1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: gaussman.lib,v 1.83 2002-07-08 07:10:34 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 | } |
---|
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 | setring(R); |
---|
587 | ideal r=-imap(G,e); |
---|
588 | kill G,gmsmaxdeg; |
---|
589 | |
---|
590 | return(r); |
---|
591 | } |
---|
592 | example |
---|
593 | { "EXAMPLE:"; echo=2; |
---|
594 | ring R=0,(x,y),ds; |
---|
595 | poly t=x5+x2y2+y5; |
---|
596 | bernstein(t); |
---|
597 | } |
---|
598 | /////////////////////////////////////////////////////////////////////////////// |
---|
599 | |
---|
600 | proc monodromy(poly t) |
---|
601 | "USAGE: monodromy(t); poly t |
---|
602 | ASSUME: characteristic 0; local degree ordering; |
---|
603 | isolated critical point 0 of t |
---|
604 | RETURN: |
---|
605 | @format |
---|
606 | list l; Jordan data jordan(M) of monodromy matrix exp(-2*pi*i*M) |
---|
607 | ideal l[1]; |
---|
608 | number l[1][i]; eigenvalue of i-th Jordan block of M |
---|
609 | intvec l[2]; |
---|
610 | int l[2][i]; size of i-th Jordan block of M |
---|
611 | intvec l[3]; |
---|
612 | int l[3][i]; multiplicity of i-th Jordan block of M |
---|
613 | @end format |
---|
614 | SEE ALSO: mondromy_lib, linalg_lib |
---|
615 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; monodromy |
---|
616 | EXAMPLE: example monodromy; shows examples |
---|
617 | " |
---|
618 | { |
---|
619 | def R=basering; |
---|
620 | int n=nvars(R)-1; |
---|
621 | def G=gmsring(t,"s"); |
---|
622 | setring(G); |
---|
623 | |
---|
624 | matrix A; |
---|
625 | module U0; |
---|
626 | ideal e; |
---|
627 | intvec m; |
---|
628 | |
---|
629 | def A0,r,H,H0,k0=saturate(); |
---|
630 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
631 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,0,0); |
---|
632 | |
---|
633 | list l=jordan(A,e,m); |
---|
634 | setring(R); |
---|
635 | list l=imap(G,l); |
---|
636 | kill G,gmsmaxdeg; |
---|
637 | |
---|
638 | return(l); |
---|
639 | } |
---|
640 | example |
---|
641 | { "EXAMPLE:"; echo=2; |
---|
642 | ring R=0,(x,y),ds; |
---|
643 | poly t=x5+x2y2+y5; |
---|
644 | monodromy(t); |
---|
645 | } |
---|
646 | /////////////////////////////////////////////////////////////////////////////// |
---|
647 | |
---|
648 | proc spectrum(poly t) |
---|
649 | "USAGE: spectrum(t); poly t |
---|
650 | ASSUME: characteristic 0; local degree ordering; |
---|
651 | isolated critical point 0 of t |
---|
652 | RETURN: |
---|
653 | @format |
---|
654 | list sp; singularity spectrum of t |
---|
655 | ideal sp[1]; |
---|
656 | number sp[1][i]; i-th spectral number |
---|
657 | intvec sp[2]; |
---|
658 | int sp[2][i]; multiplicity of i-th spectral number |
---|
659 | @end format |
---|
660 | SEE ALSO: spectrum_lib |
---|
661 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
662 | mixed Hodge structure; V-filtration; spectrum |
---|
663 | EXAMPLE: example spectrum; shows examples |
---|
664 | " |
---|
665 | { |
---|
666 | list l=vwfilt(t); |
---|
667 | return(spnf(l[1],l[3])); |
---|
668 | } |
---|
669 | example |
---|
670 | { "EXAMPLE:"; echo=2; |
---|
671 | ring R=0,(x,y),ds; |
---|
672 | poly t=x5+x2y2+y5; |
---|
673 | spprint(spectrum(t)); |
---|
674 | } |
---|
675 | /////////////////////////////////////////////////////////////////////////////// |
---|
676 | |
---|
677 | proc sppairs(poly t) |
---|
678 | "USAGE: sppairs(t); poly t |
---|
679 | ASSUME: characteristic 0; local degree ordering; |
---|
680 | isolated critical point 0 of t |
---|
681 | RETURN: |
---|
682 | @format |
---|
683 | list spp; spectral pairs of t |
---|
684 | ideal spp[1]; |
---|
685 | number spp[1][i]; V-filtration index of i-th spectral pair |
---|
686 | intvec spp[2]; |
---|
687 | int spp[2][i]; weight filtration index of i-th spectral pair |
---|
688 | intvec spp[3]; |
---|
689 | int spp[3][i]; multiplicity of i-th spectral pair |
---|
690 | @end format |
---|
691 | SEE ALSO: spectrum_lib |
---|
692 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
693 | mixed Hodge structure; V-filtration; weight filtration; |
---|
694 | spectrum; spectral pairs |
---|
695 | EXAMPLE: example sppairs; shows examples |
---|
696 | " |
---|
697 | { |
---|
698 | list l=vwfilt(t); |
---|
699 | return(list(l[1],l[2],l[3])); |
---|
700 | } |
---|
701 | example |
---|
702 | { "EXAMPLE:"; echo=2; |
---|
703 | ring R=0,(x,y),ds; |
---|
704 | poly t=x5+x2y2+y5; |
---|
705 | sppprint(sppairs(t)); |
---|
706 | } |
---|
707 | /////////////////////////////////////////////////////////////////////////////// |
---|
708 | |
---|
709 | proc spnf(ideal a,list #) |
---|
710 | "USAGE: spnf(a[,m][,V]); ideal a, intvec m, list V |
---|
711 | ASSUME: ncols(a)==size(m)==size(V); typeof(V[i])=="int" |
---|
712 | RETURN: order (a[i][,V[i]]) with multiplicity m[i] lexicographically |
---|
713 | EXAMPLE: example spnf; shows examples |
---|
714 | " |
---|
715 | { |
---|
716 | int n=ncols(a); |
---|
717 | intvec m; |
---|
718 | module v; |
---|
719 | list V; |
---|
720 | int i,j; |
---|
721 | while(i<size(#)) |
---|
722 | { |
---|
723 | i++; |
---|
724 | if(typeof(#[i])=="intvec") |
---|
725 | { |
---|
726 | m=#[i]; |
---|
727 | } |
---|
728 | if(typeof(#[i])=="module") |
---|
729 | { |
---|
730 | v=#[i]; |
---|
731 | for(j=n;j>=1;j--) |
---|
732 | { |
---|
733 | V[j]=module(v[j]); |
---|
734 | } |
---|
735 | } |
---|
736 | if(typeof(#[i])=="list") |
---|
737 | { |
---|
738 | V=#[i]; |
---|
739 | } |
---|
740 | } |
---|
741 | if(m==0) |
---|
742 | { |
---|
743 | for(i=n;i>=1;i--) |
---|
744 | { |
---|
745 | m[i]=1; |
---|
746 | } |
---|
747 | } |
---|
748 | |
---|
749 | int k; |
---|
750 | ideal a0; |
---|
751 | intvec m0; |
---|
752 | list V0; |
---|
753 | number a1; |
---|
754 | int m1; |
---|
755 | for(i=n;i>=1;i--) |
---|
756 | { |
---|
757 | if(m[i]!=0) |
---|
758 | { |
---|
759 | for(j=i-1;j>=1;j--) |
---|
760 | { |
---|
761 | if(m[j]!=0) |
---|
762 | { |
---|
763 | if(number(a[i])>number(a[j])) |
---|
764 | { |
---|
765 | a1=number(a[i]); |
---|
766 | a[i]=a[j]; |
---|
767 | a[j]=a1; |
---|
768 | m1=m[i]; |
---|
769 | m[i]=m[j]; |
---|
770 | m[j]=m1; |
---|
771 | if(size(V)>0) |
---|
772 | { |
---|
773 | v=V[i]; |
---|
774 | V[i]=V[j]; |
---|
775 | V[j]=v; |
---|
776 | } |
---|
777 | } |
---|
778 | if(number(a[i])==number(a[j])) |
---|
779 | { |
---|
780 | m[i]=m[i]+m[j]; |
---|
781 | m[j]=0; |
---|
782 | if(size(V)>0) |
---|
783 | { |
---|
784 | V[i]=V[i]+V[j]; |
---|
785 | } |
---|
786 | } |
---|
787 | } |
---|
788 | } |
---|
789 | k++; |
---|
790 | a0[k]=a[i]; |
---|
791 | m0[k]=m[i]; |
---|
792 | if(size(V)>0) |
---|
793 | { |
---|
794 | V0[k]=V[i]; |
---|
795 | } |
---|
796 | } |
---|
797 | } |
---|
798 | |
---|
799 | if(size(V0)>0) |
---|
800 | { |
---|
801 | n=size(V0); |
---|
802 | module U=std(V0[n]); |
---|
803 | for(i=n-1;i>=1;i--) |
---|
804 | { |
---|
805 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
806 | if(i>=2) |
---|
807 | { |
---|
808 | U=std(U+V0[i]); |
---|
809 | } |
---|
810 | } |
---|
811 | } |
---|
812 | |
---|
813 | list l; |
---|
814 | if(k>0) |
---|
815 | { |
---|
816 | l=a0,m0; |
---|
817 | if(size(V0)>0) |
---|
818 | { |
---|
819 | l[3]=V0; |
---|
820 | } |
---|
821 | } |
---|
822 | return(l); |
---|
823 | } |
---|
824 | example |
---|
825 | { "EXAMPLE:"; echo=2; |
---|
826 | } |
---|
827 | /////////////////////////////////////////////////////////////////////////////// |
---|
828 | |
---|
829 | proc sppnf(ideal a,intvec w,list #) |
---|
830 | "USAGE: sppnf(a,w[,m][,V]); ideal a, intvec w, intvec m, list V |
---|
831 | ASSUME: ncols(e)=size(w)=size(m)=size(V); typeof(V[i])=="module" |
---|
832 | RETURN: order (a[i][,w[i]][,V[i]]) with multiplicity m[i] lexicographically |
---|
833 | EXAMPLE: example sppnorm; shows examples |
---|
834 | " |
---|
835 | { |
---|
836 | int n=ncols(a); |
---|
837 | intvec m; |
---|
838 | module v; |
---|
839 | list V; |
---|
840 | int i,j; |
---|
841 | while(i<size(#)) |
---|
842 | { |
---|
843 | i++; |
---|
844 | if(typeof(#[i])=="intvec") |
---|
845 | { |
---|
846 | m=#[i]; |
---|
847 | } |
---|
848 | if(typeof(#[i])=="module") |
---|
849 | { |
---|
850 | v=#[i]; |
---|
851 | for(j=n;j>=1;j--) |
---|
852 | { |
---|
853 | V[j]=module(v[j]); |
---|
854 | } |
---|
855 | } |
---|
856 | if(typeof(#[i])=="list") |
---|
857 | { |
---|
858 | V=#[i]; |
---|
859 | } |
---|
860 | } |
---|
861 | if(m==0) |
---|
862 | { |
---|
863 | for(i=n;i>=1;i--) |
---|
864 | { |
---|
865 | m[i]=1; |
---|
866 | } |
---|
867 | } |
---|
868 | |
---|
869 | int k; |
---|
870 | ideal a0; |
---|
871 | intvec w0,m0; |
---|
872 | list V0; |
---|
873 | number a1; |
---|
874 | int w1,m1; |
---|
875 | for(i=n;i>=1;i--) |
---|
876 | { |
---|
877 | if(m[i]!=0) |
---|
878 | { |
---|
879 | for(j=i-1;j>=1;j--) |
---|
880 | { |
---|
881 | if(m[j]!=0) |
---|
882 | { |
---|
883 | if(number(a[i])>number(a[j])|| |
---|
884 | (number(a[i])==number(a[j])&&w[i]<w[j])) |
---|
885 | { |
---|
886 | a1=number(a[i]); |
---|
887 | a[i]=a[j]; |
---|
888 | a[j]=a1; |
---|
889 | w1=w[i]; |
---|
890 | w[i]=w[j]; |
---|
891 | w[j]=w1; |
---|
892 | m1=m[i]; |
---|
893 | m[i]=m[j]; |
---|
894 | m[j]=m1; |
---|
895 | if(size(V)>0) |
---|
896 | { |
---|
897 | v=V[i]; |
---|
898 | V[i]=V[j]; |
---|
899 | V[j]=v; |
---|
900 | } |
---|
901 | } |
---|
902 | if(number(a[i])==number(a[j])&&w[i]==w[j]) |
---|
903 | { |
---|
904 | m[i]=m[i]+m[j]; |
---|
905 | m[j]=0; |
---|
906 | if(size(V)>0) |
---|
907 | { |
---|
908 | V[i]=V[i]+V[j]; |
---|
909 | } |
---|
910 | } |
---|
911 | } |
---|
912 | } |
---|
913 | k++; |
---|
914 | a0[k]=a[i]; |
---|
915 | w0[k]=w[i]; |
---|
916 | m0[k]=m[i]; |
---|
917 | if(size(V)>0) |
---|
918 | { |
---|
919 | V0[k]=V[i]; |
---|
920 | } |
---|
921 | } |
---|
922 | } |
---|
923 | |
---|
924 | if(size(V0)>0) |
---|
925 | { |
---|
926 | n=size(V0); |
---|
927 | module U=std(V0[n]); |
---|
928 | for(i=n-1;i>=1;i--) |
---|
929 | { |
---|
930 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
931 | if(i>=2) |
---|
932 | { |
---|
933 | U=std(U+V0[i]); |
---|
934 | } |
---|
935 | } |
---|
936 | } |
---|
937 | |
---|
938 | list l; |
---|
939 | if(k>0) |
---|
940 | { |
---|
941 | l=a0,w0,m0; |
---|
942 | if(size(V0)>0) |
---|
943 | { |
---|
944 | l[4]=V0; |
---|
945 | } |
---|
946 | } |
---|
947 | return(l); |
---|
948 | } |
---|
949 | example |
---|
950 | { "EXAMPLE:"; echo=2; |
---|
951 | } |
---|
952 | /////////////////////////////////////////////////////////////////////////////// |
---|
953 | |
---|
954 | proc vfilt(poly t) |
---|
955 | "USAGE: vfilt(t); poly t |
---|
956 | ASSUME: characteristic 0; local degree ordering; |
---|
957 | isolated critical point 0 of t |
---|
958 | RETURN: |
---|
959 | @format |
---|
960 | list v; V-filtration on H''/s*H'' |
---|
961 | ideal v[1]; |
---|
962 | number v[1][i]; V-filtration index of i-th spectral number |
---|
963 | intvec v[2]; |
---|
964 | int v[2][i]; multiplicity of i-th spectral number |
---|
965 | list v[3]; |
---|
966 | module v[3][i]; vector space of i-th graded part in terms of v[4] |
---|
967 | ideal v[4]; monomial vector space basis of H''/s*H'' |
---|
968 | ideal v[5]; standard basis of Jacobian ideal |
---|
969 | @end format |
---|
970 | SEE ALSO: spectrum_lib |
---|
971 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
972 | mixed Hodge structure; V-filtration; spectrum |
---|
973 | EXAMPLE: example vfilt; shows examples |
---|
974 | " |
---|
975 | { |
---|
976 | list l=vwfilt(t); |
---|
977 | return(spnf(l[1],l[3],l[4])+list(l[5],l[6])); |
---|
978 | } |
---|
979 | example |
---|
980 | { "EXAMPLE:"; echo=2; |
---|
981 | ring R=0,(x,y),ds; |
---|
982 | poly t=x5+x2y2+y5; |
---|
983 | vfilt(t); |
---|
984 | } |
---|
985 | /////////////////////////////////////////////////////////////////////////////// |
---|
986 | |
---|
987 | proc vwfilt(poly t) |
---|
988 | "USAGE: vwfilt(t); poly t |
---|
989 | ASSUME: characteristic 0; local degree ordering; |
---|
990 | isolated critical point 0 of t |
---|
991 | RETURN: |
---|
992 | @format |
---|
993 | list vw; weighted V-filtration on H''/s*H'' |
---|
994 | ideal vw[1]; |
---|
995 | number vw[1][i]; V-filtration index of i-th spectral pair |
---|
996 | intvec vw[2]; |
---|
997 | int vw[2][i]; weight filtration index of i-th spectral pair |
---|
998 | intvec vw[3]; |
---|
999 | int vw[3][i]; multiplicity of i-th spectral pair |
---|
1000 | list vw[4]; |
---|
1001 | module vw[4][i]; vector space of i-th graded part in terms of vw[5] |
---|
1002 | ideal vw[5]; monomial vector space basis of H''/s*H'' |
---|
1003 | ideal vw[6]; standard basis of Jacobian ideal |
---|
1004 | @end format |
---|
1005 | SEE ALSO: spectrum_lib |
---|
1006 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1007 | mixed Hodge structure; V-filtration; weight filtration; |
---|
1008 | spectrum; spectral pairs |
---|
1009 | EXAMPLE: example vwfilt; shows examples |
---|
1010 | " |
---|
1011 | { |
---|
1012 | def R=basering; |
---|
1013 | int n=nvars(R)-1; |
---|
1014 | def G=gmsring(t,"s"); |
---|
1015 | setring(G); |
---|
1016 | |
---|
1017 | int mu=ncols(gmsbasis); |
---|
1018 | matrix A; |
---|
1019 | module U0; |
---|
1020 | ideal e; |
---|
1021 | intvec m; |
---|
1022 | |
---|
1023 | def A0,r,H,H0,k0=saturate(); |
---|
1024 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
1025 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,0,1); |
---|
1026 | |
---|
1027 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
1028 | list l=jordanbasis(A,e,m); |
---|
1029 | def U,v=l[1..2]; |
---|
1030 | kill l; |
---|
1031 | vector u0; |
---|
1032 | int v0; |
---|
1033 | int i,j,k,l; |
---|
1034 | for(k,l=1,1;l<=ncols(e);k,l=k+m[l],l+1) |
---|
1035 | { |
---|
1036 | for(i=k+m[l]-1;i>=k+1;i--) |
---|
1037 | { |
---|
1038 | for(j=i-1;j>=k;j--) |
---|
1039 | { |
---|
1040 | if(v[i]>v[j]) |
---|
1041 | { |
---|
1042 | v0=v[i];v[i]=v[j];v[j]=v0; |
---|
1043 | u0=U[i];U[i]=U[j];U[j]=u0; |
---|
1044 | } |
---|
1045 | } |
---|
1046 | } |
---|
1047 | } |
---|
1048 | |
---|
1049 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
1050 | matrix V=inverse(U); |
---|
1051 | A=V*A*U; |
---|
1052 | dbprint(printlevel-voice+2,"// compute normal form of H''"); |
---|
1053 | H0=std(V*H0); |
---|
1054 | U0=U0*U; |
---|
1055 | |
---|
1056 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
1057 | ideal a; |
---|
1058 | intvec w; |
---|
1059 | for(i=1;i<=mu;i++) |
---|
1060 | { |
---|
1061 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
1062 | a[i]=A[j,j]+ord(H0[i])/deg(s)-1; |
---|
1063 | w[i]=v[j]+n; |
---|
1064 | } |
---|
1065 | kill v; |
---|
1066 | module v=simplify(jet(H*U0*H0,2*k0)/s^(2*k0),1); |
---|
1067 | |
---|
1068 | kill l; |
---|
1069 | list l=sppnf(a,w,v)+list(gmsbasis,gmsstd); |
---|
1070 | setring(R); |
---|
1071 | list l=imap(G,l); |
---|
1072 | kill G,gmsmaxdeg; |
---|
1073 | attrib(l[5],"isSB",1); |
---|
1074 | |
---|
1075 | return(l); |
---|
1076 | } |
---|
1077 | example |
---|
1078 | { "EXAMPLE:"; echo=2; |
---|
1079 | ring R=0,(x,y),ds; |
---|
1080 | poly t=x5+x2y2+y5; |
---|
1081 | vwfilt(t); |
---|
1082 | } |
---|
1083 | /////////////////////////////////////////////////////////////////////////////// |
---|
1084 | |
---|
1085 | static proc commutator(matrix A) |
---|
1086 | { |
---|
1087 | int n=ncols(A); |
---|
1088 | int i,j,k; |
---|
1089 | matrix C[n^2][n^2]; |
---|
1090 | for(i=0;i<n;i++) |
---|
1091 | { |
---|
1092 | for(j=0;j<n;j++) |
---|
1093 | { |
---|
1094 | for(k=0;k<n;k++) |
---|
1095 | { |
---|
1096 | C[i*n+j+1,k*n+j+1]=C[i*n+j+1,k*n+j+1]+A[i+1,k+1]; |
---|
1097 | C[i*n+j+1,i*n+k+1]=C[i*n+j+1,i*n+k+1]-A[k+1,j+1]; |
---|
1098 | } |
---|
1099 | } |
---|
1100 | } |
---|
1101 | return(C); |
---|
1102 | } |
---|
1103 | |
---|
1104 | /////////////////////////////////////////////////////////////////////////////// |
---|
1105 | |
---|
1106 | proc tmatrix(poly t,list #) |
---|
1107 | "USAGE: tmatrix(t); poly t |
---|
1108 | ASSUME: characteristic 0; local degree ordering; |
---|
1109 | isolated critical point 0 of t |
---|
1110 | RETURN: |
---|
1111 | @format |
---|
1112 | list A; C[[s]]-matrix A[1]+s*A[2] of t on H'' |
---|
1113 | matrix A[1]; |
---|
1114 | matrix A[2]; |
---|
1115 | @end format |
---|
1116 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1117 | mixed Hodge structure; opposite Hodge filtration; V-filtration |
---|
1118 | EXAMPLE: example tmatrix; shows examples |
---|
1119 | " |
---|
1120 | { |
---|
1121 | def R=basering; |
---|
1122 | int n=nvars(R)-1; |
---|
1123 | def G=gmsring(t,"s"); |
---|
1124 | setring(G); |
---|
1125 | |
---|
1126 | int mu=ncols(gmsbasis); |
---|
1127 | matrix A; |
---|
1128 | module U0; |
---|
1129 | ideal e; |
---|
1130 | intvec m; |
---|
1131 | |
---|
1132 | def A0,r,H,H0,k0=saturate(); |
---|
1133 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
1134 | int k1=int(nmax(e)-nmin(e)); |
---|
1135 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,k0+k1,1); |
---|
1136 | |
---|
1137 | ring S=0,s,(ds,c); |
---|
1138 | matrix A=imap(G,A); |
---|
1139 | module H0=imap(G,H0); |
---|
1140 | ideal e=imap(G,e); |
---|
1141 | kill G,gmsmaxdeg; |
---|
1142 | |
---|
1143 | dbprint(printlevel-voice+2,"// transform to Jordan basis"); |
---|
1144 | module U=jordanbasis(A,e,m)[1]; |
---|
1145 | matrix V=inverse(U); |
---|
1146 | A=V*A*U; |
---|
1147 | module H=V*H0; |
---|
1148 | |
---|
1149 | dbprint(printlevel-voice+2,"// compute splitting of V-filtration"); |
---|
1150 | int i,j,k; |
---|
1151 | U=freemodule(mu); |
---|
1152 | V=matrix(0,mu,mu); |
---|
1153 | matrix v[mu^2][1]; |
---|
1154 | matrix A0=commutator(jet(A,0)); |
---|
1155 | for(k=1;k<=k0+k1;k++) |
---|
1156 | { |
---|
1157 | for(j=0;j<k;j++) |
---|
1158 | { |
---|
1159 | V=matrix(V)-(jet(A,k-j)/s^(k-j))*(jet(U,j)/s^j); |
---|
1160 | } |
---|
1161 | v=V[1..mu,1..mu]; |
---|
1162 | v=inverse(A0+k)*v; |
---|
1163 | V=v[1..mu^2,1]; |
---|
1164 | U=matrix(U)+s^k*V; |
---|
1165 | } |
---|
1166 | attrib(U,"isSB",1); |
---|
1167 | |
---|
1168 | dbprint(printlevel-voice+2,"// transform to V-splitting basis"); |
---|
1169 | A=jet(A,0); |
---|
1170 | H=std(division(H,U,(k0+k1)*deg(s))[1]); |
---|
1171 | |
---|
1172 | dbprint(printlevel-voice+2,"// compute V-leading terms of H''"); |
---|
1173 | int i0,j0; |
---|
1174 | module H1=H; |
---|
1175 | for(k=ncols(H1);k>=1;k--) |
---|
1176 | { |
---|
1177 | i0=leadexp(H1[k])[nvars(basering)+1]; |
---|
1178 | j0=ord(H1[k]);//deg(s); |
---|
1179 | H0[k]=lead(H1[k]); |
---|
1180 | H1[k]=H1[k]-lead(H1[k]); |
---|
1181 | if(H1[k]!=0) |
---|
1182 | { |
---|
1183 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
1184 | j=ord(H1[k]);//deg(s); |
---|
1185 | while(A[i,i]+j==A[i0,i0]+j0) |
---|
1186 | { |
---|
1187 | H0[k]=H0[k]+lead(H1[k]); |
---|
1188 | H1[k]=H1[k]-lead(H1[k]); |
---|
1189 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
1190 | j=ord(H1[k]);//deg(s); |
---|
1191 | } |
---|
1192 | } |
---|
1193 | } |
---|
1194 | H0=simplify(H0,1); |
---|
1195 | |
---|
1196 | dbprint(printlevel-voice+2,"// compute N"); |
---|
1197 | matrix N=A; |
---|
1198 | for(i=1;i<=ncols(N);i++) |
---|
1199 | { |
---|
1200 | N[i,i]=0; |
---|
1201 | } |
---|
1202 | |
---|
1203 | dbprint(printlevel-voice+2,"// compute splitting of Hodge filtration"); |
---|
1204 | U=0; |
---|
1205 | module U1; |
---|
1206 | module C; |
---|
1207 | list F,I; |
---|
1208 | module F0,I0,U0; |
---|
1209 | for(i0,j0=1,1;i0<=ncols(e);i0++) |
---|
1210 | { |
---|
1211 | C=matrix(0,mu,1); |
---|
1212 | for(j=m[i0];j>=1;j,j0=j-1,j0+1) |
---|
1213 | { |
---|
1214 | C=C+gen(j0); |
---|
1215 | } |
---|
1216 | F0=intersect(C,H0); |
---|
1217 | |
---|
1218 | F=list(); |
---|
1219 | j=0; |
---|
1220 | while(size(F0)>0) |
---|
1221 | { |
---|
1222 | j++; |
---|
1223 | F[j]=matrix(0,mu,1); |
---|
1224 | if(size(jet(F0,0))>0) |
---|
1225 | { |
---|
1226 | for(i=ncols(F0);i>=1;i--) |
---|
1227 | { |
---|
1228 | if(ord(F0[i])==0) |
---|
1229 | { |
---|
1230 | F[j]=F[j]+F0[i]; |
---|
1231 | } |
---|
1232 | } |
---|
1233 | } |
---|
1234 | for(i=ncols(F0);i>=1;i--) |
---|
1235 | { |
---|
1236 | F0[i]=F0[i]/s; |
---|
1237 | } |
---|
1238 | } |
---|
1239 | |
---|
1240 | I=list(); |
---|
1241 | I0=module(); |
---|
1242 | U0=std(module()); |
---|
1243 | for(i=size(F);i>=1;i--) |
---|
1244 | { |
---|
1245 | I[i]=module(); |
---|
1246 | } |
---|
1247 | for(i=1;i<=size(F);i++) |
---|
1248 | { |
---|
1249 | I0=reduce(F[i],U0); |
---|
1250 | j=i; |
---|
1251 | while(size(I0)>0) |
---|
1252 | { |
---|
1253 | U0=std(U0+I0); |
---|
1254 | I[j]=I[j]+I0; |
---|
1255 | I0=reduce(N*I0,U0); |
---|
1256 | j++; |
---|
1257 | } |
---|
1258 | } |
---|
1259 | |
---|
1260 | for(i=1;i<=size(I);i++) |
---|
1261 | { |
---|
1262 | U=U+I[i]; |
---|
1263 | } |
---|
1264 | } |
---|
1265 | |
---|
1266 | dbprint(printlevel-voice+2,"// transform to Hodge splitting basis"); |
---|
1267 | V=inverse(U); |
---|
1268 | A=V*A*U; |
---|
1269 | H=V*H; |
---|
1270 | |
---|
1271 | dbprint(printlevel-voice+2,"// compute reduced standard basis of H''"); |
---|
1272 | degBound=k0+k1+2; |
---|
1273 | option(redSB); |
---|
1274 | H=std(H); |
---|
1275 | option(noredSB); |
---|
1276 | degBound=0; |
---|
1277 | H=simplify(jet(H,k0+k1),1); |
---|
1278 | attrib(H,"isSB",1); |
---|
1279 | |
---|
1280 | dbprint(printlevel-voice+2,"// compute matrix A0+sA1 of t"); |
---|
1281 | A=division(s*A*H+s^2*diff(matrix(H),s),H,deg(s))[1]; |
---|
1282 | A0=jet(A,0); |
---|
1283 | A=jet(A,1)/s; |
---|
1284 | |
---|
1285 | setring(R); |
---|
1286 | matrix A0=imap(S,A0); |
---|
1287 | matrix A1=imap(S,A); |
---|
1288 | kill S; |
---|
1289 | return(list(A0,A1)); |
---|
1290 | } |
---|
1291 | example |
---|
1292 | { "EXAMPLE:"; echo=2; |
---|
1293 | ring R=0,(x,y),ds; |
---|
1294 | poly t=x5+x2y2+y5; |
---|
1295 | list A=tmatrix(t); |
---|
1296 | print(A[1]); |
---|
1297 | print(A[2]); |
---|
1298 | } |
---|
1299 | /////////////////////////////////////////////////////////////////////////////// |
---|
1300 | |
---|
1301 | proc endvfilt(list v) |
---|
1302 | "USAGE: endvfilt(v); list v |
---|
1303 | ASSUME: v returned by vfilt |
---|
1304 | RETURN: |
---|
1305 | @format |
---|
1306 | list ev; V-filtration on Jacobian algebra |
---|
1307 | ideal ev[1]; |
---|
1308 | number ev[1][i]; i-th V-filtration index |
---|
1309 | intvec ev[2]; |
---|
1310 | int ev[2][i]; i-th multiplicity |
---|
1311 | list ev[3]; |
---|
1312 | module ev[3][i]; vector space of i-th graded part in terms of ev[4] |
---|
1313 | ideal ev[4]; monomial vector space basis of Jacobian algebra |
---|
1314 | ideal ev[5]; standard basis of Jacobian ideal |
---|
1315 | @end format |
---|
1316 | KEYWORDS: singularities; Gauss-Manin connection; Brieskorn lattice; |
---|
1317 | mixed Hodge structure; V-filtration; endomorphism filtration |
---|
1318 | EXAMPLE: example endvfilt; shows examples |
---|
1319 | " |
---|
1320 | { |
---|
1321 | def a,d,V,m,g=v[1..5]; |
---|
1322 | attrib(g,"isSB",1); |
---|
1323 | int mu=ncols(m); |
---|
1324 | |
---|
1325 | module V0=V[1]; |
---|
1326 | for(int i=2;i<=size(V);i++) |
---|
1327 | { |
---|
1328 | V0=V0,V[i]; |
---|
1329 | } |
---|
1330 | |
---|
1331 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
1332 | list M; |
---|
1333 | module U=freemodule(ncols(m)); |
---|
1334 | for(i=ncols(m);i>=1;i--) |
---|
1335 | { |
---|
1336 | M[i]=division(coeffs(reduce(m[i]*m,g,U),m)*V0,V0)[1]; |
---|
1337 | } |
---|
1338 | |
---|
1339 | int j,k,i0,j0,i1,j1; |
---|
1340 | number b0=number(a[1]-a[ncols(a)]); |
---|
1341 | number b1,b2; |
---|
1342 | matrix M0; |
---|
1343 | module L; |
---|
1344 | list v0=freemodule(ncols(m)); |
---|
1345 | ideal a0=b0; |
---|
1346 | list l; |
---|
1347 | |
---|
1348 | while(b0<number(a[ncols(a)]-a[1])) |
---|
1349 | { |
---|
1350 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
1351 | b1=number(a[ncols(a)]-a[1]); |
---|
1352 | for(j=ncols(a);j>=1;j--) |
---|
1353 | { |
---|
1354 | for(i=ncols(a);i>=1;i--) |
---|
1355 | { |
---|
1356 | b2=number(a[i]-a[j]); |
---|
1357 | if(b2>b0&&b2<b1) |
---|
1358 | { |
---|
1359 | b1=b2; |
---|
1360 | } |
---|
1361 | else |
---|
1362 | { |
---|
1363 | if(b2<=b0) |
---|
1364 | { |
---|
1365 | i=0; |
---|
1366 | } |
---|
1367 | } |
---|
1368 | } |
---|
1369 | } |
---|
1370 | b0=b1; |
---|
1371 | |
---|
1372 | l=ideal(); |
---|
1373 | for(k=ncols(m);k>=2;k--) |
---|
1374 | { |
---|
1375 | l=l+list(ideal()); |
---|
1376 | } |
---|
1377 | |
---|
1378 | dbprint(printlevel-voice+2,"// collect conditions for EV["+string(b0)+"]"); |
---|
1379 | j=ncols(a); |
---|
1380 | j0=mu; |
---|
1381 | while(j>=1) |
---|
1382 | { |
---|
1383 | i0=1; |
---|
1384 | i=1; |
---|
1385 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
1386 | { |
---|
1387 | i0=i0+d[i]; |
---|
1388 | i++; |
---|
1389 | } |
---|
1390 | if(a[i]<a[j]+b0) |
---|
1391 | { |
---|
1392 | i0=i0+d[i]; |
---|
1393 | i++; |
---|
1394 | } |
---|
1395 | for(k=1;k<=ncols(m);k++) |
---|
1396 | { |
---|
1397 | M0=M[k]; |
---|
1398 | if(i0>1) |
---|
1399 | { |
---|
1400 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
1401 | } |
---|
1402 | } |
---|
1403 | j0=j0-d[j]; |
---|
1404 | j--; |
---|
1405 | } |
---|
1406 | |
---|
1407 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
1408 | L=transpose(module(l[1])); |
---|
1409 | for(k=2;k<=ncols(m);k++) |
---|
1410 | { |
---|
1411 | L=L,transpose(module(l[k])); |
---|
1412 | } |
---|
1413 | |
---|
1414 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
1415 | v0=v0+list(syz(L)); |
---|
1416 | a0=a0,b0; |
---|
1417 | } |
---|
1418 | |
---|
1419 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1420 | option(redSB); |
---|
1421 | for(i=1;i<size(v0);i++) |
---|
1422 | { |
---|
1423 | v0[i+1]=std(v0[i+1]); |
---|
1424 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
1425 | } |
---|
1426 | option(noredSB); |
---|
1427 | |
---|
1428 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
1429 | i=1; |
---|
1430 | while(size(v0[i])==0) |
---|
1431 | { |
---|
1432 | i++; |
---|
1433 | } |
---|
1434 | list v1=v0[i]; |
---|
1435 | intvec d1=size(v0[i]); |
---|
1436 | ideal a1=a0[i]; |
---|
1437 | i++; |
---|
1438 | while(i<=size(v0)) |
---|
1439 | { |
---|
1440 | if(size(v0[i])>0) |
---|
1441 | { |
---|
1442 | v1=v1+list(v0[i]); |
---|
1443 | d1=d1,size(v0[i]); |
---|
1444 | a1=a1,a0[i]; |
---|
1445 | } |
---|
1446 | i++; |
---|
1447 | } |
---|
1448 | return(list(a1,d1,v1,m,g)); |
---|
1449 | } |
---|
1450 | example |
---|
1451 | { "EXAMPLE:"; echo=2; |
---|
1452 | ring R=0,(x,y),ds; |
---|
1453 | poly t=x5+x2y2+y5; |
---|
1454 | endvfilt(vfilt(t)); |
---|
1455 | } |
---|
1456 | /////////////////////////////////////////////////////////////////////////////// |
---|
1457 | |
---|
1458 | proc spprint(list sp) |
---|
1459 | "USAGE: spprint(sp); list sp |
---|
1460 | RETURN: string s; spectrum sp |
---|
1461 | EXAMPLE: example spprint; shows examples |
---|
1462 | " |
---|
1463 | { |
---|
1464 | string s; |
---|
1465 | for(int i=1;i<size(sp[2]);i++) |
---|
1466 | { |
---|
1467 | s=s+"("+string(sp[1][i])+","+string(sp[2][i])+"),"; |
---|
1468 | } |
---|
1469 | s=s+"("+string(sp[1][i])+","+string(sp[2][i])+")"; |
---|
1470 | return(s); |
---|
1471 | } |
---|
1472 | example |
---|
1473 | { "EXAMPLE:"; echo=2; |
---|
1474 | ring R=0,(x,y),ds; |
---|
1475 | 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)); |
---|
1476 | spprint(sp); |
---|
1477 | } |
---|
1478 | /////////////////////////////////////////////////////////////////////////////// |
---|
1479 | |
---|
1480 | proc sppprint(list spp) |
---|
1481 | "USAGE: sppprint(spp); list spp |
---|
1482 | RETURN: string s; spectral pairs spp |
---|
1483 | EXAMPLE: example sppprint; shows examples |
---|
1484 | " |
---|
1485 | { |
---|
1486 | string s; |
---|
1487 | for(int i=1;i<size(spp[3]);i++) |
---|
1488 | { |
---|
1489 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+"),"; |
---|
1490 | } |
---|
1491 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+")"; |
---|
1492 | return(s); |
---|
1493 | } |
---|
1494 | example |
---|
1495 | { "EXAMPLE:"; echo=2; |
---|
1496 | ring R=0,(x,y),ds; |
---|
1497 | 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)); |
---|
1498 | sppprint(spp); |
---|
1499 | } |
---|
1500 | /////////////////////////////////////////////////////////////////////////////// |
---|
1501 | |
---|
1502 | proc spadd(list sp1,list sp2) |
---|
1503 | "USAGE: spadd(sp1,sp2); list sp1, list sp2 |
---|
1504 | RETURN: list sp; sum of spectra sp1 and sp2 |
---|
1505 | EXAMPLE: example spadd; shows examples |
---|
1506 | " |
---|
1507 | { |
---|
1508 | ideal s; |
---|
1509 | intvec m; |
---|
1510 | int i,i1,i2=1,1,1; |
---|
1511 | while(i1<=size(sp1[2])||i2<=size(sp2[2])) |
---|
1512 | { |
---|
1513 | if(i1<=size(sp1[2])) |
---|
1514 | { |
---|
1515 | if(i2<=size(sp2[2])) |
---|
1516 | { |
---|
1517 | if(number(sp1[1][i1])<number(sp2[1][i2])) |
---|
1518 | { |
---|
1519 | s[i]=sp1[1][i1]; |
---|
1520 | m[i]=sp1[2][i1]; |
---|
1521 | i++; |
---|
1522 | i1++; |
---|
1523 | } |
---|
1524 | else |
---|
1525 | { |
---|
1526 | if(number(sp1[1][i1])>number(sp2[1][i2])) |
---|
1527 | { |
---|
1528 | s[i]=sp2[1][i2]; |
---|
1529 | m[i]=sp2[2][i2]; |
---|
1530 | i++; |
---|
1531 | i2++; |
---|
1532 | } |
---|
1533 | else |
---|
1534 | { |
---|
1535 | if(sp1[2][i1]+sp2[2][i2]!=0) |
---|
1536 | { |
---|
1537 | s[i]=sp1[1][i1]; |
---|
1538 | m[i]=sp1[2][i1]+sp2[2][i2]; |
---|
1539 | i++; |
---|
1540 | } |
---|
1541 | i1++; |
---|
1542 | i2++; |
---|
1543 | } |
---|
1544 | } |
---|
1545 | } |
---|
1546 | else |
---|
1547 | { |
---|
1548 | s[i]=sp1[1][i1]; |
---|
1549 | m[i]=sp1[2][i1]; |
---|
1550 | i++; |
---|
1551 | i1++; |
---|
1552 | } |
---|
1553 | } |
---|
1554 | else |
---|
1555 | { |
---|
1556 | s[i]=sp2[1][i2]; |
---|
1557 | m[i]=sp2[2][i2]; |
---|
1558 | i++; |
---|
1559 | i2++; |
---|
1560 | } |
---|
1561 | } |
---|
1562 | return(list(s,m)); |
---|
1563 | } |
---|
1564 | example |
---|
1565 | { "EXAMPLE:"; echo=2; |
---|
1566 | ring R=0,(x,y),ds; |
---|
1567 | 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)); |
---|
1568 | spprint(sp1); |
---|
1569 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1570 | spprint(sp2); |
---|
1571 | spprint(spadd(sp1,sp2)); |
---|
1572 | } |
---|
1573 | /////////////////////////////////////////////////////////////////////////////// |
---|
1574 | |
---|
1575 | proc spsub(list sp1,list sp2) |
---|
1576 | "USAGE: spsub(sp1,sp2); list sp1, list sp2 |
---|
1577 | RETURN: list sp; difference of spectra sp1 and sp2 |
---|
1578 | EXAMPLE: example spsub; shows examples |
---|
1579 | " |
---|
1580 | { |
---|
1581 | return(spadd(sp1,spmul(sp2,-1))); |
---|
1582 | } |
---|
1583 | example |
---|
1584 | { "EXAMPLE:"; echo=2; |
---|
1585 | ring R=0,(x,y),ds; |
---|
1586 | 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)); |
---|
1587 | spprint(sp1); |
---|
1588 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1589 | spprint(sp2); |
---|
1590 | spprint(spsub(sp1,sp2)); |
---|
1591 | } |
---|
1592 | /////////////////////////////////////////////////////////////////////////////// |
---|
1593 | |
---|
1594 | proc spmul(list #) |
---|
1595 | "USAGE: spmul(sp0,k); list sp0, int[vec] k |
---|
1596 | RETURN: list sp; linear combination of spectra sp0 with coefficients k |
---|
1597 | EXAMPLE: example spmul; shows examples |
---|
1598 | " |
---|
1599 | { |
---|
1600 | if(size(#)==2) |
---|
1601 | { |
---|
1602 | if(typeof(#[1])=="list") |
---|
1603 | { |
---|
1604 | if(typeof(#[2])=="int") |
---|
1605 | { |
---|
1606 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1607 | } |
---|
1608 | if(typeof(#[2])=="intvec") |
---|
1609 | { |
---|
1610 | list sp0=list(ideal(),intvec(0)); |
---|
1611 | for(int i=size(#[2]);i>=1;i--) |
---|
1612 | { |
---|
1613 | sp0=spadd(sp0,spmul(#[1][i],#[2][i])); |
---|
1614 | } |
---|
1615 | return(sp0); |
---|
1616 | } |
---|
1617 | } |
---|
1618 | } |
---|
1619 | return(list(ideal(),intvec(0))); |
---|
1620 | } |
---|
1621 | example |
---|
1622 | { "EXAMPLE:"; echo=2; |
---|
1623 | ring R=0,(x,y),ds; |
---|
1624 | 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)); |
---|
1625 | spprint(sp); |
---|
1626 | spprint(spmul(sp,2)); |
---|
1627 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1628 | spprint(sp1); |
---|
1629 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1630 | spprint(sp2); |
---|
1631 | spprint(spmul(list(sp1,sp2),intvec(1,2))); |
---|
1632 | } |
---|
1633 | /////////////////////////////////////////////////////////////////////////////// |
---|
1634 | |
---|
1635 | proc spissemicont(list sp,list #) |
---|
1636 | "USAGE: spissemicont(sp[,1]); list sp, int opt |
---|
1637 | RETURN: |
---|
1638 | @format |
---|
1639 | int k= |
---|
1640 | 1; if sum of sp is positive on all intervals [a,a+1) [and (a,a+1)] |
---|
1641 | 0; if sum of sp is negative on some interval [a,a+1) [or (a,a+1)] |
---|
1642 | @end format |
---|
1643 | EXAMPLE: example spissemicont; shows examples |
---|
1644 | " |
---|
1645 | { |
---|
1646 | int opt=0; |
---|
1647 | if(size(#)>0) |
---|
1648 | { |
---|
1649 | if(typeof(#[1])=="int") |
---|
1650 | { |
---|
1651 | opt=1; |
---|
1652 | } |
---|
1653 | } |
---|
1654 | int i,j,k; |
---|
1655 | i=1; |
---|
1656 | while(i<=size(sp[2])-1) |
---|
1657 | { |
---|
1658 | j=i+1; |
---|
1659 | k=0; |
---|
1660 | while(j+1<=size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
1661 | { |
---|
1662 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
1663 | { |
---|
1664 | k=k+sp[2][j]; |
---|
1665 | } |
---|
1666 | j++; |
---|
1667 | } |
---|
1668 | if(j==size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
1669 | { |
---|
1670 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
1671 | { |
---|
1672 | k=k+sp[2][j]; |
---|
1673 | } |
---|
1674 | } |
---|
1675 | if(k<0) |
---|
1676 | { |
---|
1677 | return(0); |
---|
1678 | } |
---|
1679 | i++; |
---|
1680 | } |
---|
1681 | return(1); |
---|
1682 | } |
---|
1683 | example |
---|
1684 | { "EXAMPLE:"; echo=2; |
---|
1685 | ring R=0,(x,y),ds; |
---|
1686 | 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)); |
---|
1687 | spprint(sp1); |
---|
1688 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1689 | spprint(sp2); |
---|
1690 | spissemicont(spsub(sp1,spmul(sp2,3))); |
---|
1691 | spissemicont(spsub(sp1,spmul(sp2,4))); |
---|
1692 | } |
---|
1693 | /////////////////////////////////////////////////////////////////////////////// |
---|
1694 | |
---|
1695 | proc spsemicont(list sp0,list sp,list #) |
---|
1696 | "USAGE: spsemicont(sp0,sp,k[,1]); list sp0, list sp |
---|
1697 | RETURN: |
---|
1698 | @format |
---|
1699 | list l; |
---|
1700 | intvec l[i]; if the spectra sp0 occur with multiplicities k |
---|
1701 | in a deformation of a [quasihomogeneous] singularity |
---|
1702 | with spectrum sp then k<=l[i] |
---|
1703 | @end format |
---|
1704 | EXAMPLE: example spsemicont; shows examples |
---|
1705 | " |
---|
1706 | { |
---|
1707 | list l,l0; |
---|
1708 | int i,j,k; |
---|
1709 | while(spissemicont(sp0,#)) |
---|
1710 | { |
---|
1711 | if(size(sp)>1) |
---|
1712 | { |
---|
1713 | l0=spsemicont(sp0,list(sp[1..size(sp)-1])); |
---|
1714 | for(i=1;i<=size(l0);i++) |
---|
1715 | { |
---|
1716 | if(size(l)>0) |
---|
1717 | { |
---|
1718 | j=1; |
---|
1719 | while(j<size(l)&&l[j]!=l0[i]) |
---|
1720 | { |
---|
1721 | j++; |
---|
1722 | } |
---|
1723 | if(l[j]==l0[i]) |
---|
1724 | { |
---|
1725 | l[j][size(sp)]=k; |
---|
1726 | } |
---|
1727 | else |
---|
1728 | { |
---|
1729 | l0[i][size(sp)]=k; |
---|
1730 | l=l+list(l0[i]); |
---|
1731 | } |
---|
1732 | } |
---|
1733 | else |
---|
1734 | { |
---|
1735 | l=l0; |
---|
1736 | } |
---|
1737 | } |
---|
1738 | } |
---|
1739 | sp0=spsub(sp0,sp[size(sp)]); |
---|
1740 | k++; |
---|
1741 | } |
---|
1742 | if(size(sp)>1) |
---|
1743 | { |
---|
1744 | return(l); |
---|
1745 | } |
---|
1746 | else |
---|
1747 | { |
---|
1748 | return(list(intvec(k-1))); |
---|
1749 | } |
---|
1750 | } |
---|
1751 | example |
---|
1752 | { "EXAMPLE:"; echo=2; |
---|
1753 | ring R=0,(x,y),ds; |
---|
1754 | 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)); |
---|
1755 | spprint(sp0); |
---|
1756 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1757 | spprint(sp1); |
---|
1758 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1759 | spprint(sp2); |
---|
1760 | list sp=sp1,sp2; |
---|
1761 | list l=spsemicont(sp0,sp); |
---|
1762 | l; |
---|
1763 | spissemicont(spsub(sp0,spmul(sp,l[1]))); |
---|
1764 | spissemicont(spsub(sp0,spmul(sp,l[1]-1))); |
---|
1765 | spissemicont(spsub(sp0,spmul(sp,l[1]+1))); |
---|
1766 | } |
---|
1767 | /////////////////////////////////////////////////////////////////////////////// |
---|
1768 | |
---|
1769 | proc spmilnor(list sp) |
---|
1770 | "USAGE: spmilnor(sp); list sp |
---|
1771 | RETURN: int mu; Milnor number of spectrum sp |
---|
1772 | EXAMPLE: example spmilnor; shows examples |
---|
1773 | " |
---|
1774 | { |
---|
1775 | return(sum(sp[2])); |
---|
1776 | } |
---|
1777 | example |
---|
1778 | { "EXAMPLE:"; echo=2; |
---|
1779 | ring R=0,(x,y),ds; |
---|
1780 | 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)); |
---|
1781 | spprint(sp); |
---|
1782 | spmilnor(sp); |
---|
1783 | } |
---|
1784 | /////////////////////////////////////////////////////////////////////////////// |
---|
1785 | |
---|
1786 | proc spgeomgenus(list sp) |
---|
1787 | "USAGE: spgeomgenus(sp); list sp |
---|
1788 | RETURN: int g; geometrical genus of spectrum sp |
---|
1789 | EXAMPLE: example spgeomgenus; shows examples |
---|
1790 | " |
---|
1791 | { |
---|
1792 | int g=0; |
---|
1793 | int i=1; |
---|
1794 | while(i+1<=size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1795 | { |
---|
1796 | g=g+sp[2][i]; |
---|
1797 | i++; |
---|
1798 | } |
---|
1799 | if(i==size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1800 | { |
---|
1801 | g=g+sp[2][i]; |
---|
1802 | } |
---|
1803 | return(g); |
---|
1804 | } |
---|
1805 | example |
---|
1806 | { "EXAMPLE:"; echo=2; |
---|
1807 | ring R=0,(x,y),ds; |
---|
1808 | 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)); |
---|
1809 | spprint(sp); |
---|
1810 | spgeomgenus(sp); |
---|
1811 | } |
---|
1812 | /////////////////////////////////////////////////////////////////////////////// |
---|
1813 | |
---|
1814 | proc spgamma(list sp) |
---|
1815 | "USAGE: spgamma(sp); list sp |
---|
1816 | RETURN: number gamma; gamma invariant of spectrum sp |
---|
1817 | EXAMPLE: example spgamma; shows examples |
---|
1818 | " |
---|
1819 | { |
---|
1820 | int i,j; |
---|
1821 | number g=0; |
---|
1822 | for(i=1;i<=ncols(sp[1]);i++) |
---|
1823 | { |
---|
1824 | for(j=1;j<=sp[2][i];j++) |
---|
1825 | { |
---|
1826 | g=g+(number(sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1827 | } |
---|
1828 | } |
---|
1829 | g=-g/4+sum(sp[2])*number(sp[1][ncols(sp[1])]-sp[1][1])/48; |
---|
1830 | return(g); |
---|
1831 | } |
---|
1832 | example |
---|
1833 | { "EXAMPLE:"; echo=2; |
---|
1834 | ring R=0,(x,y),ds; |
---|
1835 | 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)); |
---|
1836 | spprint(sp); |
---|
1837 | spgamma(sp); |
---|
1838 | } |
---|
1839 | /////////////////////////////////////////////////////////////////////////////// |
---|