1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="version gmssing.lib 4.1.2.0 Feb_2019 "; // $Id$ |
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3 | category="Singularities"; |
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4 | |
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5 | info=" |
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6 | LIBRARY: gmssing.lib Gauss-Manin System of Isolated Singularities |
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7 | |
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8 | AUTHOR: Mathias Schulze, mschulze at mathematik.uni-kl.de |
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9 | |
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10 | OVERVIEW: |
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11 | A library for computing invariants related to the Gauss-Manin system of an |
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12 | isolated hypersurface singularity. |
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13 | |
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14 | REFERENCES: |
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15 | [Sch01] M. Schulze: Algorithms for the Gauss-Manin connection. J. Symb. Comp. |
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16 | 32,5 (2001), 549-564. |
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17 | [Sch02] M. Schulze: The differential structure of the Brieskorn lattice. |
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18 | In: A.M. Cohen et al.: Mathematical Software - ICMS 2002. |
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19 | World Scientific (2002). |
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20 | [Sch03] M. Schulze: Monodromy of Hypersurface Singularities. |
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21 | Acta Appl. Math. 75 (2003), 3-13. |
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22 | [Sch04] M. Schulze: A normal form algorithm for the Brieskorn lattice. |
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23 | J. Symb. Comp. 38,4 (2004), 1207-1225. |
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24 | |
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25 | PROCEDURES: |
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26 | gmsring(t,s); Gauss-Manin system of t with variable s |
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27 | gmsnf(p,K); Gauss-Manin normal form of p |
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28 | gmscoeffs(p,K); Gauss-Manin basis representation of p |
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29 | bernstein(t); Bernstein-Sato polynomial of t |
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30 | monodromy(t); Jordan data of complex monodromy of t |
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31 | spectrum(t); singularity spectrum of t |
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32 | sppairs(t); spectral pairs of t |
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33 | vfilt(t); V-filtration of t on Brieskorn lattice |
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34 | vwfilt(t); weighted V-filtration of t on Brieskorn lattice |
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35 | tmatrix(t); matrix of t w.r.t. good basis of Brieskorn lattice |
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36 | endvfilt(V); endomorphism V-filtration on Jacobian algebra |
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37 | sppnf(a,w[,m]); spectral pairs normal form of (a,w[,m]) |
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38 | sppprint(spp); print spectral pairs spp |
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39 | spadd(sp1,sp2); sum of spectra sp1 and sp2 |
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40 | spsub(sp1,sp2); difference of spectra sp1 and sp2 |
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41 | spmul(sp0,k); linear combination of spectra sp |
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42 | spissemicont(sp[,opt]); semicontinuity test of spectrum sp |
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43 | spsemicont(sp0,sp[,opt]); semicontinuous combinations of spectra sp0 in sp |
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44 | spmilnor(sp); Milnor number of spectrum sp |
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45 | spgeomgenus(sp); geometrical genus of spectrum sp |
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46 | spgamma(sp); gamma invariant of spectrum sp |
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47 | |
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48 | SEE ALSO: mondromy_lib, spectrum_lib, gmspoly_lib, dmod_lib |
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49 | |
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50 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
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51 | mixed Hodge structure; V-filtration; weight filtration; |
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52 | Bernstein-Sato polynomial; monodromy; spectrum; spectral pairs; |
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53 | good basis |
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54 | "; |
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55 | |
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56 | LIB "linalg.lib"; |
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57 | |
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58 | /////////////////////////////////////////////////////////////////////////////// |
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59 | |
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60 | static proc stdtrans(ideal I) |
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61 | { |
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62 | def @R=basering; |
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63 | |
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64 | string os=ordstr(@R); |
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65 | int j=find(os,",C"); |
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66 | if(j==0) |
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67 | { |
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68 | j=find(os,"C,"); |
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69 | } |
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70 | if(j==0) |
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71 | { |
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72 | j=find(os,",c"); |
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73 | } |
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74 | if(j==0) |
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75 | { |
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76 | j=find(os,"c,"); |
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77 | } |
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78 | if(j>0) |
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79 | { |
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80 | os[j..j+1]=" "; |
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81 | } |
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82 | |
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83 | ring @S = create_ring(ring_list(@R)[1], "(gmspoly,"+varstr(@R)+")", "(c,dp(1),"+os+")", "no_minpoly"); |
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84 | |
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85 | ideal I=homog(imap(@R,I),gmspoly); |
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86 | module M=transpose(transpose(I)+freemodule(ncols(I))); |
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87 | M=std(M); |
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88 | |
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89 | setring(@R); |
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90 | execute("map h=@S,1,"+varstr(@R)+";"); |
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91 | module M=h(M); |
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92 | |
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93 | for(int i=ncols(M);i>=1;i--) |
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94 | { |
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95 | for(j=ncols(M);j>=1;j--) |
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96 | { |
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97 | if(M[i][1]==0) |
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98 | { |
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99 | M[i]=0; |
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100 | } |
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101 | if(i!=j&&M[j][1]!=0) |
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102 | { |
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103 | if(lead(M[i][1])/lead(M[j][1])!=0) |
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104 | { |
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105 | M[i]=0; |
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106 | } |
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107 | } |
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108 | } |
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109 | } |
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110 | |
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111 | M=transpose(simplify(M,2)); |
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112 | I=ideal(M[1]); |
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113 | attrib(I,"isSB",1); |
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114 | M=M[2..ncols(M)]; |
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115 | module U=transpose(M); |
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116 | |
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117 | return(list(I,U)); |
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118 | } |
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119 | /////////////////////////////////////////////////////////////////////////////// |
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120 | |
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121 | proc gmsring(poly t,string s) |
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122 | "USAGE: gmsring(t,s); poly t, string s |
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123 | ASSUME: characteristic 0; local degree ordering; |
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124 | isolated critical point 0 of t |
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125 | RETURN: |
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126 | @format |
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127 | ring G; Gauss-Manin system of t with variable s |
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128 | poly gmspoly=t; |
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129 | ideal gmsjacob; Jacobian ideal of t |
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130 | ideal gmsstd; standard basis of Jacobian ideal |
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131 | matrix gmsmatrix; matrix(gmsjacob)*gmsmatrix==matrix(gmsstd) |
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132 | ideal gmsbasis; monomial vector space basis of Jacobian algebra |
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133 | int Gmssing::gmsmaxdeg; maximal weight of variables |
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134 | @end format |
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135 | NOTE: gmsbasis is a C[[s]]-basis of H'' and [t,s]=s^2 |
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136 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice |
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137 | EXAMPLE: example gmsring; shows examples |
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138 | " |
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139 | { |
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140 | def @R=basering; |
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141 | if((charstr(@R)!="0")&&(charstr(@R)!="QQ")) |
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142 | { |
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143 | ERROR("characteristic 0 expected"); |
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144 | } |
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145 | for(int i=nvars(@R);i>=1;i--) |
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146 | { |
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147 | if(var(i)>1) |
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148 | { |
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149 | ERROR("local ordering expected"); |
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150 | } |
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151 | } |
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152 | |
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153 | ideal dt=jacob(t); |
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154 | list l=stdtrans(dt); |
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155 | ideal g=l[1]; |
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156 | if(vdim(g)<=0) |
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157 | { |
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158 | if(vdim(g)==0) |
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159 | { |
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160 | ERROR("singularity at 0 expected"); |
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161 | } |
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162 | else |
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163 | { |
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164 | ERROR("isolated critical point 0 expected"); |
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165 | } |
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166 | } |
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167 | matrix B=l[2]; |
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168 | ideal m=kbase(g); |
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169 | |
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170 | int gmsmaxdeg; |
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171 | for(i=nvars(@R);i>=1;i--) |
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172 | { |
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173 | if(deg(var(i))>gmsmaxdeg) |
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174 | { |
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175 | gmsmaxdeg=deg(var(i)); |
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176 | } |
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177 | } |
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178 | |
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179 | string os=ordstr(@R); |
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180 | int j=find(os,",C"); |
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181 | if(j==0) |
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182 | { |
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183 | j=find(os,"C,"); |
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184 | } |
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185 | if(j==0) |
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186 | { |
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187 | j=find(os,",c"); |
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188 | } |
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189 | if(j==0) |
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190 | { |
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191 | j=find(os,"c,"); |
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192 | } |
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193 | if(j>0) |
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194 | { |
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195 | os[j..j+1]=" "; |
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196 | } |
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197 | |
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198 | ring G = create_ring(string(charstr(@R)), "("+s+","+varstr(@R)+")", "(ws("+string(deg(highcorner(g))+2*gmsmaxdeg)+"),"+os+",c)"); |
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199 | |
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200 | poly gmspoly=imap(@R,t); |
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201 | ideal gmsjacob=imap(@R,dt); |
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202 | ideal gmsstd=imap(@R,g); |
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203 | matrix gmsmatrix=imap(@R,B); |
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204 | ideal gmsbasis=imap(@R,m); |
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205 | |
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206 | attrib(gmsstd,"isSB",1); |
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207 | export gmspoly,gmsjacob,gmsstd,gmsmatrix,gmsbasis,gmsmaxdeg; |
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208 | |
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209 | return(G); |
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210 | } |
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211 | example |
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212 | { "EXAMPLE:"; echo=2; |
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213 | ring @R=0,(x,y),ds; |
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214 | poly t=x5+x2y2+y5; |
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215 | def G=gmsring(t,"s"); |
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216 | setring(G); |
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217 | gmspoly; |
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218 | print(gmsjacob); |
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219 | print(gmsstd); |
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220 | print(gmsmatrix); |
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221 | print(gmsbasis); |
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222 | Gmssing::gmsmaxdeg; |
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223 | } |
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224 | /////////////////////////////////////////////////////////////////////////////// |
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225 | |
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226 | proc gmsnf(ideal p,int K) |
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227 | "USAGE: gmsnf(p,K); poly p, int K |
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228 | ASSUME: basering returned by gmsring |
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229 | RETURN: |
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230 | list nf; |
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231 | ideal nf[1]; projection of p to <gmsbasis>C[[s]] mod s^(K+1) |
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232 | ideal nf[2]; p==nf[1]+nf[2] |
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233 | NOTE: computation can be continued by setting p=nf[2] |
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234 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice |
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235 | EXAMPLE: example gmsnf; shows examples |
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236 | " |
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237 | { |
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238 | if(system("with","gms")) |
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239 | { |
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240 | return(system("gmsnf",p,gmsstd,gmsmatrix,(K+1)*deg(var(1))-2*gmsmaxdeg,K)); |
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241 | } |
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242 | |
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243 | intvec v=1; |
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244 | v[nvars(basering)]=0; |
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245 | |
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246 | int k; |
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247 | ideal r,q; |
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248 | r[ncols(p)]=0; |
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249 | q[ncols(p)]=0; |
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250 | |
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251 | poly s; |
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252 | int i,j; |
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253 | for(k=ncols(p);k>=1;k--) |
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254 | { |
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255 | while(p[k]!=0&°(lead(p[k]),v)<=K) |
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256 | { |
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257 | i=1; |
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258 | s=lead(p[k])/lead(gmsstd[i]); |
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259 | while(i<ncols(gmsstd)&&s==0) |
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260 | { |
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261 | i++; |
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262 | s=lead(p[k])/lead(gmsstd[i]); |
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263 | } |
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264 | if(s!=0) |
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265 | { |
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266 | p[k]=p[k]-s*gmsstd[i]; |
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267 | for(j=1;j<=nrows(gmsmatrix);j++) |
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268 | { |
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269 | p[k]=p[k]+diff(s*gmsmatrix[j,i],var(j+1))*var(1); |
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270 | } |
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271 | } |
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272 | else |
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273 | { |
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274 | r[k]=r[k]+lead(p[k]); |
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275 | p[k]=p[k]-lead(p[k]); |
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276 | } |
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277 | while(deg(lead(p[k]))>(K+1)*deg(var(1))-2*gmsmaxdeg&& |
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278 | deg(lead(p[k]),v)<=K) |
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279 | { |
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280 | q[k]=q[k]+lead(p[k]); |
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281 | p[k]=p[k]-lead(p[k]); |
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282 | } |
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283 | } |
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284 | q[k]=q[k]+p[k]; |
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285 | } |
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286 | |
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287 | return(list(r,q)); |
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288 | } |
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289 | example |
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290 | { "EXAMPLE:"; echo=2; |
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291 | ring R=0,(x,y),ds; |
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292 | poly t=x5+x2y2+y5; |
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293 | def G=gmsring(t,"s"); |
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294 | setring(G); |
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295 | list l0=gmsnf(gmspoly,0); |
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296 | print(l0[1]); |
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297 | list l1=gmsnf(gmspoly,1); |
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298 | print(l1[1]); |
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299 | list l=gmsnf(l0[2],1); |
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300 | print(l[1]); |
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301 | } |
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302 | /////////////////////////////////////////////////////////////////////////////// |
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303 | |
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304 | proc gmscoeffs(ideal p,int K) |
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305 | "USAGE: gmscoeffs(p,K); poly p, int K |
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306 | ASSUME: basering constructed by gmsring |
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307 | RETURN: |
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308 | @format |
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309 | list l; |
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310 | matrix l[1]; C[[s]]-basis representation of p mod s^(K+1) |
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311 | ideal l[2]; p==matrix(gmsbasis)*l[1]+l[2] |
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312 | @end format |
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313 | NOTE: computation can be continued by setting p=l[2] |
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314 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice |
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315 | EXAMPLE: example gmscoeffs; shows examples |
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316 | " |
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317 | { |
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318 | list l=gmsnf(p,K); |
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319 | ideal r,q=l[1..2]; |
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320 | poly v=1; |
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321 | for(int i=2;i<=nvars(basering);i++) |
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322 | { |
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323 | v=v*var(i); |
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324 | } |
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325 | matrix C=coeffs(r,gmsbasis,v); |
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326 | return(list(C,q)); |
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327 | } |
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328 | example |
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329 | { "EXAMPLE:"; echo=2; |
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330 | ring R=0,(x,y),ds; |
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331 | poly t=x5+x2y2+y5; |
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332 | def G=gmsring(t,"s"); |
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333 | setring(G); |
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334 | list l0=gmscoeffs(gmspoly,0); |
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335 | print(l0[1]); |
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336 | list l1=gmscoeffs(gmspoly,1); |
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337 | print(l1[1]); |
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338 | list l=gmscoeffs(l0[2],1); |
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339 | print(l[1]); |
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340 | } |
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341 | /////////////////////////////////////////////////////////////////////////////// |
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342 | |
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343 | static proc mindegree(matrix A) |
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344 | { |
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345 | int d=-1; |
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346 | |
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347 | int i,j; |
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348 | for(i=nrows(A);i>=1;i--) |
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349 | { |
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350 | for(j=ncols(A);j>=1;j--) |
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351 | { |
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352 | if(d==-1||(ord(A[i,j])<d&&ord(A[i,j])>-1)) |
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353 | { |
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354 | d=ord(A[i,j]); |
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355 | } |
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356 | } |
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357 | } |
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358 | |
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359 | return(d); |
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360 | } |
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361 | /////////////////////////////////////////////////////////////////////////////// |
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362 | |
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363 | static proc maxdegree(matrix A) |
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364 | { |
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365 | int d=-1; |
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366 | |
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367 | int i,j; |
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368 | for(i=nrows(A);i>=1;i--) |
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369 | { |
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370 | for(j=ncols(A);j>=1;j--) |
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371 | { |
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372 | if(deg(A[i,j])>d) |
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373 | { |
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374 | d=deg(A[i,j]); |
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375 | } |
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376 | } |
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377 | } |
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378 | |
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379 | return(d); |
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380 | } |
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381 | /////////////////////////////////////////////////////////////////////////////// |
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382 | |
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383 | static proc saturate() |
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384 | { |
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385 | int mu=ncols(gmsbasis); |
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386 | ideal r=gmspoly*gmsbasis; |
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387 | matrix A0[mu][mu],C; |
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388 | module H0; |
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389 | module H,H1=freemodule(mu),freemodule(mu); |
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390 | int k=-1; |
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391 | list l; |
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392 | |
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393 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
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394 | while(size(reduce(H,std(H0*var(1)),5))>0) |
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395 | { |
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396 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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397 | k++; |
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398 | dbprint(printlevel-voice+2,"// k="+string(k)); |
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399 | l=gmscoeffs(r,k); |
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400 | C,r=l[1..2]; |
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401 | A0=A0+C; |
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402 | |
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403 | dbprint(printlevel-voice+2,"// compute saturation step"); |
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404 | H0=H; |
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405 | H1=jet(module(A0*H1+var(1)^2*diff(matrix(H1),var(1))),k); |
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406 | H=H*var(1)+H1; |
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407 | } |
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408 | |
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409 | A0=A0-k*var(1); |
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410 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
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411 | H=std(H0); |
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412 | |
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413 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
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414 | H0=division(freemodule(mu)*var(1)^k,H,k*deg(var(1)))[1]; |
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415 | |
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416 | return(A0,r,H,H0,k); |
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417 | } |
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418 | /////////////////////////////////////////////////////////////////////////////// |
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419 | |
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420 | static proc tjet(matrix A0,ideal r,module H,int K0,int K) |
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421 | { |
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422 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
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423 | dbprint(printlevel-voice+2,"// k="+string(K0+K+1)); |
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424 | list l=gmscoeffs(r,K0+K+1); |
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425 | matrix C; |
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426 | C,r=l[1..2]; |
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427 | A0=A0+C; |
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428 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
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429 | matrix A=division(A0*H+var(1)^2*diff(matrix(H),var(1)),H, |
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430 | (K+1)*deg(var(1)))[1]/var(1); |
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431 | return(A,A0,r); |
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432 | } |
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433 | /////////////////////////////////////////////////////////////////////////////// |
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434 | |
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435 | static proc eigenval(matrix A0,ideal r,module H,int K0) |
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436 | { |
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437 | dbprint(printlevel-voice+2, |
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438 | "// compute eigenvalues e with multiplicities m of A1"); |
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439 | matrix A; |
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440 | A,A0,r=tjet(A0,r,H,K0,0); |
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441 | list l=eigenvals(A); |
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442 | def e,m=l[1..2]; |
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443 | dbprint(printlevel-voice+2,"// e="+string(e)); |
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444 | dbprint(printlevel-voice+2,"// m="+string(m)); |
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445 | return(e,m,A0,r); |
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446 | } |
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447 | /////////////////////////////////////////////////////////////////////////////// |
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448 | |
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449 | static proc transform(matrix A,matrix A0,ideal r,module H,module H0,ideal e, |
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450 | intvec m,int K0,int K,int opt) |
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451 | { |
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452 | int mu=ncols(gmsbasis); |
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453 | |
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454 | int i,j,k; |
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455 | intvec d; |
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456 | d[ncols(e)]=0; |
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457 | if(opt) |
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458 | { |
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459 | dbprint(printlevel-voice+2, |
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460 | "// compute rounded maximal differences d of e"); |
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461 | for(i=1;i<=ncols(e);i++) |
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462 | { |
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463 | d[i]=int(e[ncols(e)]-e[i]); |
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464 | } |
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465 | } |
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466 | else |
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467 | { |
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468 | dbprint(printlevel-voice+2, |
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469 | "// compute maximal integer differences d of e"); |
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470 | for(i=1;i<ncols(e);i++) |
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471 | { |
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472 | for(j=i+1;j<=ncols(e);j++) |
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473 | { |
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474 | k=int(e[j]-e[i]); |
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475 | if(number(e[j]-e[i])==k) |
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476 | { |
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477 | if(k>d[i]) |
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478 | { |
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479 | d[i]=k; |
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480 | } |
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481 | if(-k>d[j]) |
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482 | { |
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483 | d[j]=-k; |
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484 | } |
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485 | } |
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486 | } |
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487 | } |
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488 | } |
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489 | dbprint(printlevel-voice+2,"// d="+string(d)); |
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490 | |
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491 | for(i,k=1,0;i<=size(d);i++) |
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492 | { |
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493 | if(k<d[i]) |
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494 | { |
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495 | k=d[i]; |
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496 | } |
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497 | } |
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498 | A,A0,r=tjet(A0,r,H,K0,K+k); |
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499 | |
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500 | module U,V; |
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501 | if(k>0) |
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502 | { |
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503 | int i0,j0,i1,j1; |
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504 | list l; |
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505 | |
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506 | while(k>0) |
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507 | { |
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508 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
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509 | U=0; |
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510 | for(i=1;i<=ncols(e);i++) |
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511 | { |
---|
512 | U=U+syz(power(jet(A,0)-e[i],m[i])); |
---|
513 | } |
---|
514 | V=inverse(U); |
---|
515 | A=V*A*U; |
---|
516 | H=matrix(H)*matrix(U); |
---|
517 | H0=matrix(V)*matrix(H0); |
---|
518 | |
---|
519 | dbprint(printlevel-voice+2,"// transform to reduce maximum of d by 1"); |
---|
520 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
521 | { |
---|
522 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
523 | { |
---|
524 | for(j0,j=1,1;j0<=ncols(e);j0++) |
---|
525 | { |
---|
526 | for(j1=1;j1<=m[j0];j1,j=j1+1,j+1) |
---|
527 | { |
---|
528 | if(d[i0]==0&&d[j0]>=1) |
---|
529 | { |
---|
530 | A[i,j]=A[i,j]*var(1); |
---|
531 | } |
---|
532 | if(d[i0]>=1&&d[j0]==0) |
---|
533 | { |
---|
534 | A[i,j]=A[i,j]/var(1); |
---|
535 | } |
---|
536 | } |
---|
537 | } |
---|
538 | } |
---|
539 | } |
---|
540 | |
---|
541 | H0=transpose(H0); |
---|
542 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
543 | { |
---|
544 | if(d[i0]>=1) |
---|
545 | { |
---|
546 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
547 | { |
---|
548 | H[i]=H[i]*var(1); |
---|
549 | } |
---|
550 | d[i0]=d[i0]-1; |
---|
551 | } |
---|
552 | else |
---|
553 | { |
---|
554 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
555 | { |
---|
556 | A[i,i]=A[i,i]-1; |
---|
557 | H0[i]=H0[i]*var(1); |
---|
558 | } |
---|
559 | e[i0]=e[i0]-1; |
---|
560 | } |
---|
561 | } |
---|
562 | H0=transpose(H0); |
---|
563 | |
---|
564 | l=sppnf(list(e,d,m)); |
---|
565 | e,d,m=l[1..3]; |
---|
566 | |
---|
567 | k--; |
---|
568 | K0++; |
---|
569 | } |
---|
570 | |
---|
571 | A=jet(A,K); |
---|
572 | } |
---|
573 | |
---|
574 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
---|
575 | U=0; |
---|
576 | for(i=1;i<=ncols(e);i++) |
---|
577 | { |
---|
578 | U=U+syz(power(jet(A,0)-e[i],m[i])); |
---|
579 | } |
---|
580 | V=inverse(U); |
---|
581 | A=V*A*U; |
---|
582 | H=matrix(H)*matrix(U); |
---|
583 | H0=matrix(V)*matrix(H0); |
---|
584 | |
---|
585 | return(A,A0,r,H,H0,e,m,K0); |
---|
586 | } |
---|
587 | /////////////////////////////////////////////////////////////////////////////// |
---|
588 | |
---|
589 | proc bernstein(poly t) |
---|
590 | "USAGE: bernstein(t); poly t |
---|
591 | ASSUME: characteristic 0; local degree ordering; |
---|
592 | isolated critical point 0 of t |
---|
593 | RETURN: |
---|
594 | @format |
---|
595 | list bs; Bernstein-Sato polynomial b(s) of t |
---|
596 | ideal bs[1]; |
---|
597 | number bs[1][i]; i-th root of b(s) |
---|
598 | intvec bs[2]; |
---|
599 | int bs[2][i]; multiplicity of i-th root of b(s) |
---|
600 | @end format |
---|
601 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
602 | Bernstein-Sato polynomial |
---|
603 | EXAMPLE: example bernstein; shows examples |
---|
604 | " |
---|
605 | { |
---|
606 | def @R=basering; |
---|
607 | int n=nvars(@R)-1; |
---|
608 | def @G=gmsring(t,"s"); |
---|
609 | setring(@G); |
---|
610 | |
---|
611 | matrix A; |
---|
612 | module U0; |
---|
613 | ideal e; |
---|
614 | intvec m; |
---|
615 | |
---|
616 | def A0,r,H,H0,K0=saturate(); |
---|
617 | A,A0,r=tjet(A0,r,H,K0,0); |
---|
618 | list l=minipoly(A); |
---|
619 | e,m=l[1..2]; |
---|
620 | e=-e; |
---|
621 | l=spnf(spadd(list(e,m),list(ideal(-1),intvec(1)))); |
---|
622 | |
---|
623 | setring(@R); |
---|
624 | list l=imap(@G,l); |
---|
625 | kill @G,gmsmaxdeg; |
---|
626 | |
---|
627 | return(l); |
---|
628 | } |
---|
629 | example |
---|
630 | { "EXAMPLE:"; echo=2; |
---|
631 | ring R=0,(x,y),ds; |
---|
632 | poly t=x5+x2y2+y5; |
---|
633 | bernstein(t); |
---|
634 | } |
---|
635 | /////////////////////////////////////////////////////////////////////////////// |
---|
636 | |
---|
637 | proc monodromy(poly t) |
---|
638 | "USAGE: monodromy(t); poly t |
---|
639 | ASSUME: characteristic 0; local degree ordering; |
---|
640 | isolated critical point 0 of t |
---|
641 | RETURN: |
---|
642 | @format |
---|
643 | list l; Jordan data jordan(M) of monodromy matrix exp(-2*pi*i*M) |
---|
644 | ideal l[1]; |
---|
645 | number l[1][i]; eigenvalue of i-th Jordan block of M |
---|
646 | intvec l[2]; |
---|
647 | int l[2][i]; size of i-th Jordan block of M |
---|
648 | intvec l[3]; |
---|
649 | int l[3][i]; multiplicity of i-th Jordan block of M |
---|
650 | @end format |
---|
651 | SEE ALSO: mondromy_lib, linalg_lib |
---|
652 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; monodromy |
---|
653 | EXAMPLE: example monodromy; shows examples |
---|
654 | " |
---|
655 | { |
---|
656 | def @R=basering; |
---|
657 | int n=nvars(@R)-1; |
---|
658 | def @G=gmsring(t,"s"); |
---|
659 | setring(@G); |
---|
660 | |
---|
661 | matrix A; |
---|
662 | module U0; |
---|
663 | ideal e; |
---|
664 | intvec m; |
---|
665 | |
---|
666 | def A0,r,H,H0,K0=saturate(); |
---|
667 | e,m,A0,r=eigenval(A0,r,H,K0); |
---|
668 | A,A0,r,H,H0,e,m,K0=transform(A,A0,r,H,H0,e,m,K0,0,0); |
---|
669 | |
---|
670 | list l=jordan(A,e,m); |
---|
671 | setring(@R); |
---|
672 | list l=imap(@G,l); |
---|
673 | kill @G,gmsmaxdeg; |
---|
674 | |
---|
675 | return(l); |
---|
676 | } |
---|
677 | example |
---|
678 | { "EXAMPLE:"; echo=2; |
---|
679 | ring R=0,(x,y),ds; |
---|
680 | poly t=x5+x2y2+y5; |
---|
681 | monodromy(t); |
---|
682 | } |
---|
683 | /////////////////////////////////////////////////////////////////////////////// |
---|
684 | |
---|
685 | proc spectrum(poly t) |
---|
686 | "USAGE: spectrum(t); poly t |
---|
687 | ASSUME: characteristic 0; local degree ordering; |
---|
688 | isolated critical point 0 of t |
---|
689 | RETURN: |
---|
690 | @format |
---|
691 | list sp; singularity spectrum of t |
---|
692 | ideal sp[1]; |
---|
693 | number sp[1][i]; i-th spectral number |
---|
694 | intvec sp[2]; |
---|
695 | int sp[2][i]; multiplicity of i-th spectral number |
---|
696 | @end format |
---|
697 | SEE ALSO: spectrum_lib |
---|
698 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
699 | mixed Hodge structure; V-filtration; spectrum |
---|
700 | EXAMPLE: example spectrum; shows examples |
---|
701 | " |
---|
702 | { |
---|
703 | list l=vwfilt(t); |
---|
704 | return(spnf(list(l[1],l[3]))); |
---|
705 | } |
---|
706 | example |
---|
707 | { "EXAMPLE:"; echo=2; |
---|
708 | ring R=0,(x,y),ds; |
---|
709 | poly t=x5+x2y2+y5; |
---|
710 | spprint(spectrum(t)); |
---|
711 | } |
---|
712 | /////////////////////////////////////////////////////////////////////////////// |
---|
713 | |
---|
714 | proc sppairs(poly t) |
---|
715 | "USAGE: sppairs(t); poly t |
---|
716 | ASSUME: characteristic 0; local degree ordering; |
---|
717 | isolated critical point 0 of t |
---|
718 | RETURN: |
---|
719 | @format |
---|
720 | list spp; spectral pairs of t |
---|
721 | ideal spp[1]; |
---|
722 | number spp[1][i]; V-filtration index of i-th spectral pair |
---|
723 | intvec spp[2]; |
---|
724 | int spp[2][i]; weight filtration index of i-th spectral pair |
---|
725 | intvec spp[3]; |
---|
726 | int spp[3][i]; multiplicity of i-th spectral pair |
---|
727 | @end format |
---|
728 | SEE ALSO: spectrum_lib |
---|
729 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
730 | mixed Hodge structure; V-filtration; weight filtration; |
---|
731 | spectrum; spectral pairs |
---|
732 | EXAMPLE: example sppairs; shows examples |
---|
733 | " |
---|
734 | { |
---|
735 | list l=vwfilt(t); |
---|
736 | return(list(l[1],l[2],l[3])); |
---|
737 | } |
---|
738 | example |
---|
739 | { "EXAMPLE:"; echo=2; |
---|
740 | ring R=0,(x,y),ds; |
---|
741 | poly t=x5+x2y2+y5; |
---|
742 | sppprint(sppairs(t)); |
---|
743 | } |
---|
744 | /////////////////////////////////////////////////////////////////////////////// |
---|
745 | |
---|
746 | proc vfilt(poly t) |
---|
747 | "USAGE: vfilt(t); poly t |
---|
748 | ASSUME: characteristic 0; local degree ordering; |
---|
749 | isolated critical point 0 of t |
---|
750 | RETURN: |
---|
751 | @format |
---|
752 | list v; V-filtration on H''/s*H'' |
---|
753 | ideal v[1]; |
---|
754 | number v[1][i]; V-filtration index of i-th spectral number |
---|
755 | intvec v[2]; |
---|
756 | int v[2][i]; multiplicity of i-th spectral number |
---|
757 | list v[3]; |
---|
758 | module v[3][i]; vector space of i-th graded part in terms of v[4] |
---|
759 | ideal v[4]; monomial vector space basis of H''/s*H'' |
---|
760 | ideal v[5]; standard basis of Jacobian ideal |
---|
761 | @end format |
---|
762 | SEE ALSO: spectrum_lib |
---|
763 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
764 | mixed Hodge structure; V-filtration; spectrum |
---|
765 | EXAMPLE: example vfilt; shows examples |
---|
766 | " |
---|
767 | { |
---|
768 | list l=vwfilt(t); |
---|
769 | return(spnf(list(l[1],l[3],l[4]))+list(l[5],l[6])); |
---|
770 | } |
---|
771 | example |
---|
772 | { "EXAMPLE:"; echo=2; |
---|
773 | ring R=0,(x,y),ds; |
---|
774 | poly t=x5+x2y2+y5; |
---|
775 | vfilt(t); |
---|
776 | } |
---|
777 | /////////////////////////////////////////////////////////////////////////////// |
---|
778 | |
---|
779 | proc vwfilt(poly t) |
---|
780 | "USAGE: vwfilt(t); poly t |
---|
781 | ASSUME: characteristic 0; local degree ordering; |
---|
782 | isolated critical point 0 of t |
---|
783 | RETURN: |
---|
784 | @format |
---|
785 | list vw; weighted V-filtration on H''/s*H'' |
---|
786 | ideal vw[1]; |
---|
787 | number vw[1][i]; V-filtration index of i-th spectral pair |
---|
788 | intvec vw[2]; |
---|
789 | int vw[2][i]; weight filtration index of i-th spectral pair |
---|
790 | intvec vw[3]; |
---|
791 | int vw[3][i]; multiplicity of i-th spectral pair |
---|
792 | list vw[4]; |
---|
793 | module vw[4][i]; vector space of i-th graded part in terms of vw[5] |
---|
794 | ideal vw[5]; monomial vector space basis of H''/s*H'' |
---|
795 | ideal vw[6]; standard basis of Jacobian ideal |
---|
796 | @end format |
---|
797 | SEE ALSO: spectrum_lib |
---|
798 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
799 | mixed Hodge structure; V-filtration; weight filtration; |
---|
800 | spectrum; spectral pairs |
---|
801 | EXAMPLE: example vwfilt; shows examples |
---|
802 | " |
---|
803 | { |
---|
804 | def @R=basering; |
---|
805 | int n=nvars(@R)-1; |
---|
806 | def @G=gmsring(t,"s"); |
---|
807 | setring(@G); |
---|
808 | |
---|
809 | int mu=ncols(gmsbasis); |
---|
810 | matrix A; |
---|
811 | ideal e; |
---|
812 | intvec m; |
---|
813 | |
---|
814 | def A0,r,H,H0,K0=saturate(); |
---|
815 | e,m,A0,r=eigenval(A0,r,H,K0); |
---|
816 | A,A0,r,H,H0,e,m,K0=transform(A,A0,r,H,H0,e,m,K0,0,1); |
---|
817 | |
---|
818 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
819 | list l=jordanbasis(A,e,m); |
---|
820 | def U,v=l[1..2]; |
---|
821 | kill l; |
---|
822 | vector u0; |
---|
823 | int v0; |
---|
824 | int i,j,k,l; |
---|
825 | for(k,l=1,1;l<=ncols(e);k,l=k+m[l],l+1) |
---|
826 | { |
---|
827 | for(i=k+m[l]-1;i>=k+1;i--) |
---|
828 | { |
---|
829 | for(j=i-1;j>=k;j--) |
---|
830 | { |
---|
831 | if(v[i]>v[j]) |
---|
832 | { |
---|
833 | v0=v[i];v[i]=v[j];v[j]=v0; |
---|
834 | u0=U[i];U[i]=U[j];U[j]=u0; |
---|
835 | } |
---|
836 | } |
---|
837 | } |
---|
838 | } |
---|
839 | |
---|
840 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
841 | matrix V=inverse(U); |
---|
842 | A=V*A*U; |
---|
843 | dbprint(printlevel-voice+2,"// compute standard basis of H''"); |
---|
844 | H=matrix(H)*matrix(U); |
---|
845 | H0=std(V*H0); |
---|
846 | |
---|
847 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
848 | ideal a; |
---|
849 | intvec w; |
---|
850 | for(i=1;i<=mu;i++) |
---|
851 | { |
---|
852 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
853 | a[i]=A[j,j]+ord(H0[i]) div deg(var(1))-1; |
---|
854 | w[i]=v[j]+n; |
---|
855 | } |
---|
856 | H=matrix(H)*matrix(H0); |
---|
857 | H=simplify(jet(H/var(1)^(mindegree(H) div deg(var(1))),0),1); |
---|
858 | |
---|
859 | kill l; |
---|
860 | list l=sppnf(list(a,w,H))+list(gmsbasis,gmsstd); |
---|
861 | setring(@R); |
---|
862 | list l=imap(@G,l); |
---|
863 | kill @G,gmsmaxdeg; |
---|
864 | attrib(l[5],"isSB",1); |
---|
865 | |
---|
866 | return(l); |
---|
867 | } |
---|
868 | example |
---|
869 | { "EXAMPLE:"; echo=2; |
---|
870 | ring R=0,(x,y),ds; |
---|
871 | poly t=x5+x2y2+y5; |
---|
872 | vwfilt(t); |
---|
873 | } |
---|
874 | /////////////////////////////////////////////////////////////////////////////// |
---|
875 | |
---|
876 | static proc fsplit(ideal e0,intvec m0,matrix A,module H,module H0) |
---|
877 | { |
---|
878 | int mu=ncols(gmsbasis); |
---|
879 | |
---|
880 | dbprint(printlevel-voice+2,"// compute standard basis of H''"); |
---|
881 | H0=std(H0); |
---|
882 | H0=simplify(H0,1); |
---|
883 | |
---|
884 | dbprint(printlevel-voice+2,"// compute Hodge filtration"); |
---|
885 | int i,j,k; |
---|
886 | ideal e; |
---|
887 | intvec m; |
---|
888 | e[mu]=0; |
---|
889 | for(i=1;i<=ncols(e0);i++) |
---|
890 | { |
---|
891 | for(j=m0[i];j>=1;j--) |
---|
892 | { |
---|
893 | k++; |
---|
894 | e[k]=e0[i]; |
---|
895 | m[k]=i; |
---|
896 | } |
---|
897 | } |
---|
898 | |
---|
899 | number n,n0; |
---|
900 | vector v,v0; |
---|
901 | list F; |
---|
902 | for(i=ncols(e0);i>=1;i--) |
---|
903 | { |
---|
904 | F[i]=module(matrix(0,mu,1)); |
---|
905 | } |
---|
906 | for(i=mu;i>=1;i--) |
---|
907 | { |
---|
908 | v=H0[i]; |
---|
909 | v0=lead(v); |
---|
910 | n0=leadcoef(e[leadexp(v0)[nvars(basering)+1]])+leadexp(v0)[1]; |
---|
911 | v=v-lead(v); |
---|
912 | while(v!=0) |
---|
913 | { |
---|
914 | n=leadcoef(e[leadexp(v)[nvars(basering)+1]])+leadexp(v)[1]; |
---|
915 | if(n==n0) |
---|
916 | { |
---|
917 | v0=v0+lead(v); |
---|
918 | v=v-lead(v); |
---|
919 | } |
---|
920 | else |
---|
921 | { |
---|
922 | v=0; |
---|
923 | } |
---|
924 | } |
---|
925 | j=m[leadexp(v0)[nvars(basering)+1]]; |
---|
926 | F[j]=F[j]+v0; |
---|
927 | } |
---|
928 | |
---|
929 | dbprint(printlevel-voice+2,"// compute splitting of Hodge filtration"); |
---|
930 | matrix A0=jet(A,0); |
---|
931 | module U,U0,U1,U2; |
---|
932 | matrix N; |
---|
933 | for(i=size(F);i>=1;i--) |
---|
934 | { |
---|
935 | N=A0-e0[i]; |
---|
936 | U0=0; |
---|
937 | while(size(F[i])>0) |
---|
938 | { |
---|
939 | U1=jet(F[i],0); |
---|
940 | k=0; |
---|
941 | while(size(U1)>0) |
---|
942 | { |
---|
943 | for(j=ncols(U1);j>=1;j--) |
---|
944 | { |
---|
945 | if(size(reduce(U1[j],std(U0),5))>0) |
---|
946 | { |
---|
947 | U0=U0+U1[j]; |
---|
948 | } |
---|
949 | } |
---|
950 | U1=N*U1; |
---|
951 | k++; |
---|
952 | } |
---|
953 | F[i]=module(F[i]/var(1)); |
---|
954 | } |
---|
955 | U=U0+U; |
---|
956 | } |
---|
957 | |
---|
958 | dbprint(printlevel-voice+2,"// transform to Hodge splitting basis"); |
---|
959 | H=matrix(H)*matrix(U); |
---|
960 | H0=lift(U,H0); |
---|
961 | A=lift(U,A*U); |
---|
962 | |
---|
963 | return(e,A,H,H0); |
---|
964 | } |
---|
965 | /////////////////////////////////////////////////////////////////////////////// |
---|
966 | |
---|
967 | static proc glift(ideal e,matrix A,module H,module H0,int K) |
---|
968 | { |
---|
969 | poly s=var(1); |
---|
970 | int mu=ncols(gmsbasis); |
---|
971 | |
---|
972 | dbprint(printlevel-voice+2,"// compute standard basis of H''"); |
---|
973 | H0=std(H0); |
---|
974 | H0=simplify(H0,1); |
---|
975 | |
---|
976 | int i,j,k; |
---|
977 | ideal v; |
---|
978 | for(i=mu;i>=1;i--) |
---|
979 | { |
---|
980 | v[i]=e[leadexp(H0[i])[nvars(basering)+1]]+leadexp(H0[i])[1]; |
---|
981 | } |
---|
982 | |
---|
983 | dbprint(printlevel-voice+2, |
---|
984 | "// compute matrix A0 of t w.r.t. good basis H0 of H''"); |
---|
985 | number c; |
---|
986 | matrix h0[mu][1]; |
---|
987 | matrix m[mu][1]; |
---|
988 | matrix a0[mu][1]; |
---|
989 | matrix A0[mu][mu]; |
---|
990 | module M=H0; |
---|
991 | module N=jet(s*A*matrix(H0)+s^2*diff(matrix(H0),s),K+1); |
---|
992 | while(size(N)>0) |
---|
993 | { |
---|
994 | j=mu; |
---|
995 | for(k=mu-1;k>=1;k--) |
---|
996 | { |
---|
997 | if(N[k]>N[j]) |
---|
998 | { |
---|
999 | j=k; |
---|
1000 | } |
---|
1001 | } |
---|
1002 | i=mu; |
---|
1003 | while(leadexp(M[i])[nvars(basering)+1]!=leadexp(N[j])[nvars(basering)+1]) |
---|
1004 | { |
---|
1005 | i--; |
---|
1006 | } |
---|
1007 | k=leadexp(N[j])[1]-leadexp(M[i])[1]; |
---|
1008 | if(k==0||i==j) |
---|
1009 | { |
---|
1010 | dbprint(printlevel-voice+3,"// compute A0["+string(i)+","+string(j)+"]"); |
---|
1011 | c=leadcoef(N[j])/leadcoef(M[i]); |
---|
1012 | A0[i,j]=A0[i,j]+c*s^k; |
---|
1013 | N[j]=jet(N[j]-c*s^k*M[i],K+1); |
---|
1014 | } |
---|
1015 | else |
---|
1016 | { |
---|
1017 | dbprint(printlevel-voice+3, |
---|
1018 | "// reduce H0["+string(j)+"] with H0["+string(i)+"]"); |
---|
1019 | c=leadcoef(N[j])/leadcoef(M[i])/(1-k-leadcoef(v[i])+leadcoef(v[j])); |
---|
1020 | H0[j]=H0[j]+c*s^(k-1)*H0[i]; |
---|
1021 | M[j]=M[j]+c*s^(k-1)*M[i]; |
---|
1022 | h0=c*s^(k-1)*H0[i]; |
---|
1023 | N[j]=N[j]+jet(s*A*h0+s^2*diff(h0,s),K+1)[1]; |
---|
1024 | m=M[i]; |
---|
1025 | a0=transpose(A0)[j]; |
---|
1026 | N=N-jet(c*s^(k-1)*m*transpose(a0),K+1); |
---|
1027 | } |
---|
1028 | } |
---|
1029 | |
---|
1030 | H0=matrix(H)*matrix(H0); |
---|
1031 | H0=H0/var(1)^(mindegree(H0) div deg(var(1))); |
---|
1032 | |
---|
1033 | return(A0,H0); |
---|
1034 | } |
---|
1035 | /////////////////////////////////////////////////////////////////////////////// |
---|
1036 | |
---|
1037 | proc tmatrix(poly t) |
---|
1038 | "USAGE: tmatrix(t); poly t |
---|
1039 | ASSUME: characteristic 0; local degree ordering; |
---|
1040 | isolated critical point 0 of t |
---|
1041 | RETURN: |
---|
1042 | @format |
---|
1043 | list l=A0,A1,T,M; |
---|
1044 | matrix A0,A1; t=A0+s*A1+s^2*(d/ds) on H'' w.r.t. C[[s]]-basis M*T |
---|
1045 | module T; C-basis of C^mu |
---|
1046 | ideal M; monomial C-basis of H''/sH'' |
---|
1047 | @end format |
---|
1048 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
1049 | mixed Hodge structure; V-filtration; weight filtration; |
---|
1050 | monodromy; spectrum; spectral pairs; good basis |
---|
1051 | EXAMPLE: example tmatrix; shows examples |
---|
1052 | " |
---|
1053 | { |
---|
1054 | def @R=basering; |
---|
1055 | int n=nvars(@R)-1; |
---|
1056 | def @G=gmsring(t,"s"); |
---|
1057 | setring(@G); |
---|
1058 | |
---|
1059 | int mu=ncols(gmsbasis); |
---|
1060 | matrix A; |
---|
1061 | module U0; |
---|
1062 | ideal e; |
---|
1063 | intvec m; |
---|
1064 | |
---|
1065 | def A0,r,H,H0,K0=saturate(); |
---|
1066 | e,m,A0,r=eigenval(A0,r,H,K0); |
---|
1067 | A,A0,r,H,H0,e,m,K0=transform(A,A0,r,H,H0,e,m,K0,K0+int(e[ncols(e)]-e[1]),1); |
---|
1068 | A,H0=glift(fsplit(e,m,A,H,H0),K0); |
---|
1069 | |
---|
1070 | A0=jet(A,0); |
---|
1071 | A=jet(A/var(1),0); |
---|
1072 | |
---|
1073 | list l=A0,A,H0,gmsbasis; |
---|
1074 | setring(@R); |
---|
1075 | list l=imap(@G,l); |
---|
1076 | kill @G,gmsmaxdeg; |
---|
1077 | |
---|
1078 | return(l); |
---|
1079 | } |
---|
1080 | example |
---|
1081 | { "EXAMPLE:"; echo=2; |
---|
1082 | ring R=0,(x,y),ds; |
---|
1083 | poly t=x5+x2y2+y5; |
---|
1084 | list l=tmatrix(t); |
---|
1085 | print(l[1]); |
---|
1086 | print(l[2]); |
---|
1087 | print(l[3]); |
---|
1088 | print(l[4]); |
---|
1089 | } |
---|
1090 | /////////////////////////////////////////////////////////////////////////////// |
---|
1091 | |
---|
1092 | proc endvfilt(list v) |
---|
1093 | "USAGE: endvfilt(v); list v |
---|
1094 | ASSUME: v returned by vfilt |
---|
1095 | RETURN: |
---|
1096 | @format |
---|
1097 | list ev; V-filtration on Jacobian algebra |
---|
1098 | ideal ev[1]; |
---|
1099 | number ev[1][i]; i-th V-filtration index |
---|
1100 | intvec ev[2]; |
---|
1101 | int ev[2][i]; i-th multiplicity |
---|
1102 | list ev[3]; |
---|
1103 | module ev[3][i]; vector space of i-th graded part in terms of ev[4] |
---|
1104 | ideal ev[4]; monomial vector space basis of Jacobian algebra |
---|
1105 | ideal ev[5]; standard basis of Jacobian ideal |
---|
1106 | @end format |
---|
1107 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
1108 | mixed Hodge structure; V-filtration; endomorphism filtration |
---|
1109 | EXAMPLE: example endvfilt; shows examples |
---|
1110 | " |
---|
1111 | { |
---|
1112 | def a,d,V,m,g=v[1..5]; |
---|
1113 | attrib(g,"isSB",1); |
---|
1114 | int mu=ncols(m); |
---|
1115 | |
---|
1116 | module V0=V[1]; |
---|
1117 | for(int i=2;i<=size(V);i++) |
---|
1118 | { |
---|
1119 | V0=V0,V[i]; |
---|
1120 | } |
---|
1121 | |
---|
1122 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
1123 | list M; |
---|
1124 | module U=freemodule(ncols(m)); |
---|
1125 | for(i=ncols(m);i>=1;i--) |
---|
1126 | { |
---|
1127 | M[i]=division(coeffs(reduce(m[i]*m,g,U),m)*V0,V0)[1]; |
---|
1128 | } |
---|
1129 | |
---|
1130 | int j,k,i0,j0,i1,j1; |
---|
1131 | number b0=number(a[1]-a[ncols(a)]); |
---|
1132 | number b1,b2; |
---|
1133 | matrix M0; |
---|
1134 | module L; |
---|
1135 | list v0=freemodule(ncols(m)); |
---|
1136 | ideal a0=b0; |
---|
1137 | list l; |
---|
1138 | |
---|
1139 | while(b0<number(a[ncols(a)]-a[1])) |
---|
1140 | { |
---|
1141 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
1142 | b1=number(a[ncols(a)]-a[1]); |
---|
1143 | for(j=ncols(a);j>=1;j--) |
---|
1144 | { |
---|
1145 | for(i=ncols(a);i>=1;i--) |
---|
1146 | { |
---|
1147 | b2=number(a[i]-a[j]); |
---|
1148 | if(b2>b0&&b2<b1) |
---|
1149 | { |
---|
1150 | b1=b2; |
---|
1151 | } |
---|
1152 | else |
---|
1153 | { |
---|
1154 | if(b2<=b0) |
---|
1155 | { |
---|
1156 | i=0; |
---|
1157 | } |
---|
1158 | } |
---|
1159 | } |
---|
1160 | } |
---|
1161 | b0=b1; |
---|
1162 | |
---|
1163 | l=ideal(); |
---|
1164 | for(k=ncols(m);k>=2;k--) |
---|
1165 | { |
---|
1166 | l=l+list(ideal()); |
---|
1167 | } |
---|
1168 | |
---|
1169 | dbprint(printlevel-voice+2,"// collect conditions for EV["+string(b0)+"]"); |
---|
1170 | j=ncols(a); |
---|
1171 | j0=mu; |
---|
1172 | while(j>=1) |
---|
1173 | { |
---|
1174 | i0=1; |
---|
1175 | i=1; |
---|
1176 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
1177 | { |
---|
1178 | i0=i0+d[i]; |
---|
1179 | i++; |
---|
1180 | } |
---|
1181 | if(a[i]<a[j]+b0) |
---|
1182 | { |
---|
1183 | i0=i0+d[i]; |
---|
1184 | i++; |
---|
1185 | } |
---|
1186 | for(k=1;k<=ncols(m);k++) |
---|
1187 | { |
---|
1188 | M0=M[k]; |
---|
1189 | if(i0>1) |
---|
1190 | { |
---|
1191 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
1192 | } |
---|
1193 | } |
---|
1194 | j0=j0-d[j]; |
---|
1195 | j--; |
---|
1196 | } |
---|
1197 | |
---|
1198 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
1199 | L=transpose(module(l[1])); |
---|
1200 | for(k=2;k<=ncols(m);k++) |
---|
1201 | { |
---|
1202 | L=L,transpose(module(l[k])); |
---|
1203 | } |
---|
1204 | |
---|
1205 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
1206 | v0=v0+list(syz(L)); |
---|
1207 | a0=a0,b0; |
---|
1208 | } |
---|
1209 | |
---|
1210 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
1211 | option(redSB); |
---|
1212 | for(i=1;i<size(v0);i++) |
---|
1213 | { |
---|
1214 | v0[i+1]=std(v0[i+1]); |
---|
1215 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
1216 | } |
---|
1217 | option(noredSB); |
---|
1218 | |
---|
1219 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
1220 | i=1; |
---|
1221 | while(size(v0[i])==0) |
---|
1222 | { |
---|
1223 | i++; |
---|
1224 | } |
---|
1225 | list v1=v0[i]; |
---|
1226 | intvec d1=size(v0[i]); |
---|
1227 | ideal a1=a0[i]; |
---|
1228 | i++; |
---|
1229 | while(i<=size(v0)) |
---|
1230 | { |
---|
1231 | if(size(v0[i])>0) |
---|
1232 | { |
---|
1233 | v1=v1+list(v0[i]); |
---|
1234 | d1=d1,size(v0[i]); |
---|
1235 | a1=a1,a0[i]; |
---|
1236 | } |
---|
1237 | i++; |
---|
1238 | } |
---|
1239 | return(list(a1,d1,v1,m,g)); |
---|
1240 | } |
---|
1241 | example |
---|
1242 | { "EXAMPLE:"; echo=2; |
---|
1243 | ring R=0,(x,y),ds; |
---|
1244 | poly t=x5+x2y2+y5; |
---|
1245 | endvfilt(vfilt(t)); |
---|
1246 | } |
---|
1247 | /////////////////////////////////////////////////////////////////////////////// |
---|
1248 | |
---|
1249 | proc sppnf(list sp) |
---|
1250 | "USAGE: sppnf(list(a,w[,m])); ideal a, intvec w, intvec m |
---|
1251 | ASSUME: ncols(a)==size(w)==size(m) |
---|
1252 | RETURN: order (a[i][,w[i]]) with multiplicity m[i] lexicographically |
---|
1253 | EXAMPLE: example sppnf; shows examples |
---|
1254 | " |
---|
1255 | { |
---|
1256 | ideal a=sp[1]; |
---|
1257 | intvec w=sp[2]; |
---|
1258 | int n=ncols(a); |
---|
1259 | intvec m; |
---|
1260 | list V; |
---|
1261 | module v; |
---|
1262 | int i,j; |
---|
1263 | for(i=3;i<=size(sp);i++) |
---|
1264 | { |
---|
1265 | if(typeof(sp[i])=="intvec") |
---|
1266 | { |
---|
1267 | m=sp[i]; |
---|
1268 | } |
---|
1269 | if(typeof(sp[i])=="module") |
---|
1270 | { |
---|
1271 | v=sp[i]; |
---|
1272 | for(j=n;j>=1;j--) |
---|
1273 | { |
---|
1274 | V[j]=module(v[j]); |
---|
1275 | } |
---|
1276 | } |
---|
1277 | if(typeof(sp[i])=="list") |
---|
1278 | { |
---|
1279 | V=sp[i]; |
---|
1280 | } |
---|
1281 | } |
---|
1282 | if(m==0) |
---|
1283 | { |
---|
1284 | for(i=n;i>=1;i--) |
---|
1285 | { |
---|
1286 | m[i]=1; |
---|
1287 | } |
---|
1288 | } |
---|
1289 | |
---|
1290 | int k; |
---|
1291 | ideal a0; |
---|
1292 | intvec w0,m0; |
---|
1293 | list V0; |
---|
1294 | number a1; |
---|
1295 | int w1,m1; |
---|
1296 | for(i=n;i>=1;i--) |
---|
1297 | { |
---|
1298 | if(m[i]!=0) |
---|
1299 | { |
---|
1300 | for(j=i-1;j>=1;j--) |
---|
1301 | { |
---|
1302 | if(m[j]!=0) |
---|
1303 | { |
---|
1304 | if(number(a[i])>number(a[j])|| |
---|
1305 | (number(a[i])==number(a[j])&&w[i]<w[j])) |
---|
1306 | { |
---|
1307 | a1=number(a[i]); |
---|
1308 | a[i]=a[j]; |
---|
1309 | a[j]=a1; |
---|
1310 | w1=w[i]; |
---|
1311 | w[i]=w[j]; |
---|
1312 | w[j]=w1; |
---|
1313 | m1=m[i]; |
---|
1314 | m[i]=m[j]; |
---|
1315 | m[j]=m1; |
---|
1316 | if(size(V)>0) |
---|
1317 | { |
---|
1318 | v=V[i]; |
---|
1319 | V[i]=V[j]; |
---|
1320 | V[j]=v; |
---|
1321 | } |
---|
1322 | } |
---|
1323 | if(number(a[i])==number(a[j])&&w[i]==w[j]) |
---|
1324 | { |
---|
1325 | m[i]=m[i]+m[j]; |
---|
1326 | m[j]=0; |
---|
1327 | if(size(V)>0) |
---|
1328 | { |
---|
1329 | V[i]=V[i]+V[j]; |
---|
1330 | } |
---|
1331 | } |
---|
1332 | } |
---|
1333 | } |
---|
1334 | k++; |
---|
1335 | a0[k]=a[i]; |
---|
1336 | w0[k]=w[i]; |
---|
1337 | m0[k]=m[i]; |
---|
1338 | if(size(V)>0) |
---|
1339 | { |
---|
1340 | V0[k]=V[i]; |
---|
1341 | } |
---|
1342 | } |
---|
1343 | } |
---|
1344 | |
---|
1345 | if(size(V0)>0) |
---|
1346 | { |
---|
1347 | n=size(V0); |
---|
1348 | module U=std(V0[n]); |
---|
1349 | for(i=n-1;i>=1;i--) |
---|
1350 | { |
---|
1351 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
1352 | if(i>=2) |
---|
1353 | { |
---|
1354 | U=std(U+V0[i]); |
---|
1355 | } |
---|
1356 | } |
---|
1357 | } |
---|
1358 | |
---|
1359 | if(k>0) |
---|
1360 | { |
---|
1361 | sp=a0,w0,m0; |
---|
1362 | if(size(V0)>0) |
---|
1363 | { |
---|
1364 | sp[4]=V0; |
---|
1365 | } |
---|
1366 | } |
---|
1367 | return(sp); |
---|
1368 | } |
---|
1369 | example |
---|
1370 | { "EXAMPLE:"; echo=2; |
---|
1371 | ring R=0,(x,y),ds; |
---|
1372 | list sp=list(ideal(-1/2,-3/10,-3/10,-1/10,-1/10,0,1/10,1/10,3/10,3/10,1/2), |
---|
1373 | intvec(2,1,1,1,1,1,1,1,1,1,0)); |
---|
1374 | sppprint(sppnf(sp)); |
---|
1375 | } |
---|
1376 | /////////////////////////////////////////////////////////////////////////////// |
---|
1377 | |
---|
1378 | proc sppprint(list spp) |
---|
1379 | "USAGE: sppprint(spp); list spp |
---|
1380 | RETURN: string s; spectral pairs spp |
---|
1381 | EXAMPLE: example sppprint; shows examples |
---|
1382 | " |
---|
1383 | { |
---|
1384 | string s; |
---|
1385 | for(int i=1;i<size(spp[3]);i++) |
---|
1386 | { |
---|
1387 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+")," |
---|
1388 | +string(spp[3][i])+"),"; |
---|
1389 | } |
---|
1390 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+")"; |
---|
1391 | return(s); |
---|
1392 | } |
---|
1393 | example |
---|
1394 | { "EXAMPLE:"; echo=2; |
---|
1395 | ring R=0,(x,y),ds; |
---|
1396 | 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), |
---|
1397 | intvec(1,2,2,1,2,2,1)); |
---|
1398 | sppprint(spp); |
---|
1399 | } |
---|
1400 | /////////////////////////////////////////////////////////////////////////////// |
---|
1401 | |
---|
1402 | proc spadd(list sp1,list sp2) |
---|
1403 | "USAGE: spadd(sp1,sp2); list sp1, list sp2 |
---|
1404 | RETURN: list sp; sum of spectra sp1 and sp2 |
---|
1405 | EXAMPLE: example spadd; shows examples |
---|
1406 | " |
---|
1407 | { |
---|
1408 | ideal s; |
---|
1409 | intvec m; |
---|
1410 | int i,i1,i2=1,1,1; |
---|
1411 | while(i1<=size(sp1[2])||i2<=size(sp2[2])) |
---|
1412 | { |
---|
1413 | if(i1<=size(sp1[2])) |
---|
1414 | { |
---|
1415 | if(i2<=size(sp2[2])) |
---|
1416 | { |
---|
1417 | if(number(sp1[1][i1])<number(sp2[1][i2])) |
---|
1418 | { |
---|
1419 | s[i]=sp1[1][i1]; |
---|
1420 | m[i]=sp1[2][i1]; |
---|
1421 | i++; |
---|
1422 | i1++; |
---|
1423 | } |
---|
1424 | else |
---|
1425 | { |
---|
1426 | if(number(sp1[1][i1])>number(sp2[1][i2])) |
---|
1427 | { |
---|
1428 | s[i]=sp2[1][i2]; |
---|
1429 | m[i]=sp2[2][i2]; |
---|
1430 | i++; |
---|
1431 | i2++; |
---|
1432 | } |
---|
1433 | else |
---|
1434 | { |
---|
1435 | if(sp1[2][i1]+sp2[2][i2]!=0) |
---|
1436 | { |
---|
1437 | s[i]=sp1[1][i1]; |
---|
1438 | m[i]=sp1[2][i1]+sp2[2][i2]; |
---|
1439 | i++; |
---|
1440 | } |
---|
1441 | i1++; |
---|
1442 | i2++; |
---|
1443 | } |
---|
1444 | } |
---|
1445 | } |
---|
1446 | else |
---|
1447 | { |
---|
1448 | s[i]=sp1[1][i1]; |
---|
1449 | m[i]=sp1[2][i1]; |
---|
1450 | i++; |
---|
1451 | i1++; |
---|
1452 | } |
---|
1453 | } |
---|
1454 | else |
---|
1455 | { |
---|
1456 | s[i]=sp2[1][i2]; |
---|
1457 | m[i]=sp2[2][i2]; |
---|
1458 | i++; |
---|
1459 | i2++; |
---|
1460 | } |
---|
1461 | } |
---|
1462 | return(list(s,m)); |
---|
1463 | } |
---|
1464 | example |
---|
1465 | { "EXAMPLE:"; echo=2; |
---|
1466 | ring R=0,(x,y),ds; |
---|
1467 | 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)); |
---|
1468 | spprint(sp1); |
---|
1469 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1470 | spprint(sp2); |
---|
1471 | spprint(spadd(sp1,sp2)); |
---|
1472 | } |
---|
1473 | /////////////////////////////////////////////////////////////////////////////// |
---|
1474 | |
---|
1475 | proc spsub(list sp1,list sp2) |
---|
1476 | "USAGE: spsub(sp1,sp2); list sp1, list sp2 |
---|
1477 | RETURN: list sp; difference of spectra sp1 and sp2 |
---|
1478 | EXAMPLE: example spsub; shows examples |
---|
1479 | " |
---|
1480 | { |
---|
1481 | return(spadd(sp1,spmul(sp2,-1))); |
---|
1482 | } |
---|
1483 | example |
---|
1484 | { "EXAMPLE:"; echo=2; |
---|
1485 | ring R=0,(x,y),ds; |
---|
1486 | 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)); |
---|
1487 | spprint(sp1); |
---|
1488 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1489 | spprint(sp2); |
---|
1490 | spprint(spsub(sp1,sp2)); |
---|
1491 | } |
---|
1492 | /////////////////////////////////////////////////////////////////////////////// |
---|
1493 | |
---|
1494 | proc spmul(list #) |
---|
1495 | "USAGE: spmul(sp0,k); list sp0, int[vec] k |
---|
1496 | RETURN: list sp; linear combination of spectra sp0 with coefficients k |
---|
1497 | EXAMPLE: example spmul; shows examples |
---|
1498 | " |
---|
1499 | { |
---|
1500 | if(size(#)==2) |
---|
1501 | { |
---|
1502 | if(typeof(#[1])=="list") |
---|
1503 | { |
---|
1504 | if(typeof(#[2])=="int") |
---|
1505 | { |
---|
1506 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1507 | } |
---|
1508 | if(typeof(#[2])=="intvec") |
---|
1509 | { |
---|
1510 | list sp0=list(ideal(),intvec(0)); |
---|
1511 | for(int i=size(#[2]);i>=1;i--) |
---|
1512 | { |
---|
1513 | sp0=spadd(sp0,spmul(#[1][i],#[2][i])); |
---|
1514 | } |
---|
1515 | return(sp0); |
---|
1516 | } |
---|
1517 | } |
---|
1518 | } |
---|
1519 | return(list(ideal(),intvec(0))); |
---|
1520 | } |
---|
1521 | example |
---|
1522 | { "EXAMPLE:"; echo=2; |
---|
1523 | ring R=0,(x,y),ds; |
---|
1524 | list sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1525 | spprint(sp); |
---|
1526 | spprint(spmul(sp,2)); |
---|
1527 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1528 | spprint(sp1); |
---|
1529 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1530 | spprint(sp2); |
---|
1531 | spprint(spmul(list(sp1,sp2),intvec(1,2))); |
---|
1532 | } |
---|
1533 | /////////////////////////////////////////////////////////////////////////////// |
---|
1534 | |
---|
1535 | proc spissemicont(list sp,list #) |
---|
1536 | "USAGE: spissemicont(sp[,1]); list sp, int opt |
---|
1537 | RETURN: |
---|
1538 | @format |
---|
1539 | int k= |
---|
1540 | 1; if sum of sp is positive on all intervals [a,a+1) [and (a,a+1)] |
---|
1541 | 0; if sum of sp is negative on some interval [a,a+1) [or (a,a+1)] |
---|
1542 | @end format |
---|
1543 | EXAMPLE: example spissemicont; shows examples |
---|
1544 | " |
---|
1545 | { |
---|
1546 | int opt=0; |
---|
1547 | if(size(#)>0) |
---|
1548 | { |
---|
1549 | if(typeof(#[1])=="int") |
---|
1550 | { |
---|
1551 | opt=1; |
---|
1552 | } |
---|
1553 | } |
---|
1554 | int i,j,k; |
---|
1555 | i=1; |
---|
1556 | while(i<=size(sp[2])-1) |
---|
1557 | { |
---|
1558 | j=i+1; |
---|
1559 | k=0; |
---|
1560 | while(j+1<=size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
1561 | { |
---|
1562 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
1563 | { |
---|
1564 | k=k+sp[2][j]; |
---|
1565 | } |
---|
1566 | j++; |
---|
1567 | } |
---|
1568 | if(j==size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
1569 | { |
---|
1570 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
1571 | { |
---|
1572 | k=k+sp[2][j]; |
---|
1573 | } |
---|
1574 | } |
---|
1575 | if(k<0) |
---|
1576 | { |
---|
1577 | return(0); |
---|
1578 | } |
---|
1579 | i++; |
---|
1580 | } |
---|
1581 | return(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 | spissemicont(spsub(sp1,spmul(sp2,3))); |
---|
1591 | spissemicont(spsub(sp1,spmul(sp2,4))); |
---|
1592 | } |
---|
1593 | /////////////////////////////////////////////////////////////////////////////// |
---|
1594 | |
---|
1595 | proc spsemicont(list sp0,list sp,list #) |
---|
1596 | "USAGE: spsemicont(sp0,sp,k[,1]); list sp0, list sp |
---|
1597 | RETURN: |
---|
1598 | @format |
---|
1599 | list l; |
---|
1600 | intvec l[i]; if the spectra sp0 occur with multiplicities k |
---|
1601 | in a deformation of a [quasihomogeneous] singularity |
---|
1602 | with spectrum sp then k<=l[i] |
---|
1603 | @end format |
---|
1604 | EXAMPLE: example spsemicont; shows examples |
---|
1605 | " |
---|
1606 | { |
---|
1607 | list l,l0; |
---|
1608 | int i,j,k; |
---|
1609 | while(spissemicont(sp0,#)) |
---|
1610 | { |
---|
1611 | if(size(sp)>1) |
---|
1612 | { |
---|
1613 | l0=spsemicont(sp0,list(sp[1..size(sp)-1])); |
---|
1614 | for(i=1;i<=size(l0);i++) |
---|
1615 | { |
---|
1616 | if(size(l)>0) |
---|
1617 | { |
---|
1618 | j=1; |
---|
1619 | while(j<size(l)&&l[j]!=l0[i]) |
---|
1620 | { |
---|
1621 | j++; |
---|
1622 | } |
---|
1623 | if(l[j]==l0[i]) |
---|
1624 | { |
---|
1625 | l[j][size(sp)]=k; |
---|
1626 | } |
---|
1627 | else |
---|
1628 | { |
---|
1629 | l0[i][size(sp)]=k; |
---|
1630 | l=l+list(l0[i]); |
---|
1631 | } |
---|
1632 | } |
---|
1633 | else |
---|
1634 | { |
---|
1635 | l=l0; |
---|
1636 | } |
---|
1637 | } |
---|
1638 | } |
---|
1639 | sp0=spsub(sp0,sp[size(sp)]); |
---|
1640 | k++; |
---|
1641 | } |
---|
1642 | if(size(sp)>1) |
---|
1643 | { |
---|
1644 | return(l); |
---|
1645 | } |
---|
1646 | else |
---|
1647 | { |
---|
1648 | return(list(intvec(k-1))); |
---|
1649 | } |
---|
1650 | } |
---|
1651 | example |
---|
1652 | { "EXAMPLE:"; echo=2; |
---|
1653 | ring R=0,(x,y),ds; |
---|
1654 | 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)); |
---|
1655 | spprint(sp0); |
---|
1656 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1657 | spprint(sp1); |
---|
1658 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1659 | spprint(sp2); |
---|
1660 | list sp=sp1,sp2; |
---|
1661 | list l=spsemicont(sp0,sp); |
---|
1662 | l; |
---|
1663 | spissemicont(spsub(sp0,spmul(sp,l[1]))); |
---|
1664 | spissemicont(spsub(sp0,spmul(sp,l[1]-1))); |
---|
1665 | spissemicont(spsub(sp0,spmul(sp,l[1]+1))); |
---|
1666 | } |
---|
1667 | /////////////////////////////////////////////////////////////////////////////// |
---|
1668 | |
---|
1669 | proc spmilnor(list sp) |
---|
1670 | "USAGE: spmilnor(sp); list sp |
---|
1671 | RETURN: int mu; Milnor number of spectrum sp |
---|
1672 | EXAMPLE: example spmilnor; shows examples |
---|
1673 | " |
---|
1674 | { |
---|
1675 | return(sum(sp[2])); |
---|
1676 | } |
---|
1677 | example |
---|
1678 | { "EXAMPLE:"; echo=2; |
---|
1679 | ring R=0,(x,y),ds; |
---|
1680 | 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)); |
---|
1681 | spprint(sp); |
---|
1682 | spmilnor(sp); |
---|
1683 | } |
---|
1684 | /////////////////////////////////////////////////////////////////////////////// |
---|
1685 | |
---|
1686 | proc spgeomgenus(list sp) |
---|
1687 | "USAGE: spgeomgenus(sp); list sp |
---|
1688 | RETURN: int g; geometrical genus of spectrum sp |
---|
1689 | EXAMPLE: example spgeomgenus; shows examples |
---|
1690 | " |
---|
1691 | { |
---|
1692 | int g=0; |
---|
1693 | int i=1; |
---|
1694 | while(i+1<=size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1695 | { |
---|
1696 | g=g+sp[2][i]; |
---|
1697 | i++; |
---|
1698 | } |
---|
1699 | if(i==size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
1700 | { |
---|
1701 | g=g+sp[2][i]; |
---|
1702 | } |
---|
1703 | return(g); |
---|
1704 | } |
---|
1705 | example |
---|
1706 | { "EXAMPLE:"; echo=2; |
---|
1707 | ring R=0,(x,y),ds; |
---|
1708 | 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)); |
---|
1709 | spprint(sp); |
---|
1710 | spgeomgenus(sp); |
---|
1711 | } |
---|
1712 | /////////////////////////////////////////////////////////////////////////////// |
---|
1713 | |
---|
1714 | proc spgamma(list sp) |
---|
1715 | "USAGE: spgamma(sp); list sp |
---|
1716 | RETURN: number gamma; gamma invariant of spectrum sp |
---|
1717 | EXAMPLE: example spgamma; shows examples |
---|
1718 | " |
---|
1719 | { |
---|
1720 | int i,j; |
---|
1721 | number g=0; |
---|
1722 | for(i=1;i<=ncols(sp[1]);i++) |
---|
1723 | { |
---|
1724 | for(j=1;j<=sp[2][i];j++) |
---|
1725 | { |
---|
1726 | g=g+(number(sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1727 | } |
---|
1728 | } |
---|
1729 | g=-g/4+sum(sp[2])*number(sp[1][ncols(sp[1])]-sp[1][1])/48; |
---|
1730 | return(g); |
---|
1731 | } |
---|
1732 | example |
---|
1733 | { "EXAMPLE:"; echo=2; |
---|
1734 | ring R=0,(x,y),ds; |
---|
1735 | 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)); |
---|
1736 | spprint(sp); |
---|
1737 | spgamma(sp); |
---|
1738 | } |
---|
1739 | /////////////////////////////////////////////////////////////////////////////// |
---|