1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | |
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3 | version="$Id: mondromy.lib,v 1.3 1999-06-22 15:38:54 mschulze Exp $"; |
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4 | info=" |
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5 | LIBRARY: mondromy.lib PROCEDURES TO COMPUTE THE MONODROMY OF A SINGULARITY |
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6 | AUTHOR: Mathias Schulze, email: mschulze@mathematik.uni-kl.de |
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7 | |
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8 | invunit(u,n); series inverse of polynomial u up to order n |
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9 | detadj(U); determinant and adjoint matrix of square matrix U |
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10 | jacoblift(f); lifts f^kappa in jacob(f) with minimal kappa |
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11 | monodromy(f[,opt]); monodromy of isolated hypersurface singularity f |
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12 | H''basis(f); basis of Brieskorn lattice H'' |
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13 | "; |
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14 | |
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15 | LIB "ring.lib"; |
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16 | LIB "sing.lib"; |
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17 | LIB "jordan.lib"; |
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18 | /////////////////////////////////////////////////////////////////////////////// |
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19 | |
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20 | static proc pcvladdl(list l1,list l2) |
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21 | { |
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22 | return(system("pcvLAddL",l1,l2)); |
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23 | } |
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24 | |
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25 | static proc pcvpmull(poly p,list l) |
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26 | { |
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27 | return(system("pcvPMulL",p,l)); |
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28 | } |
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29 | |
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30 | static proc pcvmindeg(list #) |
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31 | { |
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32 | return(system("pcvMinDeg",#[1])); |
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33 | } |
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34 | |
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35 | static proc pcvp2cv(list l,int i0,int i1) |
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36 | { |
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37 | return(system("pcvP2CV",l,i0,i1)); |
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38 | } |
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39 | |
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40 | static proc pcvcv2p(list l,int i0,int i1) |
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41 | { |
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42 | return(system("pcvCV2P",l,i0,i1)); |
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43 | } |
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44 | |
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45 | static proc pcvdim(int i0,int i1) |
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46 | { |
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47 | return(system("pcvDim",i0,i1)); |
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48 | } |
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49 | |
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50 | static proc pcvbasis(int i0,int i1) |
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51 | { |
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52 | return(system("pcvBasis",i0,i1)); |
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53 | } |
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54 | |
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55 | static proc pcvinit() |
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56 | { |
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57 | if(system("with","DynamicLoading")) |
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58 | { |
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59 | pcvlal=Pcv::LAddL; |
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60 | pcvpmull=Pcv::PMulL; |
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61 | pcvmindeg=Pcv::MinDeg; |
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62 | pcvp2cv=Pcv::P2CV; |
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63 | pcvcv2p=Pcv::CV2P; |
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64 | pcvdim=Pcv::Dim; |
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65 | pcvbasis=Pcv::Basis; |
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66 | } |
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67 | } |
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68 | |
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69 | /////////////////////////////////////////////////////////////////////////////// |
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70 | |
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71 | static proc min(intvec v) |
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72 | { |
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73 | int m=v[1]; |
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74 | int i; |
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75 | for(i=2;i<=size(v);i++) |
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76 | { |
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77 | if(m>v[i]) |
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78 | { |
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79 | m=v[i]; |
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80 | } |
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81 | } |
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82 | return(m); |
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83 | } |
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84 | /////////////////////////////////////////////////////////////////////////////// |
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85 | |
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86 | static proc max(intvec v) |
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87 | { |
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88 | int m=v[1]; |
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89 | int i; |
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90 | for(i=2;i<=size(v);i++) |
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91 | { |
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92 | if(m<v[i]) |
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93 | { |
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94 | m=v[i]; |
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95 | } |
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96 | } |
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97 | return(m); |
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98 | } |
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99 | /////////////////////////////////////////////////////////////////////////////// |
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100 | |
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101 | static proc mdivp(matrix m,poly p) |
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102 | { |
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103 | int i,j; |
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104 | for(i=nrows(m);i>=1;i--) |
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105 | { |
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106 | for(j=ncols(m);j>=1;j--) |
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107 | { |
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108 | m[i,j]=m[i,j]/p; |
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109 | } |
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110 | } |
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111 | return(m); |
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112 | } |
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113 | /////////////////////////////////////////////////////////////////////////////// |
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114 | |
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115 | proc codimV(list V,int N) |
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116 | { |
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117 | int codim=pcvdim(0,N); |
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118 | if(size(V)>0) |
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119 | { |
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120 | dbprint(printlevel-voice+2,"//vector space dimension: "+string(codim)); |
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121 | dbprint(printlevel-voice+2, |
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122 | "//number of subspace generators: "+string(size(V))); |
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123 | int t=timer; |
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124 | codim=codim-ncols(interred(module(V[1..size(V)]))); |
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125 | dbprint(printlevel-voice+2,"//codimension: "+string(codim)); |
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126 | } |
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127 | return(codim); |
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128 | } |
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129 | /////////////////////////////////////////////////////////////////////////////// |
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130 | |
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131 | proc quotV(list V,int N) |
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132 | { |
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133 | module Q=freemodule(pcvdim(0,N)); |
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134 | if(size(V)>0) |
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135 | { |
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136 | dbprint(printlevel-voice+2,"//vector space dimension: "+string(nrows(Q))); |
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137 | dbprint(printlevel-voice+2, |
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138 | "//number of subspace generators: "+string(size(V))); |
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139 | int t=timer; |
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140 | Q=interred(reduce(std(Q),std(module(V[1..size(V)])))); |
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141 | } |
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142 | return(list(Q[1..size(Q)])); |
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143 | } |
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144 | /////////////////////////////////////////////////////////////////////////////// |
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145 | |
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146 | proc invunit(poly u,int n) |
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147 | "USAGE: invunit(u,n); u poly, n int |
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148 | ASSUME: The polynomial u is a series unit. |
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149 | RETURN: The procedure returns the series inverse of u up to order n |
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150 | or a zero polynomial if u is no series unit. |
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151 | DISPLAY: The procedure displays comments if printlevel>=1. |
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152 | EXAMPLE: example invunit; shows an example. |
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153 | " |
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154 | { |
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155 | if(pcvmindeg(u)==0) |
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156 | { |
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157 | def br=basering; |
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158 | changeord("pr","dp"); |
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159 | qring qr=std(maxideal(n+1)); |
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160 | kill pr; |
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161 | |
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162 | dbprint(printlevel-voice+2,"//computing inverse..."); |
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163 | int t=timer; |
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164 | poly v=lift(fetch(br,u),1)[1,1]; |
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165 | dbprint(printlevel-voice+2,"//...inverse computed ["+string(timer-t)+ |
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166 | " secs, "+string(memory(1))+" bytes]"); |
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167 | |
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168 | setring br; |
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169 | return(fetch(qr,v)); |
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170 | } |
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171 | else |
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172 | { |
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173 | print("//no series unit"); |
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174 | return(poly(0)); |
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175 | } |
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176 | } |
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177 | example |
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178 | { "EXAMPLE:"; echo=2; |
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179 | ring R=0,x,dp; |
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180 | invunit(1+x,10); |
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181 | } |
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182 | /////////////////////////////////////////////////////////////////////////////// |
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183 | |
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184 | proc detadj(module U) |
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185 | "USAGE: detadj(U); U matrix |
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186 | ASSUME: U is a square matrix with non zero determinant. |
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187 | RETURN: The procedure returns a list with at most 2 entries. |
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188 | If U is not a sqaure matrix, the list is empty. |
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189 | If U is a sqaure matrix, then the first entry is the determinant of U. |
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190 | If U is a square matrix and the determinant of U not zero, |
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191 | then the second entry is the adjoint matrix of U. |
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192 | DISPLAY: The procedure displays comments if printlevel>=1. |
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193 | EXAMPLE: example detadj; shows an example. |
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194 | " |
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195 | { |
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196 | if(nrows(U)==ncols(U)) |
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197 | { |
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198 | dbprint(printlevel-voice+2,"//computing determinant..."); |
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199 | int t=timer; |
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200 | poly detU=det(U); |
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201 | dbprint(printlevel-voice+2,"//...determinant computed ["+string(timer-t)+ |
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202 | " secs, "+string(memory(1))+" bytes]"); |
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203 | |
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204 | if(detU==0) |
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205 | { |
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206 | print("//determinant zero"); |
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207 | return(list(detU)); |
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208 | } |
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209 | else |
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210 | { |
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211 | def br=basering; |
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212 | changeord("pr","dp"); |
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213 | matrix U=fetch(br,U); |
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214 | poly detU=fetch(br,detU); |
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215 | |
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216 | dbprint(printlevel-voice+2,"//computing adjoint matrix..."); |
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217 | t=timer; |
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218 | matrix adjU=lift(U,detU*freemodule(nrows(U))); |
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219 | dbprint(printlevel-voice+2,"//...adjoint matrix computed [" |
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220 | +string(timer-t)+" secs, "+string(memory(1))+" bytes]"); |
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221 | |
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222 | setring br; |
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223 | matrix adjU=fetch(pr,adjU); |
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224 | kill pr; |
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225 | } |
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226 | } |
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227 | else |
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228 | { |
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229 | print("//no square matrix"); |
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230 | return(list()); |
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231 | } |
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232 | |
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233 | return(list(detU,adjU)); |
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234 | } |
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235 | example |
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236 | { "EXAMPLE:"; echo=2; |
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237 | ring R=0,x,dp; |
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238 | matrix U[2][2]=1,1+x,1+x2,1+x3; |
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239 | list daU=detadj(U); |
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240 | daU[1]; |
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241 | print(daU[2]); |
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242 | } |
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243 | /////////////////////////////////////////////////////////////////////////////// |
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244 | |
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245 | proc jacoblift(poly f) |
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246 | "USAGE: jacoblift(f); f poly |
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247 | ASSUME: The polynomial f in a series ring (local ordering) defines |
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248 | an isolated hypersurface singularity. |
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249 | RETURN: The procedure returns a list with entries kappa, xi, u of type |
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250 | int, vector, poly such that kappa is minimal with f^kappa in jacob(f), |
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251 | u is a unit, and u*f^kappa=(matrix(jacob(f))*xi)[1,1]. |
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252 | DISPLAY: The procedure displays comments if printlevel>=1. |
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253 | EXAMPLE: example jacoblift; shows an example. |
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254 | " |
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255 | { |
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256 | dbprint(printlevel-voice+2,"//computing kappa..."); |
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257 | int t=timer; |
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258 | ideal jf=jacob(f); |
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259 | ideal sjf=std(jf); |
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260 | int kappa=1; |
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261 | poly fkappa=f; |
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262 | while(reduce(fkappa,sjf)!=0) |
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263 | { |
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264 | dbprint(printlevel-voice+2,"//kappa="+string(kappa)); |
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265 | kappa++; |
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266 | fkappa=fkappa*f; |
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267 | } |
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268 | dbprint(printlevel-voice+2,"//kappa="+string(kappa)); |
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269 | dbprint(printlevel-voice+2,"//...kappa computed ["+string(timer-t)+" secs, " |
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270 | +string(memory(1))+" bytes]"); |
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271 | |
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272 | dbprint(printlevel-voice+2,"//computing xi..."); |
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273 | t=timer; |
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274 | vector xi=lift(jf,fkappa)[1]; |
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275 | dbprint(printlevel-voice+2,"//...xi computed ["+string(timer-t)+" secs, " |
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276 | +string(memory(1))+" bytes]"); |
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277 | |
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278 | dbprint(printlevel-voice+2,"//computing u..."); |
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279 | t=timer; |
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280 | poly u=(matrix(jf)*xi)[1,1]/fkappa; |
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281 | dbprint(printlevel-voice+2,"//...u computed ["+string(timer-t)+" secs, " |
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282 | +string(memory(1))+" bytes]"); |
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283 | |
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284 | return(list(kappa,xi,u)); |
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285 | } |
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286 | example |
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287 | { "EXAMPLE:"; echo=2; |
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288 | ring R=0,(x,y),ds; |
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289 | poly f=x2y2+x6+y6; |
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290 | jacoblift(f); |
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291 | } |
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292 | /////////////////////////////////////////////////////////////////////////////// |
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293 | |
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294 | static proc getdeltaP1(poly f,int K,int N,int dN) |
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295 | { |
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296 | return(pcvpmull(f^K,pcvbasis(0,N+dN-K*pcvmindeg(f)))); |
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297 | } |
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298 | /////////////////////////////////////////////////////////////////////////////// |
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299 | |
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300 | static proc getdeltaP2(poly f,int N,int dN) |
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301 | { |
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302 | def of,jf=pcvmindeg(f),jacob(f); |
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303 | list b=pcvbasis(N-of+2,N+dN-of+2); |
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304 | list P2; |
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305 | P2[size(b)*((nvars(basering)-1)*nvars(basering))/2]=0; |
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306 | int i,j,k,l; |
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307 | intvec alpha; |
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308 | for(k,l=1,1;k<=size(b);k++) |
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309 | { |
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310 | alpha=leadexp(b[k]); |
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311 | for(i=nvars(basering)-1;i>=1;i--) |
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312 | { |
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313 | for(j=nvars(basering);j>i;j--) |
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314 | { |
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315 | P2[l]=alpha[i]*jf[j]*(b[k]/var(i))-alpha[j]*jf[i]*(b[k]/var(j)); |
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316 | l++; |
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317 | } |
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318 | } |
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319 | } |
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320 | return(P2); |
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321 | } |
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322 | /////////////////////////////////////////////////////////////////////////////// |
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323 | |
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324 | static proc getdeltaPe(poly f,list e,int K,int dK) |
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325 | { |
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326 | int k; |
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327 | list Pe,fke; |
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328 | for(k,fke=K,pcvpmull(f^K,e);k<K+dK;k,fke=k+1,pcvpmull(f,fke)) |
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329 | { |
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330 | Pe=Pe+fke; |
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331 | } |
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332 | return(Pe); |
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333 | } |
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334 | /////////////////////////////////////////////////////////////////////////////// |
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335 | |
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336 | static proc incK(poly f,int mu,int K,int deltaK,int N, |
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337 | list e,list P1,list P2,list Pe,list V1,list V2,list Ve) |
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338 | { |
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339 | int deltaN=deltaK*pcvmindeg(f); |
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340 | |
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341 | list deltaP1; |
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342 | P1=pcvpmull(f^deltaK,P1); |
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343 | V1=pcvp2cv(P1,0,N+deltaN); |
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344 | |
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345 | list deltaP2=getdeltaP2(f,N,deltaN); |
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346 | V2=pcvladdl(V2,pcvp2cv(P2,N,N+deltaN))+pcvp2cv(deltaP2,0,N+deltaN); |
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347 | P2=P2+deltaP2; |
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348 | |
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349 | list deltaPe=getdeltaPe(f,e,K,deltaK); |
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350 | Ve=pcvladdl(Ve,pcvp2cv(Pe,N,N+deltaN))+pcvp2cv(deltaPe,0,N+deltaN); |
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351 | Pe=Pe+deltaPe; |
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352 | |
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353 | K=K+deltaK; |
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354 | dbprint(printlevel-voice+2,"//K="+string(K)); |
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355 | |
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356 | N=N+deltaN; |
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357 | dbprint(printlevel-voice+2,"//N="+string(N)); |
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358 | |
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359 | deltaN=1; |
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360 | dbprint(printlevel-voice+2,"//computing codimension of"); |
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361 | dbprint(printlevel-voice+2,"//df^dOmega^(n-1)+f^K*Omega^(n+1) in " |
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362 | +"Omega^(n+1) mod m^N*Omega^(n+1)..."); |
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363 | int t=timer; |
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364 | while(codimV(V1+V2,N)<K*mu) |
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365 | { |
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366 | dbprint(printlevel-voice+2,"//...codimension computed ["+string(timer-t) |
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367 | +" secs, "+string(memory(1))+" bytes]"); |
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368 | |
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369 | deltaP1=getdeltaP1(f,K,N,deltaN); |
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370 | V1=pcvladdl(V1,pcvp2cv(P1,N,N+deltaN))+pcvp2cv(deltaP1,0,N+deltaN); |
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371 | P1=P1+deltaP1; |
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372 | |
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373 | deltaP2=getdeltaP2(f,N,deltaN); |
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374 | V2=pcvladdl(V2,pcvp2cv(P2,N,N+deltaN))+pcvp2cv(deltaP2,0,N+deltaN); |
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375 | P2=P2+deltaP2; |
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376 | |
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377 | Ve=pcvladdl(Ve,pcvp2cv(Pe,N,N+deltaN)); |
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378 | |
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379 | N=N+deltaN; |
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380 | dbprint(printlevel-voice+2,"//N="+string(N)); |
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381 | |
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382 | dbprint(printlevel-voice+2,"//computing codimension of"); |
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383 | dbprint(printlevel-voice+2,"//df^dOmega^(n-1)+f^K*Omega^(n+1) in " |
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384 | +"Omega^(n+1) mod m^N*Omega^(n+1)..."); |
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385 | t=timer; |
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386 | } |
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387 | dbprint(printlevel-voice+2,"//...codimension computed ["+string(timer-t) |
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388 | +" secs, "+string(memory(1))+" bytes]"); |
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389 | |
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390 | return(K,N,P1,P2,Pe,V1,V2,Ve); |
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391 | } |
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392 | /////////////////////////////////////////////////////////////////////////////// |
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393 | |
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394 | static proc nablaK(poly f,int kappa,vector xi,poly u,int N,int prevN, |
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395 | list Vnablae,list e) |
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396 | { |
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397 | xi=jet(xi,N); |
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398 | u=invunit(u,N); |
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399 | poly fkappa=kappa*f^(kappa-1); |
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400 | |
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401 | poly p,q; |
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402 | list nablae; |
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403 | int i,j; |
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404 | for(i=1;i<=size(e);i++) |
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405 | { |
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406 | for(j,p=nvars(basering),0;j>=1;j--) |
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407 | { |
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408 | q=jet(e[i]*xi[j],N); |
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409 | if(q!=0) |
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410 | { |
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411 | p=p+diff(q*jet(u,N-pcvmindeg(q)),var(j)); |
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412 | } |
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413 | } |
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414 | nablae=nablae+list(p-jet(fkappa*e[i],N-1)); |
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415 | } |
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416 | |
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417 | return(pcvladdl(Vnablae,pcvp2cv(nablae,prevN,N-prevN))); |
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418 | } |
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419 | /////////////////////////////////////////////////////////////////////////////// |
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420 | |
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421 | static proc MK(poly f,int mu,int kappa,vector xi,poly u, |
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422 | int K,int N,int prevN,list e,list V1,list V2,list Ve,list Vnablae) |
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423 | { |
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424 | dbprint(printlevel-voice+2,"//computing nabla(e)..."); |
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425 | int t=timer; |
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426 | Vnablae=nablaK(f,kappa,xi,u,N,prevN,Vnablae,e); |
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427 | dbprint(printlevel-voice+2,"//...nabla(e) computed ["+string(timer-t) |
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428 | +" secs, "+string(memory(1))+" bytes]"); |
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429 | |
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430 | dbprint(printlevel-voice+2, |
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431 | "//lifting nabla(e) to C-basis of H''/t^KH''..."); |
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432 | list V=Ve+V1+V2; |
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433 | module W=module(V[1..size(V)]); |
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434 | dbprint(printlevel-voice+2,"//vector space dimension: "+string(nrows(W))); |
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435 | dbprint(printlevel-voice+2,"//number of generators: "+string(ncols(W))); |
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436 | t=timer; |
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437 | matrix C=lift(W,module(Vnablae[1..size(Vnablae)])); |
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438 | dbprint(printlevel-voice+2,"//...nabla(e) lifted ["+string(timer-t) |
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439 | +" secs, "+string(memory(1))+" bytes]"); |
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440 | |
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441 | dbprint(printlevel-voice+2,"//computing e-lift of nabla(e)..."); |
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442 | t=timer; |
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443 | int i1,i2,j,k; |
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444 | matrix M[mu][mu]; |
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445 | for(j=1;j<=mu;j++) |
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446 | { |
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447 | for(k,i2=0,1;k<K;k++) |
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448 | { |
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449 | for(i1=1;i1<=mu;i1,i2=i1+1,i2+1) |
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450 | { |
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451 | M[i1,j]=M[i1,j]+C[i2,j]*var(1)^k; |
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452 | } |
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453 | } |
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454 | } |
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455 | dbprint(printlevel-voice+2,"//...e-lift of nabla(e) computed [" |
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456 | +string(timer-t)+" secs, "+string(memory(1))+" bytes]"); |
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457 | |
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458 | return(M,N,Vnablae); |
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459 | } |
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460 | /////////////////////////////////////////////////////////////////////////////// |
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461 | |
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462 | static proc mid(ideal l) |
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463 | { |
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464 | int i,j,id; |
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465 | int mid=0; |
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466 | for(i=size(l);i>=1;i--) |
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467 | { |
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468 | for(j=i-1;j>=1;j--) |
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469 | { |
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470 | id=int(l[i]-l[j]); |
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471 | id=max(intvec(id,-id)); |
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472 | mid=max(intvec(id,mid)); |
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473 | } |
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474 | } |
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475 | return(mid); |
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476 | } |
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477 | /////////////////////////////////////////////////////////////////////////////// |
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478 | |
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479 | static proc decmide(matrix M,ideal eM0,list bM0) |
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480 | { |
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481 | matrix M0=jet(M,0); |
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482 | |
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483 | dbprint(printlevel-voice+2, |
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484 | "//computing basis U of generalized eigenspaces of M0..."); |
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485 | int t=timer; |
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486 | int i,j; |
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487 | matrix U,M0e; |
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488 | matrix E=freemodule(nrows(M)); |
---|
489 | for(i=ncols(eM0);i>=1;i--) |
---|
490 | { |
---|
491 | M0e=E; |
---|
492 | for(j=max(bM0[i]);j>=1;j--) |
---|
493 | { |
---|
494 | M0e=M0e*(M0-eM0[i]*E); |
---|
495 | } |
---|
496 | U=syz(M0e)+U; |
---|
497 | } |
---|
498 | dbprint(printlevel-voice+2,"//...U computed ["+string(timer-t)+" secs, " |
---|
499 | +string(memory(1))+" bytes]"); |
---|
500 | |
---|
501 | dbprint(printlevel-voice+2,"//transforming M to U..."); |
---|
502 | t=timer; |
---|
503 | list daU=detadj(U); |
---|
504 | daU[2]=(1/number(daU[1]))*daU[2]; |
---|
505 | M=daU[2]*M*U; |
---|
506 | dbprint(printlevel-voice+2,"//...M transformed ["+string(timer-t)+" secs, " |
---|
507 | +string(memory(1))+" bytes]"); |
---|
508 | |
---|
509 | dbprint(printlevel-voice+2, |
---|
510 | "//computing integer differences of eigenvalues of M0..."); |
---|
511 | t=timer; |
---|
512 | int k; |
---|
513 | intvec ideM0; |
---|
514 | ideM0[ncols(eM0)]=0; |
---|
515 | for(i=ncols(eM0);i>=1;i--) |
---|
516 | { |
---|
517 | for(j=ncols(eM0);j>=1;j--) |
---|
518 | { |
---|
519 | k=int(eM0[i]-eM0[j]); |
---|
520 | if(k) |
---|
521 | { |
---|
522 | if(k>0) |
---|
523 | { |
---|
524 | ideM0[i]=max(intvec(k,ideM0[i])); |
---|
525 | } |
---|
526 | else |
---|
527 | { |
---|
528 | ideM0[j]=max(intvec(-k,ideM0[j])); |
---|
529 | } |
---|
530 | } |
---|
531 | } |
---|
532 | } |
---|
533 | for(i,k=size(bM0),nrows(M);i>=1;i--) |
---|
534 | { |
---|
535 | for(j=sum(bM0[i]);j>=1;j--) |
---|
536 | { |
---|
537 | ideM0[k]=ideM0[i]; |
---|
538 | k--; |
---|
539 | } |
---|
540 | } |
---|
541 | dbprint(printlevel-voice+2, |
---|
542 | "//...integer differences of eigenvalues of M0 computed ["+string(timer-t) |
---|
543 | +" secs, "+string(memory(1))+" bytes]"); |
---|
544 | |
---|
545 | dbprint(printlevel-voice+2,"//transforming M..."); |
---|
546 | t=timer; |
---|
547 | for(i=nrows(M);i>=1;i--) |
---|
548 | { |
---|
549 | if(!ideM0[i]) |
---|
550 | { |
---|
551 | M[i,i]=M[i,i]+1; |
---|
552 | } |
---|
553 | for(j=ncols(M);j>=1;j--) |
---|
554 | { |
---|
555 | if(ideM0[i]&&!ideM0[j]) |
---|
556 | { |
---|
557 | M[i,j]=M[i,j]*var(1); |
---|
558 | } |
---|
559 | else |
---|
560 | { |
---|
561 | if(!ideM0[i]&&ideM0[j]) |
---|
562 | { |
---|
563 | M[i,j]=M[i,j]/var(1); |
---|
564 | } |
---|
565 | } |
---|
566 | } |
---|
567 | } |
---|
568 | dbprint(printlevel-voice+2,"//...M transformed ["+string(timer-t)+" secs, " |
---|
569 | +string(memory(1))+" bytes]"); |
---|
570 | |
---|
571 | return(M); |
---|
572 | } |
---|
573 | /////////////////////////////////////////////////////////////////////////////// |
---|
574 | |
---|
575 | static proc nonqhmonodromy(poly f,int mu,int opt) |
---|
576 | { |
---|
577 | pcvinit(); |
---|
578 | |
---|
579 | dbprint(printlevel-voice+2,"//computing kappa, xi and u with "+ |
---|
580 | "u*f^kappa=(matrix(jacob(f))*xi)[1,1]..."); |
---|
581 | list jl=jacoblift(f); |
---|
582 | def kappa,xi,u=jl[1..3]; |
---|
583 | dbprint(printlevel-voice+2,"//...kappa, xi and u computed"); |
---|
584 | dbprint(printlevel-voice+2,"//kappa="+string(kappa)); |
---|
585 | if(kappa==1) |
---|
586 | { |
---|
587 | dbprint(printlevel-voice+2, |
---|
588 | "//f quasihomogenous with respect to suitable coordinates"); |
---|
589 | } |
---|
590 | else |
---|
591 | { |
---|
592 | dbprint(printlevel-voice+2, |
---|
593 | "//f not quasihomogenous for any choice of coordinates"); |
---|
594 | } |
---|
595 | dbprint(printlevel-voice+2,"//xi="); |
---|
596 | dbprint(printlevel-voice+2,xi); |
---|
597 | dbprint(printlevel-voice+2,"//u="+string(u)); |
---|
598 | |
---|
599 | int K,N,prevN; |
---|
600 | list e,P1,P2,Pe,V1,V2,Ve,Vnablae; |
---|
601 | |
---|
602 | dbprint(printlevel-voice+2,"//increasing K and N..."); |
---|
603 | K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,1,N,e,P1,P2,Pe,V1,V2,Ve); |
---|
604 | dbprint(printlevel-voice+2,"//...K and N increased"); |
---|
605 | |
---|
606 | dbprint(printlevel-voice+2,"//computing C{f}-basis e of Brieskorn lattice " |
---|
607 | +"H''=Omega^(n+1)/df^dOmega^(n-1)..."); |
---|
608 | int t=timer; |
---|
609 | e=pcvcv2p(quotV(V1+V2,N),0,N); |
---|
610 | dbprint(printlevel-voice+2,"//...e computed ["+string(timer-t)+" secs, " |
---|
611 | +string(memory(1))+" bytes]"); |
---|
612 | |
---|
613 | dbprint(printlevel-voice+2,"//e="); |
---|
614 | dbprint(printlevel-voice+2,e); |
---|
615 | |
---|
616 | Pe=e; |
---|
617 | Ve=pcvp2cv(Pe,0,N); |
---|
618 | |
---|
619 | if(kappa==1) |
---|
620 | { |
---|
621 | dbprint(printlevel-voice+2, |
---|
622 | "//computing 0-jet M of e-matrix of t*nabla..."); |
---|
623 | matrix M=list(MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae))[1]; |
---|
624 | dbprint(printlevel-voice+2,"//...M computed"); |
---|
625 | } |
---|
626 | else |
---|
627 | { |
---|
628 | dbprint(printlevel-voice+2, |
---|
629 | "//computing transformation matrix U to simple pole..."); |
---|
630 | |
---|
631 | dbprint(printlevel-voice+2,"//computing t*nabla-stable lattice..."); |
---|
632 | matrix M,prevU; |
---|
633 | matrix U=freemodule(mu)*var(1)^((mu-1)*(kappa-1)); |
---|
634 | int i; |
---|
635 | dbprint(printlevel-voice+2,"//comparing with previous lattice..."); |
---|
636 | t=timer; |
---|
637 | for(i=mu-1;i>=1&&size(reduce(U,std(prevU)))>0;i--) |
---|
638 | { |
---|
639 | dbprint(printlevel-voice+2,"//...compared with previous lattice [" |
---|
640 | +string(timer-t)+" secs, "+string(memory(1))+" bytes]"); |
---|
641 | |
---|
642 | dbprint(printlevel-voice+2,"//increasing K and N..."); |
---|
643 | K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,kappa-1,N,e,P1,P2,Pe,V1,V2,Ve); |
---|
644 | dbprint(printlevel-voice+2,"//...K and N increased"); |
---|
645 | |
---|
646 | dbprint(printlevel-voice+2, |
---|
647 | "//computing (K-1)-jet M of e-matrix of t^kappa*nabla..."); |
---|
648 | M,prevN,Vnablae=MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae); |
---|
649 | dbprint(printlevel-voice+2,"//...M computed"); |
---|
650 | |
---|
651 | prevU=U; |
---|
652 | |
---|
653 | dbprint(printlevel-voice+2,"//enlarging lattice..."); |
---|
654 | t=timer; |
---|
655 | U=interred(jet(module(U)+module(var(1)*diff(U,var(1)))+ |
---|
656 | module(mdivp(M*U,var(1)^(kappa-1))),(kappa-1)*(mu-1))); |
---|
657 | dbprint(printlevel-voice+2,"//...lattice enlarged ["+string(timer-t) |
---|
658 | +" secs, "+string(memory(1))+" bytes]"); |
---|
659 | |
---|
660 | dbprint(printlevel-voice+2,"//comparing with previous lattice..."); |
---|
661 | t=timer; |
---|
662 | } |
---|
663 | dbprint(printlevel-voice+2,"//...compared with previous lattice [" |
---|
664 | +string(timer-t)+" secs, "+string(memory(1))+" bytes]"); |
---|
665 | dbprint(printlevel-voice+2,"//...t*nabla-stable lattice computed"); |
---|
666 | |
---|
667 | if(ncols(U)>nrows(U)) |
---|
668 | { |
---|
669 | dbprint(printlevel-voice+2, |
---|
670 | "//computing C{f}-basis of t*nabla-stable lattice..."); |
---|
671 | t=timer; |
---|
672 | U=minbase(U); |
---|
673 | dbprint(printlevel-voice+2, |
---|
674 | "//...C{f}-basis of t*nabla-stable lattice computed ["+string(timer-t) |
---|
675 | +" secs, "+string(memory(1))+" bytes]"); |
---|
676 | } |
---|
677 | |
---|
678 | U=mdivp(U,var(1)^pcvmindeg(U)); |
---|
679 | |
---|
680 | dbprint(printlevel-voice+2,"//...U computed"); |
---|
681 | |
---|
682 | dbprint(printlevel-voice+2, |
---|
683 | "//computing determinant and adjoint matrix of U..."); |
---|
684 | list daU=detadj(U); |
---|
685 | poly p=var(1)^min(intvec(pcvmindeg(daU[2]),pcvmindeg(daU[1]))); |
---|
686 | daU[1]=daU[1]/p; |
---|
687 | daU[2]=mdivp(daU[2],p); |
---|
688 | dbprint(printlevel-voice+2, |
---|
689 | "//...determinant and adjoint matrix of U computed"); |
---|
690 | |
---|
691 | if(K<kappa+pcvmindeg(daU[1])) |
---|
692 | { |
---|
693 | dbprint(printlevel-voice+2,"//increasing K and N..."); |
---|
694 | K,N,P1,P2,Pe,V1,V2,Ve= |
---|
695 | incK(f,mu,K,kappa+pcvmindeg(daU[1])-K,N,e,P1,P2,Pe,V1,V2,Ve); |
---|
696 | dbprint(printlevel-voice+2,"//...K and N increased"); |
---|
697 | |
---|
698 | dbprint(printlevel-voice+2,"//computing M..."); |
---|
699 | M,prevN,Vnablae=MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae); |
---|
700 | dbprint(printlevel-voice+2,"//...M computed"); |
---|
701 | } |
---|
702 | |
---|
703 | dbprint(printlevel-voice+2,"//transforming M/t^kappa to simple pole..."); |
---|
704 | t=timer; |
---|
705 | M=mdivp(daU[2]*(var(1)^kappa*diff(U,var(1))+M*U), |
---|
706 | leadcoef(daU[1])*var(1)^(kappa+pcvmindeg(daU[1])-1)); |
---|
707 | dbprint(printlevel-voice+2,"//...M/t^kappa transformed to simple pole [" |
---|
708 | +string(timer-t)+" secs, "+string(memory(1))+" bytes]"); |
---|
709 | } |
---|
710 | |
---|
711 | if(opt==0) |
---|
712 | { |
---|
713 | dbprint(printlevel-voice+2, |
---|
714 | "//...computing maximal integer difference delta of eigenvalues of M0"); |
---|
715 | t=timer; |
---|
716 | list jd=jordan(M); |
---|
717 | def eM0,bM0=jd[1..2]; |
---|
718 | int delta=mid(eM0); |
---|
719 | dbprint(printlevel-voice+2,"//...delta computed ["+string(timer-t) |
---|
720 | +" secs, "+string(memory(1))+" bytes]"); |
---|
721 | |
---|
722 | dbprint(printlevel-voice+2,"//delta="+string(delta)); |
---|
723 | |
---|
724 | if(delta>0) |
---|
725 | { |
---|
726 | dbprint(printlevel-voice+2,"//increasing K and N..."); |
---|
727 | if(kappa==1) |
---|
728 | { |
---|
729 | K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,1+delta-K,N,e,P1,P2,Pe,V1,V2,Ve); |
---|
730 | } |
---|
731 | else |
---|
732 | { |
---|
733 | K,N,P1,P2,Pe,V1,V2,Ve= |
---|
734 | incK(f,mu,K,kappa+pcvmindeg(daU[1])+delta-K,N,e,P1,P2,Pe,V1,V2,Ve); |
---|
735 | } |
---|
736 | dbprint(printlevel-voice+2,"//...K and N increased"); |
---|
737 | |
---|
738 | dbprint(printlevel-voice+2,"//computing M..."); |
---|
739 | M,prevN,Vnablae=MK(f,mu,kappa,xi,u,K,N,prevN,e,V1,V2,Ve,Vnablae); |
---|
740 | dbprint(printlevel-voice+2,"//...M computed"); |
---|
741 | |
---|
742 | if(kappa>1) |
---|
743 | { |
---|
744 | dbprint(printlevel-voice+2, |
---|
745 | "//transforming M/t^kappa to simple pole..."); |
---|
746 | t=timer; |
---|
747 | M=mdivp(invunit(daU[1]/var(1)^pcvmindeg(daU[1]),delta)* |
---|
748 | daU[2]*(var(1)^kappa*diff(U,var(1))+M*U), |
---|
749 | var(1)^(kappa+pcvmindeg(daU[1])-1)); |
---|
750 | dbprint(printlevel-voice+2, |
---|
751 | "//...M/t^kappa transformed to simple pole ["+string(timer-t) |
---|
752 | +" secs, "+string(memory(1))+" bytes]"); |
---|
753 | } |
---|
754 | |
---|
755 | dbprint(printlevel-voice+2,"//decreasing delta..."); |
---|
756 | M=decmide(M,eM0,bM0); |
---|
757 | delta--; |
---|
758 | dbprint(printlevel-voice+2,"//delta="+string(delta)); |
---|
759 | |
---|
760 | while(delta>0) |
---|
761 | { |
---|
762 | jd=jordan(M); |
---|
763 | eM0,bM0=jd[1..2]; |
---|
764 | M=decmide(M,eM0,bM0); |
---|
765 | delta--; |
---|
766 | dbprint(printlevel-voice+2,"//delta="+string(delta)); |
---|
767 | } |
---|
768 | dbprint(printlevel-voice+2,"//...delta decreased"); |
---|
769 | } |
---|
770 | } |
---|
771 | |
---|
772 | dbprint(printlevel-voice+2,"//computing 0-jet M0 of M..."); |
---|
773 | matrix M0=jet(M,0); |
---|
774 | dbprint(printlevel-voice+2,"//...M0 computed"); |
---|
775 | |
---|
776 | return(M0); |
---|
777 | } |
---|
778 | /////////////////////////////////////////////////////////////////////////////// |
---|
779 | |
---|
780 | static proc qhmonodromy(poly f,intvec w) |
---|
781 | { |
---|
782 | dbprint(printlevel-voice+2,"//computing basis e of Milnor algebra..."); |
---|
783 | int t=timer; |
---|
784 | ideal e=kbase(std(jacob(f))); |
---|
785 | dbprint(printlevel-voice+2,"//...e computed ["+string(timer-t)+" secs, " |
---|
786 | +string(memory(1))+" bytes]"); |
---|
787 | |
---|
788 | dbprint(printlevel-voice+2, |
---|
789 | "//computing Milnor number mu and quasihomogeneous degree d..."); |
---|
790 | int mu,d=size(e),(transpose(leadexp(f))*w)[1]; |
---|
791 | dbprint(printlevel-voice+2,"...mu and d computed"); |
---|
792 | |
---|
793 | dbprint(printlevel-voice+2,"//computing te-matrix M of t*nabla..."); |
---|
794 | matrix M[mu][mu]; |
---|
795 | int i; |
---|
796 | for(i=mu;i>=1;i--) |
---|
797 | { |
---|
798 | M[i,i]=number((transpose(leadexp(e[i])+1)*w)[1])/d; |
---|
799 | } |
---|
800 | dbprint(printlevel-voice+2,"//...M computed"); |
---|
801 | |
---|
802 | return(M); |
---|
803 | } |
---|
804 | /////////////////////////////////////////////////////////////////////////////// |
---|
805 | |
---|
806 | proc monodromy(poly f, list #) |
---|
807 | "USAGE: monodromy(f[,opt]); f poly, opt int |
---|
808 | ASSUME: The polynomial f in a series ring (local ordering) defines |
---|
809 | an isolated hypersurface singularity. |
---|
810 | RETURN: The procedure returns a residue matrix M of the meromorphic |
---|
811 | Gauss-Manin connection of the singularity defined by f |
---|
812 | or an empty matrix if the assumptions are not fulfilled. |
---|
813 | If opt=0 (default), exp(2*pi*i*M) is a monodromy matrix of f, |
---|
814 | else, only the characteristic polynomial of exp(2*pi*i*M) coincides |
---|
815 | with the characteristic polynomial of the monodromy of f. |
---|
816 | THEORY: The procedure uses an algorithm by Brieskorn (See E. Brieskorn, |
---|
817 | manuscipta math. 2 (1970), 103-161) to compute a connection matrix of |
---|
818 | the meromorphic Gauss-Manin connection up to arbitrarily high order, |
---|
819 | and an algorithm of Gerard and Levelt (See R. Gerard, A.H.M. Levelt, |
---|
820 | Ann. Inst. Fourier, Grenoble 23,1 (1973), pp. 157-195) to transform |
---|
821 | it to a simple pole. |
---|
822 | DISPLAY: The procedure displays more comments for higher printlevel. |
---|
823 | EXAMPLE: example monodromy; shows an example. |
---|
824 | " |
---|
825 | { |
---|
826 | int opt; |
---|
827 | if(size(#)>0) |
---|
828 | { |
---|
829 | if(typeof(opt)=="int") |
---|
830 | { |
---|
831 | opt=#[1]; |
---|
832 | } |
---|
833 | else |
---|
834 | { |
---|
835 | print("\\second parameter no int"); |
---|
836 | return(); |
---|
837 | } |
---|
838 | |
---|
839 | } |
---|
840 | |
---|
841 | dbprint(printlevel-voice+2,"//basering="+string(basering)); |
---|
842 | |
---|
843 | int i; |
---|
844 | for(i=nvars(basering);i>=1;i--) |
---|
845 | { |
---|
846 | if(1<var(i)) |
---|
847 | { |
---|
848 | i=-1; |
---|
849 | } |
---|
850 | } |
---|
851 | |
---|
852 | if(i<0) |
---|
853 | { |
---|
854 | print("//no series ring (local ordering)"); |
---|
855 | |
---|
856 | matrix M[1][0]; |
---|
857 | return(M); |
---|
858 | } |
---|
859 | else |
---|
860 | { |
---|
861 | dbprint(printlevel-voice+2,"//f="+string(f)); |
---|
862 | |
---|
863 | dbprint(printlevel-voice+2,"//computing milnor number mu of f..."); |
---|
864 | int t=timer; |
---|
865 | int mu=milnor(f); |
---|
866 | dbprint(printlevel-voice+2,"//...mu computed ["+string(timer-t)+" secs, " |
---|
867 | +string(memory(1))+" bytes]"); |
---|
868 | |
---|
869 | dbprint(printlevel-voice+2,"//mu="+string(mu)); |
---|
870 | |
---|
871 | if(mu<=0) |
---|
872 | { |
---|
873 | if(mu==0) |
---|
874 | { |
---|
875 | print("//no singularity"); |
---|
876 | } |
---|
877 | else |
---|
878 | { |
---|
879 | print("//non isolated singularity"); |
---|
880 | } |
---|
881 | |
---|
882 | matrix M[1][0]; |
---|
883 | return(M); |
---|
884 | } |
---|
885 | else |
---|
886 | { |
---|
887 | dbprint(printlevel-voice+2,"//computing weight vector w..."); |
---|
888 | intvec w=qhweight(f); |
---|
889 | dbprint(printlevel-voice+2,"//...w computed"); |
---|
890 | |
---|
891 | dbprint(printlevel-voice+2,"//w="+string(w)); |
---|
892 | |
---|
893 | if(w==0) |
---|
894 | { |
---|
895 | dbprint(printlevel-voice+2, |
---|
896 | "//f not quasihomogeneous with respect to given coordinates"); |
---|
897 | return(nonqhmonodromy(f,mu,opt)); |
---|
898 | } |
---|
899 | else |
---|
900 | { |
---|
901 | dbprint(printlevel-voice+2, |
---|
902 | "//f quasihomogeneous with respect to given coordinates"); |
---|
903 | return(qhmonodromy(f,w)); |
---|
904 | } |
---|
905 | } |
---|
906 | } |
---|
907 | } |
---|
908 | example |
---|
909 | { "EXAMPLE:"; echo=2; |
---|
910 | ring R=0,(x,y),ds; |
---|
911 | poly f=x2y2+x6+y6; |
---|
912 | matrix M=monodromy(f); |
---|
913 | print(M); |
---|
914 | } |
---|
915 | /////////////////////////////////////////////////////////////////////////////// |
---|
916 | |
---|
917 | proc H''basis(poly f) |
---|
918 | "USAGE: H''basis(f); f poly |
---|
919 | ASSUME: The polynomial f in a series ring (local ordering) defines |
---|
920 | an isolated hypersurface singularity. |
---|
921 | RETURN: The procedure returns a list of representatives of a C{f}-basis of the |
---|
922 | Brieskorn lattice H''=Omega^(n+1)/df^dOmega^(n-1). |
---|
923 | THEORY: H'' is a free C{f}-module of rank milnor(f). |
---|
924 | DISPLAY: The procedure displays more comments for higher printlevel. |
---|
925 | EXAMPLE: example H''basis; shows an example. |
---|
926 | " |
---|
927 | { |
---|
928 | pcvinit(); |
---|
929 | |
---|
930 | dbprint(printlevel-voice+2,"//basering="+string(basering)); |
---|
931 | |
---|
932 | int i; |
---|
933 | for(i=nvars(basering);i>=1;i--) |
---|
934 | { |
---|
935 | if(1<var(i)) |
---|
936 | { |
---|
937 | i=-1; |
---|
938 | } |
---|
939 | } |
---|
940 | |
---|
941 | if(i<0) |
---|
942 | { |
---|
943 | print("//no series ring (local ordering)"); |
---|
944 | |
---|
945 | return(list()); |
---|
946 | } |
---|
947 | else |
---|
948 | { |
---|
949 | dbprint(printlevel-voice+2,"//f="+string(f)); |
---|
950 | |
---|
951 | dbprint(printlevel-voice+2,"//computing milnor number mu of f..."); |
---|
952 | int t=timer; |
---|
953 | int mu=milnor(f); |
---|
954 | dbprint(printlevel-voice+2,"//...mu computed ["+string(timer-t)+" secs, " |
---|
955 | +string(memory(1))+" bytes]"); |
---|
956 | |
---|
957 | dbprint(printlevel-voice+2,"//mu="+string(mu)); |
---|
958 | |
---|
959 | if(mu<=0) |
---|
960 | { |
---|
961 | if(mu==0) |
---|
962 | { |
---|
963 | print("//no singularity"); |
---|
964 | } |
---|
965 | else |
---|
966 | { |
---|
967 | print("//non isolated singularity"); |
---|
968 | } |
---|
969 | |
---|
970 | return(list()); |
---|
971 | } |
---|
972 | else |
---|
973 | { |
---|
974 | dbprint(printlevel-voice+2,"//computing kappa, xi and u with "+ |
---|
975 | "u*f^kappa=(matrix(jacob(f))*xi)[1,1]..."); |
---|
976 | list jl=jacoblift(f); |
---|
977 | def kappa,xi,u=jl[1..3]; |
---|
978 | dbprint(printlevel-voice+2,"//...kappa, xi and u computed"); |
---|
979 | dbprint(printlevel-voice+2,"//kappa="+string(kappa)); |
---|
980 | if(kappa==1) |
---|
981 | { |
---|
982 | dbprint(printlevel-voice+2, |
---|
983 | "//f quasihomogenous with respect to suitable coordinates"); |
---|
984 | } |
---|
985 | else |
---|
986 | { |
---|
987 | dbprint(printlevel-voice+2, |
---|
988 | "//f not quasihomogenous for any choice of coordinates"); |
---|
989 | } |
---|
990 | dbprint(printlevel-voice+2,"//xi="); |
---|
991 | dbprint(printlevel-voice+2,xi); |
---|
992 | dbprint(printlevel-voice+2,"//u="+string(u)); |
---|
993 | |
---|
994 | int K,N,prevN; |
---|
995 | list e,P1,P2,Pe,V1,V2,Ve,Vnablae; |
---|
996 | |
---|
997 | dbprint(printlevel-voice+2,"//increasing K and N..."); |
---|
998 | K,N,P1,P2,Pe,V1,V2,Ve=incK(f,mu,K,1,N,e,P1,P2,Pe,V1,V2,Ve); |
---|
999 | dbprint(printlevel-voice+2,"//...K and N increased"); |
---|
1000 | |
---|
1001 | dbprint(printlevel-voice+2, |
---|
1002 | "//computing C{f}-basis e of Brieskorn lattice " |
---|
1003 | +"H''=Omega^(n+1)/df^dOmega^(n-1)..."); |
---|
1004 | t=timer; |
---|
1005 | e=pcvcv2p(quotV(V1+V2,N),0,N); |
---|
1006 | dbprint(printlevel-voice+2,"//...e computed ["+string(timer-t)+" secs, " |
---|
1007 | +string(memory(1))+" bytes]"); |
---|
1008 | |
---|
1009 | dbprint(printlevel-voice+2,"//e="); |
---|
1010 | dbprint(printlevel-voice+2,e); |
---|
1011 | |
---|
1012 | return(e); |
---|
1013 | } |
---|
1014 | } |
---|
1015 | } |
---|
1016 | example |
---|
1017 | { "EXAMPLE:"; echo=2; |
---|
1018 | ring R=0,(x,y),ds; |
---|
1019 | poly f=x2y2+x6+y6; |
---|
1020 | H''basis(f); |
---|
1021 | } |
---|
1022 | /////////////////////////////////////////////////////////////////////////////// |
---|