
1252 | U0=std(module()); |
---|
1253 | for(i=size(F);i>=1;i--) |
---|
1254 | { |
---|
1255 | I[i]=module(); |
---|
1256 | } |
---|
1257 | for(i=1;i<=size(F);i++) |
---|
1258 | { |
---|
1259 | I0=reduce(F[i],U0); |
---|
1260 | j=i; |
---|
1261 | while(size(I0)>0) |
---|
1262 | { |
---|
1263 | U0=std(U0+I0); |
---|
1264 | I[j]=I[j]+I0; |
---|
1265 | I0=reduce(N*I0,U0); |
---|
1266 | j++; |
---|
1267 | } |
---|
1268 | } |
---|
1269 | |
---|
1270 | for(i=1;i<=size(I);i++) |
---|
1271 | { |
---|
1272 | U=U+I[i]; |
---|
1273 | } |
---|
1274 | } |
---|
1275 | |
---|
1276 | dbprint(printlevel-voice+2,"// transform to Hodge splitting basis"); |
---|
1277 | V=inverse(U); |
---|
1278 | A=V*A*U; |
---|
1279 | H=V*H; |
---|
1280 | |
---|
1281 | dbprint(printlevel-voice+2,"// compute reduced standard basis of H''"); |
---|
1282 | degBound=k0+k1+2; |
---|
1283 | option(redSB); |
---|
1284 | H=std(H); |
---|
1285 | option(noredSB); |
---|
1286 | degBound=0; |
---|
1287 | H=simplify(jet(H,k0+k1),1); |
---|
1288 | attrib(H,"isSB",1); |
---|
1289 | |
---|
1290 | dbprint(printlevel-voice+2,"// compute matrix A0+sA1 of t"); |
---|
1291 | A=division(s*A*H+s^2*diff(matrix(H),s),H,deg(s))[1]; |
---|
1292 | A0=jet(A,0); |
---|
1293 | A=jet(A,1)/s; |
---|
1294 | |
---|
1295 | setring(R); |
---|
1296 | matrix A0=imap(S,A0); |
---|
1297 | matrix A1=imap(S,A); |
---|
1298 | kill S; |
---|
1299 | return(list(A0,A1)); |
---|
1300 | } |
---|
1301 | example |
---|
1302 | { "EXAMPLE:"; echo=2; |
---|
1303 | ring R=0,(x,y),ds; |
---|
1304 | poly t=x5+x2y2+y5; |
---|
1305 | list A=tmatrix(t); |
---|
1306 | print(A[1]); |
---|
1307 | print(A[2]); |
---|
1308 | } |
---|
1309 | /////////////////////////////////////////////////////////////////////////////// |
---|
1310 | |
---|
1311 | proc endvfilt(list v) |
---|
1312 | "USAGE: endvfilt(v); list v |
---|
1313 | ASSUME: v returned by vfilt |
---|
1314 | RETURN: |
---|
1315 | @format |
---|
1316 | list ev; V-filtration on Jacobian algebra |
---|
1317 | ideal ev[1]; |
---|
1318 | number ev[1][i]; i-th V-filtration index |
---|
1319 | intvec ev[2]; |
---|
1320 | int ev[2][i]; i-th multiplicity |
---|
1321 | list ev[3]; |
---|
1322 | module ev[3][i]; vector space of i-th graded part in terms of ev[4] |
---|
1323 | ideal ev[4]; monomial vector space basis of Jacobian algebra |
---|
1324 | ideal ev[5]; standard basis of Jacobian ideal |
---|
1325 | @end format |
---|
1326 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1327 | mixed Hodge structure; V-filtration; endomorphism filtration |
---|
1328 | EXAMPLE: example endvfilt; shows examples |
---|
1329 | " |
---|
1330 | { |
---|
1331 | def a,d,V,m,g=v[1..5]; |
---|
1332 | attrib(g,"isSB",1); |
---|
1333 | int mu=ncols(m); |
---|
1334 | |
---|
1335 | module V0=V[1]; |
---|
1336 | for(int i=2;i<=size(V);i++) |
---|
1337 | { |
---|
1338 | V0=V0,V[i]; |
---|
1339 | } |
---|
1340 | |
---|
1341 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
1342 | list M; |
---|
1343 | module U=freemodule(ncols(m)); |
---|
1344 | for(i=ncols(m);i>=1;i--) |
---|
1345 | { |
---|
1346 | M[i]=division(coeffs(reduce(m[i]*m,g,U),m)*V0,V0)[1]; |
---|
1347 | } |
---|
1348 | |
---|
1349 | int j,k,i0,j0,i1,j1; |
---|
1350 | number b0=number(a[1]-a[ncols(a)]); |
---|
1351 | number b1,b2; |
---|
1352 | matrix M0; |
---|
1353 | module L; |
---|
1354 | list v0=freemodule(ncols(m)); |
---|
1355 | ideal a0=b0; |
---|
1356 | list l; |
---|
1357 | |
---|
1358 | while(b0<number(a[ncols(a)]-a[1])) |
---|
1359 | { |
---|
1360 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
1361 | b1=number(a[ncols(a)]-a[1]); |
---|
1362 | for(j=ncols(a);j>=1;j--) |
---|
1363 | { |
---|
1364 | for(i=ncols(a);i>=1;i--) |
---|
1365 | { |
---|
1366 | b2=number(a[i]-a[j]); |
---|
1367 | if(b2>b0&&b2<b1) |
---|
1368 | { |
---|
1369 | b1=b2; |
---|
1370 | } |
---|
1371 | else |
---|
1372 | { |
---|
1373 | if(b2<=b0) |
---|
1374 | { |
---|
1375 | i=0; |
---|
1376 | } |
---|
1377 | } |
---|
1378 | } |
---|
1379 | } |
---|
1380 | b0=b1; |
---|
1381 | |
---|
1382 | l=ideal(); |
---|
1383 | for(k=ncols(m);k>=2;k--) |
---|
1384 | { |
---|
1385 | l=l+list(ideal()); |
---|
1386 | } |
---|
1387 | |
---|
1388 | dbprint(printlevel-voice+2,"// collect conditions for EV["+string(b0)+"]"); |
---|
1389 | j=ncols(a); |
---|
1390 | j0=mu; |
---|
1391 | while(j>=1) |
---|
1392 | { |
---|
1393 | i0=1; |
---|
1394 | i=1; |
---|
1395 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
1396 | { |
---|
1397 | i0=i0+d[i]; |
---|
1398 | i++; |
---|
1399 | } |
---|
1400 | if(a[i]<a[j]+b0) |
---|
1401 | { |
---|
1402 | i0=i0+d[i]; |
---|
1403 | i++; |
---|
1404 | } |
---|
1405 | for(k=1;k<=ncols(m);k++) |
---|
1406 | { |
---|
1407 | M0=M[k]; |
---|
1408 | if(i0>1) |
---|
1409 | { |
---|
1410 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
1411 | } |
---|
1412 | } |
---|
1413 | j0=j0-d[j]; |
---|
1414 | j--; |
---|
1415 | } |
---|
1416 | |
---|
1417 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
1418 | L=transpose(module(l[1])); |
---|
1419 | for(k=2;k<=ncols(m);k++) |
---|
1420 | { |
---|
1421 | L=L,transpose(module(l[k])); |
---|
1422 | } |
---|
1423 | |
---|
1424 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
1425 | v0=v0+list(syz(L)); |
---|
1426 | a0=a0,b0; |
---|
1427 | } |
---|
1428 | |
---|
1429 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1430 | option(redSB); |
---|
1431 | for(i=1;i<size(v0);i++) |
---|
1432 | { |
---|
1433 | v0[i+1]=std(v0[i+1]); |
---|
1434 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
1435 | } |
---|
1436 | option(noredSB); |
---|
1437 | |
---|
1438 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
1439 | i=1; |
---|
1440 | while(size(v0[i])==0) |
---|
1441 | { |
---|
1442 | i++; |
---|
1443 | } |
---|
1444 | list v1=v0[i]; |
---|
1445 | intvec d1=size(v0[i]); |
---|
1446 | ideal a1=a0[i]; |
---|
1447 | i++; |
---|
1448 | while(i<=size(v0)) |
---|
1449 | { |
---|
1450 | if(size(v0[i])>0) |
---|
1451 | { |
---|
1452 | v1=v1+list(v0[i]); |
---|
1453 | d1=d1,size(v0[i]); |
---|
1454 | a1=a1,a0[i]; |
---|
1455 | } |
---|
1456 | i++; |
---|
1457 | } |
---|
1458 | return(list(a1,d1,v1,m,g)); |
---|
1459 | } |
---|
1460 | example |
---|
1461 | { "EXAMPLE:"; echo=2; |
---|
1462 | ring R=0,(x,y),ds; |
---|
1463 | poly t=x5+x2y2+y5; |
---|
1464 | endvfilt(vfilt(t)); |
---|
1465 | } |
---|
1466 | /////////////////////////////////////////////////////////////////////////////// |
---|
1467 | |
---|
1468 | proc spprint(list sp) |
---|
1469 | "USAGE: spprint(sp); list sp |
---|
1470 | RETURN: string s; spectrum sp |
---|
1471 | EXAMPLE: example spprint; shows examples |
---|
1472 | " |
---|
1473 | { |
---|
1474 | string s; |
---|
1475 | for(int i=1;i<size(sp[2]);i++) |
---|
1476 | { |
---|
1477 | s=s+"("+string(sp[1][i])+","+string(sp[2][i])+"),"; |
---|
1478 | } |
---|
1479 | s=s+"("+string(sp[1][i])+","+string(sp[2][i])+")"; |
---|
1480 | return(s); |
---|
1481 | } |
---|
1482 | example |
---|
1483 | { "EXAMPLE:"; echo=2; |
---|
1484 | ring R=0,(x,y),ds; |
---|
1485 | 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)); |
---|
1486 | spprint(sp); |
---|
1487 | } |
---|
1488 | /////////////////////////////////////////////////////////////////////////////// |
---|
1489 | |
---|
1490 | proc sppprint(list spp) |
---|
1491 | "USAGE: sppprint(spp); list spp |
---|
1492 | RETURN: string s; spectral pairs spp |
---|
1493 | EXAMPLE: example sppprint; shows examples |
---|
1494 | " |
---|
1495 | { |
---|
1496 | string s; |
---|
1497 | for(int i=1;i<size(spp[3]);i++) |
---|
1498 | { |
---|
1499 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+"),"; |
---|
1500 | } |
---|
1501 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+")"; |
---|
1502 | return(s); |
---|
1503 | } |
---|
1504 | example |
---|
1505 | { "EXAMPLE:"; echo=2; |
---|
1506 | ring R=0,(x,y),ds; |
---|
1507 | 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)); |
---|
1508 | sppprint(spp); |
---|
1509 | } |
---|
1510 | /////////////////////////////////////////////////////////////////////////////// |
---|
1511 | |
---|
1512 | proc spadd(list sp1,list sp2) |
---|
1513 | "USAGE: spadd(sp1,sp2); list sp1, list sp2 |
---|
1514 | RETURN: list sp; sum of spectra sp1 and sp2 |
---|
1515 | EXAMPLE: example spadd; shows examples |
---|
1516 | " |
---|
1517 | { |
---|
1518 | ideal s; |
---|
1519 | intvec m; |
---|
1520 | int i,i1,i2=1,1,1; |
---|
1521 | while(i1<=size(sp1[2])||i2<=size(sp2[2])) |
---|
1522 | { |
---|
1523 | if(i1<=size(sp1[2])) |
---|
1524 | { |
---|
1525 | if(i2<=size(sp2[2])) |
---|
1526 | { |
---|
1527 | if(number(sp1[1][i1])<number(sp2[1][i2])) |
---|
1528 | { |
---|
1529 | s[i]=sp1[1][i1]; |
---|
1530 | m[i]=sp1[2][i1]; |
---|
1531 | i++; |
---|
1532 | i1++; |
---|
1533 | } |
---|
1534 | else |
---|
1535 | { |
---|
1536 | if(number(sp1[1][i1])>number(sp2[1][i2])) |
---|
1537 | { |
---|
1538 | s[i]=sp2[1][i2]; |
---|
1539 | m[i]=sp2[2][i2]; |
---|
1540 | i++; |
---|
1541 | i2++; |
---|
1542 | } |
---|
1543 | else |
---|
1544 | { |
---|
1545 | if(sp1[2][i1]+sp2[2][i2]!=0) |
---|
1546 | { |
---|
1547 | s[i]=sp1[1][i1]; |
---|
1548 | m[i]=sp1[2][i1]+sp2[2][i2]; |
---|
1549 | i++; |
---|
1550 | } |
---|
1551 | i1++; |
---|
1552 | i2++; |
---|
1553 | } |
---|
1554 | } |
---|
1555 | } |
---|
1556 | else |
---|
1557 | { |
---|
1558 | s[i]=sp1[1][i1]; |
---|
1559 | m[i]=sp1[2][i1]; |
---|
1560 | i++; |
---|
1561 | i1++; |
---|
1562 | } |
---|
1563 | } |
---|
1564 | else |
---|
1565 | { |
---|
1566 | s[i]=sp2[1][i2]; |
---|
1567 | m[i]=sp2[2][i2]; |
---|
1568 | i++; |
---|
1569 | i2++; |
---|
1570 | } |
---|
1571 | } |
---|
1572 | return(list(s,m)); |
---|
1573 | } |
---|
1574 | example |
---|
1575 | { "EXAMPLE:"; echo=2; |
---|
1576 | ring R=0,(x,y),ds; |
---|
1577 | 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)); |
---|
1578 | spprint(sp1); |
---|
1579 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1580 | spprint(sp2); |
---|
1581 | spprint(spadd(sp1,sp2)); |
---|
1582 | } |
---|
1583 | /////////////////////////////////////////////////////////////////////////////// |
---|
1584 | |
---|
1585 | proc spsub(list sp1,list sp2) |
---|
1586 | "USAGE: spsub(sp1,sp2); list sp1, list sp2 |
---|
1587 | RETURN: list sp; difference of spectra sp1 and sp2 |
---|
1588 | EXAMPLE: example spsub; shows examples |
---|
1589 | " |
---|
1590 | { |
---|
1591 | return(spadd(sp1,spmul(sp2,-1))); |
---|
1592 | } |
---|
1593 | example |
---|
1594 | { "EXAMPLE:"; echo=2; |
---|
1595 | ring R=0,(x,y),ds; |
---|
1596 | 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)); |
---|
1597 | spprint(sp1); |
---|
1598 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1599 | spprint(sp2); |
---|
1600 | spprint(spsub(sp1,sp2)); |
---|
1601 | } |
---|
1602 | /////////////////////////////////////////////////////////////////////////////// |
---|
1603 | |
---|
1604 | proc spmul(list #) |
---|
1605 | "USAGE: spmul(sp0,k); list sp0, int[vec] k |
---|
1606 | RETURN: list sp; linear combination of spectra sp0 with coefficients k |
---|
1607 | EXAMPLE: example spmul; shows examples |
---|
1608 | " |
---|
1609 | { |
---|
1610 | if(size(#)==2) |
---|
1611 | { |
---|
1612 | if(typeof(#[1])=="list") |
---|
1613 | { |
---|
1614 | if(typeof(#[2])=="int") |
---|
1615 | { |
---|
1616 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1617 | } |
---|
1618 | if(typeof(#[2])=="intvec") |
---|
1619 | { |
---|
1620 | list sp0=list(ideal(),intvec(0)); |
---|
1621 | for(int i=size(#[2]);i>=1;i--) |
---|
1622 | { |
---|
1623 | sp0=spadd(sp0,spmul(#[1][i],#[2][i])); |
---|
1624 | } |
---|
1625 | return(sp0); |
---|
1626 | } |
---|
1627 | } |
---|
1628 | } |
---|
1629 | return(list(ideal(),intvec(0))); |
---|
1630 | } |
---|
1631 | example |
---|
1632 | { "EXAMPLE:"; echo=2; |
---|
1633 | ring R=0,(x,y),ds; |
---|
1634 | 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)); |
---|
1635 | spprint(sp); |
---|
1636 | spprint(spmul(sp,2)); |
---|
1637 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1638 | spprint(sp1); |
---|
1639 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1640 | spprint(sp2); |
---|
1641 | spprint(spmul(list(sp1,sp2),intvec(1,2))); |
---|
1642 | } |
---|
1643 | /////////////////////////////////////////////////////////////////////////////// |
---|
1644 | |
---|
1645 | proc spissemicont(list sp,list #) |
---|
1646 | "USAGE: spissemicont(sp[,1]); list sp, int opt |
---|
1647 | RETURN: |
---|
1648 | @format |
---|
1649 | int k= |
---|
1650 | 1; if sum of sp is positive on all intervals [a,a+1) [and (a,a+1)] |
---|
1651 | 0; if sum of sp is negative on some interval [a,a+1) [or (a,a+1)] |
---|
1652 | @end format |
---|
1653 | EXAMPLE: example spissemicont; shows examples |
---|
1654 | " |
---|
1655 | { |
---|
1656 | int opt=0; |
---|
1657 | if(size(#)>0) |
---|
1658 | { |
---|
1659 | if(typeof(#[1])=="int") |
---|
1660 | { |
---|
1661 | opt=1; |
---|
1662 | } |
---|
1663 | } |
---|
1664 | int i,j,k; |
---|
1665 | i=1; |
---|
1666 | while(i<=size(sp[2])-1) |
---|
1667 | { |
---|
1668 | j=i+1; |
---|
1669 | k=0; |
---|
1670 | while(j+1<=size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
1671 | { |
---|
1672 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
1673 | { |
---|
1674 | k=k+sp[2][j]; |
---|
1675 | } |
---|
1676 | j++; |
---|
1677 | } |
---|
1678 | if(j==size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
1679 | { |
---|
1680 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
1681 | { |
---|
1682 | k=k+sp[2][j]; |
---|
1683 | } |
---|
1684 | } |
---|
1685 | if(k<0) |
---|
1686 | { |
---|
1687 | return(0); |
---|
1688 | } |
---|
1689 | i++; |
---|
1690 | } |
---|
1691 | return(1); |
---|
1692 | } |
---|
1693 | example |
---|
1694 | { "EXAMPLE:"; echo=2; |
---|
1695 | ring R=0,(x,y),ds; |
---|
1696 | 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)); |
---|
1697 | spprint(sp1); |
---|
1698 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1699 | spprint(sp2); |
---|
1700 | spissemicont(spsub(sp1,spmul(sp2,3))); |
---|
1701 | spissemicont(spsub(sp1,spmul(sp2,4))); |
---|
1702 | } |
---|
1703 | /////////////////////////////////////////////////////////////////////////////// |
---|
1704 | |
---|
1705 | proc spsemicont(list sp0,list sp,list #) |
---|
1706 | "USAGE: spsemicont(sp0,sp,k[,1]); list sp0, list sp |
---|
1707 | RETURN: |
---|
1708 | @format |
---|
1709 | list l; |
---|
1710 | intvec l[i]; if the spectra sp0 occur with multiplicities k |
---|
1711 | in a deformation of a [quasihomogeneous] singularity |
---|
1712 | with spectrum sp then k<=l[i] |
---|
1713 | @end format |
---|
1714 | EXAMPLE: example spsemicont; shows examples |
---|
1715 | " |
---|
1716 | { |
---|
1717 | list l,l0; |
---|
1718 | int i,j,k; |
---|
1719 | while(spissemicont(sp0,#)) |
---|
1720 | { |
---|
1721 | if(size(sp)>1) |
---|
1722 | { |
---|
1723 | l0=spsemicont(sp0,list(sp[1..size(sp)-1])); |
---|
1724 | for(i=1;i<=size(l0);i++) |
---|
1725 | { |
---|
1726 | if(size(l)>0) |
---|
1727 | { |
---|
1728 | j=1; |
---|
1729 | while(j<size(l)&&l[j]!=l0[i]) |
---|
1730 | { |
---|
1731 | j++; |
---|
1732 | } |
---|
1733 | if(l[j]==l0[i]) |
---|
1734 | { |
---|
1735 | l[j][size(sp)]=k; |
---|
1736 | } |
---|
1737 | else |
---|
1738 | { |
---|
1739 | l0[i][size(sp)]=k; |
---|
1740 | l=l+list(l0[i]); |
---|
1741 | } |
---|
1742 | } |
---|
1743 | else |
---|
1744 | { |
---|
1745 | l=l0; |
---|
1746 | } |
---|
1747 | } |
---|
1748 | } |
---|
1749 | sp0=spsub(sp0,sp[size(sp)]); |
---|
1750 | k++; |
---|
1751 | } |
---|
1752 | if(size(sp)>1) |
---|
1753 | { |
---|
1754 | return(l); |
---|
1755 | } |
---|
1756 | else |
---|
1757 | { |
---|
1758 | return(list(intvec(k-1))); |
---|
1759 | } |
---|
1760 | } |
---|
1761 | example |
---|
1762 | { "EXAMPLE:"; echo=2; |
---|
1763 | ring R=0,(x,y),ds; |
---|
1764 | 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)); |
---|
1765 | spprint(sp0); |
---|
1766 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1767 | spprint(sp1); |
---|
1768 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1769 | spprint(sp2); |
---|
1770 | list sp=sp1,sp2; |
---|
1771 | list l=spsemicont(sp0,sp); |
---|
1772 | l; |
---|
1773 | spissemicont(spsub(sp0,spmul(sp,l[1]))); |
---|
1774 | spissemicont(spsub(sp0,spmul(sp,l[1]-1))); |
---|
1775 | spissemicont(spsub(sp0,spmul(sp,l[1]+1))); |
---|
1776 | } |
---|
1777 | /////////////////////////////////////////////////////////////////////////////// |
---|
1778 | |
---|
1779 | proc spmilnor(list sp) |
---|
1780 | "USAGE: spmilnor(sp); list sp |
---|
1781 | RETURN: int mu; Milnor number of spectrum sp |
---|
1782 | EXAMPLE: example spmilnor; shows examples |
---|
1783 | " |
---|
1784 | { |
---|
1785 | return(sum(sp[2])); |
---|
1786 | } |
---|
1787 | example |
---|
1788 | { "EXAMPLE:"; echo=2; |
---|
1789 | ring R=0,(x,y),ds; |
---|
1790 | 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)); |
---|
1791 | spprint(sp); |
---|
1792 | spmilnor(sp); |
---|
1793 | } |
---|
1794 | /////////////////////////////////////////////////////////////////////////////// |
---|
1795 | |
---|
1796 | proc spgeomgenus(list sp) |
---|
1797 | "USAGE: spgeomgenus(sp); list sp |
---|
1798 | RETURN: int g; geometrical genus of spectrum sp |
---|
1799 | EXAMPLE: example spgeomgenus; shows examples |
---|
1800 | " |
---|
1801 | { |
---|
1802 | int g=0; |
---|
1803 | int i=1; |
---|
1804 | while(i+1<=size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1805 | { |
---|
1806 | g=g+sp[2][i]; |
---|
1807 | i++; |
---|
1808 | } |
---|
1809 | if(i==size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1810 | { |
---|
1811 | g=g+sp[2][i]; |
---|
1812 | } |
---|
1813 | return(g); |
---|
1814 | } |
---|
1815 | example |
---|
1816 | { "EXAMPLE:"; echo=2; |
---|
1817 | ring R=0,(x,y),ds; |
---|
1818 | 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)); |
---|
1819 | spprint(sp); |
---|
1820 | spgeomgenus(sp); |
---|
1821 | } |
---|
1822 | /////////////////////////////////////////////////////////////////////////////// |
---|
1823 | |
---|
1824 | proc spgamma(list sp) |
---|
1825 | "USAGE: spgamma(sp); list sp |
---|
1826 | RETURN: number gamma; gamma invariant of spectrum sp |
---|
1827 | EXAMPLE: example spgamma; shows examples |
---|
1828 | " |
---|
1829 | { |
---|
1830 | int i,j; |
---|
1831 | number g=0; |
---|
1832 | for(i=1;i<=ncols(sp[1]);i++) |
---|
1833 | { |
---|
1834 | for(j=1;j<=sp[2][i];j++) |
---|
1835 | { |
---|
1836 | g=g+(number(sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1837 | } |
---|
1838 | } |
---|
1839 | g=-g/4+sum(sp[2])*number(sp[1][ncols(sp[1])]-sp[1][1])/48; |
---|
1840 | return(g); |
---|
1841 | } |
---|
1842 | example |
---|
1843 | { "EXAMPLE:"; echo=2; |
---|
1844 | ring R=0,(x,y),ds; |
---|
1845 | 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)); |
---|
1846 | spprint(sp); |
---|
1847 | spgamma(sp); |
---|
1848 | } |
---|
1849 | /////////////////////////////////////////////////////////////////////////////// |
---|