1 | //////////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id$"; |
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3 | category="Noncommutative"; |
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4 | info=" |
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5 | LIBRARY: derham.lib Computation of deRham cohomology |
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6 | |
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7 | AUTHORS: Cornelia Rottner, rottner@mathematik.uni-kl.de |
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8 | |
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9 | OVERVIEW: |
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10 | A library for computing the de Rham cohomology of complements of complex affine |
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11 | varieties. |
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12 | |
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13 | |
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14 | REFERENCES: |
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15 | [OT] Oaku, T.; Takayama, N.: Algorithms of D-modules - restriction, tensor product, |
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16 | localzation, and local cohomology groups, J. Pure Appl. Algebra 156, 267-308 |
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17 | (2001) |
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18 | [R] Rottner, C.: Computing de Rham Cohomology,diploma thesis (2012) |
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19 | [W1] Walther, U.: Algorithmic computation of local cohomology modules and the local |
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20 | cohomological dimension of algebraic varieties, J. Pure Appl. Algebra 139, |
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21 | 303-321 (1999) |
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22 | [W2] Walther, U.: Algorithmic computation of de Rham Cohomology of Complements of |
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23 | Complex Affine Varieties, J. Symbolic Computation 29, 796-839 (2000) |
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24 | |
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25 | |
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26 | PROCEDURES: |
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27 | |
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28 | deRhamCohomology(list[,opt]); computes the de Rham cohomology |
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29 | MVComplex(list); computes the Mayer-Vietoris complex |
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30 | "; |
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31 | |
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32 | LIB "nctools.lib"; |
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33 | LIB "matrix.lib"; |
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34 | LIB "qhmoduli.lib"; |
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35 | LIB "general.lib"; |
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36 | LIB "dmod.lib"; |
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37 | LIB "bfun.lib"; |
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38 | LIB "dmodapp.lib"; |
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39 | LIB "poly.lib"; |
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40 | LIB "schreyer.lib"; |
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41 | |
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42 | //////////////////////////////////////////////////////////////////////////////////// |
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43 | |
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44 | proc deRhamCohomology(list L,list #) |
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45 | "USAGE: deRhamCohomology(L[,choices]); L a list consisting of polynomials, choices |
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46 | optional list consisting of one up to three strings @* |
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47 | The optional strings may be one of the strings@* |
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48 | -'Vdres': compute the Cartan-Eilenberg resolutions via V_d-homogenization |
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49 | and without using Schreyer's method @* |
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50 | -'Sres': compute the Cartan-Eilenberg resolutions in the homogenized Weyl |
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51 | algebra using Schreyer's method@* |
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52 | one of the strings@* |
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53 | -'iterativeloc': compute localizations by factorizing the polynomials and |
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54 | sucessive localization of the factors @* |
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55 | -'no iterativeloc': compute localizations by directly localizing the |
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56 | product@* |
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57 | and one of the strings |
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58 | -'onlybounds': computes bounds for the minimal and maximal interger roots |
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59 | of the global b-function |
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60 | -'exactroots' computes the minimal and maximal integer root of the global |
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61 | b-function |
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62 | The default is 'Sres', 'iterativeloc' and 'onlybounds'. |
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63 | ASSUME: -The basering must be a polynomial ring over the field of rational numbers@* |
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64 | RETURN: list, where the ith entry is the (i-1)st de Rham cohomology group of the |
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65 | complement of the complex affine variety given by the polynomials in L |
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66 | EXAMPLE:example deRhamCohomology; shows an example |
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67 | " |
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68 | { |
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69 | intvec saveoptions=option(get); |
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70 | intvec i1,i2; |
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71 | option(none); |
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72 | int recursiveloc=1; |
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73 | int i,j,nr,nc; |
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74 | def R=basering; |
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75 | poly islcm, forlcm; |
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76 | int n=nvars(R); |
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77 | int le=size(L)+n; |
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78 | string Syzstring="Sres"; |
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79 | int onlybounds=1; |
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80 | for (i=1; i<=size(#); i++) |
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81 | { |
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82 | if (#[i]=="Vdres") |
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83 | { |
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84 | Syzstring="Vdres"; |
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85 | } |
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86 | if (#[i]=="noiterativeloc") |
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87 | { |
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88 | recursiveloc=0; |
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89 | } |
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90 | if (#[i]=="exactroots") |
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91 | { |
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92 | onlybounds=0; |
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93 | } |
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94 | } |
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95 | for (i=1; i<=size(L); i++) |
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96 | { |
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97 | if (L[i]==0) |
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98 | { |
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99 | L=delete(L,i); |
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100 | i=i-1; |
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101 | } |
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102 | } |
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103 | if (size(L)==0) |
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104 | { |
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105 | return (list(0)); |
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106 | } |
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107 | for (i=1; i<= size(L); i++) |
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108 | { |
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109 | if (leadcoef(L[i])-L[i]==0) |
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110 | { |
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111 | return(list(1)); |
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112 | } |
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113 | } |
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114 | if (size(L)==0) |
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115 | { |
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116 | /*the complement of the variety given by the input is the whole space*/ |
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117 | return(list(1)); |
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118 | } |
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119 | for (i=1; i<=size(L); i++) |
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120 | { |
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121 | if (typeof(L[i])!="poly") |
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122 | { |
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123 | print("The input list must consist of polynomials"); |
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124 | retrun(); |
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125 | } |
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126 | } |
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127 | /* 1st step: compute the Mayer-Vietoris Complex and its Fourier transform*/ |
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128 | def W=MVComplex(L,recursiveloc);//new ring that contains the MV complex |
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129 | setring W; |
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130 | list fortoVdstrict=MV; |
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131 | ideal IFourier=var(n+1); |
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132 | for (i=2;i<=n;i++) |
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133 | { |
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134 | IFourier=IFourier,var(n+i); |
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135 | } |
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136 | for (i=1; i<=n;i++) |
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137 | { |
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138 | IFourier=IFourier,-var(i); |
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139 | } |
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140 | map cFourier=W,IFourier; |
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141 | matrix sup; |
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142 | for (i=1; i<=size(MV); i++) |
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143 | { |
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144 | sup=fortoVdstrict[i]; |
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145 | /*takes the Fourier transform of the MV complex*/ |
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146 | fortoVdstrict[i]=cFourier(sup); |
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147 | } |
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148 | /* 2nd step: Compute a V_d-strict free complex that is quasi-isomorphic to the |
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149 | complex fortoVdstrict |
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150 | The 1st entry of the list rem will be the quasi-isomorphic complex, the 2nd |
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151 | entry contains the cohomology modules and is needed for the computation of the |
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152 | global b-function*/ |
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153 | list rem=toVdStrictFreeComplex(fortoVdstrict,Syzstring); |
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154 | list newcomplex=rem[1]; |
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155 | //////////////////////////////////////////////////////////////////////////////////// |
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156 | /* 3rd step: Compute the bounds for the minimal and maximal integer root of the |
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157 | global b-function of newcomplex(i.e. compute the lcm of the b-functions of its |
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158 | cohomology modules)(if onlybouns=1). Else we compute the minimal and maximal |
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159 | integer root. |
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160 | |
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161 | If we compute only the bounds, we omit additional Groebner basis computations. |
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162 | However this leads to a higher-dimensional truncated complex. |
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163 | |
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164 | Note that the cohomology modules are already contained in rem[2]. |
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165 | minmaxk[1] and minmaxk[2] will contain the bounds resp exact roots.*/ |
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166 | if (onlybounds==1) |
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167 | { |
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168 | list minmaxk=globalBFun(rem[2],Syzstring); |
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169 | } |
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170 | else |
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171 | { |
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172 | list minmaxk=exactGlobalBFun(rem[2],Syzstring); |
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173 | } |
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174 | if (size(minmaxk)==0) |
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175 | { |
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176 | return (0); |
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177 | } |
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178 | /*4th step: Truncate the complex D_n/(x_1,...,x_n)\otimes C, (where |
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179 | C=(C^i[m^i],d^i) is given by newcomplex, i.e. C^i=D_n^newcomplex[3*i-2], |
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180 | m^i=newcomplex[3*i-1], d^i=newcomplex[3*i]), using Thm 5.7 in [W1]: |
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181 | The truncated module D_n/(x_1,..,x_n)\otimes C[i] is generated by the set |
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182 | (0,...,P_(i_j),0,...), where P_(i_j) is a monomial in C[D(1),...,D(n)] and |
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183 | if it is placed in component k it holds that |
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184 | minmaxk[1]-m^i[k]<=deg(P_(i_j))<=minmaxk[2]-m^i[k]*/ |
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185 | int k,l; |
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186 | list truncatedcomplex,shorten,upto; |
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187 | for (i=1; i<=size(newcomplex) div 3; i++) |
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188 | { |
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189 | shorten[3*i-1]=list(); |
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190 | for (j=1; j<=size(newcomplex[3*i-1]); j++) |
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191 | { |
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192 | /*shorten[3*i-1][j][k]=minmaxk[k]-m^i[j]+1 (for k=1,2) if this value is |
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193 | positive otherwise we will set it to be list(); |
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194 | we added +1, because we will use a list, where we put in position l |
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195 | polys of degree l+1*/ |
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196 | shorten[3*i-1][j]=list(minmaxk[1]-newcomplex[3*i-1][j]+1); |
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197 | shorten[3*i-1][j][2]=minmaxk[2]-newcomplex[3*i-1][j]+1; |
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198 | upto[size(upto)+1]=shorten[3*i-1][j][2]; |
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199 | if (shorten[3*i-1][j][2]<=0) |
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200 | { |
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201 | shorten[3*i-1][j]=list(); |
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202 | } |
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203 | else |
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204 | { |
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205 | if (shorten[3*i-1][j][1]<=0) |
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206 | { |
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207 | shorten[3*i-1][j][1]=1; |
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208 | } |
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209 | } |
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210 | } |
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211 | } |
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212 | int iupto=Max(upto);//maximal degree +1 of the polynomials we have to consider |
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213 | if (iupto<=0) |
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214 | { |
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215 | return(list(0)); |
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216 | } |
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217 | list allpolys; |
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218 | /*allpolys[i] will consist list of all monomials in D(1),...,D(n) of degree i-1*/ |
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219 | allpolys[1]=list(1); |
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220 | list minvar; |
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221 | minvar[1]=list(1); |
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222 | for (i=1; i<=iupto-1; i++) |
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223 | { |
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224 | allpolys[i+1]=list(); |
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225 | minvar[i+1]=list(); |
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226 | for (k=1; k<=size(allpolys[i]); k++) |
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227 | { |
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228 | for (j=minvar[i][k]; j<=nvars(W) div 2; j++) |
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229 | { |
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230 | allpolys[i+1][size(allpolys[i+1])+1]=allpolys[i][k]*D(j); |
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231 | minvar[i+1][size(minvar[i+1])+1]=j; |
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232 | } |
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233 | } |
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234 | } |
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235 | list keepformatrix,sizetruncom,fortrun,fst; |
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236 | int count,stc; |
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237 | intvec v,forin; |
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238 | matrix subm; |
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239 | /*now we compute the truncation*/ |
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240 | for (i=1; i<=size(newcomplex) div 3; i++) |
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241 | { |
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242 | /*truncatedcomplex[2*i-1] will contain all the generators for the truncation |
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243 | of D_n/(x(1),..,x(n))\otimes C[i]*/ |
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244 | truncatedcomplex[2*i-1]=list(); |
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245 | sizetruncom[2*i-1]=list(); |
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246 | sizetruncom[2*i]=list(); |
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247 | /*truncatedcomplex[2*i] will be the map trunc(D_n/(x(1),..,x(n))\otimes C[i]) |
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248 | ->trunc(D_n/(x(1),..,x(n))\otimes C[i+1])*/ |
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249 | truncatedcomplex[2*i]=newcomplex[3*i]; |
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250 | v=0;count=0; |
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251 | sizetruncom[2*i][1]=0; |
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252 | for (j=1; j<=newcomplex[3*i-2]; j++) |
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253 | { |
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254 | if (size(shorten[3*i-1][j])!=0) |
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255 | { |
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256 | fortrun=sublist(allpolys,shorten[3*i-1][j][1],shorten[3*i-1][j][2]); |
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257 | truncatedcomplex[2*i-1][size(truncatedcomplex[2*i-1])+1]=fortrun[1]; |
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258 | count=count+fortrun[2]; |
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259 | fst=list(int(shorten[3*i-1][j][1])-1,int(shorten[3*i-1][j][2])-1); |
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260 | sizetruncom[2*i-1][size(sizetruncom[2*i-1])+1]=fst; |
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261 | sizetruncom[2*i][size(sizetruncom[2*i])+1]=count; |
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262 | if (v!=0) |
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263 | { |
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264 | v[size(v)+1]=j; |
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265 | } |
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266 | else |
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267 | { |
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268 | v[1]=j; |
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269 | } |
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270 | } |
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271 | } |
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272 | if (v!=0) |
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273 | { |
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274 | subm=submat(truncatedcomplex[2*i],v,1..ncols(truncatedcomplex[2*i])); |
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275 | truncatedcomplex[2*i]=subm; |
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276 | if (i!=1) |
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277 | { |
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278 | i1=1..nrows(truncatedcomplex[2*(i-1)]); |
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279 | subm=submat(truncatedcomplex[2*(i-1)],i1,v); |
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280 | truncatedcomplex[2*(i-1)]=subm; |
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281 | } |
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282 | } |
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283 | else |
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284 | { |
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285 | truncatedcomplex[2*i]=matrix(0,1,ncols(truncatedcomplex[2*i])); |
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286 | if (i!=1) |
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287 | { |
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288 | nr=nrows(truncatedcomplex[2*(i-1)]); |
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289 | truncatedcomplex[2*(i-1)]=matrix(0,nr,1); |
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290 | } |
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291 | } |
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292 | } |
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293 | matrix M; |
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294 | int st,pi,pj; |
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295 | poly ptc; |
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296 | int b,d,ideg,kplus,lplus; |
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297 | poly form,lform,nform; |
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298 | /*computation of the maps*/ |
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299 | for (i=1; i<size(truncatedcomplex) div 2; i++) |
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300 | { |
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301 | nr=max(1,sizetruncom[2*i][size(sizetruncom[2*i])]); |
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302 | nc=max(1,sizetruncom[2*i+2][size(sizetruncom[2*i+2])]); |
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303 | M=matrix(0,nr,nc); |
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304 | for (k=1; k<=size(truncatedcomplex[2*i-1]);k++) |
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305 | { |
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306 | for (l=1; l<=size(truncatedcomplex[2*(i+1)-1]); l++) |
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307 | { |
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308 | if (size(sizetruncom[2*i])!=1) |
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309 | { |
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310 | for (j=1; j<=size(truncatedcomplex[2*i-1][k]); j++) |
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311 | { |
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312 | for (b=1; b<=size(truncatedcomplex[2*i-1][k][j]); b++) |
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313 | { |
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314 | form=truncatedcomplex[2*i-1][k][j][b][1]; |
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315 | form=form*truncatedcomplex[2*i][k,l]; |
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316 | while (form!=0) |
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317 | { |
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318 | lform=lead(form); |
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319 | v=leadexp(lform); |
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320 | v=v[1..n]; |
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321 | if (v==(0:n)) |
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322 | { |
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323 | ideg=deg(lform)-sizetruncom[2*(i+1)-1][l][1]; |
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324 | if (ideg>=0) |
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325 | { |
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326 | nr=ideg+1; |
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327 | st=size(truncatedcomplex[2*(i+1)-1][l][nr]); |
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328 | for (d=1; d<=st;d++) |
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329 | { |
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330 | nc=2*(i+1)-1; |
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331 | ptc=truncatedcomplex[nc][l][ideg+1][d][1]; |
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332 | if (leadmonom(lform)==ptc) |
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333 | { |
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334 | nr=2*i-1; |
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335 | pi=truncatedcomplex[nr][k][j][b][2]; |
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336 | pi=pi+sizetruncom[2*i][k]; |
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337 | nc=2*(i+1)-1; |
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338 | nr=ideg+1; |
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339 | pj=truncatedcomplex[nc][l][nr][d][2]; |
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340 | pj=pj+sizetruncom[2*(i+1)][l]; |
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341 | M[pi,pj]=leadcoef(lform); |
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342 | break; |
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343 | } |
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344 | } |
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345 | } |
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346 | } |
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347 | form=form-lform; |
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348 | } |
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349 | } |
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350 | } |
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351 | } |
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352 | } |
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353 | } |
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354 | truncatedcomplex[2*i]=M; |
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355 | truncatedcomplex[2*i-1]=sizetruncom[2*i][size(sizetruncom[2*i])]; |
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356 | } |
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357 | truncatedcomplex[2*i-1]=sizetruncom[2*i][size(sizetruncom[2*i])]; |
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358 | if (truncatedcomplex[2*i-1]!=0) |
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359 | { |
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360 | truncatedcomplex[2*i]=matrix(0,truncatedcomplex[2*i-1],1); |
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361 | } |
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362 | setring R; |
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363 | list truncatedcomplex=imap(W,truncatedcomplex); |
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364 | /*computes the cohomology of the complex (D^i,d^i) given by truncatedcomplex, |
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365 | i.e. D^i=C^truncatedcomplex[2*i-1] and d^i=truncatedcomplex[2*i]*/ |
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366 | list derhamhom=findCohomology(truncatedcomplex,le); |
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367 | option(set,saveoptions); |
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368 | return (derhamhom); |
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369 | } |
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370 | |
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371 | example |
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372 | { "EXAMPLE:"; |
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373 | ring r = 0,(x,y,z),dp; |
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374 | list L=(xy,xz); |
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375 | deRhamCohomology(L); |
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376 | } |
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377 | |
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378 | //////////////////////////////////////////////////////////////////////////////////// |
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379 | //COMPUTATION OF THE MAYER-VIETORIS COMPLEX |
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380 | //////////////////////////////////////////////////////////////////////////////////// |
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381 | |
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382 | proc MVComplex(list L,list #) |
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383 | "USAGE:MVComplex(L); L a list of polynomials |
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384 | ASSUME: -Basering is a polynomial ring with n vwariables and rational coefficients |
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385 | -L is a list of non-constant polynomials |
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386 | RETURN: ring W: the nth Weyl algebra @* |
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387 | W contains a list MV, which represents the Mayer-Vietrois complex (C^i,d^i) of the |
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388 | polynomials contained in L as follows:@* |
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389 | the C^i are given by D_n^ncols(C[2*i-1])/im(C[2*i-1]) and the differentials |
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390 | d^i are given by C[2*i] |
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391 | EXAMPLE:example MVComplex; shows an example |
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392 | " |
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393 | { |
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394 | /* We follow algorithm 3.2.5 in [R],if #!=0 we use also Remark 3.2.6 in [R] for |
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395 | an additional iterative localization*/ |
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396 | def R=basering; |
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397 | int i; |
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398 | int iterative=1; |
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399 | if (size(#)!=0) |
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400 | { |
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401 | iterative=#[1]; |
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402 | } |
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403 | for (i=1; i<=size(L); i++) |
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404 | { |
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405 | if (L[i]==0) |
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406 | { |
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407 | print("localization with respect to 0 not possible"); |
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408 | return(); |
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409 | } |
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410 | if (leadcoef(L[i])-L[i]==0) |
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411 | { |
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412 | print("polynomials must be non-constant"); |
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413 | return(); |
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414 | } |
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415 | } |
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416 | if (iterative==1) |
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417 | { |
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418 | /*compute the localizations by factorizing the polynomials and iterative |
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419 | localization of the factors */ |
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420 | for (i=1; i<=size(L); i++) |
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421 | { |
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422 | L[i]=factorize(L[i],1); |
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423 | } |
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424 | } |
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425 | int r=size(L); |
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426 | int n=nvars(basering); |
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427 | int le=size(L)+n; |
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428 | /*construct the ring Ws*/ |
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429 | def W=makeWeyl(n); |
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430 | setring W; |
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431 | list man=ringlist(W); |
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432 | if (n==1) |
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433 | { |
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434 | man[2][1]="x(1)"; |
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435 | man[2][2]="D(1)"; |
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436 | def Wi=ring(man); |
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437 | setring Wi; |
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438 | kill W; |
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439 | def W=Wi; |
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440 | setring W; |
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441 | list man=ringlist(W); |
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442 | } |
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443 | man[2][size(man[2])+1]="s";; |
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444 | man[3][3]=man[3][2]; |
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445 | man[3][2]=list("dp",intvec(1)); |
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446 | matrix N=UpOneMatrix(size(man[2])); |
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447 | man[5]=N; |
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448 | matrix M[1][1]; |
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449 | man[6]=transpose(concat(transpose(concat(man[6],M)),M)); |
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450 | def Ws=ring(man); |
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451 | setring Ws; |
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452 | int j,k,l,c; |
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453 | list L=fetch(R,L); |
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454 | list Cech; |
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455 | ideal J=var(1+n); |
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456 | for (i=2; i<=n; i++) |
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457 | { |
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458 | J=J,var(i+n); |
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459 | } |
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460 | Cech[1]=list(J); |
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461 | list Theta, remminroots; |
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462 | Theta[1]=list(list(list(),1,1)); |
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463 | list rem,findminintroot,diffmaps; |
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464 | int minroot,st,sk; |
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465 | intvec k1; |
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466 | poly fred,forfetch; |
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467 | matrix subm; |
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468 | int rmr; |
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469 | if (iterative==0) |
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470 | {/*computation of the modules of the MV complex*/ |
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471 | for (i=1; i<=r; i++) |
---|
472 | { |
---|
473 | findminintroot=list(); |
---|
474 | Cech[i+1]=list(); |
---|
475 | Theta[i+1]=list(); |
---|
476 | k1=1; |
---|
477 | for (j=1; j<=i; j++) |
---|
478 | { |
---|
479 | k1[size(k1)+1]=size(Theta[j+1]); |
---|
480 | for (k=1; k<=k1[j]; k++) |
---|
481 | { |
---|
482 | Theta[j+1][size(Theta[j+1])+1]=list(Theta[j][k][1]+list(i)); |
---|
483 | Theta[j+1][size(Theta[j+1])][2]=Theta[j][k][2]*L[i]; |
---|
484 | /*We compute the s-parametric annihilator J(s) and the b-function |
---|
485 | of the polynomial L[i] and Cech[i][k] to localize the module |
---|
486 | D_n/(D(1),...,D(n))[L[i]^(-1)]\otimes D_n^c/im(Cech[i][k]), |
---|
487 | where c=ncols(Cech[i][k]) and the im(Cech[i][k]) is generated by |
---|
488 | the rows of the matrix. |
---|
489 | If we plug the minimal integer root r(or a smaller integer |
---|
490 | value)in J(s), then D_n^ncols(J(s))/im(J(r)) is isomorphic to |
---|
491 | the above localization*/ |
---|
492 | rem=SannfsIBM(L[i],Cech[j][k]); |
---|
493 | Cech[j+1][size(Cech[j+1])+1]=rem[1]; |
---|
494 | findminintroot[size(findminintroot)+1]=rem[2]; |
---|
495 | } |
---|
496 | } |
---|
497 | /* we compute the minimal root of all b-functions of L[i] computed above, |
---|
498 | because we want to plug in the same root r in all s-parametric |
---|
499 | annihilators we computed for L[i] ->this will ensure we can compute |
---|
500 | the maps of the MV complex*/ |
---|
501 | minroot=minIntRoot(findminintroot); |
---|
502 | for (j=1; j<=i; j++) |
---|
503 | { |
---|
504 | for (k=1; k<=k1[j]; k++) |
---|
505 | { |
---|
506 | sk=size(Cech[j+1])+1-k; |
---|
507 | Cech[j+1][size(Cech[j+1])+1-k]=subst(Cech[j+1][sk],s,minroot); |
---|
508 | } |
---|
509 | } |
---|
510 | remminroots[i]=minroot; |
---|
511 | } |
---|
512 | Cech=delete(Cech,1); |
---|
513 | Theta=delete(Theta,1); |
---|
514 | list zw; |
---|
515 | poly reme; |
---|
516 | /*computation of the maps of the MV complex*/ |
---|
517 | for (i=1; i<r; i++) |
---|
518 | { |
---|
519 | diffmaps[i]=matrix(0,size(Cech[i]),size(Cech[i+1])); |
---|
520 | for (j=1; j<=size(Cech[i]); j++) |
---|
521 | { |
---|
522 | for (k=1; k<=size(Cech[i+1]); k++) |
---|
523 | { |
---|
524 | zw=LMSubset(Theta[i][j][1],Theta[i+1][k][1]); |
---|
525 | if (zw[2]!=0) |
---|
526 | { |
---|
527 | rmr=-remminroots[zw[1]]; |
---|
528 | reme=zw[2]*(Theta[i+1][k][2]/Theta[i][j][2])^(rmr); |
---|
529 | zw[2]=zw[2]*(Theta[i+1][k][2]/Theta[i][j][2])^(rmr); |
---|
530 | diffmaps[i][j,k]=zw[2]; |
---|
531 | } |
---|
532 | } |
---|
533 | } |
---|
534 | } |
---|
535 | diffmaps[r]=matrix(0,1,1); |
---|
536 | } |
---|
537 | if (iterative==1) |
---|
538 | { |
---|
539 | for (i=1; i<=r;i++) |
---|
540 | { |
---|
541 | Cech[i+1]=list(); |
---|
542 | Theta[i+1]=list(); |
---|
543 | k1=1; |
---|
544 | for (c=1; c<=size(L[i]); c++) |
---|
545 | { |
---|
546 | findminintroot=list(); |
---|
547 | for (j=1; j<=i; j++) |
---|
548 | { |
---|
549 | if (c==1) |
---|
550 | { |
---|
551 | k1[size(k1)+1]=size(Theta[j+1]); |
---|
552 | } |
---|
553 | for (k=1; k<=k1[j]; k++) |
---|
554 | { |
---|
555 | /*We compute the s-parametric annihilator J(s) und the b- |
---|
556 | function of the polynomial L[i][c] and Cech[i][k] to |
---|
557 | localize the module D_n/(D(1),...,D(n))[L[i][c]^(-1)]\otimes |
---|
558 | D_n^c/im(Cech[i][k]), where c=ncols(Cech[i][k]). |
---|
559 | If we plug the minimal integer root r(or a smaller integer |
---|
560 | value)in J(s), then D_n^ncols(J(s))/im(J(r)) is isomorphic |
---|
561 | to the above localization*/ |
---|
562 | if (c==1) |
---|
563 | { |
---|
564 | rmr=size(Theta[j+1])+1; |
---|
565 | Theta[j+1][rmr]=list(Theta[j][k][1]+list(i)); |
---|
566 | Theta[j+1][size(Theta[j+1])][2]=Theta[j][k][2]*L[i][c]; |
---|
567 | rem=SannfsIBM(L[i][c],Cech[j][k]); |
---|
568 | Cech[j+1][size(Cech[j+1])+1]=rem[1]; |
---|
569 | findminintroot[size(findminintroot)+1]=rem[2]; |
---|
570 | } |
---|
571 | else |
---|
572 | { |
---|
573 | st=size(Theta[j+1])-k1[j]+k; |
---|
574 | Theta[j+1][st][2]=Theta[j+1][st][2]*L[i][c]; |
---|
575 | rem=SannfsIBM(L[i][c],Cech[j+1][size(Cech[j+1])-k1[j]+k]); |
---|
576 | Cech[j+1][size(Cech[j+1])-k1[j]+k]=rem[1]; |
---|
577 | findminintroot[size(findminintroot)+1]=rem[2]; |
---|
578 | } |
---|
579 | } |
---|
580 | } |
---|
581 | /* we compute the minimal root of all b-functions of L[i][c] |
---|
582 | computed above,because we want to plug in the same root r in all |
---|
583 | s-parametric annihilators we computed for L[i] ->this will |
---|
584 | ensure we can compute the maps of the MV complex*/ |
---|
585 | minroot=minIntRoot(findminintroot); |
---|
586 | for (j=1; j<=i; j++) |
---|
587 | { |
---|
588 | for (k=1; k<=k1[j]; k++) |
---|
589 | { |
---|
590 | st=size(Cech[j+1])+1-k; |
---|
591 | Cech[j+1][st]=subst(Cech[j+1][st],s,minroot); |
---|
592 | } |
---|
593 | } |
---|
594 | if (c==1) |
---|
595 | { |
---|
596 | remminroots[i]=list(); |
---|
597 | } |
---|
598 | remminroots[i][c]=minroot; |
---|
599 | } |
---|
600 | } |
---|
601 | Cech=delete(Cech,1); |
---|
602 | Theta=delete(Theta,1); |
---|
603 | list zw; |
---|
604 | poly reme; |
---|
605 | /*maps of the MV Complex*/ |
---|
606 | for (i=1; i<r; i++) |
---|
607 | { |
---|
608 | diffmaps[i]=matrix(0,size(Cech[i]),size(Cech[i+1])); |
---|
609 | for (j=1; j<=size(Cech[i]); j++) |
---|
610 | { |
---|
611 | for (k=1; k<=size(Cech[i+1]); k++) |
---|
612 | { |
---|
613 | zw=LMSubset(Theta[i][j][1],Theta[i+1][k][1]); |
---|
614 | if (zw[2]!=0) |
---|
615 | { |
---|
616 | reme=1; |
---|
617 | for (c=1; c<=size(L[zw[1]]);c++) |
---|
618 | { |
---|
619 | reme=reme*L[zw[1]][c]^(-remminroots[zw[1]][c]); |
---|
620 | } |
---|
621 | diffmaps[i][j,k]=zw[2]*reme; |
---|
622 | } |
---|
623 | } |
---|
624 | } |
---|
625 | } |
---|
626 | diffmaps[r]=matrix(0,1,1); |
---|
627 | } |
---|
628 | setring W; |
---|
629 | /*map the modules and maps to the Weyl algebra*/ |
---|
630 | list diffmaps=imap(Ws,diffmaps); |
---|
631 | list Cechmodules=imap(Ws,Cech); |
---|
632 | list Cech; |
---|
633 | matrix sup; |
---|
634 | for (i=1; i<=r; i++) |
---|
635 | { |
---|
636 | sup=transpose(matrix(Cechmodules[i][1])); |
---|
637 | Cech[2*i-1]=sup; |
---|
638 | for (j=2; j<=size(Cechmodules[i]); j++) |
---|
639 | { |
---|
640 | sup=transpose(matrix(Cechmodules[i][j])); |
---|
641 | Cech[2*i-1]=dsum(Cech[2*i-1],sup); |
---|
642 | } |
---|
643 | sup=matrix(diffmaps[i]); |
---|
644 | Cech[2*i]=sup; |
---|
645 | } |
---|
646 | list MV=Cech; |
---|
647 | export MV; |
---|
648 | return (W); |
---|
649 | } |
---|
650 | |
---|
651 | example |
---|
652 | { "EXAMPLE:"; |
---|
653 | ring r = 0,(x,y,z),dp; |
---|
654 | list L=xy,xz; |
---|
655 | def C=MVComplex(L); |
---|
656 | setring C; |
---|
657 | MV; |
---|
658 | } |
---|
659 | |
---|
660 | //////////////////////////////////////////////////////////////////////////////////// |
---|
661 | |
---|
662 | static proc SannfsIBM(poly F,ideal myJ) |
---|
663 | "USAGE: SannfsIBM(f,J), F poly, J ideal |
---|
664 | ASSUME: basering is D_n[s], where D_n is the Weyl algebra and s and extra |
---|
665 | commutative variable@* |
---|
666 | f is a polynomial in the variables x(1),...,x(n) with rational coefficients |
---|
667 | @* |
---|
668 | J is holonomic and f-saturated |
---|
669 | RETURN AlList of the form (K,g), where K is an ideal and g a univariant polynomial |
---|
670 | in the variable s. K is the s-parametric annihilator of F and J and g is |
---|
671 | the b-function of F and J. |
---|
672 | " |
---|
673 | { |
---|
674 | /*modified version of the procedure SannfsBM from the library dmod.lib: SannfsBM |
---|
675 | computes the s-parametric annihilator for J=(x_1,...,x_n)*/ |
---|
676 | /* We use Algorithm 3.1.12 in[R] to compute the s-parametric |
---|
677 | annihilator. Then we use the s-parametric annihilator to compute the b-function |
---|
678 | via Algorithm 4.7 in [W1].*/ |
---|
679 | /* We assume that the basering the the nth Weyl algebra D_n. We create the ring |
---|
680 | D_n[s,t], where t*s=s*t-t*/ |
---|
681 | def save = basering; |
---|
682 | int N = nvars(basering)-1; |
---|
683 | int Nnew = N+2; |
---|
684 | int i,j; |
---|
685 | string s; |
---|
686 | list RL = ringlist(basering); |
---|
687 | list L, Lord; |
---|
688 | list tmp; |
---|
689 | intvec iv; |
---|
690 | L[1] = RL[1]; |
---|
691 | L[4] = RL[4]; |
---|
692 | list Name = RL[2]; |
---|
693 | Name=delete(Name,size(Name)); |
---|
694 | list RName; |
---|
695 | RName[1] = "t"; |
---|
696 | RName[2] = "s"; |
---|
697 | list DName; |
---|
698 | for(i=1;i<=N div 2;i++) |
---|
699 | { |
---|
700 | DName[i] = var(N div 2+i); |
---|
701 | Name=delete(Name,N div 2+1); |
---|
702 | } |
---|
703 | tmp[1] = "t"; |
---|
704 | tmp[2] = "s"; |
---|
705 | list NName = tmp +Name+DName; |
---|
706 | L[2] = NName; |
---|
707 | kill NName; |
---|
708 | tmp[1] = "lp"; |
---|
709 | iv = 1,1; |
---|
710 | tmp[2] = iv; |
---|
711 | Lord[1] = tmp; |
---|
712 | tmp[1] = "dp"; |
---|
713 | s = "iv="; |
---|
714 | for(i=1;i<=Nnew;i++) |
---|
715 | { |
---|
716 | s = s+"1,"; |
---|
717 | } |
---|
718 | s[size(s)]= ";"; |
---|
719 | execute(s); |
---|
720 | kill s; |
---|
721 | tmp[2] = iv; |
---|
722 | Lord[2] = tmp; |
---|
723 | tmp[1] = "C"; |
---|
724 | iv = 0; |
---|
725 | tmp[2] = iv; |
---|
726 | Lord[3] = tmp; |
---|
727 | tmp = 0; |
---|
728 | L[3] = Lord; |
---|
729 | def @R@ = ring(L); |
---|
730 | setring @R@; |
---|
731 | matrix @D[Nnew][Nnew]; |
---|
732 | @D[1,2]=t; |
---|
733 | for(i=1; i<=N div 2; i++) |
---|
734 | { |
---|
735 | @D[2+i, N div 2+2+i]=1; |
---|
736 | } |
---|
737 | def @R = nc_algebra(1,@D); |
---|
738 | setring @R; |
---|
739 | kill @R@; |
---|
740 | /*we start with the computation of the s-parametric annihilator*/ |
---|
741 | poly F = imap(save,F); |
---|
742 | ideal myJ=imap(save,myJ); |
---|
743 | for (i=1; i<=N div 2; i++) |
---|
744 | { |
---|
745 | myJ=subst(myJ,D(i),D(i)+diff(F,x(i))*t); |
---|
746 | } |
---|
747 | ideal I = t*F+s; |
---|
748 | I=I,myJ;//the s-parametric annihilator in D_n[s,t] |
---|
749 | /*we compute the intersection of I and D_n[s]*/ |
---|
750 | ideal J = slimgb(I); |
---|
751 | ideal K = nselect(J,1); |
---|
752 | K = slimgb(K);//the s-parametric annihilator |
---|
753 | /*we use K to compute the b-function*/ |
---|
754 | ideal B=K,F; |
---|
755 | B=slimgb(B); |
---|
756 | vector p=pIntersect(s,B); |
---|
757 | poly f=vec2poly(p,2); |
---|
758 | setring save; |
---|
759 | poly f=imap(@R,f); |
---|
760 | ideal K=imap(@R,K); |
---|
761 | return (list(K,f)); |
---|
762 | } |
---|
763 | |
---|
764 | //////////////////////////////////////////////////////////////////////////////////// |
---|
765 | //COMPUTATION OF QA UASI-ISOMORPHIC V_D-STRICT FREE COMPLEX |
---|
766 | //////////////////////////////////////////////////////////////////////////////////// |
---|
767 | |
---|
768 | static proc toVdStrictFreeComplex(list L,string Syzstring,list #) |
---|
769 | "USAGE: toVdStrictFreeComplex(L, Syzstring [,d]); L a list of the form |
---|
770 | (M_1,f_1,...,M_s,f_s), where the M_i and f_i are matrices, Syzstring a |
---|
771 | string, d an optional integer |
---|
772 | ASSUME: Basering is the Weyl algebra D_n @* |
---|
773 | (M_1,f_1,...,M_s,f_s) represents a complex 0->D_n^(r_1)/im(M_1)-> |
---|
774 | D_n^(r_2)/im(M_2)->...->D_n^(r_s)->0 with differentials f_i, where im(M_i) |
---|
775 | is generated by the rows of M_i. In particular it hold:@* |
---|
776 | - The M_i are m_i x r_i-matrices and the f_iare r_i x r_(i+1)-matrices @* |
---|
777 | -the image of M_1*f_i is contained in the image of M_(i+1) @* |
---|
778 | d is an integer between 1 and n. If no value for d is given, it is assumed |
---|
779 | to be n @* |
---|
780 | Syzstring is either: @* |
---|
781 | -'Sres' (computes the resolutions and Groebner bases in the homogenized |
---|
782 | Weyl algebra using Schreyer's method)@* |
---|
783 | or @* |
---|
784 | -'Vdres' (computes the resolutions via V_d-homogenization and without |
---|
785 | Schreyer's method)@* |
---|
786 | RETURN: list of the form (L_1,L_2), were L_1 and L_2 are lists @* |
---|
787 | L_1 is of the form (i_(-n-1),g_(-n-1),m_(-n-1),...,i_s,g_s,m_s) such that:@* |
---|
788 | -the i_j are integers, the g_j are i_j x i_(j+1)-matrices, the m_j intvecs |
---|
789 | of size i_j@* |
---|
790 | -D_n^(i_(-n-1))[m_(-n-1)]->...->D_n^(i_s)[m_s]->0 is a V_d-strict complex |
---|
791 | with differentials m_i that is quasi-isomorphic to the complex given by L@* |
---|
792 | L_2 is of the form (H_1,n_1,...,H_s,n_s), where the H_i are matrices and |
---|
793 | the n_i are shift vectors such that:@* |
---|
794 | -coker(H_i) is the ith cohomology group of the complex given by L_1@* |
---|
795 | -the n_i are the shift vectors of the coker(H_i) |
---|
796 | THEORY: We follow Algorithm 3.8 in [W2] |
---|
797 | " |
---|
798 | { |
---|
799 | def B=basering; |
---|
800 | int n=nvars(B) div 2+2; |
---|
801 | int d=nvars(B) div 2; |
---|
802 | intvec v; |
---|
803 | list out, outall; |
---|
804 | int i,j,k,indi,nc,nr; |
---|
805 | matrix mem; |
---|
806 | intvec i1,i2; |
---|
807 | if (size(#)!=0) |
---|
808 | { |
---|
809 | for (i=1; i<=size(#); i++) |
---|
810 | { |
---|
811 | if (typeof(#[i])=="int") |
---|
812 | { |
---|
813 | if (#[1]>=1 and #[1]<=n) |
---|
814 | { |
---|
815 | d=#[i]; |
---|
816 | } |
---|
817 | } |
---|
818 | } |
---|
819 | } |
---|
820 | /* If size(L)=2, our complex consists for only one non-trivial module. |
---|
821 | Therefore, we just have to compute a V_d-strict resolution of this module.*/ |
---|
822 | if (size(L)==2) |
---|
823 | { |
---|
824 | v=(0:ncols(L[1])); |
---|
825 | out[3*n-1]=v; |
---|
826 | out[3*n-2]=ncols(L[1]); |
---|
827 | out[3*n]=L[2]; |
---|
828 | if (Syzstring=="Vdres") |
---|
829 | { |
---|
830 | /*if Syzstring="Vdres", we compute a V_d-strict Groebner basis of L[1] |
---|
831 | using F-homogenization (Prop. 3.9 in [OT]); then we compute the syzygies |
---|
832 | and make them V_d-strict using Prop 3.9[OT] and so on*/ |
---|
833 | out[3*n-3]=VdStrictGB(L[1],d,v); |
---|
834 | for (i=n-1; i>=1; i--) |
---|
835 | { |
---|
836 | out[3*i-2]=nrows(out[3*i]); |
---|
837 | v=0; |
---|
838 | for (j=1; j<=out[3*i-2]; j++) |
---|
839 | { |
---|
840 | mem=submat(out[3*i],j,intvec(1..ncols(out[3*i]))); |
---|
841 | v[j]=VdDeg(mem,d, out[3*i+2]);//next shift vector |
---|
842 | } |
---|
843 | out[3*i-1]=v; |
---|
844 | if (i!=1) |
---|
845 | { |
---|
846 | /*next step in the resolution*/ |
---|
847 | out[3*i-3]=transpose(syz(transpose(out[3*i]))); |
---|
848 | if (out[3*i-3]!=matrix(0,nrows(out[3*i-3]),ncols(out[3*i-3]))) |
---|
849 | { |
---|
850 | /*makes the resolution V_d-strict*/ |
---|
851 | out[3*i-3]=VdStrictGB(out[3*i-3],d,out[3*i-1]); |
---|
852 | } |
---|
853 | else |
---|
854 | { |
---|
855 | /*resolution is already computed*/ |
---|
856 | out[3*i-3]=matrix(0,1,ncols(out[3*i-3])); |
---|
857 | out[3*i-4]=intvec(0); |
---|
858 | out[3*i-5]=int(0); |
---|
859 | for (j=i-2; j>=1; j--) |
---|
860 | { |
---|
861 | out[3*j]=matrix(0,1,1); |
---|
862 | out[3*j-1]=intvec(0); |
---|
863 | out[3*j-2]=int(0); |
---|
864 | } |
---|
865 | break; |
---|
866 | } |
---|
867 | } |
---|
868 | } |
---|
869 | } |
---|
870 | else |
---|
871 | { |
---|
872 | /*in the case Syzstring!="Vdres" we compute the resolution in the |
---|
873 | homogenized Weyl algebra using Thm 9.10 in[OT]*/ |
---|
874 | def HomWeyl=makeHomogenizedWeyl(d); |
---|
875 | setring HomWeyl; |
---|
876 | list L=fetch(B,L); |
---|
877 | L[1]=nHomogenize(L[1]); |
---|
878 | list out=fetch(B,out); |
---|
879 | out[3*n-3]=L[1]; |
---|
880 | /*computes a ring with a list RES; RES is a V_d-strict resolution of |
---|
881 | L[1]*/ |
---|
882 | def ringofSyz=Sres(transpose(L[1]),d); |
---|
883 | setring ringofSyz; |
---|
884 | int logens=2; |
---|
885 | matrix mem; |
---|
886 | list out=fetch(HomWeyl,out); |
---|
887 | out[3*n-3]=transpose(matrix(RES[2])); |
---|
888 | out[3*n-3]=subst(out[3*n-3],h,1); |
---|
889 | for (i=n-1; i>=1; i--) |
---|
890 | { |
---|
891 | out[3*i-2]=nrows(out[3*i]); |
---|
892 | v=0; |
---|
893 | for (j=1; j<=out[3*i-2]; j++) |
---|
894 | { |
---|
895 | mem=submat(out[3*i],j,intvec(1..ncols(out[3*i]))); |
---|
896 | v[j]=VdDeg(mem,d, out[3*i+2]); |
---|
897 | } |
---|
898 | out[3*i-1]=v;//shift vector such that the resolution RES is V_d-strict |
---|
899 | if (i!=1) |
---|
900 | { |
---|
901 | indi=0; |
---|
902 | if (size(RES)>=n-i+2) |
---|
903 | { |
---|
904 | nr=nrows(matrix(RES[n-i+2])); |
---|
905 | mem=matrix(0,nr,ncols(matrix(RES[n-i+2]))); |
---|
906 | if (matrix(RES[n-i+2])!=mem) |
---|
907 | { |
---|
908 | indi=1; |
---|
909 | out[3*i-3]=(matrix(RES[n-i+2])); |
---|
910 | if (nrows(out[3*i-3])-logens+1!=nrows(out[3*i])) |
---|
911 | { |
---|
912 | mem=out[3*i-3]; |
---|
913 | out[3*i-3]=matrix(mem,nrows(mem)+logens-1,ncols(mem)); |
---|
914 | } |
---|
915 | mem=out[3*i-3]; |
---|
916 | i1=intvec(logens..nrows(mem)); |
---|
917 | mem=submat(mem,i1,intvec(1..ncols(mem))); |
---|
918 | out[3*i-3]=transpose(mem); |
---|
919 | out[3*i-3]=subst(out[3*i-3],h,1); |
---|
920 | logens=logens+ncols(out[3*i-3]); |
---|
921 | } |
---|
922 | } |
---|
923 | if(indi==0) |
---|
924 | { |
---|
925 | out[3*i-3]=matrix(0,1,nrows(out[3*i])); |
---|
926 | out[3*i-4]=intvec(0); |
---|
927 | out[3*i-5]=int(0); |
---|
928 | for (j=i-2; j>=1; j--) |
---|
929 | { |
---|
930 | out[3*j]=matrix(0,1,1); |
---|
931 | out[3*j-1]=intvec(0); |
---|
932 | out[3*j-2]=int(0); |
---|
933 | } |
---|
934 | break; |
---|
935 | } |
---|
936 | } |
---|
937 | } |
---|
938 | setring B; |
---|
939 | out=fetch(ringofSyz,out);//contains the V_d-strict resolution |
---|
940 | kill ringofSyz; |
---|
941 | } |
---|
942 | outall[1]=out; |
---|
943 | outall[2]=list(list(out[3*n-3],out[3*n-1])); |
---|
944 | return(outall); |
---|
945 | } |
---|
946 | /*case size(L)>2: We compute a quasi-isomorphic free complex following Alg 3.8 in |
---|
947 | [W2]*/ |
---|
948 | /* We denote the complex given by L as (C^i,d^i). |
---|
949 | We start by computing in the proc shortExaxtPieces representations for the |
---|
950 | short exact sequences B^i->Z^i->H^i and Z^i->C^i->B^(i+1), where the B^i, Z^i |
---|
951 | and H^i are coboundaries, cocycles and cohomology groups, respectively.*/ |
---|
952 | out=shortExactPieces(L); |
---|
953 | list rem; |
---|
954 | /* shortExactpiecesToVdStrict makes the sequences B^i->Z^i->H^i and |
---|
955 | Z^i->C^i->B^(i+1) V_d-strict*/ |
---|
956 | rem=shortExactPiecesToVdStrict(out,d,Syzstring); |
---|
957 | /*VdStrictDoubleComplexes computes V_d-strict resolutions over the seqeunces from |
---|
958 | proc shortExactPiecesToVdstrict*/ |
---|
959 | out=VdStrictDoubleComplexes(rem[1],d,Syzstring); |
---|
960 | for (i=1;i<=size(out); i++) |
---|
961 | { |
---|
962 | rem[2][i][1]=out[i][1][5][1]; |
---|
963 | rem[2][i][2]=out[i][1][8][1]; |
---|
964 | } |
---|
965 | /* AssemblingDoubleComplexes puts the resolution of the C^i (from the sequences |
---|
966 | Z^i->C^i->B^(i+1)) together to obtain a Cartan-Eilenberg resolution of |
---|
967 | (C^i,d^i)*/ |
---|
968 | out=assemblingDoubleComplexes(out); |
---|
969 | /*the proc totalComplex takes the total complex of the double complex from the |
---|
970 | proc assemblingDoubleComplexes*/ |
---|
971 | out=totalComplex(out); |
---|
972 | outall[1]=out; |
---|
973 | outall[2]=rem[2];//contains the cohomology groups and their shift vectors |
---|
974 | return (outall); |
---|
975 | } |
---|
976 | |
---|
977 | //////////////////////////////////////////////////////////////////////////////////// |
---|
978 | |
---|
979 | |
---|
980 | static proc sublist(list L,int m,int n) |
---|
981 | { |
---|
982 | list out; |
---|
983 | int i; int j; |
---|
984 | int count; |
---|
985 | for (i=m; i<=n; i++) |
---|
986 | { |
---|
987 | out[size(out)+1]=list(); |
---|
988 | for (j=1; j<=size(L[i]); j++) |
---|
989 | { |
---|
990 | count=count+1; |
---|
991 | out[size(out)][j]=list(L[i][j],count); |
---|
992 | } |
---|
993 | } |
---|
994 | list o=list(out,count); |
---|
995 | return(o); |
---|
996 | } |
---|
997 | |
---|
998 | //////////////////////////////////////////////////////////////////////////////////// |
---|
999 | |
---|
1000 | static proc LMSubset(list L,list M) |
---|
1001 | { |
---|
1002 | int i; |
---|
1003 | int j=1; |
---|
1004 | list position=(M[size(M)],(-1)^(size(L))); |
---|
1005 | for (i=1; i<=size(L); i++) |
---|
1006 | { |
---|
1007 | if (L[i]!=M[j]) |
---|
1008 | { |
---|
1009 | if (L[i]!=M[i+1] or j!=i) |
---|
1010 | { |
---|
1011 | return (L[i],0); |
---|
1012 | } |
---|
1013 | else |
---|
1014 | { |
---|
1015 | position=(M[i],(-1)^(i-1)); |
---|
1016 | j=j+1; |
---|
1017 | } |
---|
1018 | } |
---|
1019 | j=j+1; |
---|
1020 | |
---|
1021 | } |
---|
1022 | return (position); |
---|
1023 | } |
---|
1024 | |
---|
1025 | //////////////////////////////////////////////////////////////////////////////////// |
---|
1026 | |
---|
1027 | static proc shortExactPieces(list L) |
---|
1028 | { |
---|
1029 | /*we follow Section 3.3 in [W2]*/ |
---|
1030 | /* we assume that L=(M_1,f_1,...,M_s,f_s) defines the complex C=(C^i,d^i) |
---|
1031 | as in the procedure toVdstrictcomplex*/ |
---|
1032 | matrix Bnew= divdr(L[2],L[3]); |
---|
1033 | matrix Bold=Bnew; |
---|
1034 | matrix Z=divdr(Bnew,L[1]); |
---|
1035 | list bzh,zcb; |
---|
1036 | bzh=list(list(),list(),Z,unitmat(ncols(Z)),Z); |
---|
1037 | zcb=(Z, Bnew, L[1], unitmat(ncols(L[1])), Bnew); |
---|
1038 | list sep; |
---|
1039 | /* the list sep will be of size s such that |
---|
1040 | -sep[i]=(sep[i][1],sep[i][2]) is a list of two lists |
---|
1041 | -sep[i][1]=(B^i,f^(BZi),Z^i,f_^(ZHi),H^i) such that coker(B^i)->coker(Z^i) |
---|
1042 | ->coker(H^i) represents the short exact seqeuence B^i(C)->Z^i(C)->H^i(C) |
---|
1043 | -sep[i][2]=(Z^i,f^(ZCi),C^i,f^(CBi),B^(i+1)) such that coker(Z^i)->coker(C^i)-> |
---|
1044 | coker(B^(i+1)) represents the short exact seqeuence Z^i(C)->C^i->B^(i+1)(C)*/ |
---|
1045 | sep[1]=list(bzh,zcb); |
---|
1046 | int i; |
---|
1047 | list out; |
---|
1048 | for (i=3; i<=size(L)-2; i=i+2) |
---|
1049 | { |
---|
1050 | /*the proc bzhzcb computes representations for the short exact seqeunces */ |
---|
1051 | out=bzhzcb(Bold, L[i-1] , L[i], L[i+1], L[i+2]); |
---|
1052 | sep[size(sep)+1]=out[1]; |
---|
1053 | Bold=out[2]; |
---|
1054 | } |
---|
1055 | bzh=(divdr(L[size(L)-2], L[size(L)-1]),L[size(L)-2], L[size(L)-1]); |
---|
1056 | bzh[4]=unitmat(ncols(L[size(L)-1])); |
---|
1057 | bzh[5]=transpose(concat(transpose(L[size(L)-2]),transpose(L[size(L)-1]))); |
---|
1058 | zcb=(L[size(L)-1], unitmat(ncols(L[size(L)-1])), L[size(L)-1],list(),list()); |
---|
1059 | sep[size(sep)+1]=list(bzh,zcb); |
---|
1060 | return(sep); |
---|
1061 | } |
---|
1062 | |
---|
1063 | //////////////////////////////////////////////////////////////////////////////////// |
---|
1064 | |
---|
1065 | static proc bzhzcb (matrix Bold,matrix f0,matrix C1,matrix f1,matrix C2) |
---|
1066 | { |
---|
1067 | matrix Bnew=divdr(f1,C2); |
---|
1068 | matrix Z= divdr(Bnew,C1); |
---|
1069 | matrix lift1= matrixLift(Bnew,f0); |
---|
1070 | matrix H=transpose(concat(transpose(lift1),transpose(Z))); |
---|
1071 | list bzh=(Bold, lift1, Z, unitmat(ncols(Z)),H); |
---|
1072 | list zcb=(Z, Bnew, C1, unitmat(ncols(C1)),Bnew); |
---|
1073 | list out=(list(bzh, zcb), Bnew); |
---|
1074 | return(out); |
---|
1075 | } |
---|
1076 | |
---|
1077 | //////////////////////////////////////////////////////////////////////////////////// |
---|
1078 | |
---|
1079 | static proc shortExactPiecesToVdStrict(list C,int d,list #) |
---|
1080 | {/* We transform the short exact pieces from procedure shortExactPieces to V_d- |
---|
1081 | strict short exact sequences. For this, we use Algorithm 3.11 and Lemma 4.2 in |
---|
1082 | [W2].*/ |
---|
1083 | /* If we compute our Groebner bases in the homogenized Weyl algebra, we already |
---|
1084 | compute some resolutions it omit additional Groebner basis computations later |
---|
1085 | on.*/ |
---|
1086 | int s =size(C);int i; int j; |
---|
1087 | string Syzstring="Sres"; |
---|
1088 | intvec v=0:ncols(C[s][1][5]); |
---|
1089 | if (size(#)!=0) |
---|
1090 | { |
---|
1091 | for (i=1; i<=size(#); i++) |
---|
1092 | { |
---|
1093 | if (typeof(#[i])=="string") |
---|
1094 | { |
---|
1095 | Syzstring=#[i]; |
---|
1096 | } |
---|
1097 | if (typeof(#[i])=="intvec") |
---|
1098 | { |
---|
1099 | v=#[i]; |
---|
1100 | } |
---|
1101 | } |
---|
1102 | } |
---|
1103 | list out; |
---|
1104 | list forout; |
---|
1105 | if (Syzstring=="Vdres") |
---|
1106 | { |
---|
1107 | out[s]=list(toVdStrictSequence(C[s][1],d,v, Syzstring,s)); |
---|
1108 | } |
---|
1109 | else |
---|
1110 | { |
---|
1111 | forout=toVdStrictSequence(C[s][1],d,v, Syzstring,s); |
---|
1112 | list resolutionofA=forout[9]; |
---|
1113 | list resolutionofC=forout[10]; |
---|
1114 | forout=delete(forout,10); |
---|
1115 | forout=delete(forout,9); |
---|
1116 | out[s]=list(forout); |
---|
1117 | for (i=1; i<=size(resolutionofC); i++) |
---|
1118 | { |
---|
1119 | out[s][1][5][i+1]=resolutionofC[i];//save the resolutions |
---|
1120 | out[s][1][1][i+1]=resolutionofA[i]; |
---|
1121 | } |
---|
1122 | } |
---|
1123 | out[s][2]=list(list(out[s][1][3][1])); |
---|
1124 | out[s][2][2]=list(unitmat(ncols(out[s][1][3][1]))); |
---|
1125 | out[s][2][3]=list(out[s][1][3][1]); |
---|
1126 | out[s][2][4]=list(list()); |
---|
1127 | out[s][2][5]=list(list()); |
---|
1128 | out[s][2][6]=list(out[s][1][7][1]); |
---|
1129 | out[s][2][7]=list(out[s][2][6][1]); |
---|
1130 | out[s][2][8]=list(list()); |
---|
1131 | list resolutionofD; |
---|
1132 | list resolutionofF; |
---|
1133 | for (i=s-1; i>=2; i--) |
---|
1134 | { |
---|
1135 | C[i][2][5]=out[i+1][1][1][1]; |
---|
1136 | forout=toVdStrictSequences(C[i],d,out[i+1][1][6][1],Syzstring,s); |
---|
1137 | if (Syzstring=="Sres") |
---|
1138 | { |
---|
1139 | resolutionofD=forout[3];//save the resolutions |
---|
1140 | resolutionofF=forout[4]; |
---|
1141 | forout=delete(forout,4); |
---|
1142 | forout=delete(forout,3); |
---|
1143 | } |
---|
1144 | out[i]=forout; |
---|
1145 | if(Syzstring=="Sres") |
---|
1146 | { |
---|
1147 | for (j=2; j<=size(out[i+1][1][1]); j++) |
---|
1148 | { |
---|
1149 | out[i][2][5][j]=out[i+1][1][1][j]; |
---|
1150 | } |
---|
1151 | for (j=1; j<=size(resolutionofD);j++) |
---|
1152 | { |
---|
1153 | out[i][1][1][j+1]=resolutionofD[j]; |
---|
1154 | out[i][1][5][j+1]=resolutionofF[j]; |
---|
1155 | } |
---|
1156 | } |
---|
1157 | } |
---|
1158 | out[1]=list(list());//initalize our list |
---|
1159 | C[1][2][5]=out[2][1][1][1]; |
---|
1160 | /*Compute the last V_d-strict seqeunce*/ |
---|
1161 | if (Syzstring=="Vdres") |
---|
1162 | { |
---|
1163 | out[1][2]=toVdStrictSequence(C[1][2],d,out[2][1][6][1],Syzstring,s,"J_Agiv"); |
---|
1164 | } |
---|
1165 | else |
---|
1166 | { |
---|
1167 | forout=toVdStrictSequence(C[1][2],d,out[2][1][6][1],Syzstring,s,"J_Agiv"); |
---|
1168 | out[1][2]=delete(forout,9); |
---|
1169 | list resolutionofA2=forout[9]; |
---|
1170 | for (i=1; i<=size(out[2][1][1]); i++) |
---|
1171 | { |
---|
1172 | /*put the modules for the resolutions in the right spot*/ |
---|
1173 | out[1][2][5][i]=out[2][1][1][i]; |
---|
1174 | } |
---|
1175 | for (i=1; i<=size(resolutionofA2); i++) |
---|
1176 | { |
---|
1177 | out[1][2][1][i+1]=resolutionofA2[i]; |
---|
1178 | } |
---|
1179 | } |
---|
1180 | out[1][1][3]=list(out[1][2][1][1]); |
---|
1181 | out[1][1][5]=list(out[1][2][1][1]); |
---|
1182 | out[1][1][4]=list(unitmat(ncols(out[1][1][3][1]))); |
---|
1183 | out[1][1][7]=list(out[1][2][6][1]); |
---|
1184 | out[1][1][8]=list(out[1][2][6][1]); |
---|
1185 | out[1][1][1]=list(list()); |
---|
1186 | out[1][1][2]=list(list()); |
---|
1187 | out[1][1][6]=list(list()); |
---|
1188 | if (Syzstring=="Sres") |
---|
1189 | { |
---|
1190 | for (i=1; i<=size(out[1][2][1]); i++) |
---|
1191 | { |
---|
1192 | out[1][1][3][i]=out[1][2][1][i]; |
---|
1193 | out[1][1][5][i]=out[1][2][1][i]; |
---|
1194 | } |
---|
1195 | } |
---|
1196 | list Hi; |
---|
1197 | for (i=1; i<=size(out); i++) |
---|
1198 | { |
---|
1199 | Hi[i]=list(out[i][1][5][1],out[i][1][8][1]); |
---|
1200 | } |
---|
1201 | list outall; |
---|
1202 | outall[1]=out; |
---|
1203 | outall[2]=Hi; |
---|
1204 | return(outall); |
---|
1205 | } |
---|
1206 | |
---|
1207 | //////////////////////////////////////////////////////////////////////////////////// |
---|
1208 | |
---|
1209 | static proc toVdStrictSequence(list C,int n,intvec v,string Syzstring,int si,list #) |
---|
1210 | { |
---|
1211 | /*this is the Algorithm 3.11 in [W2]*/ |
---|
1212 | int omitemptylist; |
---|
1213 | int lengthofres=si+n-1; |
---|
1214 | int i,j,logens; |
---|
1215 | def B=basering; |
---|
1216 | matrix bi=slimgb(transpose(C[5])); |
---|
1217 | /* Computation of a V_d-strict Groebner basis of C[5]: |
---|
1218 | -if Syzstring=="Vdres" this is done using the method of weighted homogenization |
---|
1219 | (Prop. 3.9 [OT]) |
---|
1220 | -else we use the homogenized Weyl algebra for Groebner basis computations |
---|
1221 | (Prop 9.9 [OT]), |
---|
1222 | in this case we already compute someresolutions (Thm. 9.10 [OT]) to omit |
---|
1223 | extra Groebner basis computations later on*/ |
---|
1224 | int nr,nc; |
---|
1225 | intvec i1,i2; |
---|
1226 | if (Syzstring=="Vdres") |
---|
1227 | { |
---|
1228 | if(size(#)==0) |
---|
1229 | { |
---|
1230 | matrix J_C=VdStrictGB(C[5],n,list(v)); |
---|
1231 | } |
---|
1232 | else |
---|
1233 | { |
---|
1234 | matrix J_C=C[5];//C[5] is already a V_d-strict Groebner basis |
---|
1235 | } |
---|
1236 | } |
---|
1237 | else |
---|
1238 | { |
---|
1239 | if (size(#)==0) |
---|
1240 | { |
---|
1241 | matrix MC=C[5]; |
---|
1242 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, v); |
---|
1243 | setring HomWeyl; |
---|
1244 | matrix J_C=fetch(B,MC); |
---|
1245 | J_C=nHomogenize(J_C); |
---|
1246 | /*computation of V_d-strict resolution of C[5]->needed for proc |
---|
1247 | VdstrictDoubleComplexes*/ |
---|
1248 | def ringofSyz=Sres(transpose(J_C),lengthofres); |
---|
1249 | setring ringofSyz; |
---|
1250 | matrix J_C=transpose(matrix(RES[2])); |
---|
1251 | J_C=subst(J_C,h,1); |
---|
1252 | logens=ncols(J_C)+1; |
---|
1253 | matrix zerom; |
---|
1254 | list rofC;//will contain resolution of C |
---|
1255 | for (i=3; i<=n+si+1; i++) |
---|
1256 | { |
---|
1257 | if (size(RES)>=i) |
---|
1258 | { |
---|
1259 | zerom=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i]))); |
---|
1260 | if (RES[i]!=zerom) |
---|
1261 | { |
---|
1262 | rofC[i-2]=(matrix(RES[i])); |
---|
1263 | |
---|
1264 | if (i==3) |
---|
1265 | { |
---|
1266 | if (nrows(rofC[i-2])-logens+1!=nrows(J_C)) |
---|
1267 | { |
---|
1268 | //build the resolution |
---|
1269 | nr=nrows(J_C)+logens-1; |
---|
1270 | nc=ncols(rofC[i-2]); |
---|
1271 | rofC[i-2]=matrix(rofC[i-2],nr,nc); |
---|
1272 | } |
---|
1273 | |
---|
1274 | } |
---|
1275 | if (i!=3) |
---|
1276 | { |
---|
1277 | if (nrows(rofC[i-2])-logens+1!=nrows(rofC[i-3])) |
---|
1278 | { |
---|
1279 | nr=nrows(rofC[i-3])+logens-1; |
---|
1280 | nc=ncols(rofC[i-2]); |
---|
1281 | rofC[i-2]=matrix(rofC[i-2],nr,nc); |
---|
1282 | } |
---|
1283 | } |
---|
1284 | i1=intvec(logens..nrows(rofC[i-2])); |
---|
1285 | i2=intvec(1..ncols(rofC[i-2])); |
---|
1286 | rofC[i-2]=transpose(submat(rofC[i-2],i1,i2)); |
---|
1287 | logens=logens+ncols(rofC[i-2]); |
---|
1288 | rofC[i-2]=subst(rofC[i-2],h,1); |
---|
1289 | } |
---|
1290 | else |
---|
1291 | { |
---|
1292 | rofC[i-2]=list(); |
---|
1293 | } |
---|
1294 | } |
---|
1295 | else |
---|
1296 | { |
---|
1297 | rofC[i-2]=list(); |
---|
1298 | } |
---|
1299 | } |
---|
1300 | if(size(rofC[1])==0) |
---|
1301 | { |
---|
1302 | omitemptylist=1; |
---|
1303 | } |
---|
1304 | setring B; |
---|
1305 | matrix J_C=fetch(ringofSyz,J_C); |
---|
1306 | if (omitemptylist!=1) |
---|
1307 | { |
---|
1308 | list rofC=fetch(ringofSyz,rofC); |
---|
1309 | } |
---|
1310 | omitemptylist=0; |
---|
1311 | kill HomWeyl; |
---|
1312 | kill ringofSyz; |
---|
1313 | } |
---|
1314 | else |
---|
1315 | { |
---|
1316 | matrix J_C=C[5];//C[5] is already a V_d-strict Groebner basis |
---|
1317 | } |
---|
1318 | } |
---|
1319 | /* we compute a V_d-strict Groebner basis for C[3]*/ |
---|
1320 | matrix J_A=C[1]; |
---|
1321 | matrix f_CB=C[4]; |
---|
1322 | matrix f_ACB=transpose(concat(transpose(C[2]),transpose(f_CB))); |
---|
1323 | matrix J_AC=divdr(f_ACB,C[3]); |
---|
1324 | matrix P=matrixLift(J_AC * prodr(ncols(C[1]),ncols(C[5])) ,J_C); |
---|
1325 | list storePi; |
---|
1326 | matrix Pi[1][ncols(J_AC)]; |
---|
1327 | for (i=1; i<=nrows(J_C); i++) |
---|
1328 | { |
---|
1329 | for (j=1; j<=nrows(J_AC);j++) |
---|
1330 | { |
---|
1331 | Pi=Pi+P[i,j]*submat(J_AC,j,intvec(1..ncols(J_AC))); |
---|
1332 | } |
---|
1333 | storePi[i]=Pi; |
---|
1334 | Pi=0; |
---|
1335 | } |
---|
1336 | /*we compute the shift vector for C[1]*/ |
---|
1337 | intvec m_a; |
---|
1338 | list findMin; |
---|
1339 | int comMin; |
---|
1340 | for (i=1; i<=ncols(J_A); i++) |
---|
1341 | { |
---|
1342 | for (j=1; j<=size(storePi);j++) |
---|
1343 | { |
---|
1344 | if (storePi[j][1,i]!=0) |
---|
1345 | { |
---|
1346 | comMin=VdDeg(storePi[j]*prodr(ncols(J_A),ncols(C[5])),n,v); |
---|
1347 | comMin=comMin-VdDeg(storePi[j][1,i],n,intvec(0)); |
---|
1348 | findMin[size(findMin)+1]=comMin; |
---|
1349 | } |
---|
1350 | } |
---|
1351 | if (size(findMin)!=0) |
---|
1352 | { |
---|
1353 | m_a[i]=Min(findMin); |
---|
1354 | findMin=list(); |
---|
1355 | } |
---|
1356 | else |
---|
1357 | { |
---|
1358 | m_a[i]=0; |
---|
1359 | } |
---|
1360 | } |
---|
1361 | matrix zero[ncols(J_A)][ncols(J_C)]; |
---|
1362 | matrix g_AB=concat(unitmat(ncols(J_A)),zero); |
---|
1363 | matrix g_BC= transpose(concat(transpose(zero),transpose(unitmat(ncols(J_C))))); |
---|
1364 | intvec m_b=m_a,v; |
---|
1365 | /* computation of a V_d-strict Groebner basis of C[1] (and resolution if |
---|
1366 | Syzstring=='Vdres') */ |
---|
1367 | if (Syzstring=="Vdres") |
---|
1368 | { |
---|
1369 | J_A=VdStrictGB(J_A,n,m_a); |
---|
1370 | } |
---|
1371 | else |
---|
1372 | { |
---|
1373 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, m_a); |
---|
1374 | setring HomWeyl; |
---|
1375 | matrix J_A=fetch(B,J_A); |
---|
1376 | J_A=nHomogenize(J_A); |
---|
1377 | def ringofSyz=Sres(transpose(J_A),lengthofres); |
---|
1378 | setring ringofSyz; |
---|
1379 | matrix J_A=transpose(matrix(RES[2])); |
---|
1380 | matrix zerom; |
---|
1381 | J_A=subst(J_A,h,1); |
---|
1382 | logens=ncols(J_A)+1; |
---|
1383 | list rofA; |
---|
1384 | for (i=3; i<=n+si+1; i++) |
---|
1385 | { |
---|
1386 | if (size(RES)>=i) |
---|
1387 | { |
---|
1388 | zerom=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i]))); |
---|
1389 | if (RES[i]!=zerom) |
---|
1390 | { |
---|
1391 | rofA[i-2]=matrix(RES[i]);// resolution for C[1] |
---|
1392 | if (i==3) |
---|
1393 | { |
---|
1394 | if (nrows(rofA[i-2])-logens+1!=nrows(J_A)) |
---|
1395 | { |
---|
1396 | nr=nrows(J_A)+logens-1; |
---|
1397 | nc=ncols(rofA[i-2]); |
---|
1398 | rofA[i-2]=matrix(rofA[i-2],nr,nc); |
---|
1399 | } |
---|
1400 | } |
---|
1401 | if (i!=3) |
---|
1402 | { |
---|
1403 | if (nrows(rofA[i-2])-logens+1!=nrows(rofA[i-3])) |
---|
1404 | { |
---|
1405 | nr=nrows(rofA[i-3])+logens-1; |
---|
1406 | nc=ncols(rofA[i-2]); |
---|
1407 | rofA[i-2]=matrix(rofA[i-2],nr,nc); |
---|
1408 | } |
---|
1409 | } |
---|
1410 | i1=intvec(logens..nrows(rofA[i-2])); |
---|
1411 | i2=intvec(1..ncols(rofA[i-2])); |
---|
1412 | rofA[i-2]=transpose(submat(rofA[i-2],i1,i2)); |
---|
1413 | logens=logens+ncols(rofA[i-2]); |
---|
1414 | rofA[i-2]=subst(rofA[i-2],h,1); |
---|
1415 | } |
---|
1416 | else |
---|
1417 | { |
---|
1418 | rofA[i-2]=list(); |
---|
1419 | } |
---|
1420 | } |
---|
1421 | else |
---|
1422 | { |
---|
1423 | rofA[i-2]=list(); |
---|
1424 | } |
---|
1425 | } |
---|
1426 | if(size(rofA[1])==0) |
---|
1427 | { |
---|
1428 | omitemptylist=1; |
---|
1429 | } |
---|
1430 | setring B; |
---|
1431 | J_A=fetch(ringofSyz,J_A); |
---|
1432 | if (omitemptylist!=1) |
---|
1433 | { |
---|
1434 | list rofA=fetch(ringofSyz,rofA); |
---|
1435 | } |
---|
1436 | omitemptylist=0; |
---|
1437 | kill HomWeyl; |
---|
1438 | kill ringofSyz; |
---|
1439 | } |
---|
1440 | J_AC=transpose(storePi[1]); |
---|
1441 | for (i=2; i<= size(storePi); i++) |
---|
1442 | { |
---|
1443 | J_AC=concat(J_AC, transpose(storePi[i])); |
---|
1444 | } |
---|
1445 | J_AC=transpose(concat(transpose(matrix(J_A,nrows(J_A),nrows(J_AC))),J_AC)); |
---|
1446 | list Vdstrict=(list(J_A),list(g_AB),list(J_AC),list(g_BC),list(J_C),list(m_a)); |
---|
1447 | Vdstrict[7]=list(m_b); |
---|
1448 | Vdstrict[8]=list(v); |
---|
1449 | if(Syzstring=="Sres") |
---|
1450 | { |
---|
1451 | Vdstrict[9]=rofA; |
---|
1452 | if(size(#)==0) |
---|
1453 | { |
---|
1454 | Vdstrict[10]=rofC; |
---|
1455 | } |
---|
1456 | } |
---|
1457 | return (Vdstrict); |
---|
1458 | } |
---|
1459 | |
---|
1460 | //////////////////////////////////////////////////////////////////////////////////// |
---|
1461 | |
---|
1462 | static proc toVdStrictSequences (list L,int d,intvec v,string Syzstring,int sizeL) |
---|
1463 | { |
---|
1464 | /* this is Argorithm 3.11 combined with Lemma 4.2 in [W2] for two short exact |
---|
1465 | pieces. |
---|
1466 | We asume that we are given two sequences of the form coker(L[i][1])-> |
---|
1467 | coker(L[i][3])->coker(L[i][5]) with differentials L[i][2] and L[i][4] such |
---|
1468 | that L[1][3]=L[2][1].We are going to transform them to V_d-strict sequences |
---|
1469 | J_D->J_A->J_F and J_A->J_B->J_C*/ |
---|
1470 | int omitemptylist; |
---|
1471 | int lengthofres=sizeL+d-1; |
---|
1472 | int logens; |
---|
1473 | def B=basering; |
---|
1474 | matrix J_F=L[1][5]; |
---|
1475 | matrix J_D=L[1][1]; |
---|
1476 | matrix f_FA=L[1][4]; |
---|
1477 | /*We find new presentations coker(J_DF) and coker(J_DFC) for L[1][4]=L[2][1] |
---|
1478 | and L[2][4],resp. such that ncols(L[i][1])+ncols(L[i][5])=ncols(L[i][3]) */ |
---|
1479 | matrix f_DFA=transpose(concat(transpose(L[1][2]),transpose(f_FA))); |
---|
1480 | matrix J_DF=divdr(f_DFA,L[1][3]);//coker(J_DF) is isomorphic to coker(L[2][1]); |
---|
1481 | matrix J_C=L[2][5]; |
---|
1482 | matrix f_CB=L[2][4]; |
---|
1483 | matrix f_DFCB=transpose(concat(transpose(f_DFA*L[2][2]),transpose(f_CB))); |
---|
1484 | matrix J_DFC=divdr(f_DFCB,L[2][3]);//coker(J_DFC) are coker(L[2][3)]) isomorphic |
---|
1485 | /* find a shift vector on the range of J_F such that the first sequence is |
---|
1486 | exact*/ |
---|
1487 | matrix P=matrixLift(J_DFC*prodr(ncols(J_DF),ncols(L[2][5])),J_C); |
---|
1488 | list storePi; |
---|
1489 | matrix Pi[1][ncols(J_DFC)]; |
---|
1490 | int i; int j; |
---|
1491 | for (i=1; i<=nrows(J_C); i++) |
---|
1492 | { |
---|
1493 | for (j=1; j<=nrows(J_DFC);j++) |
---|
1494 | { |
---|
1495 | Pi=Pi+P[i,j]*submat(J_DFC,j,intvec(1..ncols(J_DFC))); |
---|
1496 | } |
---|
1497 | storePi[i]=Pi; |
---|
1498 | Pi=0; |
---|
1499 | } |
---|
1500 | intvec m_a; |
---|
1501 | list findMin; |
---|
1502 | list noMin; |
---|
1503 | int comMin; |
---|
1504 | int nr,nc; |
---|
1505 | intvec i1,i2; |
---|
1506 | for (i=1; i<=ncols(J_DF); i++) |
---|
1507 | { |
---|
1508 | for (j=1; j<=size(storePi);j++) |
---|
1509 | { |
---|
1510 | if (storePi[j][1,i]!=0) |
---|
1511 | { |
---|
1512 | comMin=VdDeg(storePi[j]*prodr(ncols(J_DF),ncols(J_C)),d,v); |
---|
1513 | comMin=comMin-VdDeg(storePi[j][1,i],d,intvec(0)); |
---|
1514 | findMin[size(findMin)+1]=comMin; |
---|
1515 | } |
---|
1516 | } |
---|
1517 | if (size(findMin)!=0) |
---|
1518 | { |
---|
1519 | m_a[i]=Min(findMin);// shift vector for L[2][1] |
---|
1520 | findMin=list(); |
---|
1521 | noMin[i]=0; |
---|
1522 | } |
---|
1523 | else |
---|
1524 | { |
---|
1525 | noMin[i]=1; |
---|
1526 | } |
---|
1527 | } |
---|
1528 | if (size(m_a) < ncols(J_DF)) |
---|
1529 | { |
---|
1530 | m_a[ncols(J_DF)]=0; |
---|
1531 | } |
---|
1532 | intvec m_f=m_a[ncols(J_D)+1..size(m_a)]; |
---|
1533 | /* Computation of a V_d-strict Groebner basis of J_F=L[1][5]: |
---|
1534 | if Syzstring=="Vdres" this is done using the method of weighted homogenization |
---|
1535 | (Prop. 3.9 [OT]) |
---|
1536 | else we use the homogenized Weyl algerba for Groebner basis computations |
---|
1537 | (Prop 9.9 [OT]), in this case we already compute resolutions |
---|
1538 | (Thm. 9.10 in [OT]) to omit extra Groebner basis computations later on*/ |
---|
1539 | if (Syzstring=="Vdres") |
---|
1540 | { |
---|
1541 | J_F=VdStrictGB(J_F,d,m_f); |
---|
1542 | } |
---|
1543 | else |
---|
1544 | { |
---|
1545 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, m_f); |
---|
1546 | setring HomWeyl; |
---|
1547 | matrix J_F=fetch(B,J_F); |
---|
1548 | J_F=nHomogenize(J_F); |
---|
1549 | def ringofSyz=Sres(transpose(J_F),lengthofres); |
---|
1550 | setring ringofSyz; |
---|
1551 | matrix J_F=transpose(matrix(RES[2])); |
---|
1552 | J_F=subst(J_F,h,1); |
---|
1553 | logens=ncols(J_F)+1; |
---|
1554 | list rofF; |
---|
1555 | for (i=3; i<=d+sizeL+1; i++) |
---|
1556 | { |
---|
1557 | if (size(RES)>=i) |
---|
1558 | { |
---|
1559 | if (RES[i]!=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i])))) |
---|
1560 | { |
---|
1561 | rofF[i-2]=(matrix(RES[i]));// resolution for J_F |
---|
1562 | if (i==3) |
---|
1563 | { |
---|
1564 | if (nrows(rofF[i-2])-logens+1!=nrows(J_F)) |
---|
1565 | { |
---|
1566 | nr=nrows(J_F)+logens-1; |
---|
1567 | nc=ncols(rofF[i-2]); |
---|
1568 | rofF[i-2]=matrix(rofF[i-2],nr,nc); |
---|
1569 | } |
---|
1570 | } |
---|
1571 | if (i!=3) |
---|
1572 | { |
---|
1573 | if (nrows(rofF[i-2])-logens+1!=nrows(rofF[i-3])) |
---|
1574 | { |
---|
1575 | nr=nrows(rofF[i-3])+logens-1; |
---|
1576 | rofF[i-2]=matrix(rofF[i-2],nr,ncols(rofF[i-2])); |
---|
1577 | } |
---|
1578 | } |
---|
1579 | i1=intvec(logens..nrows(rofF[i-2])); |
---|
1580 | i2=intvec(1..ncols(rofF[i-2])); |
---|
1581 | rofF[i-2]=transpose(submat(rofF[i-2],i1,i2)); |
---|
1582 | logens=logens+ncols(rofF[i-2]); |
---|
1583 | rofF[i-2]=subst(rofF[i-2],h,1); |
---|
1584 | } |
---|
1585 | else |
---|
1586 | { |
---|
1587 | rofF[i-2]=list(); |
---|
1588 | } |
---|
1589 | } |
---|
1590 | else |
---|
1591 | { |
---|
1592 | rofF[i-2]=list(); |
---|
1593 | } |
---|
1594 | } |
---|
1595 | if(size(rofF[1])==0) |
---|
1596 | { |
---|
1597 | omitemptylist=1; |
---|
1598 | } |
---|
1599 | setring B; |
---|
1600 | J_F=fetch(ringofSyz,J_F); |
---|
1601 | if (omitemptylist!=1) |
---|
1602 | { |
---|
1603 | list rofF=fetch(ringofSyz,rofF); |
---|
1604 | } |
---|
1605 | omitemptylist=0; |
---|
1606 | kill HomWeyl; |
---|
1607 | kill ringofSyz; |
---|
1608 | } |
---|
1609 | /*find shift vectors on the range of J_D*/ |
---|
1610 | P=matrixLift(J_DF * prodr(ncols(L[1][1]),ncols(L[1][5])) ,J_F); |
---|
1611 | list storePinew; |
---|
1612 | matrix Pidf[1][ncols(J_DF)]; |
---|
1613 | for (i=1; i<=nrows(J_F); i++) |
---|
1614 | { |
---|
1615 | for (j=1; j<=nrows(J_DF);j++) |
---|
1616 | { |
---|
1617 | Pidf=Pidf+P[i,j]*submat(J_DF,j,intvec(1..ncols(J_DF))); |
---|
1618 | } |
---|
1619 | storePinew[i]=Pidf; |
---|
1620 | Pidf=0; |
---|
1621 | } |
---|
1622 | intvec m_d; |
---|
1623 | for (i=1; i<=ncols(J_D); i++) |
---|
1624 | { |
---|
1625 | for (j=1; j<=size(storePinew);j++) |
---|
1626 | { |
---|
1627 | if (storePinew[j][1,i]!=0) |
---|
1628 | { |
---|
1629 | comMin=VdDeg(storePinew[j]*prodr(ncols(J_D),ncols(L[1][5])),d,m_f); |
---|
1630 | comMin=comMin-VdDeg(storePinew[j][1,i],d,intvec(0)); |
---|
1631 | findMin[size(findMin)+1]=comMin; |
---|
1632 | } |
---|
1633 | } |
---|
1634 | if (size(findMin)!=0) |
---|
1635 | { |
---|
1636 | if (noMin[i]==0) |
---|
1637 | { |
---|
1638 | m_d[i]=Min(insert(findMin,m_a[i])); |
---|
1639 | m_a[i]=m_d[i]; |
---|
1640 | } |
---|
1641 | else |
---|
1642 | { |
---|
1643 | m_d[i]=Min(findMin); |
---|
1644 | m_a[i]=m_d[i]; |
---|
1645 | } |
---|
1646 | } |
---|
1647 | else |
---|
1648 | { |
---|
1649 | m_d[i]=m_a[i]; |
---|
1650 | } |
---|
1651 | findMin=list(); |
---|
1652 | } |
---|
1653 | /* compute a V_d-strict Groebner basis (and resolution of J_D if |
---|
1654 | Syzstring!='Vdres') for J_D*/ |
---|
1655 | if (Syzstring=="Vdres") |
---|
1656 | { |
---|
1657 | J_D=VdStrictGB(J_D,d,m_d); |
---|
1658 | } |
---|
1659 | else |
---|
1660 | { |
---|
1661 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, m_d); |
---|
1662 | setring HomWeyl; |
---|
1663 | matrix J_D=fetch(B,J_D); |
---|
1664 | J_D=nHomogenize(J_D); |
---|
1665 | def ringofSyz=Sres(transpose(J_D),lengthofres); |
---|
1666 | setring ringofSyz; |
---|
1667 | matrix J_D=transpose(matrix(RES[2])); |
---|
1668 | J_D=subst(J_D,h,1); |
---|
1669 | logens=ncols(J_D)+1; |
---|
1670 | list rofD; |
---|
1671 | for (i=3; i<=d+sizeL+1; i++) |
---|
1672 | { |
---|
1673 | if (size(RES)>=i) |
---|
1674 | { |
---|
1675 | if (RES[i]!=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i])))) |
---|
1676 | { |
---|
1677 | rofD[i-2]=(matrix(RES[i]));// resolution for J_D |
---|
1678 | if (i==3) |
---|
1679 | { |
---|
1680 | if (nrows(rofD[i-2])-logens+1!=nrows(J_D)) |
---|
1681 | { |
---|
1682 | nr=nrows(J_D)+logens-1; |
---|
1683 | rofD[i-2]=matrix(rofD[i-2],nr,ncols(rofD[i-2])); |
---|
1684 | } |
---|
1685 | } |
---|
1686 | if (i!=3) |
---|
1687 | { |
---|
1688 | if (nrows(rofD[i-2])-logens+1!=nrows(rofD[i-3])) |
---|
1689 | { |
---|
1690 | nr=nrows(rofD[i-3])+logens-1; |
---|
1691 | rofD[i-2]=matrix(rofD[i-2],nr,ncols(rofD[i-2])); |
---|
1692 | } |
---|
1693 | } |
---|
1694 | i1=intvec(logens..nrows(rofD[i-2])); |
---|
1695 | i2=intvec(1..ncols(rofD[i-2])); |
---|
1696 | rofD[i-2]=transpose(submat(rofD[i-2],i1,i2)); |
---|
1697 | logens=logens+ncols(rofD[i-2]); |
---|
1698 | rofD[i-2]=subst(rofD[i-2],h,1); |
---|
1699 | } |
---|
1700 | else |
---|
1701 | { |
---|
1702 | rofD[i-2]=list(); |
---|
1703 | } |
---|
1704 | } |
---|
1705 | else |
---|
1706 | { |
---|
1707 | rofD[i-2]=list(); |
---|
1708 | } |
---|
1709 | } |
---|
1710 | if(size(rofD[1])==0) |
---|
1711 | { |
---|
1712 | omitemptylist=1; |
---|
1713 | } |
---|
1714 | setring B; |
---|
1715 | J_D=fetch(ringofSyz,J_D); |
---|
1716 | if (omitemptylist!=1) |
---|
1717 | { |
---|
1718 | list rofD=fetch(ringofSyz,rofD); |
---|
1719 | } |
---|
1720 | omitemptylist=0; |
---|
1721 | kill HomWeyl; |
---|
1722 | kill ringofSyz; |
---|
1723 | } |
---|
1724 | /* compute new matrices for J_A and J_B such that their rows form a V_d-strict |
---|
1725 | Groebner basis and nrows(J_A)=nrows(J_D)+nrows(J_F) and |
---|
1726 | nrows(J_B)=nrows(J_A)+nrows(J_C)*/ |
---|
1727 | J_DF=transpose(storePinew[1]); |
---|
1728 | for (i=2; i<=nrows(J_F); i++) |
---|
1729 | { |
---|
1730 | J_DF=concat(J_DF,transpose(storePinew[i])); |
---|
1731 | } |
---|
1732 | J_DF=transpose(concat(transpose(matrix(J_D,nrows(J_D),nrows(J_DF))),J_DF)); |
---|
1733 | J_DFC=transpose(storePi[1]); |
---|
1734 | for (i=2; i<=nrows(J_C); i++) |
---|
1735 | { |
---|
1736 | J_DFC=concat(J_DFC,transpose(storePi[i])); |
---|
1737 | } |
---|
1738 | J_DFC=transpose(concat(transpose(matrix(J_DF,nrows(J_DF),nrows(J_DFC))),J_DFC)); |
---|
1739 | intvec m_b=m_a,v; |
---|
1740 | matrix zero[ncols(J_D)][ncols(J_F)]; |
---|
1741 | matrix g_DA=concat(unitmat(ncols(J_D)),zero); |
---|
1742 | matrix g_AF=transpose(concat(transpose(zero),unitmat(ncols(J_F)))); |
---|
1743 | matrix zero1[ncols(J_DF)][ncols(J_C)]; |
---|
1744 | matrix g_AB=concat(unitmat(ncols(J_DF)),zero1); |
---|
1745 | matrix g_BC=transpose(concat(transpose(zero1),unitmat(ncols(J_C)))); |
---|
1746 | list out; |
---|
1747 | out[1]=list(list(J_D),list(g_DA),list(J_DF),list(g_AF),list(J_F)); |
---|
1748 | out[1]=out[1]+list(list(m_d),list(m_a),list(m_f)); |
---|
1749 | out[2]=list(list(J_DF),list(g_AB),list(J_DFC),list(g_BC),list(J_C)); |
---|
1750 | out[2]=out[2]+list(list(m_a),list(m_b),list(v)); |
---|
1751 | if (Syzstring=="Sres") |
---|
1752 | { |
---|
1753 | out[3]=rofD; |
---|
1754 | out[4]=rofF; |
---|
1755 | } |
---|
1756 | return(out); |
---|
1757 | } |
---|
1758 | |
---|
1759 | //////////////////////////////////////////////////////////////////////////////////// |
---|
1760 | |
---|
1761 | static proc VdStrictDoubleComplexes(list L,int d,string Syzstring) |
---|
1762 | { |
---|
1763 | /* We compute V_d-strict resolutions over the V_d-strict short exact pieces from |
---|
1764 | the procedure shortExactPiecesToVdStrict. |
---|
1765 | We use Algorithms 3.14 and 3.15 in [W2]*/ |
---|
1766 | int i,k,c,j,l,totaldeg,comparedegs,SBcom,verk; |
---|
1767 | intvec fordegs; |
---|
1768 | intvec n_b,i1,i2; |
---|
1769 | matrix rem,forML,subm,zerom,unitm,subm2; |
---|
1770 | matrix J_B; |
---|
1771 | list store; |
---|
1772 | int t=size(L)+d; |
---|
1773 | int vd1,vd2,nr,nc; |
---|
1774 | def B=basering; |
---|
1775 | int n=nvars(B) div 2; |
---|
1776 | intvec v; |
---|
1777 | list forhW; |
---|
1778 | if (Syzstring=="Sres") |
---|
1779 | { |
---|
1780 | /*we already computed some of the resolutions in the procedure |
---|
1781 | shortExactPiecesToVdStrict*/ |
---|
1782 | matrix Pold,Pnew,Picombined; intvec containsndeg; matrix Pinew; |
---|
1783 | for (k=1; k<=(size(L)+d-1); k++) |
---|
1784 | { |
---|
1785 | L[1][1][1][k+1]=list(); |
---|
1786 | L[1][1][2][k+1]=list(); |
---|
1787 | L[1][1][6][k+1]=list(); |
---|
1788 | } |
---|
1789 | L[1][1][6][size(L)+d+1]=list(); |
---|
1790 | matrix mem; |
---|
1791 | for (i=2; i<=d+size(L)+1; i++) |
---|
1792 | {; |
---|
1793 | v=0; |
---|
1794 | if(size(L[1][1][3][i-1])!=0) |
---|
1795 | { |
---|
1796 | if(i!=d+size(L)+1) |
---|
1797 | { |
---|
1798 | /*horizontal differential*/ |
---|
1799 | L[1][1][4][i-1]=unitmat(nrows(L[1][1][3][i-1])); |
---|
1800 | } |
---|
1801 | for (j=1; j<=nrows(L[1][1][3][i-1]); j++) |
---|
1802 | { |
---|
1803 | mem=submat(L[1][1][3][i-1],j,intvec(1..ncols(L[1][1][3][i-1]))); |
---|
1804 | v[j]=VdDeg(mem,d,L[1][1][7][i-1]); |
---|
1805 | } |
---|
1806 | L[1][1][7][i]=v;//new shift vector |
---|
1807 | L[1][1][8][i]=v; |
---|
1808 | L[1][2][6][i]=v; |
---|
1809 | } |
---|
1810 | else |
---|
1811 | { |
---|
1812 | if (i!=d+size(L)+1) |
---|
1813 | { |
---|
1814 | L[1][1][4][i-1]=list(); |
---|
1815 | } |
---|
1816 | L[1][1][7][i]=list(); |
---|
1817 | L[1][1][8][i]=list(); |
---|
1818 | L[1][2][6][i]=list(); |
---|
1819 | } |
---|
1820 | } |
---|
1821 | if (size(L[1][1][3][d+size(L)])!=0) |
---|
1822 | { |
---|
1823 | /*horizontal differential*/ |
---|
1824 | L[1][1][4][d+size(L)]=unitmat(nrows(L[1][1][3][d+size(L)])); |
---|
1825 | } |
---|
1826 | else |
---|
1827 | { |
---|
1828 | L[1][1][4][d+size(L)]=list(); |
---|
1829 | } |
---|
1830 | for (k=1; k<size(L); k++) |
---|
1831 | { |
---|
1832 | /* We build a V_d-strict resolution for coker(L[k][2][1][1])-> |
---|
1833 | coker(L[k][2][3][1])->coker(L[k][2][5][1]) using the resolution |
---|
1834 | obtained for coker(L[k][1][3][1]). |
---|
1835 | L[k][2][i][j] will be the jth module in the resolution of L[k][2][i][1] |
---|
1836 | for i=1,3,5. |
---|
1837 | L[k][2][i+5][j] will be the jth shift vector in the resolution of |
---|
1838 | L[k][2][i][1](this holds also for the case Syzstring=="Vdres")*/ |
---|
1839 | for (i=2; i<=d+size(L); i++) |
---|
1840 | { |
---|
1841 | v=0; |
---|
1842 | if (size(L[k][2][5][i-1])!=0) |
---|
1843 | { |
---|
1844 | for (j=1; j<=nrows(L[k][2][5][i-1]); j++) |
---|
1845 | { |
---|
1846 | i1=intvec(1..ncols(L[k][2][5][i-1])); |
---|
1847 | mem=submat(L[k][2][5][i-1],j,i1); |
---|
1848 | v[j]=VdDeg(mem,d,L[k][2][8][i-1]); |
---|
1849 | } |
---|
1850 | /*next shift vector in th resolution of coker(L[k][2][5][1])*/ |
---|
1851 | L[k][2][8][i]=v; |
---|
1852 | } |
---|
1853 | else |
---|
1854 | { |
---|
1855 | L[k][2][8][i]=list(); |
---|
1856 | } |
---|
1857 | /* we build step by step a resolution for coker(L[k][2][5][1]) using |
---|
1858 | the resolutions of coker(L[k][2][1][1]) and coker(L[k][2][5][1])*/ |
---|
1859 | if (size(L[k][2][5][i])!=0) |
---|
1860 | { |
---|
1861 | if (size(L[k][2][1][i])!=0 or size(L[k][2][1][i-1])!=0) |
---|
1862 | { |
---|
1863 | L[k][2][3][i]=transpose(syz(transpose(L[k][2][3][i-1]))); |
---|
1864 | nr= nrows(L[k][2][1][i-1]); |
---|
1865 | nc=ncols(L[k][2][5][i]); |
---|
1866 | Pold=matrixLift(L[k][2][3][i]*prodr(nr,nc), L[k][2][5][i]); |
---|
1867 | matrix Pi[1][ncols(L[k][2][3][i])]; |
---|
1868 | for (l=1; l<=nrows(L[k][2][5][i]); l++) |
---|
1869 | { |
---|
1870 | for (j=1; j<=nrows(L[k][2][3][i]); j++) |
---|
1871 | { |
---|
1872 | i2=intvec(1..ncols(L[k][2][3][i])); |
---|
1873 | Pi=Pi+Pold[l,j]*submat(L[k][2][3][i],j,i2); |
---|
1874 | } |
---|
1875 | if (l==1) |
---|
1876 | { |
---|
1877 | Picombined=transpose(Pi); |
---|
1878 | } |
---|
1879 | else |
---|
1880 | { |
---|
1881 | Picombined=concat(Picombined,transpose(Pi)); |
---|
1882 | } |
---|
1883 | Pi=0; |
---|
1884 | } |
---|
1885 | kill Pi; |
---|
1886 | Picombined=transpose(Picombined); |
---|
1887 | if (size(L[k][2][1][i])!=0) |
---|
1888 | { |
---|
1889 | if (i==2) |
---|
1890 | { |
---|
1891 | containsndeg=(0:ncols(L[k][2][1][1])); |
---|
1892 | } |
---|
1893 | containsndeg=nDeg(L[k][2][1][i-1],containsndeg); |
---|
1894 | forhW=list(L[k][2][6][i],containsndeg); |
---|
1895 | def HomWeyl=makeHomogenizedWeyl(n,forhW); |
---|
1896 | setring HomWeyl; |
---|
1897 | list L=fetch(B,L); |
---|
1898 | matrix M=L[k][2][1][i]; |
---|
1899 | list forM=nHomogenize(M,containsndeg,1); |
---|
1900 | M=forM[1]; |
---|
1901 | totaldeg=forM[2]; |
---|
1902 | kill forM; |
---|
1903 | matrix Maorig=fetch(B,Picombined); |
---|
1904 | matrix Ma=submat(Maorig,(1..nrows(Maorig)),(1..ncols(M))); |
---|
1905 | matrix mem,subm,zerom; |
---|
1906 | matrix Pinew; |
---|
1907 | M=transpose(M); |
---|
1908 | SBcom=0; |
---|
1909 | for (l=1; l<=nrows(Ma); l++) |
---|
1910 | { |
---|
1911 | zerom=matrix(0,1,(ncols(Maorig)-ncols(Ma))); |
---|
1912 | i1=(ncols(Ma)+1..ncols(Maorig)); |
---|
1913 | if (submat(Maorig,l,i1)==zerom) |
---|
1914 | { |
---|
1915 | for (cc=1; cc<=ncols(Ma); cc++) |
---|
1916 | { |
---|
1917 | Maorig[l,cc]=0; |
---|
1918 | } |
---|
1919 | } |
---|
1920 | i2=(ncols(Ma)+1..ncols(Maorig)); |
---|
1921 | i1=(1..ncols(Ma)); |
---|
1922 | if (VdDeg(submat(Maorig,l,i1),d,L[k][2][6][i])> |
---|
1923 | VdDeg(submat(Maorig,l,i2),d,L[k][2][8][i]) and |
---|
1924 | submat(Maorig,l,i1)!=matrix(0,1,ncols(Ma))) |
---|
1925 | { |
---|
1926 | /*V_d-Grad is to big--> we make it smaller using |
---|
1927 | Vdnormal form computations*/ |
---|
1928 | if (SBcom==0) |
---|
1929 | { |
---|
1930 | M=slimgb(M); |
---|
1931 | SBcom=1; |
---|
1932 | } |
---|
1933 | //print("Reduzierung des V_d-Grades(Stelle1)"); |
---|
1934 | i2=(ncols(Ma)+1..ncols(Maorig)); |
---|
1935 | vd1=VdDeg(submat(Maorig,l,i2),d,L[k][2][8][i]); |
---|
1936 | mem=submat(Ma,l,(1..ncols(Ma))); |
---|
1937 | mem=nHomogenize(mem,containsndeg); |
---|
1938 | mem=h^totaldeg*mem; |
---|
1939 | mem=transpose(mem); |
---|
1940 | mem=reduce(mem,M); |
---|
1941 | matrix jt=transpose(subst(mem,h,1)); |
---|
1942 | setring B; |
---|
1943 | matrix jt=fetch(HomWeyl,jt); |
---|
1944 | matrix need=fetch(HomWeyl,Maorig); |
---|
1945 | need=submat(need,l,(1..ncols(need))); |
---|
1946 | i1=L[k][2][6][i]; |
---|
1947 | i2=L[k][2][8][i]; |
---|
1948 | jt=VdNormalForm(need,L[k][2][1][i],d,i1,i2); |
---|
1949 | setring HomWeyl; |
---|
1950 | mem=fetch(B,jt); |
---|
1951 | mem=transpose(mem); |
---|
1952 | if (l==1) |
---|
1953 | { |
---|
1954 | Pinew=mem; |
---|
1955 | } |
---|
1956 | else |
---|
1957 | { |
---|
1958 | Pinew=concat(Pinew,mem); |
---|
1959 | } |
---|
1960 | vd2=VdDeg(transpose(mem),d,L[k][2][6][i]); |
---|
1961 | if (vd2>vd1 and mem!=matrix(0,nrows(mem),ncols(mem))) |
---|
1962 | {//should not happen!! |
---|
1963 | //print("Reduzierung fehlgeschlagen!!(Stelle1)"); |
---|
1964 | } |
---|
1965 | } |
---|
1966 | else |
---|
1967 | { |
---|
1968 | if (l==1) |
---|
1969 | { |
---|
1970 | Pinew=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
1971 | } |
---|
1972 | else |
---|
1973 | { |
---|
1974 | subm=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
1975 | Pinew=concat(Pinew,subm); |
---|
1976 | } |
---|
1977 | } |
---|
1978 | } |
---|
1979 | Pinew=subst(Pinew,h,1); |
---|
1980 | Pinew=transpose(Pinew); |
---|
1981 | setring B; |
---|
1982 | Pinew=fetch(HomWeyl,Pinew); |
---|
1983 | kill HomWeyl; |
---|
1984 | L[k][2][3][i]=concat(Pinew,L[k][2][5][i]); |
---|
1985 | subm=transpose(L[k][2][3][i]); |
---|
1986 | subm=concat(transpose(L[k][2][1][i]),subm); |
---|
1987 | L[k][2][3][i]=transpose(subm); |
---|
1988 | } |
---|
1989 | else |
---|
1990 | { |
---|
1991 | L[k][2][3][i]=Picombined; |
---|
1992 | } |
---|
1993 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
1994 | nr=nrows(L[k][2][1][i-1]); |
---|
1995 | nc=ncols(L[k][2][5][i]); |
---|
1996 | L[k][2][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
1997 | L[k][2][4][i]=prodr(nrows(L[k][2][1][i-1]),nc); |
---|
1998 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
1999 | L[k][2][7][i]=v; |
---|
2000 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
2001 | } |
---|
2002 | else |
---|
2003 | { |
---|
2004 | L[k][2][3][i]=L[k][2][5][i]; |
---|
2005 | L[k][2][2][i]=list(); |
---|
2006 | L[k][2][7][i]=L[k][2][8][i]; |
---|
2007 | L[k][2][4][i]=unitmat(nrows(L[k][2][5][i-1])); |
---|
2008 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
2009 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
2010 | } |
---|
2011 | } |
---|
2012 | else |
---|
2013 | { |
---|
2014 | if (size(L[k][2][1][i])!=0) |
---|
2015 | { |
---|
2016 | if (size(L[k][2][5][i-1])!=0) |
---|
2017 | { |
---|
2018 | nr=nrows(L[k][2][5][i-1]); |
---|
2019 | L[k][2][3][i]=concat(L[k][2][1][i],matrix(0,1,nr)); |
---|
2020 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
2021 | L[k][2][7][i]=v; |
---|
2022 | nc=nrows(L[k][2][1][i-1]); |
---|
2023 | L[k][2][2][i]=concat(unitmat(nc),matrix(0,nc,nr)); |
---|
2024 | L[k][2][4][i]=prodr(nrows(L[k][2][1][i-1]),nr); |
---|
2025 | } |
---|
2026 | else |
---|
2027 | { |
---|
2028 | L[k][2][3][i]=L[k][2][1][i]; |
---|
2029 | L[k][2][7][i]=L[k][2][6][i]; |
---|
2030 | L[k][2][2][i]=unitmat(nrows(L[k][2][1][i-1])); |
---|
2031 | L[k][2][4][i]=list(); |
---|
2032 | } |
---|
2033 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
2034 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
2035 | } |
---|
2036 | else |
---|
2037 | { |
---|
2038 | L[k][2][3][i]=list(); |
---|
2039 | if (size(L[k][2][6][i])!=0) |
---|
2040 | { |
---|
2041 | if (size(L[k][2][8][i])!=0) |
---|
2042 | { |
---|
2043 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
2044 | L[k][2][7][i]=v; |
---|
2045 | nr=nrows(L[k][2][1][i-1]); |
---|
2046 | nc=nrows(L[k][2][5][i-1]); |
---|
2047 | L[k][2][2][i]=concat(unitmat(nc),matrix(0,nr,nc)); |
---|
2048 | L[k][2][4][i]=prodr(nr,nrows(L[k][2][5][i-1])); |
---|
2049 | } |
---|
2050 | else |
---|
2051 | { |
---|
2052 | L[k][2][7][i]=L[k][2][6][i]; |
---|
2053 | L[k][2][2][i]=unitmat(nrows(L[k][2][1][i-1])); |
---|
2054 | L[k][2][4][i]=list(); |
---|
2055 | } |
---|
2056 | } |
---|
2057 | else |
---|
2058 | { |
---|
2059 | if (size(L[k][2][8][i])!=0) |
---|
2060 | { |
---|
2061 | L[k][2][7][i]=L[k][2][8][i]; |
---|
2062 | L[k][2][2][i]=list(); |
---|
2063 | L[k][2][4][i]=unitmat(nrows(L[k][2][5][i-1])); |
---|
2064 | } |
---|
2065 | else |
---|
2066 | { |
---|
2067 | L[k][2][7][i]=list(); |
---|
2068 | L[k][2][2][i]=list(); |
---|
2069 | L[k][2][4][i]=list(); |
---|
2070 | } |
---|
2071 | } |
---|
2072 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
2073 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
2074 | } |
---|
2075 | } |
---|
2076 | } |
---|
2077 | i=d+size(L)+1; |
---|
2078 | v=0; |
---|
2079 | if (size(L[k][2][5][i-1])!=0) |
---|
2080 | { |
---|
2081 | for (j=1; j<=nrows(L[k][2][5][i-1]); j++) |
---|
2082 | { |
---|
2083 | mem=submat(L[k][2][5][i-1],j,intvec(1..ncols(L[k][2][5][i-1]))); |
---|
2084 | v[j]=VdDeg(mem,d,L[k][2][8][i-1]); |
---|
2085 | } |
---|
2086 | L[k][2][8][i]=v; |
---|
2087 | if (size(L[k][2][6][i])!=0) |
---|
2088 | { |
---|
2089 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
2090 | L[k][2][7][i]=v; |
---|
2091 | } |
---|
2092 | else |
---|
2093 | { |
---|
2094 | L[k][2][7][i]=L[k][2][8][i]; |
---|
2095 | } |
---|
2096 | } |
---|
2097 | else |
---|
2098 | { |
---|
2099 | L[k][2][8][i]=list(); |
---|
2100 | L[k][2][7][i]=L[k][2][6][i]; |
---|
2101 | } |
---|
2102 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
2103 | /* now we build V_d-strict resolutions for the sequences |
---|
2104 | coker(L[k+1][1][1][1])->coker(L[k+1][1][3][1])->coker(L[k+1][1][5][i]) |
---|
2105 | using the resolutions for coker(L[k][2][5][1]) we just obtained |
---|
2106 | (works exactly the same as above)*/ |
---|
2107 | for (i=2; i<=d+size(L); i++) |
---|
2108 | { |
---|
2109 | v=0; |
---|
2110 | if (size(L[k+1][1][5][i-1])!=0) |
---|
2111 | { |
---|
2112 | for (j=1; j<=nrows(L[k+1][1][5][i-1]); j++) |
---|
2113 | { |
---|
2114 | i1=intvec(1..ncols(L[k+1][1][5][i-1])); |
---|
2115 | mem=submat(L[k+1][1][5][i-1],j,i1); |
---|
2116 | v[j]=VdDeg(mem,d,L[k+1][1][8][i-1]); |
---|
2117 | } |
---|
2118 | L[k+1][1][8][i]=v; |
---|
2119 | } |
---|
2120 | else |
---|
2121 | { |
---|
2122 | L[k+1][1][8][i]=list(); |
---|
2123 | } |
---|
2124 | if (size(L[k+1][1][5][i])!=0) |
---|
2125 | { |
---|
2126 | if (size(L[k+1][1][1][i])!=0 or size(L[k+1][1][1][i-1])!=0) |
---|
2127 | { |
---|
2128 | L[k+1][1][3][i]=transpose(syz(transpose(L[k+1][1][3][i-1]))); |
---|
2129 | nr=nrows(L[k+1][1][1][i-1]); |
---|
2130 | nc=ncols(L[k+1][1][5][i]); |
---|
2131 | Pold=matrixLift(L[k+1][1][3][i]*prodr(nr,nc),L[k+1][1][5][i]); |
---|
2132 | matrix Pi[1][ncols(L[k+1][1][3][i])]; |
---|
2133 | for (l=1; l<=nrows(L[k+1][1][5][i]); l++) |
---|
2134 | { |
---|
2135 | for (j=1; j<=nrows(L[k+1][1][3][i]); j++) |
---|
2136 | { |
---|
2137 | i2=intvec(1..ncols(L[k+1][1][3][i])); |
---|
2138 | Pi=Pi+Pold[l,j]*submat(L[k+1][1][3][i],j,i2); |
---|
2139 | } |
---|
2140 | if (l==1) |
---|
2141 | { |
---|
2142 | Picombined=transpose(Pi); |
---|
2143 | } |
---|
2144 | else |
---|
2145 | { |
---|
2146 | Picombined=concat(Picombined,transpose(Pi)); |
---|
2147 | } |
---|
2148 | Pi=0; |
---|
2149 | } |
---|
2150 | kill Pi; |
---|
2151 | Picombined=transpose(Picombined); |
---|
2152 | if(size(L[k+1][1][1][i])!=0) |
---|
2153 | { |
---|
2154 | if (i==2) |
---|
2155 | { |
---|
2156 | containsndeg=(0:ncols(L[k+1][1][1][i-1])); |
---|
2157 | } |
---|
2158 | containsndeg=nDeg(L[k+1][1][1][i-1],containsndeg); |
---|
2159 | forhW=list(L[k+1][1][6][i], containsndeg); |
---|
2160 | def HomWeyl=makeHomogenizedWeyl(n,forhW); |
---|
2161 | setring HomWeyl; |
---|
2162 | list L=fetch(B,L); |
---|
2163 | matrix M=L[k+1][1][1][i]; |
---|
2164 | list forM=nHomogenize(M,containsndeg,1); |
---|
2165 | M=forM[1]; |
---|
2166 | totaldeg=forM[2]; |
---|
2167 | kill forM; |
---|
2168 | matrix Maorig=fetch(B,Picombined); |
---|
2169 | matrix Ma=submat(Maorig,(1..nrows(Maorig)),(1..ncols(M))); |
---|
2170 | Ma=nHomogenize(Ma,containsndeg); |
---|
2171 | matrix mem,subm,zerom,subm2; |
---|
2172 | matrix Pinew; |
---|
2173 | M=transpose(M); |
---|
2174 | SBcom=0; |
---|
2175 | for (l=1; l<=nrows(Ma); l++) |
---|
2176 | { |
---|
2177 | i2=(ncols(Ma)+1..ncols(Maorig)); |
---|
2178 | nc=ncols(Maorig)-ncols(Ma); |
---|
2179 | if (submat(Maorig,l,i2)==matrix(0,1,nc)) |
---|
2180 | { |
---|
2181 | for (cc=1; cc<=ncols(Ma); cc++) |
---|
2182 | { |
---|
2183 | Maorig[l,cc]=0; |
---|
2184 | } |
---|
2185 | } |
---|
2186 | i1=(1..ncols(Ma)); |
---|
2187 | i2=L[k+1][1][8][i]; |
---|
2188 | subm=submat(Maorig,l,i1); |
---|
2189 | subm2=submat(Maorig,l,(ncols(Ma)+1..ncols(Maorig))); |
---|
2190 | if (VdDeg(subm,d,L[k+1][1][6][i])>VdDeg(subm2,d,i2) |
---|
2191 | and subm!=matrix(0,1,ncols(Ma))) |
---|
2192 | { |
---|
2193 | //print("Reduzierung des Vd-Grades (Stelle2)"); |
---|
2194 | if (SBcom==0) |
---|
2195 | { |
---|
2196 | M=slimgb(M); |
---|
2197 | SBcom=1; |
---|
2198 | } |
---|
2199 | vd1=VdDeg(subm2,d,L[k+1][1][8][i]); |
---|
2200 | mem=submat(Ma,l,(1..ncols(Ma))); |
---|
2201 | mem=nHomogenize(mem,containsndeg); |
---|
2202 | mem=h^totaldeg*mem; |
---|
2203 | mem=transpose(mem); |
---|
2204 | mem=reduce(mem,M); |
---|
2205 | if (l==1) |
---|
2206 | { |
---|
2207 | Pinew=mem; |
---|
2208 | } |
---|
2209 | else |
---|
2210 | { |
---|
2211 | Pinew=concat(Pinew,mem); |
---|
2212 | } |
---|
2213 | vd2=VdDeg(transpose(mem),d,L[k+1][1][6][i]); |
---|
2214 | if (vd2>vd1 and mem!=matrix(0,nrows(mem),ncols(mem))) |
---|
2215 | {//should not happen |
---|
2216 | //print("Reduzierung fehlgeschlagen!!!!(Stelle2)"); |
---|
2217 | } |
---|
2218 | } |
---|
2219 | else |
---|
2220 | { |
---|
2221 | if (l==1) |
---|
2222 | { |
---|
2223 | Pinew=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
2224 | } |
---|
2225 | else |
---|
2226 | { |
---|
2227 | subm=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
2228 | Pinew=concat(Pinew,subm); |
---|
2229 | } |
---|
2230 | } |
---|
2231 | } |
---|
2232 | Pinew=subst(Pinew,h,1); |
---|
2233 | Pinew=transpose(Pinew); |
---|
2234 | setring B; |
---|
2235 | Pinew=fetch(HomWeyl,Pinew); |
---|
2236 | kill HomWeyl; |
---|
2237 | L[k+1][1][3][i]=concat(Pinew,L[k+1][1][5][i]); |
---|
2238 | subm=transpose(L[k+1][1][1][i]); |
---|
2239 | subm2=transpose(L[k+1][1][3][i]); |
---|
2240 | L[k+1][1][3][i]=transpose(concat(subm,subm2)); |
---|
2241 | } |
---|
2242 | else |
---|
2243 | { |
---|
2244 | L[k+1][1][3][i]=Picombined; |
---|
2245 | } |
---|
2246 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
2247 | nr=nrows(L[k+1][1][1][i-1]); |
---|
2248 | nc=ncols(L[k+1][1][5][i]); |
---|
2249 | L[k+1][1][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
2250 | L[k+1][1][4][i]=prodr(nr,nc); |
---|
2251 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
2252 | L[k+1][1][7][i]=v; |
---|
2253 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
2254 | } |
---|
2255 | else |
---|
2256 | { |
---|
2257 | L[k+1][1][3][i]=L[k+1][1][5][i]; |
---|
2258 | L[k+1][1][2][i]=list(); |
---|
2259 | L[k+1][1][4][i]=unitmat(nrows(L[k+1][1][5][i-1])); |
---|
2260 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
2261 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
2262 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
2263 | } |
---|
2264 | } |
---|
2265 | else |
---|
2266 | { |
---|
2267 | if (size(L[k+1][1][1][i])!=0) |
---|
2268 | { |
---|
2269 | if (size(L[k+1][1][5][i-1])!=0) |
---|
2270 | { |
---|
2271 | zerom=matrix(0,1,nrows(L[k+1][1][5][i-1])); |
---|
2272 | L[k+1][1][3][i]=concat(L[k+1][1][1][i],zerom); |
---|
2273 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
2274 | L[k+1][1][7][i]=v; |
---|
2275 | nr=nrows(L[k+1][1][1][i-1]); |
---|
2276 | nc=nrows(L[k+1][1][5][i-1]); |
---|
2277 | L[k+1][1][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
2278 | L[k+1][1][4][i]=prodr(nr,nc); |
---|
2279 | } |
---|
2280 | else |
---|
2281 | { |
---|
2282 | L[k+1][1][3][i]=L[k+1][1][1][i]; |
---|
2283 | L[k+1][1][7][i]=L[k+1][1][6][i]; |
---|
2284 | L[k+1][1][2][i]=unitmat(nrows(L[k+1][1][1][i-1])); |
---|
2285 | L[k+1][1][4][i]=list(); |
---|
2286 | } |
---|
2287 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
2288 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
2289 | } |
---|
2290 | else |
---|
2291 | { |
---|
2292 | L[k+1][1][3][i]=list(); |
---|
2293 | if (size(L[k+1][1][6][i])!=0) |
---|
2294 | { |
---|
2295 | if (size(L[k+1][1][8][i])!=0) |
---|
2296 | { |
---|
2297 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
2298 | L[k+1][1][7][i]=v; |
---|
2299 | nr=nrows(L[k+1][1][1][i-1]); |
---|
2300 | nc=nrows(L[k+1][1][5][i-1]); |
---|
2301 | L[k+1][1][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
2302 | L[k+1][1][4][i]=prodr(nr,nrows(L[k+1][1][5][i-1])); |
---|
2303 | } |
---|
2304 | else |
---|
2305 | { |
---|
2306 | L[k+1][1][7][i]=L[k+1][1][6][i]; |
---|
2307 | L[k+1][1][2][i]=unitmat(nrows(L[k+1][1][1][i-1])); |
---|
2308 | L[k+1][1][4][i]=list(); |
---|
2309 | } |
---|
2310 | } |
---|
2311 | else |
---|
2312 | { |
---|
2313 | if (size(L[k+1][1][8][i])!=0) |
---|
2314 | { |
---|
2315 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
2316 | L[k+1][1][2][i]=list(); |
---|
2317 | L[k+1][1][4][i]=unitmat(nrows(L[k+1][1][5][i-1])); |
---|
2318 | } |
---|
2319 | else |
---|
2320 | { |
---|
2321 | L[k+1][1][7][i]=list(); |
---|
2322 | L[k+1][1][2][i]=list(); |
---|
2323 | L[k+1][1][4][i]=list(); |
---|
2324 | } |
---|
2325 | } |
---|
2326 | |
---|
2327 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
2328 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
2329 | } |
---|
2330 | } |
---|
2331 | } |
---|
2332 | i=size(L)+d+1; |
---|
2333 | v=0; |
---|
2334 | if (size(L[k+1][1][5][i-1])!=0) |
---|
2335 | { |
---|
2336 | for (j=1; j<=nrows(L[k+1][1][5][i-1]); j++) |
---|
2337 | { |
---|
2338 | i1=intvec(1..ncols(L[k+1][1][5][i-1])); |
---|
2339 | mem=submat(L[k+1][1][5][i-1],j,i1); |
---|
2340 | v[j]=VdDeg(mem,d,L[k+1][1][8][i-1]); |
---|
2341 | } |
---|
2342 | L[k+1][1][8][i]=v; |
---|
2343 | if (size(L[k+1][1][6][i])!=0) |
---|
2344 | { |
---|
2345 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
2346 | L[k+1][1][7][i]=v; |
---|
2347 | } |
---|
2348 | else |
---|
2349 | { |
---|
2350 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
2351 | } |
---|
2352 | } |
---|
2353 | else |
---|
2354 | { |
---|
2355 | L[k+1][1][8][i]=list(); |
---|
2356 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
2357 | } |
---|
2358 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
2359 | } |
---|
2360 | for (k=1; k<=(size(L)+d); k++) |
---|
2361 | { |
---|
2362 | L[size(L)][2][5][k]=list(); |
---|
2363 | L[size(L)][2][4][k]=list(); |
---|
2364 | L[size(L)][2][8][k]=list(); |
---|
2365 | L[size(L)][2][3][k]=L[size(L)][2][1][k]; |
---|
2366 | L[size(L)][2][7][k]=L[size(L)][2][6][k]; |
---|
2367 | } |
---|
2368 | L[size(L)][2][7][size(L)+d+1]=L[size(L)][2][6][size(L)+d+1]; |
---|
2369 | L[size(L)][2][8][size(L)+d+1]=list(); |
---|
2370 | /* building the resolution of the last short exact piece*/ |
---|
2371 | for (i=2; i<=d+size(L); i++) |
---|
2372 | { |
---|
2373 | v=0; |
---|
2374 | if(size(L[size(L)][2][1][i-1])!=0) |
---|
2375 | { |
---|
2376 | L[size(L)][2][2][i]=unitmat(nrows(L[size(L)][2][1][i-1])); |
---|
2377 | } |
---|
2378 | else |
---|
2379 | { |
---|
2380 | L[size(L)][2][2][i-1]=list(); |
---|
2381 | } |
---|
2382 | } |
---|
2383 | return(L); |
---|
2384 | } |
---|
2385 | /*case Syzstring=="Vdres"*/ |
---|
2386 | list forVd; |
---|
2387 | for (k=1; k<=(size(L)+d); k++)//????? |
---|
2388 | { |
---|
2389 | /* we compute a V_d-strict resolution for the first short exact piece*/ |
---|
2390 | L[1][1][1][k+1]=list(); |
---|
2391 | L[1][1][2][k+1]=list(); |
---|
2392 | L[1][1][6][k+1]=list(); |
---|
2393 | if (size(L[1][1][3][k])!=0) |
---|
2394 | { |
---|
2395 | for (i=1; i<=nrows(L[1][1][3][k]); i++) |
---|
2396 | { |
---|
2397 | rem=submat(L[1][1][3][k],i,(1..ncols(L[1][1][3][k]))); |
---|
2398 | n_b[i]=VdDeg(rem,d,L[1][1][7][k]); |
---|
2399 | } |
---|
2400 | J_B=transpose(syz(transpose(L[1][1][3][k]))); |
---|
2401 | L[1][1][7][k+1]=n_b; |
---|
2402 | L[1][1][8][k+1]=n_b; |
---|
2403 | L[1][1][4][k+1]=unitmat(nrows(L[1][1][3][k])); |
---|
2404 | if (J_B!=matrix(0,nrows(J_B),ncols(J_B))) |
---|
2405 | { |
---|
2406 | J_B=VdStrictGB(J_B,d,n_b); |
---|
2407 | L[1][1][3][k+1]=J_B; |
---|
2408 | L[1][1][5][k+1]=J_B; |
---|
2409 | } |
---|
2410 | else |
---|
2411 | { |
---|
2412 | L[1][1][3][k+1]=list(); |
---|
2413 | L[1][1][5][k+1]=list(); |
---|
2414 | } |
---|
2415 | n_b=0; |
---|
2416 | } |
---|
2417 | else |
---|
2418 | { |
---|
2419 | L[1][1][3][k+1]=list(); |
---|
2420 | L[1][1][5][k+1]=list(); |
---|
2421 | L[1][1][7][k+1]=list(); |
---|
2422 | L[1][1][8][k+1]=list(); |
---|
2423 | L[1][1][4][k+1]=list(); |
---|
2424 | } |
---|
2425 | /* we compute step by step V_d-strict resolutions over |
---|
2426 | coker(L[i][2][1][1])->coker(L[i][2][3][1])->coker(L[i][2][1][5]) |
---|
2427 | and coker(L[i+1][1][1][1])->coker(L[i+1][1][3][1])->coker(L[i+1][1][1][5]) |
---|
2428 | using the already computed resolutions for coker(L[i][2][1][1])= |
---|
2429 | coker(L[i][1][3][1]) and coker(L[i+1][1][1][1])=coker(L[i][2][5][1])*/ |
---|
2430 | for (i=1; i<size(L); i++) |
---|
2431 | { |
---|
2432 | forVd[1]=L[i][2][1][k]; |
---|
2433 | forVd[2]=L[i][2][2][k]; |
---|
2434 | forVd[3]=L[i][2][3][k]; |
---|
2435 | forVd[4]=L[i][2][4][k]; |
---|
2436 | forVd[5]=L[i][2][5][k]; |
---|
2437 | forVd[6]=L[i][2][6][k]; |
---|
2438 | forVd[7]=L[i][2][7][k]; |
---|
2439 | forVd[8]=L[i][2][8][k]; |
---|
2440 | store=toVdStrict2x3Complex(forVd,d,L[i][1][3][k+1],L[i][1][7][k+1]); |
---|
2441 | for (j=1; j<=8; j++) |
---|
2442 | { |
---|
2443 | L[i][2][j][k+1]=store[j]; |
---|
2444 | } |
---|
2445 | forVd[1]=L[i+1][1][1][k]; |
---|
2446 | forVd[2]=L[i+1][1][2][k]; |
---|
2447 | forVd[3]=L[i+1][1][3][k]; |
---|
2448 | forVd[4]=L[i+1][1][4][k]; |
---|
2449 | forVd[5]=L[i+1][1][5][k]; |
---|
2450 | forVd[6]=L[i+1][1][6][k]; |
---|
2451 | forVd[7]=L[i+1][1][7][k]; |
---|
2452 | forVd[8]=L[i+1][1][8][k]; |
---|
2453 | store=toVdStrict2x3Complex(forVd,d,L[i][2][5][k+1],L[i][2][8][k+1]); |
---|
2454 | for (j=1; j<=8; j++) |
---|
2455 | { |
---|
2456 | L[i+1][1][j][k+1]=store[j]; |
---|
2457 | } |
---|
2458 | } |
---|
2459 | if (size(L[size(L)][1][7][k+1])!=0) |
---|
2460 | { |
---|
2461 | L[size(L)][2][4][k+1]=list(); |
---|
2462 | L[size(L)][2][5][k+1]=list(); |
---|
2463 | L[size(L)][2][6][k+1]=L[size(L)][1][7][k+1]; |
---|
2464 | L[size(L)][2][7][k+1]=L[size(L)][1][7][k+1]; |
---|
2465 | L[size(L)][2][8][k+1]=list(); |
---|
2466 | L[size(L)][2][2][k+1]=unitmat(size(L[size(L)][1][7][k+1])); |
---|
2467 | if (size(L[size(L)][1][3][k+1])!=0) |
---|
2468 | { |
---|
2469 | L[size(L)][2][1][k+1]=L[size(L)][1][3][k+1]; |
---|
2470 | L[size(L)][2][3][k+1]=L[size(L)][1][3][k+1]; |
---|
2471 | } |
---|
2472 | else |
---|
2473 | { |
---|
2474 | L[size(L)][2][1][k+1]=list(); |
---|
2475 | L[size(L)][2][3][k+1]=list(); |
---|
2476 | } |
---|
2477 | } |
---|
2478 | else |
---|
2479 | { |
---|
2480 | for (j=1; j<=8; j++) |
---|
2481 | { |
---|
2482 | L[size(L)][2][j][k+1]=list(); |
---|
2483 | } |
---|
2484 | } |
---|
2485 | } |
---|
2486 | k=t; |
---|
2487 | intvec n_c; |
---|
2488 | intvec vn_b; |
---|
2489 | list N_b; |
---|
2490 | int n; |
---|
2491 | /*computation of the shift vectors*/ |
---|
2492 | for (i=1; i<=size(L); i++) |
---|
2493 | { |
---|
2494 | for (n=1; n<=2; n++) |
---|
2495 | { |
---|
2496 | if (i==1 and n==1) |
---|
2497 | { |
---|
2498 | L[i][n][6][k+1]=list(); |
---|
2499 | } |
---|
2500 | else |
---|
2501 | { |
---|
2502 | if (n==1) |
---|
2503 | { |
---|
2504 | L[i][1][6][k+1]=L[i-1][2][8][k+1]; |
---|
2505 | } |
---|
2506 | else |
---|
2507 | { |
---|
2508 | L[i][2][6][k+1]=L[i][1][7][k+1]; |
---|
2509 | } |
---|
2510 | } |
---|
2511 | N_b[1]=L[i][n][6][k+1]; |
---|
2512 | if (size(L[i][n][5][k])!=0) |
---|
2513 | { |
---|
2514 | for (j=1; j<=nrows(L[i][n][5][k]); j++) |
---|
2515 | { |
---|
2516 | rem=submat(L[i][n][5][k],j,(1..ncols(L[i][n][5][k]))); |
---|
2517 | n_c[j]=VdDeg(rem,d,L[i][n][8][k]); |
---|
2518 | } |
---|
2519 | L[i][n][8][k+1]=n_c; |
---|
2520 | } |
---|
2521 | else |
---|
2522 | { |
---|
2523 | L[i][n][8][k+1]=list(); |
---|
2524 | } |
---|
2525 | N_b[2]=L[i][n][8][k+1]; |
---|
2526 | n_c=0; |
---|
2527 | if (size(N_b[1])!=0) |
---|
2528 | { |
---|
2529 | vn_b=N_b[1]; |
---|
2530 | if (size(N_b[2])!=0) |
---|
2531 | { |
---|
2532 | vn_b=vn_b,N_b[2]; |
---|
2533 | } |
---|
2534 | L[i][n][7][k+1]=vn_b; |
---|
2535 | } |
---|
2536 | else |
---|
2537 | { |
---|
2538 | if (size(N_b[2])!=0) |
---|
2539 | { |
---|
2540 | L[i][n][7][k+1]=N_b[2]; |
---|
2541 | } |
---|
2542 | else |
---|
2543 | { |
---|
2544 | L[i][n][7][k+1]=list(); |
---|
2545 | } |
---|
2546 | } |
---|
2547 | } |
---|
2548 | } |
---|
2549 | return(L); |
---|
2550 | } |
---|
2551 | |
---|
2552 | //////////////////////////////////////////////////////////////////////////////////// |
---|
2553 | |
---|
2554 | static proc toVdStrict2x3Complex(list L,int d,list #) |
---|
2555 | { |
---|
2556 | /* We build a one-step free resolution over a V_d-strict short exact piece |
---|
2557 | (Algorithm 3.14 in [W2]). |
---|
2558 | This procedure is called from the procedure VdStrictDoubleComplexes |
---|
2559 | if Syzstring=='Vdres'*/ |
---|
2560 | matrix rem; |
---|
2561 | int i,j,cc; |
---|
2562 | list J_A=list(list()); |
---|
2563 | list J_B=list(list()); |
---|
2564 | list J_C=list(list()); |
---|
2565 | list g_AB=list(list()); |
---|
2566 | list g_BC=list(list()); |
---|
2567 | list n_a=list(list()); |
---|
2568 | list n_b=list(list()); |
---|
2569 | list n_c=list(list()); |
---|
2570 | intvec n_b1; |
---|
2571 | matrix fromnf; |
---|
2572 | intvec i1,i2; |
---|
2573 | /* compute a one step V_d-strict resolution for L[5]*/ |
---|
2574 | if (size(L[5])!=0) |
---|
2575 | { |
---|
2576 | intvec n_c1; |
---|
2577 | for (i=1; i<=nrows(L[5]); i++) |
---|
2578 | { |
---|
2579 | rem=submat(L[5],i,intvec(1..ncols(L[5]))); |
---|
2580 | n_c1[i]=VdDeg(rem,d, L[8]);//new shift vector |
---|
2581 | } |
---|
2582 | n_c[1]=n_c1; |
---|
2583 | J_C[1]=transpose(syz(transpose(L[5]))); |
---|
2584 | if (J_C[1]!=matrix(0,nrows(J_C[1]),ncols(J_C[1]))) |
---|
2585 | { |
---|
2586 | J_C[1]=VdStrictGB(J_C[1],d,n_c1); |
---|
2587 | if (size(#[2])!=0)// new shift vector for the resolution of L[1] |
---|
2588 | { |
---|
2589 | n_a[1]=#[2]; |
---|
2590 | n_b1=n_a[1],n_c[1]; |
---|
2591 | n_b[1]=n_b1; |
---|
2592 | matrix zero[nrows(L[1])][nrows(L[5])]; |
---|
2593 | g_AB=concat(unitmat(nrows(L[1])),matrix(0,nrows(L[1]),nrows(L[5]))); |
---|
2594 | if (size(#[1])!=0) |
---|
2595 | { |
---|
2596 | J_A=#[1];// one step V_d-strict resolution for L[1] |
---|
2597 | /* use resolutions of L[1] and L[5] to build a resolution for |
---|
2598 | L[3]*/ |
---|
2599 | J_B[1]=transpose(matrix(syz(transpose(L[3])))); |
---|
2600 | matrix P=matrixLift(J_B[1]*prodr(nrows(L[1]),nrows(L[5])),J_C[1]); |
---|
2601 | matrix Pi[1][ncols(J_B[1])]; |
---|
2602 | matrix Picombined; |
---|
2603 | for (i=1; i<=nrows(J_C[1]); i++) |
---|
2604 | { |
---|
2605 | for (j=1; j<=nrows(J_B[1]);j++) |
---|
2606 | { |
---|
2607 | Pi=Pi+P[i,j]*submat(J_B[1],j,intvec(1..ncols(J_B[1]))); |
---|
2608 | } |
---|
2609 | if (i==1) |
---|
2610 | { |
---|
2611 | Picombined=transpose(Pi); |
---|
2612 | } |
---|
2613 | else |
---|
2614 | { |
---|
2615 | Picombined=concat(Picombined,transpose(Pi)); |
---|
2616 | } |
---|
2617 | Pi=0; |
---|
2618 | } |
---|
2619 | Picombined=transpose(Picombined); |
---|
2620 | fromnf=VdNormalForm(Picombined,J_A[1],d,n_a[1],n_c[1]); |
---|
2621 | i1=intvec(1..nrows(Picombined)); |
---|
2622 | i2=intvec((ncols(J_A[1])+1)..ncols(Picombined)); |
---|
2623 | Picombined=concat(fromnf,submat(Picombined,i1,i2)); |
---|
2624 | J_B[1]=transpose(matrix(J_A[1],nrows(J_A[1]),ncols(J_B[1]))); |
---|
2625 | J_B[1]=transpose(concat(J_B[1],transpose(Picombined))); |
---|
2626 | g_BC=transpose(concat(transpose(zero),unitmat(nrows(L[5])))); |
---|
2627 | } |
---|
2628 | else//L[1] is already a resolution |
---|
2629 | { |
---|
2630 | //compute a resolution for L[3] |
---|
2631 | J_B=transpose(matrix(syz(transpose(L[3])))); |
---|
2632 | matrix P=matrixLift(J_B[1]*prodr(nrows(L[1]),nrows(L[5])),J_C[1]); |
---|
2633 | matrix Pi[1][ncols(J_B[1])]; |
---|
2634 | matrix Picombined; |
---|
2635 | for (i=1; i<=nrows(J_C[1]); i++) |
---|
2636 | { |
---|
2637 | for (j=1; j<=nrows(J_B[1]);j++) |
---|
2638 | { |
---|
2639 | Pi=Pi+P[i,j]*submat(J_B[1],j,intvec(1..ncols(J_B[1]))); |
---|
2640 | } |
---|
2641 | if (i==1) |
---|
2642 | { |
---|
2643 | Picombined=transpose(Pi); |
---|
2644 | } |
---|
2645 | else |
---|
2646 | { |
---|
2647 | Picombined=concat(Picombined,transpose(Pi)); |
---|
2648 | } |
---|
2649 | Pi=0; |
---|
2650 | } |
---|
2651 | Picombined=transpose(Picombined); |
---|
2652 | J_B[1]=Picombined; |
---|
2653 | g_BC=transpose(concat(transpose(zero),unitmat(nrows(L[5])))); |
---|
2654 | } |
---|
2655 | } |
---|
2656 | else |
---|
2657 | { |
---|
2658 | n_b=n_c[1]; |
---|
2659 | J_B[1]=J_C[1]; |
---|
2660 | g_BC=unitmat(ncols(J_C[1])); |
---|
2661 | } |
---|
2662 | } |
---|
2663 | else |
---|
2664 | { |
---|
2665 | J_C=list(list());// L[5] is already a resolution |
---|
2666 | if (size(#[2])!=0) |
---|
2667 | { |
---|
2668 | matrix zero[nrows(L[1])][nrows(L[5])]; |
---|
2669 | g_BC=transpose(concat(transpose(zero),unitmat(nrows(L[5])))); |
---|
2670 | n_a[1]=#[2]; |
---|
2671 | n_b1=n_a[1],n_c[1]; |
---|
2672 | n_b[1]=n_b1; |
---|
2673 | g_AB=concat(unitmat(nrows(L[1])),matrix(0,nrows(L[1]),nrows(L[5]))); |
---|
2674 | if (size(#[1])!=0) |
---|
2675 | { |
---|
2676 | J_A=#[1]; |
---|
2677 | /*resolution of L[3]*/ |
---|
2678 | nr=nrows(J_A[1]); |
---|
2679 | J_B=concat(J_A[1],matrix(0,nr,nrows(L[3])-nrows(L[1]))); |
---|
2680 | } |
---|
2681 | } |
---|
2682 | else |
---|
2683 | { |
---|
2684 | n_b=n_c[1]; |
---|
2685 | g_BC=unitmat(ncols(L[5])); |
---|
2686 | } |
---|
2687 | } |
---|
2688 | } |
---|
2689 | else// L[5]=list(); |
---|
2690 | { |
---|
2691 | if (size(#[2])!=0) |
---|
2692 | { |
---|
2693 | n_a[1]=#[2]; |
---|
2694 | n_b=n_a[1]; |
---|
2695 | g_AB=unitmat(size(n_b[1])); |
---|
2696 | if (size(#[1])!=0) |
---|
2697 | { |
---|
2698 | J_A=#[1]; |
---|
2699 | J_B[1]=J_A[1];// resolution of L[3] equals that of L[1] |
---|
2700 | } |
---|
2701 | } |
---|
2702 | } |
---|
2703 | list out=(J_A[1],g_AB[1],J_B[1],g_BC[1],J_C[1],n_a[1],n_b[1],n_c[1]); |
---|
2704 | return (out); |
---|
2705 | } |
---|
2706 | |
---|
2707 | //////////////////////////////////////////////////////////////////////////////////// |
---|
2708 | |
---|
2709 | static proc assemblingDoubleComplexes(list L) |
---|
2710 | { |
---|
2711 | /* The input is the output of VdStrictDoubleComplexes, we assemble the |
---|
2712 | resolutions of the L[i][2][3][1] to obtain a V_d-strict free Cartan-Eilenberg |
---|
2713 | resolution with modules P^i_j (1<=i<=size(L), j>=0) for the seqeunce |
---|
2714 | coker(L[1][2][3][1])->...->coker(L[size(L)][2][3][1])*/ |
---|
2715 | list out; |
---|
2716 | int i,j,k,l,oldj,newj,nr,nc; |
---|
2717 | for (i=1; i<=size(L); i++) |
---|
2718 | { |
---|
2719 | out[i]=list(list()); |
---|
2720 | out[i][1][1]=ncols(L[i][2][3][1]);//rank of module P^i_0 |
---|
2721 | if (size(L[i][2][5][1])!=0) |
---|
2722 | { |
---|
2723 | /*horizontal differential P^i_0->P^(i+1)_0*/ |
---|
2724 | nc=ncols(L[i][2][5][1]); |
---|
2725 | out[i][1][4]=prodr(ncols(L[i][2][3][1])-ncols(L[i][2][5][1]),nc); |
---|
2726 | } |
---|
2727 | else |
---|
2728 | { |
---|
2729 | /*horizontal differential P^i_0->0*/ |
---|
2730 | out[i][1][4]=matrix(0,ncols(L[i][2][3][1]),1); |
---|
2731 | } |
---|
2732 | oldj=newj; |
---|
2733 | for (j=1; j<=size(L[i][2][3]);j++) |
---|
2734 | { |
---|
2735 | out[i][j][2]=L[i][2][7][j];//shift vector of P^i_{j-1} |
---|
2736 | if (size(L[i][2][3][j])==0) |
---|
2737 | { |
---|
2738 | newj =j; |
---|
2739 | break; |
---|
2740 | } |
---|
2741 | out[i][j+1]=list(); |
---|
2742 | out[i][j+1][1]=nrows(L[i][2][3][j]);//rank of the module P^i_j |
---|
2743 | out[i][j+1][3]=L[i][2][3][j];//vertical differential P^i_j->P^(i+1)_j |
---|
2744 | if (size(L[i][2][5][j])!=0) |
---|
2745 | { |
---|
2746 | //horizonal differential P^i_j->P^(i-1)_j |
---|
2747 | nr=nrows(L[i][2][3][j])-nrows(L[i][2][5][j]); |
---|
2748 | out[i][j+1][4]=(-1)^j*prodr(nr,nrows(L[i][2][5][j])); |
---|
2749 | } |
---|
2750 | else |
---|
2751 | { |
---|
2752 | /*horizontal differential P^i_j->P^(i-1)_j*/ |
---|
2753 | out[i][j+1][4]=matrix(0,nrows(L[i][2][3][j]),1); |
---|
2754 | } |
---|
2755 | if(j==size(L[i][2][3])) |
---|
2756 | { |
---|
2757 | out[i][j+1][2]=L[i][2][7][j+1];//shift vector of P^i_j |
---|
2758 | newj=j+1; |
---|
2759 | } |
---|
2760 | } |
---|
2761 | if (i>1) |
---|
2762 | { |
---|
2763 | |
---|
2764 | for (k=1; k<=Min(list(oldj,newj)); k++) |
---|
2765 | { |
---|
2766 | /*horizonal differential P^(i-1)_(k-1)->P^i_(k-1)*/ |
---|
2767 | nr=nrows(out[i-1][k][4]); |
---|
2768 | out[i-1][k][4]=matrix(out[i-1][k][4],nr,out[i][k][1]); |
---|
2769 | } |
---|
2770 | for (k=newj+1; k<=oldj; k++) |
---|
2771 | { |
---|
2772 | /*no differential needed*/ |
---|
2773 | out[i-1][k]=delete(out[i-1][k],4); |
---|
2774 | } |
---|
2775 | } |
---|
2776 | } |
---|
2777 | return (out); |
---|
2778 | } |
---|
2779 | |
---|
2780 | //////////////////////////////////////////////////////////////////////////////////// |
---|
2781 | |
---|
2782 | static proc totalComplex(list L); |
---|
2783 | { |
---|
2784 | /* Input is the output of assemblingDoubleComplexes. |
---|
2785 | We obtain a complex C^1[m^1]->...->C^(r)[m^r] with differentials d^i and |
---|
2786 | shift vectors m^i (where C^r is placed in degree size(L)-1). |
---|
2787 | This complex is dercribed in the list out as follows: |
---|
2788 | rank(C^i)=out[3*i-2]; m_i=out[3*i-1] and d^i=out[3*i]*/ |
---|
2789 | list out;intvec rem1;intvec v; list remsize; int emp; |
---|
2790 | int i; int j; int c; int d; matrix M; int k; int l; |
---|
2791 | int n=nvars(basering) div 2; |
---|
2792 | list K; |
---|
2793 | for (i=1; i<=n+1; i++) |
---|
2794 | { |
---|
2795 | K[i]=list(); |
---|
2796 | } |
---|
2797 | L=K+L; |
---|
2798 | for (i=1; i<=size(L); i++) |
---|
2799 | { |
---|
2800 | emp=0; |
---|
2801 | if (size(L[i])!=0) |
---|
2802 | { |
---|
2803 | out[3*i-2]=L[i][1][1]; |
---|
2804 | v=L[i][1][1]; |
---|
2805 | rem1=L[i][1][2]; |
---|
2806 | emp=1; |
---|
2807 | } |
---|
2808 | else |
---|
2809 | { |
---|
2810 | out[3*i-2]=0; |
---|
2811 | v=0; |
---|
2812 | } |
---|
2813 | for (j=i+1; j<=size(L); j++) |
---|
2814 | { |
---|
2815 | if (size(L[j])>=j-i+1) |
---|
2816 | { |
---|
2817 | out[3*i-2]=out[3*i-2]+L[j][j-i+1][1]; |
---|
2818 | if (emp==0) |
---|
2819 | { |
---|
2820 | rem1=L[j][j-i+1][2]; |
---|
2821 | emp=1; |
---|
2822 | } |
---|
2823 | else |
---|
2824 | { |
---|
2825 | rem1=rem1,L[j][j-i+1][2]; |
---|
2826 | } |
---|
2827 | v[size(v)+1]=L[j][j-i+1][1]; |
---|
2828 | } |
---|
2829 | else |
---|
2830 | { |
---|
2831 | v[size(v)+1]=0; |
---|
2832 | } |
---|
2833 | } |
---|
2834 | out[3*i-1]=rem1; |
---|
2835 | v[size(v)+1]=0; |
---|
2836 | remsize[i]=v; |
---|
2837 | } |
---|
2838 | int o1; |
---|
2839 | int o2; |
---|
2840 | for (i=1; i<=size(L)-1; i++) |
---|
2841 | { |
---|
2842 | o1=1; |
---|
2843 | o2=1; |
---|
2844 | if (size(out[3*i-2])!=0) |
---|
2845 | { |
---|
2846 | o1=out[3*i-2]; |
---|
2847 | } |
---|
2848 | if (size(out[3*i+1])!=0) |
---|
2849 | { |
---|
2850 | o2=out[3*i+1]; |
---|
2851 | } |
---|
2852 | M=matrix(0,o1,o2); |
---|
2853 | if (size(L[i])!=0) |
---|
2854 | { |
---|
2855 | if (size(L[i][1][4])!=0) |
---|
2856 | { |
---|
2857 | M=matrix(L[i][1][4],o1,o2); |
---|
2858 | } |
---|
2859 | } |
---|
2860 | c=remsize[i][1]; |
---|
2861 | for (j=i+1; j<=size(L); j++) |
---|
2862 | { |
---|
2863 | if (remsize[i][j-i+1]!=0) |
---|
2864 | { |
---|
2865 | for (k=c+1; k<=c+remsize[i][j-i+1]; k++) |
---|
2866 | { |
---|
2867 | for (l=d+1; l<=d+remsize[i+1][j-i];l++) |
---|
2868 | { |
---|
2869 | M[k,l]=L[j][j-i+1][3][(k-c),(l-d)]; |
---|
2870 | } |
---|
2871 | } |
---|
2872 | d=d+remsize[i+1][j-i]; |
---|
2873 | if (remsize[i+1][j-i+1]!=0) |
---|
2874 | { |
---|
2875 | for (k=c+1; k<=c+remsize[i][j-i+1]; k++) |
---|
2876 | { |
---|
2877 | for (l=d+1; l<=d+remsize[i+1][j-i+1];l++) |
---|
2878 | { |
---|
2879 | M[k,l]=L[j][j-i+1][4][k-c,l-d]; |
---|
2880 | } |
---|
2881 | } |
---|
2882 | c=c+remsize[i][j-i+1]; |
---|
2883 | } |
---|
2884 | } |
---|
2885 | else |
---|
2886 | { |
---|
2887 | d=d+remsize[i+1][j-i]; |
---|
2888 | } |
---|
2889 | } |
---|
2890 | out[3*i]=M; |
---|
2891 | d=0; c=0; |
---|
2892 | } |
---|
2893 | out[3*size(L)]=matrix(0,out[3*size(L)-2],1); |
---|
2894 | return (out); |
---|
2895 | |
---|
2896 | } |
---|
2897 | |
---|
2898 | //////////////////////////////////////////////////////////////////////////////////// |
---|
2899 | //COMPUTATION OF THE BLOBAL B-FUNCTION |
---|
2900 | //////////////////////////////////////////////////////////////////////////////////// |
---|
2901 | |
---|
2902 | static proc globalBFun(list L,list #) |
---|
2903 | { |
---|
2904 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
2905 | where L[i]=(L[i][1],L[i][2]) and L[i][1] is a m_i x n_i-matrix and L[i][2] an |
---|
2906 | intvec of size n_i. |
---|
2907 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
2908 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
2909 | 6.1.1 in [R]) by combining Algorithm 6.1.6 in [R] and the method of principal |
---|
2910 | intersection (cf. Remark 6.1.7 in [R] 2012). |
---|
2911 | This works ONLY IF ALL B-FUNCTIONS ARE NON-ZERO, but this is the case since this |
---|
2912 | proc is only called from the procedure deRhamCohomology and the input comes |
---|
2913 | originally from the procedure toVdstrictFreeComplex*/ |
---|
2914 | if (size(#)==0)//# may contain the Syzstring |
---|
2915 | { |
---|
2916 | string Syzstring="Sres"; |
---|
2917 | } |
---|
2918 | else |
---|
2919 | { |
---|
2920 | string Syzstring=#[1]; |
---|
2921 | } |
---|
2922 | int i,j; |
---|
2923 | def W=basering; |
---|
2924 | int n=nvars(W) div 2; |
---|
2925 | list G0; |
---|
2926 | ideal I; |
---|
2927 | for (j=1; j<=size(L); j++) |
---|
2928 | { |
---|
2929 | G0[j]=list(); |
---|
2930 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
2931 | { |
---|
2932 | G0[j][i]=I; |
---|
2933 | } |
---|
2934 | } |
---|
2935 | list out; |
---|
2936 | ideal I; poly f; |
---|
2937 | intvec i1; |
---|
2938 | for (j=1; j<=size(L); j++) |
---|
2939 | { |
---|
2940 | /*if the shift vector L[j][2] is non-zero we have to compute a V_d-strict |
---|
2941 | Groebner basis of L[j][1] with respect to the zero shift; otherwise L[i][1] |
---|
2942 | is already a V_d-strict Groebner basis, because it was obtained by the |
---|
2943 | procedure toVdStrictFreeComplex*/ |
---|
2944 | if (L[j][2]!=intvec(0:size(L[j][2]))) |
---|
2945 | { |
---|
2946 | if (Syzstring=="Vdres") |
---|
2947 | { |
---|
2948 | L[j][1]=VdStrictGB(L[j][1],n); |
---|
2949 | } |
---|
2950 | else |
---|
2951 | { |
---|
2952 | def HomWeyl=makeHomogenizedWeyl(n); |
---|
2953 | setring HomWeyl; |
---|
2954 | list L=fetch(W,L); |
---|
2955 | L[j][1]=nHomogenize(L[j][1]); |
---|
2956 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
2957 | L[j][1]=subst(L[j][1],h,1); |
---|
2958 | setring W; |
---|
2959 | L=fetch(HomWeyl,L); |
---|
2960 | kill HomWeyl; |
---|
2961 | } |
---|
2962 | } |
---|
2963 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
2964 | { |
---|
2965 | G0[j][i]=I; |
---|
2966 | } |
---|
2967 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
2968 | { |
---|
2969 | /*computes the terms of maximal V_d-degree of the biggest non-zero |
---|
2970 | component of submat(L[j][1],i,(1..ncols(L[j][1])))*/ |
---|
2971 | i1=(1..ncols(L[j][1])); |
---|
2972 | out=VdDeg(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1); |
---|
2973 | f=L[j][1][i,out[2]]; |
---|
2974 | G0[j][out[2]]=G0[j][out[2]],f; |
---|
2975 | G0[j][out[2]]=compress(G0[j][out[2]]); |
---|
2976 | } |
---|
2977 | } |
---|
2978 | list save; |
---|
2979 | int l; |
---|
2980 | list weights; |
---|
2981 | /*bFctIdealModified computes the intersection of G0[j][i] and |
---|
2982 | x(1)D(1)+...+x(n)D(n) using the method of principal intersection*/ |
---|
2983 | for (j=1; j<=size(G0); j++) |
---|
2984 | { |
---|
2985 | for (i=1; i<=size(G0[j]); i++) |
---|
2986 | { |
---|
2987 | G0[j][i]=bFctIdealModified(G0[j][i]); |
---|
2988 | } |
---|
2989 | for (i=1; i<=size(G0[j]); i++) |
---|
2990 | { |
---|
2991 | weights=list(); |
---|
2992 | if (size(G0[j][i])!=0) |
---|
2993 | { |
---|
2994 | for (l=i; l<=size(G0[j]); l++) |
---|
2995 | { |
---|
2996 | if (size(G0[j][l])!=0) |
---|
2997 | { |
---|
2998 | weights[size(weights)+1]=L[j][2][l]; |
---|
2999 | } |
---|
3000 | } |
---|
3001 | G0[j][i]=list(G0[j][i][1]+Min(weights),G0[j][i][2]+Max(weights)); |
---|
3002 | } |
---|
3003 | } |
---|
3004 | } |
---|
3005 | list allmin; |
---|
3006 | list allmax; |
---|
3007 | for (j=1; j<=size(G0); j++) |
---|
3008 | { |
---|
3009 | for (i=1; i<=size(G0[j]); i++) |
---|
3010 | { |
---|
3011 | if (size(G0[j][i])!=0) |
---|
3012 | { |
---|
3013 | allmin[size(allmin)+1]=G0[j][i][1]; |
---|
3014 | allmax[size(allmax)+1]=G0[j][i][2]; |
---|
3015 | } |
---|
3016 | } |
---|
3017 | } |
---|
3018 | list minmax=list(Min(allmin),Max(allmax)); |
---|
3019 | return(minmax); |
---|
3020 | } |
---|
3021 | |
---|
3022 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3023 | |
---|
3024 | static proc exactGlobalBFun(list L,list #) |
---|
3025 | { |
---|
3026 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
3027 | where L[i]=(L[i][1],L[i][2]) and L[i][1] is a m_i x n_i-matrix and L[i][2] an |
---|
3028 | intvec of size n_i. |
---|
3029 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
3030 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
3031 | 6.1.1 in [R]) by combining Algorithm 6.1.6 in [R] and the method of principal |
---|
3032 | intersection (cf. Remark 6.1.7 in [R] 2012). |
---|
3033 | This works ONLY IF ALL B-FUNCTIONS ARE NON-ZERO, but this is the case since this |
---|
3034 | proc is only called from the procedure deRhamCohomology and the input comes |
---|
3035 | originally from the procedure toVdstrictFreeComplex*/ |
---|
3036 | if (size(#)==0)//# may contain the Syzstring |
---|
3037 | { |
---|
3038 | string Syzstring="Sres"; |
---|
3039 | } |
---|
3040 | else |
---|
3041 | { |
---|
3042 | string Syzstring=#[1]; |
---|
3043 | } |
---|
3044 | int i,j,k; |
---|
3045 | def W=basering; |
---|
3046 | int n=nvars(W) div 2; |
---|
3047 | list G0; |
---|
3048 | ideal I; |
---|
3049 | for (j=1; j<=size(L); j++) |
---|
3050 | { |
---|
3051 | G0[j]=list(); |
---|
3052 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
3053 | { |
---|
3054 | G0[j][i]=I; |
---|
3055 | } |
---|
3056 | } |
---|
3057 | list out; |
---|
3058 | matrix M; |
---|
3059 | ideal I; poly f; |
---|
3060 | intvec i1; |
---|
3061 | for (j=1; j<=size(L); j++) |
---|
3062 | { |
---|
3063 | M=L[j][1]; |
---|
3064 | /*if the shift vector L[j][2] is non-zero we have to compute a V_d-strict |
---|
3065 | Groebner basis of L[j][1] with respect to the zero shift; otherwise L[i][1] |
---|
3066 | is already a V_d-strict Groebner basis, because it was obtained by the |
---|
3067 | procedure toVdStrictFreeComplex*/ |
---|
3068 | for (k=1; k<=ncols(L[j][1]); k++) |
---|
3069 | { |
---|
3070 | L[j][1]=permcol(M,1,k); |
---|
3071 | if (Syzstring=="Vdres") |
---|
3072 | { |
---|
3073 | L[j][1]=VdStrictGB(L[j][1],n); |
---|
3074 | } |
---|
3075 | else |
---|
3076 | { |
---|
3077 | def HomWeyl=makeHomogenizedWeyl(n); |
---|
3078 | setring HomWeyl; |
---|
3079 | list L=fetch(W,L); |
---|
3080 | L[j][1]=nHomogenize(L[j][1]); |
---|
3081 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
3082 | L[j][1]=subst(L[j][1],h,1); |
---|
3083 | setring W; |
---|
3084 | L=fetch(HomWeyl,L); |
---|
3085 | kill HomWeyl; |
---|
3086 | } |
---|
3087 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
3088 | { |
---|
3089 | /*computes the terms of maximal V_d-degree of the biggest non-zero |
---|
3090 | component of submat(L[j][1],i,(1..ncols(L[j][1])))*/ |
---|
3091 | i1=(1..ncols(L[j][1])); |
---|
3092 | out=VdDeg(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1); |
---|
3093 | f=L[j][1][i,out[2]]; |
---|
3094 | if (out[2]==1) |
---|
3095 | { |
---|
3096 | G0[j][k]=G0[j][k],f; |
---|
3097 | G0[j][k]=compress(G0[j][k]); |
---|
3098 | } |
---|
3099 | } |
---|
3100 | } |
---|
3101 | } |
---|
3102 | list save; |
---|
3103 | int l; |
---|
3104 | list weights; |
---|
3105 | /*bFctIdealModified computes the intersection of G0[j][i] and |
---|
3106 | x(1)D(1)+...+x(n)D(n) using the method of principal intersection*/ |
---|
3107 | for (j=1; j<=size(G0); j++) |
---|
3108 | { |
---|
3109 | for (i=1; i<=size(G0[j]); i++) |
---|
3110 | { |
---|
3111 | G0[j][i]=bFctIdealModified(G0[j][i]); |
---|
3112 | } |
---|
3113 | for (i=1; i<=size(G0[j]); i++) |
---|
3114 | { |
---|
3115 | if (size(G0[j][i])!=0) |
---|
3116 | { |
---|
3117 | G0[j][i]=list(G0[j][i][1]+L[j][2][i],G0[j][i][2]+L[j][2][i]); |
---|
3118 | } |
---|
3119 | } |
---|
3120 | } |
---|
3121 | list allmin; |
---|
3122 | list allmax; |
---|
3123 | for (j=1; j<=size(G0); j++) |
---|
3124 | { |
---|
3125 | for (i=1; i<=size(G0[j]); i++) |
---|
3126 | { |
---|
3127 | if (size(G0[j][i])!=0) |
---|
3128 | { |
---|
3129 | allmin[size(allmin)+1]=G0[j][i][1]; |
---|
3130 | allmax[size(allmax)+1]=G0[j][i][2]; |
---|
3131 | } |
---|
3132 | } |
---|
3133 | } |
---|
3134 | list minmax=list(Min(allmin),Max(allmax)); |
---|
3135 | return(minmax); |
---|
3136 | } |
---|
3137 | |
---|
3138 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3139 | |
---|
3140 | static proc bFctIdealModified (ideal I) |
---|
3141 | {/*modified version of the procedure bfunIdeal from bfun.lib*/ |
---|
3142 | def B= basering; |
---|
3143 | int n = nvars(B) div 2; |
---|
3144 | intvec w=(1:n); |
---|
3145 | I= initialIdealW(I,-w,w); |
---|
3146 | poly s; int i; |
---|
3147 | for (i=1; i<=n; i++) |
---|
3148 | { |
---|
3149 | s=s+x(i)*D(i); |
---|
3150 | } |
---|
3151 | /*pIntersect computes the intersection on s and I*/ |
---|
3152 | vector b = pIntersect(s,I); |
---|
3153 | list RL = ringlist(B); RL = RL[1..4]; |
---|
3154 | RL[2] = list(safeVarName("s")); |
---|
3155 | RL[3] = list(list("dp",intvec(1)),list("C",intvec(0))); |
---|
3156 | def @S = ring(RL); setring @S; |
---|
3157 | vector b = imap(B,b); |
---|
3158 | poly bs = vec2poly(b); |
---|
3159 | ring r=0,s,dp; |
---|
3160 | poly bs=imap(@S,bs); |
---|
3161 | /*find minimal and maximal integer root*/ |
---|
3162 | ideal allfac=factorize(bs,1); |
---|
3163 | list allfacs; |
---|
3164 | for (i=1; i<=ncols(allfac); i++) |
---|
3165 | { |
---|
3166 | allfacs[i]=allfac[i]; |
---|
3167 | } |
---|
3168 | number testzero; |
---|
3169 | list zeros; |
---|
3170 | for (i=1; i<=size(allfacs); i++) |
---|
3171 | { |
---|
3172 | if (deg(allfacs[i])==1) |
---|
3173 | { |
---|
3174 | testzero=number(subst(allfacs[i],s,0))/leadcoef(allfacs[i]); |
---|
3175 | if (testzero-int(testzero)==0) |
---|
3176 | { |
---|
3177 | zeros[size(zeros)+1]=int(-1)*int(testzero); |
---|
3178 | } |
---|
3179 | } |
---|
3180 | } |
---|
3181 | if (size(zeros)!=0) |
---|
3182 | { |
---|
3183 | list minmax=(Min(zeros),Max(zeros)); |
---|
3184 | } |
---|
3185 | else |
---|
3186 | { |
---|
3187 | list minmax=list(); |
---|
3188 | } |
---|
3189 | setring B; |
---|
3190 | return(minmax); |
---|
3191 | } |
---|
3192 | |
---|
3193 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3194 | |
---|
3195 | static proc safeVarName (string s) |
---|
3196 | {/* from the library "bfun.lib"*/ |
---|
3197 | string S = "," + charstr(basering) + "," + varstr(basering) + ","; |
---|
3198 | s = "," + s + ","; |
---|
3199 | while (find(S,s) <> 0) |
---|
3200 | { |
---|
3201 | s[1] = "@"; |
---|
3202 | s = "," + s; |
---|
3203 | } |
---|
3204 | s = s[2..size(s)-1]; |
---|
3205 | return(s) |
---|
3206 | } |
---|
3207 | |
---|
3208 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3209 | |
---|
3210 | static proc globalBFunOT(list L,list #) |
---|
3211 | { |
---|
3212 | /*this proc is currently not used since globalBFun computes the same output and is |
---|
3213 | faster, however globalBFun works only for non-zero b-functions!*/ |
---|
3214 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
3215 | where L[i]=(L[i][1],L[i][2]) and L[i][1] is a m_i x n_i-matrix and L[i][2] an |
---|
3216 | intvec of size n_i. |
---|
3217 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
3218 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
3219 | 6.1.1 in [R]) using Algorithm 6.1.6 in [R].*/ |
---|
3220 | if (size(#)==0) |
---|
3221 | { |
---|
3222 | string Syzstring="Sres"; |
---|
3223 | } |
---|
3224 | else |
---|
3225 | { |
---|
3226 | string Syzstring=#[1]; |
---|
3227 | } |
---|
3228 | int i; int j; |
---|
3229 | def W=basering; |
---|
3230 | int n=nvars(W) div 2; |
---|
3231 | list G0; |
---|
3232 | ideal I; |
---|
3233 | intvec i1; |
---|
3234 | for (j=1; j<=size(L); j++) |
---|
3235 | { |
---|
3236 | G0[j]=list(); |
---|
3237 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
3238 | { |
---|
3239 | G0[j][i]=I; |
---|
3240 | } |
---|
3241 | } |
---|
3242 | list out; |
---|
3243 | for (j=1; j<=size(L); j++) |
---|
3244 | { |
---|
3245 | if (L[j][2]!=intvec(0:size(L[j][2]))) |
---|
3246 | { |
---|
3247 | if (Syzstring=="Vdres") |
---|
3248 | { |
---|
3249 | L[j][1]=VdStrictGB(L[j][1],n); |
---|
3250 | } |
---|
3251 | else |
---|
3252 | { |
---|
3253 | def HomWeyl=makeHomogenizedWeyl(n); |
---|
3254 | setring HomWeyl; |
---|
3255 | list L=fetch(W,L); |
---|
3256 | L[j][1]=nHomogenize(L[j][1]); |
---|
3257 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
3258 | L[j][1]=subst(L[j][1],h,1); |
---|
3259 | setring W; |
---|
3260 | L=fetch(HomWeyl,L); |
---|
3261 | kill HomWeyl; |
---|
3262 | } |
---|
3263 | } |
---|
3264 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
3265 | { |
---|
3266 | i1=(1..ncols(L[j][1])); |
---|
3267 | out=VdDeg(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1); |
---|
3268 | G0[j][out[2]][size(G0[j][out[2]])+1]=(out[1]); |
---|
3269 | } |
---|
3270 | } |
---|
3271 | list Data=ringlist(W); |
---|
3272 | for (i=1; i<=n; i++) |
---|
3273 | { |
---|
3274 | Data[2][2*n+i]=Data[2][i]; |
---|
3275 | Data[2][3*n+i]=Data[2][n+i]; |
---|
3276 | Data[2][i]="v("+string(i)+")"; |
---|
3277 | Data[2][n+i]="w("+string(i)+")"; |
---|
3278 | } |
---|
3279 | Data[3][1][1]="M"; |
---|
3280 | intvec mord=(0:16*n^2); |
---|
3281 | mord[1..2*n]=(1:2*n); |
---|
3282 | mord[6*n+1..8*n]=(1:2*n); |
---|
3283 | for (i=0; i<=2*n-2; i++) |
---|
3284 | { |
---|
3285 | mord[(3+i)*4*n-i]=-1; |
---|
3286 | mord[(2*n+2+i)*4*n-2*n-i]=-1; |
---|
3287 | } |
---|
3288 | Data[3][1][2]=mord; |
---|
3289 | matrix Ones=UpOneMatrix(4*n); |
---|
3290 | Data[5]=Ones; |
---|
3291 | matrix con[2*n][2*n]; |
---|
3292 | Data[6]=transpose(concat(con,transpose(concat(con,Data[6])))); |
---|
3293 | def Wuv=ring(Data); |
---|
3294 | setring Wuv; |
---|
3295 | list G0=imap(W,G0); list G3; poly lterm;intvec lexp; |
---|
3296 | list G1,G2,LL; |
---|
3297 | intvec e,f; |
---|
3298 | int kapp,k,l; |
---|
3299 | poly h; |
---|
3300 | ideal I; |
---|
3301 | for (l=1; l<=size(G0); l++) |
---|
3302 | { |
---|
3303 | G1[l]=list(); G2[l]=list(); G3[l]=list(); |
---|
3304 | for (i=1; i<=size(G0[l]); i++) |
---|
3305 | { |
---|
3306 | for (j=1; j<=ncols(G0[l][i]);j++) |
---|
3307 | { |
---|
3308 | G0[l][i][j]=mHom(G0[l][i][j]); |
---|
3309 | } |
---|
3310 | for (j=1; j<=nvars(Wuv) div 4; j++) |
---|
3311 | { |
---|
3312 | G0[l][i][size(G0[l][i])+1]=1-v(j)*w(j); |
---|
3313 | } |
---|
3314 | G1[l][i]=slimgb(G0[l][i]); |
---|
3315 | G2[l][i]=I; |
---|
3316 | G3[l][i]=list(); |
---|
3317 | for (j=1; j<=ncols(G1[l][i]); j++) |
---|
3318 | { |
---|
3319 | e=leadexp(G1[l][i][j]); |
---|
3320 | f=e[1..2*n]; |
---|
3321 | if (f==intvec(0:(2*n))) |
---|
3322 | { |
---|
3323 | for (k=1; k<=n; k++) |
---|
3324 | { |
---|
3325 | kapp=-e[2*n+k]+e[3*n+k]; |
---|
3326 | if (kapp>0) |
---|
3327 | { |
---|
3328 | G1[l][i][j]=(x(k)^kapp)*G1[l][i][j]; |
---|
3329 | } |
---|
3330 | if (kapp<0) |
---|
3331 | { |
---|
3332 | G1[l][i][j]=(D(k)^(-kapp))*G1[l][i][j]; |
---|
3333 | } |
---|
3334 | } |
---|
3335 | G2[l][i][size(G2[l][i])+1]=G1[l][i][j]; |
---|
3336 | G3[l][i][size(G3[l][i])+1]=list(); |
---|
3337 | while (G1[l][i][j]!=0) |
---|
3338 | { |
---|
3339 | lterm=lead(G1[l][i][j]); |
---|
3340 | G1[l][i][j]=G1[l][i][j]-lterm; |
---|
3341 | lexp=leadexp(lterm); |
---|
3342 | lexp=lexp[2*n+1..3*n]; |
---|
3343 | LL=list(lexp,leadcoef(lterm)); |
---|
3344 | G3[l][i][size(G3[l][i])][size(G3[l][i][size(G3[l][i])])+1]=LL; |
---|
3345 | } |
---|
3346 | } |
---|
3347 | } |
---|
3348 | } |
---|
3349 | } |
---|
3350 | ring r=0,(s(1..n)),dp; |
---|
3351 | ideal I; |
---|
3352 | map G3forr=Wuv,I; |
---|
3353 | list G3=G3forr(G3); |
---|
3354 | poly fs,gs; |
---|
3355 | int a; |
---|
3356 | list G4; |
---|
3357 | for (l=1; l<=size(G3); l++) |
---|
3358 | { |
---|
3359 | G4[l]=list(); |
---|
3360 | for (i=1; i<=size(G3[l]);i++) |
---|
3361 | { |
---|
3362 | G4[l][i]=I; |
---|
3363 | |
---|
3364 | for (j=1; j<=size(G3[l][i]); j++) |
---|
3365 | { |
---|
3366 | fs=0; |
---|
3367 | for (k=1; k<=size(G3[l][i][j]); k++) |
---|
3368 | { |
---|
3369 | gs=1; |
---|
3370 | for (a=1; a<=n; a++) |
---|
3371 | { |
---|
3372 | if (G3[l][i][j][k][1][a]!=0) |
---|
3373 | { |
---|
3374 | gs=gs*permuteVar(list(G3[l][i][j][k][1][a]),a); |
---|
3375 | } |
---|
3376 | } |
---|
3377 | gs=gs*G3[l][i][j][k][2]; |
---|
3378 | fs=fs+gs; |
---|
3379 | } |
---|
3380 | G4[l][i]=G4[l][i],fs; |
---|
3381 | } |
---|
3382 | } |
---|
3383 | } |
---|
3384 | if (n==1) |
---|
3385 | { |
---|
3386 | ring rnew=0,t,dp; |
---|
3387 | } |
---|
3388 | else |
---|
3389 | { |
---|
3390 | ring rnew=0,(t,s(2..n)),dp; |
---|
3391 | } |
---|
3392 | ideal Iformap; |
---|
3393 | Iformap[1]=t; |
---|
3394 | poly forel=1; |
---|
3395 | for (i=2; i<=n; i++) |
---|
3396 | { |
---|
3397 | Iformap[1]=Iformap[1]-s(i); |
---|
3398 | Iformap[i]=s(i); |
---|
3399 | forel=forel*s(i); |
---|
3400 | } |
---|
3401 | map rtornew=r,Iformap; |
---|
3402 | list G4=rtornew(G4); |
---|
3403 | list getintvecs=fetch(W,L); |
---|
3404 | ideal J; |
---|
3405 | option(redSB); |
---|
3406 | for (l=1; l<=size(G4); l++) |
---|
3407 | { |
---|
3408 | J=1; |
---|
3409 | for (i=1; i<=size(G4[l]); i++) |
---|
3410 | { |
---|
3411 | G4[l][i]=eliminate(G4[l][i],forel); |
---|
3412 | J=intersect(J,G4[l][i]); |
---|
3413 | } |
---|
3414 | G4[l]=poly(std(J)[1]); |
---|
3415 | } |
---|
3416 | list minmax; |
---|
3417 | list mini=list(); |
---|
3418 | list maxi=list(); |
---|
3419 | list L=fetch(W,L); |
---|
3420 | for (i=1; i<=size(G4); i++) |
---|
3421 | { |
---|
3422 | minmax[i]=minIntRoot(G4[i],1); |
---|
3423 | if (size(minmax[i])!=0) |
---|
3424 | { |
---|
3425 | mini=insert(mini,minmax[i][1]+Min(L[i][2])); |
---|
3426 | maxi=insert(maxi,minmax[i][2]+Max(L[i][2])); |
---|
3427 | } |
---|
3428 | } |
---|
3429 | mini=Min(mini); |
---|
3430 | maxi=Max(maxi); |
---|
3431 | minmax=list(mini[1],maxi[1]); |
---|
3432 | option(none); |
---|
3433 | return(minmax); |
---|
3434 | } |
---|
3435 | |
---|
3436 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3437 | //COMPUTATION OF THE COHOMOLOGY |
---|
3438 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3439 | |
---|
3440 | static proc findCohomology(list L,int le) |
---|
3441 | { |
---|
3442 | /*computes the cohomology of the complex (D^i,d^i) given by D^i=C^L[2*i-1] and |
---|
3443 | d^i=L[2*i]*/ |
---|
3444 | def R=basering; |
---|
3445 | ring r=0,(x),dp; |
---|
3446 | list L=imap(R,L); |
---|
3447 | list out; |
---|
3448 | int i, ker, im; |
---|
3449 | matrix S; |
---|
3450 | option(returnSB); |
---|
3451 | option(redSB); |
---|
3452 | for (i=2; i<=size(L); i=i+2) |
---|
3453 | { |
---|
3454 | if (L[i-1]==0) |
---|
3455 | { |
---|
3456 | out[i div 2]=0; |
---|
3457 | im=0; |
---|
3458 | } |
---|
3459 | else |
---|
3460 | { |
---|
3461 | S=matrix(syz(transpose(L[i]))); |
---|
3462 | if (S!=matrix(0,nrows(S),ncols(S))) |
---|
3463 | { |
---|
3464 | ker=ncols(S); |
---|
3465 | out[i div 2]=ker-im; |
---|
3466 | im=L[i-1]-ker; |
---|
3467 | } |
---|
3468 | else |
---|
3469 | { |
---|
3470 | out[i-1]=0; |
---|
3471 | im=L[i-1]; |
---|
3472 | } |
---|
3473 | } |
---|
3474 | } |
---|
3475 | option(none); |
---|
3476 | while (size(out)>le) |
---|
3477 | { |
---|
3478 | out=delete(out,1); |
---|
3479 | } |
---|
3480 | setring R; |
---|
3481 | return(out); |
---|
3482 | } |
---|
3483 | |
---|
3484 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3485 | //AUXILIARY PROCEDURES |
---|
3486 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3487 | |
---|
3488 | static proc divdr(matrix m,matrix n) |
---|
3489 | { |
---|
3490 | m=transpose(m); |
---|
3491 | n=transpose(n); |
---|
3492 | matrix con=concat(m,n); |
---|
3493 | matrix s=syz(con); |
---|
3494 | s=submat(s,1..ncols(m),1..ncols(s)); |
---|
3495 | s=transpose(compress(s)); |
---|
3496 | return(s); |
---|
3497 | } |
---|
3498 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3499 | |
---|
3500 | static proc matrixLift(matrix M,matrix N) |
---|
3501 | { |
---|
3502 | intvec v=option(get); |
---|
3503 | option(none); |
---|
3504 | matrix l=transpose(lift(transpose(M),transpose(N))); |
---|
3505 | option(set,v); |
---|
3506 | return(l); |
---|
3507 | } |
---|
3508 | |
---|
3509 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3510 | |
---|
3511 | static proc VdStrictGB (matrix M,int d,list #) |
---|
3512 | "USAGE:VdStrictGB(M,d[,v]); M a matrix, d an integer, v an optional intvec |
---|
3513 | ASSUME:-basering is the nth Weyl algebra D_n @* |
---|
3514 | -1<=d<=n @* |
---|
3515 | -v (if given) is the shift vector on the range of M (in particular, |
---|
3516 | size(v)=ncols(M)); otherwise v is assumed to be the zero shift vector |
---|
3517 | RETURN:matrix N; the rows of N form a V_d-strict Groebner basis with respect to v |
---|
3518 | for the module generated by the rows of M |
---|
3519 | " |
---|
3520 | { |
---|
3521 | if (M==matrix(0,nrows(M),ncols(M))) |
---|
3522 | { |
---|
3523 | return (matrix(0,1,ncols(M))); |
---|
3524 | } |
---|
3525 | intvec op=option(get); |
---|
3526 | def W =basering; |
---|
3527 | int ncM=ncols(M); |
---|
3528 | list Data=ringlist(W); |
---|
3529 | Data[2]=list("nhv")+Data[2]; |
---|
3530 | Data[3][3]=Data[3][1]; |
---|
3531 | Data[3][1]=list("dp",intvec(1)); |
---|
3532 | matrix re[size(Data[2])][size(Data[2])]=UpOneMatrix(size(Data[2])); |
---|
3533 | Data[5]=re; |
---|
3534 | int k,l; |
---|
3535 | Data[6]=transpose(concat(matrix(0,1,1),transpose(concat(matrix(0,1,1),Data[6])))); |
---|
3536 | def Whom=ring(Data);// D_n[nhv] with the new commuative variable nhv |
---|
3537 | setring Whom; |
---|
3538 | matrix Mnew=imap(W,M); |
---|
3539 | intvec v; |
---|
3540 | if (size(#)!=0) |
---|
3541 | { |
---|
3542 | v=#[1]; |
---|
3543 | } |
---|
3544 | if (size(v) < ncM) |
---|
3545 | { |
---|
3546 | v=v,0:(ncM-size(v)); |
---|
3547 | } |
---|
3548 | Mnew=homogenize(Mnew, d, v);//homogenization of M with respect to the new variable |
---|
3549 | Mnew=transpose(Mnew); |
---|
3550 | Mnew=slimgb(Mnew);// computes a Groebner basis of the homogenzition of M |
---|
3551 | Mnew=subst(Mnew,nhv,1);// substitution of 1 gives V_d-strict Groebner basis of M |
---|
3552 | Mnew=compress(Mnew); |
---|
3553 | Mnew=transpose(Mnew); |
---|
3554 | setring W; |
---|
3555 | M=imap(Whom,Mnew); |
---|
3556 | option(set,op); |
---|
3557 | return(M); |
---|
3558 | } |
---|
3559 | |
---|
3560 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3561 | |
---|
3562 | static proc VdNormalForm(matrix F,matrix M,int d,intvec v,list #) |
---|
3563 | "USAGE:VdNormalForm(F,M,d,v[,w]); F and M matrices, d int, v intvec, w an optional |
---|
3564 | intvec |
---|
3565 | ASSUME:-basering is the nth Weyl algebra D_n @* |
---|
3566 | -F a n_1 x n_2-matrix and M a m_1 x m_2-matrix with m_2<=n_2 @* |
---|
3567 | -d is an integer between 1 and n @* |
---|
3568 | -v is a shift vector for D_n^(m_2) and hence size(v)=m_2 @* |
---|
3569 | -w is a shift vector for D_n^(m_1-m_2) and hence size(v)=m_1-m_2 @* |
---|
3570 | RETURN:a n_1 x n_2-matrix N such that:@* |
---|
3571 | -If no optional intvec w is given:(N[i,1],..,N[i,m_2]) is a V_d-strict normal |
---|
3572 | form of (F[i,1],...,F[i,m_2]) with respect to a V_d-strict Groebner basis of |
---|
3573 | the rows of M and the shift vector v |
---|
3574 | -If w is given:(N[i,1],..,N[i,m_2]) is chosen such that |
---|
3575 | Vddeg((N[i,1],...,N[i,m_2])[v])<=Vddeg((F[i,m_2+1],...,F[i,m_1])[v]); |
---|
3576 | -N[i,j]=F[i,j] for j>m_2 |
---|
3577 | " |
---|
3578 | { |
---|
3579 | int SBcom; |
---|
3580 | def W =basering; |
---|
3581 | int c=ncols(M); |
---|
3582 | matrix keepF=F; |
---|
3583 | if (size(#)!=0) |
---|
3584 | { |
---|
3585 | intvec w=#[1]; |
---|
3586 | } |
---|
3587 | F=submat(F,intvec(1..nrows(F)),intvec(1..c)); |
---|
3588 | list Data=ringlist(W); |
---|
3589 | Data[2]=list("nhv")+Data[2]; |
---|
3590 | Data[3][3]=Data[3][1]; |
---|
3591 | Data[3][1]=list("dp",intvec(1)); |
---|
3592 | matrix re[size(Data[2])][size(Data[2])]=UpOneMatrix(size(Data[2])); |
---|
3593 | Data[5]=re; |
---|
3594 | int k,l,nr,nc; |
---|
3595 | matrix rep[size(Data[2])][size(Data[2])]; |
---|
3596 | for (l=size(Data[2])-1;l>=1; l--) |
---|
3597 | { |
---|
3598 | for (k=l-1; k>=1;k--) |
---|
3599 | { |
---|
3600 | rep[k+1,l+1]=Data[6][k,l]; |
---|
3601 | } |
---|
3602 | } |
---|
3603 | Data[6]=rep; |
---|
3604 | def Whom=ring(Data);//new ring D_n[nvh] this new commuative variable nhv |
---|
3605 | setring Whom; |
---|
3606 | matrix Mnew=imap(W,M); |
---|
3607 | list forMnew=homogenize(Mnew,d,v,1);//commputes homogenization of M; |
---|
3608 | Mnew=forMnew[1]; |
---|
3609 | int rightexp=forMnew[2]; |
---|
3610 | matrix Fnew=imap(W,F); |
---|
3611 | matrix keepF=imap(W,keepF); |
---|
3612 | matrix Fb; |
---|
3613 | int cc; |
---|
3614 | intvec i1,i2; |
---|
3615 | matrix zeromat,subm1,subm2,zeromat2; |
---|
3616 | for (l=1; l<=nrows(Fnew); l++) |
---|
3617 | { |
---|
3618 | if (size(#)!=0) |
---|
3619 | { |
---|
3620 | subm2=submat(keepF,l,((ncols(Fnew)+1)..ncols(keepF))); |
---|
3621 | zeromat2=matrix(0,1,ncols(subm2)); |
---|
3622 | if (submat(keepF,l,((ncols(Fnew)+1)..ncols(keepF)))==zeromat2) |
---|
3623 | { |
---|
3624 | for (cc=1; cc<=ncols(Fnew); c++) |
---|
3625 | { |
---|
3626 | Fnew[l,cc]=0; |
---|
3627 | } |
---|
3628 | } |
---|
3629 | i1=intvec(1..ncols(Fnew)); |
---|
3630 | subm1=submat(Fnew,l,i1); |
---|
3631 | subm2=submat(keepF,l,(ncols(Fnew)+1)..ncols(keepF)); |
---|
3632 | zeromat=matrix(0,1,ncols(Fnew)); |
---|
3633 | if (VdDegnhv(subm1,d,v)>VdDegnhv(subm2,d,w) |
---|
3634 | and submat(Fnew,l,intvec(1..ncols(Fnew)))!=zeromat) |
---|
3635 | { |
---|
3636 | //print("Reduzierung des V_d-Grades nötig"); |
---|
3637 | /*We need to reduce the V_d-degree. First we homogenize the |
---|
3638 | lth row of Fnew*/ |
---|
3639 | Fb=homogenize(subm1,d,v)*(nhv^rightexp); |
---|
3640 | if (SBcom==0) |
---|
3641 | { |
---|
3642 | /*computes a V_d-strict standard basis*/ |
---|
3643 | Mnew=slimgb(transpose(Mnew));// |
---|
3644 | SBcom=1; |
---|
3645 | } |
---|
3646 | /*computes a V_d-strict normal form for FB*/ |
---|
3647 | Fb=transpose(reduce(transpose(Fb),Mnew)); |
---|
3648 | if (VdDegnhv(Fb,d,v)> VdDegnhv(subm2,d,w) |
---|
3649 | and Fb!=matrix(0,nrows(Fb),ncols(Fb)))//should not happen |
---|
3650 | { |
---|
3651 | //print("Reduzierung fehlgeschlagen!!!!!!!!!!!!!!!!"); |
---|
3652 | } |
---|
3653 | } |
---|
3654 | else |
---|
3655 | { |
---|
3656 | /*condition on V_ddeg already satisfied -> no normal form |
---|
3657 | computation is needed*/ |
---|
3658 | Fb=submat(Fnew,l,intvec(1..ncols(Fnew))); |
---|
3659 | } |
---|
3660 | } |
---|
3661 | else |
---|
3662 | { |
---|
3663 | Fb=homogenize(submat(Fnew,l,intvec(1..ncols(Fnew))),d,v); |
---|
3664 | if (SBcom==0) |
---|
3665 | { |
---|
3666 | Mnew=slimgb(transpose(Mnew));// computes a V_d-strict Groebner basis |
---|
3667 | SBcom=1; |
---|
3668 | } |
---|
3669 | Fb=transpose(reduce(transpose(Fb),Mnew));//normal form |
---|
3670 | } |
---|
3671 | for (k=1; k<=ncols(Fnew);k++) |
---|
3672 | { |
---|
3673 | Fnew[l,k]=Fb[1,k]; |
---|
3674 | } |
---|
3675 | } |
---|
3676 | Fnew=subst(Fnew,nhv,1);//obtain normal form in D_n |
---|
3677 | setring W; |
---|
3678 | F=imap(Whom,Fnew); |
---|
3679 | return(F); |
---|
3680 | } |
---|
3681 | |
---|
3682 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3683 | |
---|
3684 | static proc homogenize (matrix M,int d,intvec v,list #) |
---|
3685 | { |
---|
3686 | /* we compute the F[v]-homogenization of each row of M (cf. Def. 3.4 in [OT])*/ |
---|
3687 | if (M==matrix(0,nrows(M),ncols(M))) |
---|
3688 | { |
---|
3689 | return(M); |
---|
3690 | } |
---|
3691 | int i,l,s, kmin, nhvexp; |
---|
3692 | poly f; |
---|
3693 | intvec vnm; |
---|
3694 | list findmin,maxnhv,rempoly,remk,rem1,rem2; |
---|
3695 | int n=(nvars(basering)-1) div 2; |
---|
3696 | for (int k=1; k<=nrows(M); k++) |
---|
3697 | { |
---|
3698 | for (l=1; l<=ncols (M); l++) |
---|
3699 | { |
---|
3700 | f=M[k,l]; |
---|
3701 | s=size(f); |
---|
3702 | for (i=1; i<=s; i++) |
---|
3703 | { |
---|
3704 | vnm=leadexp(f); |
---|
3705 | vnm=vnm[n+2..n+d+1]-vnm[2..d+1]; |
---|
3706 | kmin=sum(vnm)+v[l]; |
---|
3707 | rem1[size(rem1)+1]=lead(f); |
---|
3708 | rem2[size(rem2)+1]=kmin; |
---|
3709 | findmin=insert(findmin,kmin); |
---|
3710 | f=f-lead(f); |
---|
3711 | } |
---|
3712 | rempoly[l]=rem1; |
---|
3713 | remk[l]=rem2; |
---|
3714 | rem1=list(); |
---|
3715 | rem2=list(); |
---|
3716 | } |
---|
3717 | if (size(findmin)!=0) |
---|
3718 | { |
---|
3719 | kmin=Min(findmin); |
---|
3720 | } |
---|
3721 | for (l=1; l<=ncols(M); l++) |
---|
3722 | { |
---|
3723 | if (M[k,l]!=0) |
---|
3724 | { |
---|
3725 | M[k,l]=0; |
---|
3726 | for (i=1; i<=size(rempoly[l]);i++) |
---|
3727 | { |
---|
3728 | nhvexp=remk[l][i]-kmin; |
---|
3729 | M[k,l]=M[k,l]+nhv^(nhvexp)*rempoly[l][i]; |
---|
3730 | maxnhv[size(maxnhv)+1]=nhvexp; |
---|
3731 | } |
---|
3732 | } |
---|
3733 | } |
---|
3734 | rempoly=list(); |
---|
3735 | remk=list(); |
---|
3736 | findmin=list(); |
---|
3737 | } |
---|
3738 | maxnhv=Max(maxnhv); |
---|
3739 | nhvexp=maxnhv[1]; |
---|
3740 | if (size(#)!=0) |
---|
3741 | { |
---|
3742 | return(list(M,nhvexp));//only needed for normal form computations |
---|
3743 | } |
---|
3744 | return(M); |
---|
3745 | } |
---|
3746 | |
---|
3747 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3748 | |
---|
3749 | static proc soldr (matrix M,matrix N) |
---|
3750 | { |
---|
3751 | /* We compute a ncols(M) x nrows(M)-matrix C such that |
---|
3752 | C[i,1]M_1+...+C[i,nrows(M)]M_(nrows(M))= e_i mod im(N), |
---|
3753 | where e_i is the ith basis element on the range of M, M_j denotes the jth row |
---|
3754 | of M and im(N) is generated by the rows of N */ |
---|
3755 | int n=nrows(M); |
---|
3756 | int q=ncols(M); |
---|
3757 | matrix S=concat(transpose(M),transpose(N)); |
---|
3758 | def W=basering; |
---|
3759 | list Data=ringlist(W); |
---|
3760 | list Save=Data[3]; |
---|
3761 | Data[3]=list(list("c",0),list("dp",intvec(1..nvars(W)))); |
---|
3762 | def Wmod=ring(Data); |
---|
3763 | setring Wmod; |
---|
3764 | matrix Smod=imap(W,S); |
---|
3765 | matrix E[q][1]; |
---|
3766 | matrix Smod2,Smodnew; |
---|
3767 | option(returnSB); |
---|
3768 | int i,j; |
---|
3769 | for (i=1;i<=q;i++) |
---|
3770 | { |
---|
3771 | E[i,1]=1; |
---|
3772 | Smod2=concat(E,Smod); |
---|
3773 | Smod2=syz(Smod2); |
---|
3774 | E[i,1]=0; |
---|
3775 | for (j=1;j<=ncols(Smod2);j++) |
---|
3776 | { |
---|
3777 | if (Smod2[1,j]==1) |
---|
3778 | { |
---|
3779 | Smodnew=concat(Smodnew,(-1)*(submat(Smod2,intvec(2..n+1),j))); |
---|
3780 | break; |
---|
3781 | } |
---|
3782 | } |
---|
3783 | } |
---|
3784 | Smodnew=transpose(submat(Smodnew,intvec(1..n),intvec(2..q+1))); |
---|
3785 | setring W; |
---|
3786 | matrix Snew=imap(Wmod,Smodnew); |
---|
3787 | option(none); |
---|
3788 | return (Snew); |
---|
3789 | } |
---|
3790 | |
---|
3791 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3792 | |
---|
3793 | static proc prodr (int k,int l) |
---|
3794 | { |
---|
3795 | if (k==0) |
---|
3796 | { |
---|
3797 | matrix P=unitmat(l); |
---|
3798 | return (P); |
---|
3799 | } |
---|
3800 | matrix O[l][k]; |
---|
3801 | matrix P=transpose(concat(O,unitmat(l))); |
---|
3802 | return (P); |
---|
3803 | } |
---|
3804 | |
---|
3805 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3806 | |
---|
3807 | static proc VdDeg(matrix M,int d,intvec v,list #) |
---|
3808 | { |
---|
3809 | /* We assume that the basering it the nth Weyl algebra and that M is a 1 x r- |
---|
3810 | matrix. |
---|
3811 | We compute the V_d-deg of M with respect to the shift vector v, |
---|
3812 | i.e V_ddeg(M)=max (V_ddeg(M_i)+v[i]), where k=V_ddeg(M_i) if k is the minimal |
---|
3813 | integer, such that M_i can be expressed as a sum of operators |
---|
3814 | x(1)^(a_1)*...*x(n)^(a_n)*D(1)^(b_1)*...*D(n)^(b_n) with |
---|
3815 | a_1+..+a_d+k>=b_1+..+b_d*/ |
---|
3816 | int i, j, etoint; |
---|
3817 | int n=nvars(basering) div 2; |
---|
3818 | intvec e; |
---|
3819 | list findmax; |
---|
3820 | int c=ncols(M); |
---|
3821 | poly l; |
---|
3822 | list positionpoly,positionVd; |
---|
3823 | for (i=1; i<=c; i++) |
---|
3824 | { |
---|
3825 | positionpoly[i]=list(); |
---|
3826 | positionVd[i]=list(); |
---|
3827 | while (M[1,i]!=0) |
---|
3828 | { |
---|
3829 | l=lead(M[1,i]); |
---|
3830 | positionpoly[i][size(positionpoly[i])+1]=l; |
---|
3831 | e=leadexp(l); |
---|
3832 | e=-e[1..d]+e[n+1..n+d]; |
---|
3833 | e=sum(e)+v[i]; |
---|
3834 | etoint=e[1]; |
---|
3835 | positionVd[i][size(positionVd[i])+1]=etoint; |
---|
3836 | findmax[size(findmax)+1]=etoint; |
---|
3837 | M[1,i]=M[1,i]-l; |
---|
3838 | } |
---|
3839 | } |
---|
3840 | if (size(findmax)!=0) |
---|
3841 | { |
---|
3842 | int maxVd=Max(findmax); |
---|
3843 | if (size(#)==0) |
---|
3844 | { |
---|
3845 | return (maxVd); |
---|
3846 | } |
---|
3847 | } |
---|
3848 | else // M is 0-modul |
---|
3849 | { |
---|
3850 | return(int(0)); |
---|
3851 | } |
---|
3852 | l=0; |
---|
3853 | for (i=c; i>=1; i--) |
---|
3854 | { |
---|
3855 | for (j=1; j<=size(positionVd[i]); j++) |
---|
3856 | { |
---|
3857 | if (positionVd[i][j]==maxVd) |
---|
3858 | { |
---|
3859 | l=l+positionpoly[i][j]; |
---|
3860 | } |
---|
3861 | } |
---|
3862 | if (l!=0) |
---|
3863 | { |
---|
3864 | /*returns the largest component that has maximal V_d-degree |
---|
3865 | and its terms of maximal V_d-deg (needed for globalBFun)*/ |
---|
3866 | return (list(l,i)); |
---|
3867 | } |
---|
3868 | } |
---|
3869 | } |
---|
3870 | |
---|
3871 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3872 | |
---|
3873 | static proc VdDegnhv(matrix M,int d,intvec v,list #) |
---|
3874 | { |
---|
3875 | /* As the procedure VdDeg, but the basering is the nth Weyl algebra |
---|
3876 | with a commutative variable nhv*/ |
---|
3877 | int i,j,etoint; |
---|
3878 | int n=nvars(basering) div 2; |
---|
3879 | intvec e; |
---|
3880 | int etoint; |
---|
3881 | list findmax; |
---|
3882 | int c=ncols(M); |
---|
3883 | poly l; |
---|
3884 | list positionpoly; |
---|
3885 | list positionVd; |
---|
3886 | for (i=1; i<=c; i++) |
---|
3887 | { |
---|
3888 | positionpoly[i]=list(); |
---|
3889 | positionVd[i]=list(); |
---|
3890 | while (M[1,i]!=0) |
---|
3891 | { |
---|
3892 | l=lead(M[1,i]); |
---|
3893 | positionpoly[i][size(positionpoly[i])+1]=l; |
---|
3894 | e=leadexp(l); |
---|
3895 | e=-e[2..d+1]+e[n+2..n+d+1]; |
---|
3896 | e=sum(e)+v[i]; |
---|
3897 | etoint=e[1]; |
---|
3898 | positionVd[i][size(positionVd[i])+1]=etoint; |
---|
3899 | findmax[size(findmax)+1]=etoint; |
---|
3900 | M[1,i]=M[1,i]-l; |
---|
3901 | } |
---|
3902 | } |
---|
3903 | if (size(findmax)!=0) |
---|
3904 | { |
---|
3905 | int maxVd=Max(findmax); |
---|
3906 | if (size(#)==0) |
---|
3907 | { |
---|
3908 | return (maxVd); |
---|
3909 | } |
---|
3910 | } |
---|
3911 | else // M is 0-modul |
---|
3912 | { |
---|
3913 | return(int(0)); |
---|
3914 | } |
---|
3915 | } |
---|
3916 | |
---|
3917 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3918 | |
---|
3919 | static proc deletecol(matrix M,int l) |
---|
3920 | { |
---|
3921 | int s=ncols(M); |
---|
3922 | if (l==1) |
---|
3923 | { |
---|
3924 | M=submat(M,(1..nrows(M)),(2..ncols(M))); |
---|
3925 | return(M); |
---|
3926 | } |
---|
3927 | if (l==s) |
---|
3928 | { |
---|
3929 | M=submat(M,(1..nrows(M)),(1..(ncols(M)-1))); |
---|
3930 | return(M); |
---|
3931 | } |
---|
3932 | intvec v=(1..(l-1)),((l+1)..s); |
---|
3933 | M=submat(M,(1..nrows(M)),v); |
---|
3934 | return(M); |
---|
3935 | } |
---|
3936 | |
---|
3937 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3938 | |
---|
3939 | static proc mHom(poly f) |
---|
3940 | {/*for globalBFunOT*/ |
---|
3941 | poly g; |
---|
3942 | poly l; |
---|
3943 | poly add; |
---|
3944 | intvec e; |
---|
3945 | list minint; |
---|
3946 | list remf; |
---|
3947 | int i; |
---|
3948 | int j; |
---|
3949 | int n=nvars(basering) div 4; |
---|
3950 | if (f==0) |
---|
3951 | { |
---|
3952 | return(f); |
---|
3953 | } |
---|
3954 | while (f!=0) |
---|
3955 | { |
---|
3956 | l=lead(f); |
---|
3957 | e=leadexp(l); |
---|
3958 | remf[size(remf)+1]=list(); |
---|
3959 | remf[size(remf)][1]=l; |
---|
3960 | for (i=1; i<=n; i++) |
---|
3961 | { |
---|
3962 | remf[size(remf)][i+1]=-e[2*n+i]+e[3*n+i]; |
---|
3963 | if (size(minint)<i) |
---|
3964 | { |
---|
3965 | minint[i]=list(); |
---|
3966 | } |
---|
3967 | minint[i][size(minint[i])+1]=-e[2*n+i]+e[3*n+i]; |
---|
3968 | } |
---|
3969 | f=f-l; |
---|
3970 | } |
---|
3971 | for (i=1; i<=n; i++) |
---|
3972 | { |
---|
3973 | minint[i]=Min(minint[i]); |
---|
3974 | } |
---|
3975 | for (i=1; i<=size(remf); i++) |
---|
3976 | { |
---|
3977 | add=remf[i][1]; |
---|
3978 | for (j=1; j<=n; j++) |
---|
3979 | { |
---|
3980 | add=v(j)^(remf[i][j+1]-minint[j])*add; |
---|
3981 | } |
---|
3982 | g=g+add; |
---|
3983 | } |
---|
3984 | return (g); |
---|
3985 | } |
---|
3986 | |
---|
3987 | //////////////////////////////////////////////////////////////////////////////////// |
---|
3988 | |
---|
3989 | static proc permuteVar(list L,int n) |
---|
3990 | {/*for globalBFunOT*/ |
---|
3991 | if (typeof(L[1])=="intvec") |
---|
3992 | { |
---|
3993 | intvec v=L[1]; |
---|
3994 | } |
---|
3995 | else |
---|
3996 | { |
---|
3997 | intvec v=(1:L[1]),(0:L[1]); |
---|
3998 | } |
---|
3999 | int i;int k; int indi=0; |
---|
4000 | int j; |
---|
4001 | int s=size(v); |
---|
4002 | poly e; |
---|
4003 | intvec fore; |
---|
4004 | for (i=2; i<=size(v); i=i+2) |
---|
4005 | { |
---|
4006 | |
---|
4007 | if (v[i]!=0) |
---|
4008 | { |
---|
4009 | j=i+1; |
---|
4010 | while (v[j]!=0) |
---|
4011 | { |
---|
4012 | j=j+1; |
---|
4013 | } |
---|
4014 | v[i]=0; |
---|
4015 | v[j]=1; |
---|
4016 | fore=0; |
---|
4017 | indi=0; |
---|
4018 | for (k=1; k<=size(v); k++) |
---|
4019 | { |
---|
4020 | if (k!=i and k!=j) |
---|
4021 | { |
---|
4022 | if (indi==0) |
---|
4023 | { |
---|
4024 | indi=1; |
---|
4025 | fore[1]=v[k]; |
---|
4026 | } |
---|
4027 | else |
---|
4028 | { |
---|
4029 | fore[size(fore)+1]=v[k]; |
---|
4030 | } |
---|
4031 | } |
---|
4032 | } |
---|
4033 | e=e-(j-i)*permutevar(list(fore),n); |
---|
4034 | } |
---|
4035 | } |
---|
4036 | e=e+s(n)^(size(v) div 2); |
---|
4037 | return (e); |
---|
4038 | } |
---|
4039 | |
---|
4040 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4041 | |
---|
4042 | static proc makeHomogenizedWeyl(int n,list #) |
---|
4043 | { |
---|
4044 | /*modified version of the procedure makeWeyl() from the library nctools.lib*/ |
---|
4045 | /*Creates the nth homogenized Weyl algebra with variables x(1),..,x(n),D(1),.., |
---|
4046 | D(n) and homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
4047 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
4048 | if (n<1) |
---|
4049 | { |
---|
4050 | print("Incorrect input"); |
---|
4051 | return(); |
---|
4052 | } |
---|
4053 | if (n ==1) |
---|
4054 | { |
---|
4055 | ring @rr = 0,(x(1),D(1),h),dp; |
---|
4056 | } |
---|
4057 | else |
---|
4058 | { |
---|
4059 | ring @rr = 0,(x(1..n),D(1..n),h),dp; |
---|
4060 | } |
---|
4061 | setring @rr; |
---|
4062 | if (size(#)==0) |
---|
4063 | { |
---|
4064 | def @rrr = homogenizedWeyl(); |
---|
4065 | } |
---|
4066 | else |
---|
4067 | { |
---|
4068 | def @rrr=homogenizedWeyl(#); |
---|
4069 | } |
---|
4070 | return(@rrr); |
---|
4071 | } |
---|
4072 | |
---|
4073 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4074 | |
---|
4075 | static proc homogenizedWeyl (list #) |
---|
4076 | { |
---|
4077 | /*modified version of the procedure Weyl() from the library nctools.lib*/ |
---|
4078 | /*Creates a homogenized Weyl algebra structure on the basering. We assume |
---|
4079 | n=nvars(basering) is odd. The first (n-1)/2 variables will be treated as the |
---|
4080 | x(i), the next (n-1)/2 as the corresponding differentials D(i) and the last as |
---|
4081 | the homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
4082 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
4083 | string rname=nameof(basering); |
---|
4084 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
4085 | { |
---|
4086 | "You have to call the procedure from the ring"; |
---|
4087 | return(); |
---|
4088 | } |
---|
4089 | int nv = nvars(basering); |
---|
4090 | int N = (nv-1) div 2; |
---|
4091 | if (((nv-1) % 2) != 0) |
---|
4092 | { |
---|
4093 | "Cannot create homogenized Weyl structure for an even number of generators"; |
---|
4094 | return(); |
---|
4095 | } |
---|
4096 | matrix @D[nv][nv]; |
---|
4097 | int i; |
---|
4098 | for ( i=1; i<=N; i++ ) |
---|
4099 | { |
---|
4100 | @D[i,N+i]=h^2; |
---|
4101 | } |
---|
4102 | def @R = nc_algebra(1,@D); |
---|
4103 | setring @R; |
---|
4104 | list RL=ringlist(@R); |
---|
4105 | intvec v; |
---|
4106 | /*we need this ordering for Groebner basis computations*/ |
---|
4107 | for (i=1; i<=N; i++) |
---|
4108 | { |
---|
4109 | v[i]=-1; |
---|
4110 | v[N+i]=1; |
---|
4111 | } |
---|
4112 | v[nv]=0; |
---|
4113 | /* we assign weights to module components*/ |
---|
4114 | if (size(#)!=0) |
---|
4115 | { |
---|
4116 | if (typeof(#[1])=="intvec") |
---|
4117 | { |
---|
4118 | intvec m=#[1]; |
---|
4119 | for (i=1; i<=size(m); i++) |
---|
4120 | { |
---|
4121 | v[size(v)+1]=m[i];//assigns weight m[i] to the ith module component |
---|
4122 | } |
---|
4123 | RL[3]=insert(RL[3],list("am",v)); |
---|
4124 | } |
---|
4125 | else |
---|
4126 | { |
---|
4127 | RL[3]=insert(RL[3],list("a",v)); |
---|
4128 | } |
---|
4129 | } |
---|
4130 | else |
---|
4131 | { |
---|
4132 | RL[3]=insert(RL[3],list("a",v)); |
---|
4133 | } |
---|
4134 | intvec w=(1:nv); |
---|
4135 | if (size(#)>=2) |
---|
4136 | { |
---|
4137 | if (typeof(#[2])=="intvec") |
---|
4138 | { |
---|
4139 | intvec n=#[2]; |
---|
4140 | for (i=1; i<=size(n); i++) |
---|
4141 | { |
---|
4142 | w[size(w)+1]=n[i]; |
---|
4143 | } |
---|
4144 | RL[3]=insert(RL[3],list("am",w)); |
---|
4145 | } |
---|
4146 | else |
---|
4147 | { |
---|
4148 | RL[3]=insert(RL[3],list("a",w)); |
---|
4149 | } |
---|
4150 | } |
---|
4151 | else |
---|
4152 | { |
---|
4153 | RL[3]=insert(RL[3],list("a",w)); |
---|
4154 | } |
---|
4155 | /*this ordering is needed for globalBFun and globalBFunOT*/ |
---|
4156 | list saveord=RL[3][3]; |
---|
4157 | RL[3][3]=RL[3][4]; |
---|
4158 | RL[3][4]=saveord; |
---|
4159 | intvec notforh=(1:(size(RL[3][4][2])-1)); |
---|
4160 | RL[3][4][2]=notforh; |
---|
4161 | RL[3][5]=list("dp",1); |
---|
4162 | def @@R=ring(RL); |
---|
4163 | return(@@R); |
---|
4164 | } |
---|
4165 | |
---|
4166 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4167 | |
---|
4168 | static proc nHomogenize (matrix M,list #) |
---|
4169 | { |
---|
4170 | /* # may contain an intvec v, if no intvec is given, we assume that v=(0:ncols(M)) |
---|
4171 | We compute the h[v]-homogenization of the rows of M as in Definition 9.2 [OT]*/ |
---|
4172 | int l; poly f; int s; int i; intvec vnm;int kmin; list findmax; |
---|
4173 | int n=(nvars(basering)-1) div 2; |
---|
4174 | list rempoly; |
---|
4175 | list remk; |
---|
4176 | list rem1; |
---|
4177 | list rem2; |
---|
4178 | list maxhexp; |
---|
4179 | int hexp; |
---|
4180 | intvec v=(0:ncols(M)); |
---|
4181 | if (size(#)!=0) |
---|
4182 | { |
---|
4183 | if (typeof(#[1])=="intvec") |
---|
4184 | { |
---|
4185 | v=#[1]; |
---|
4186 | } |
---|
4187 | } |
---|
4188 | if (size(v)<ncols(M)) |
---|
4189 | { |
---|
4190 | for (i=size(v)+1; i<=ncols(M); i++) |
---|
4191 | { |
---|
4192 | v[i]=0; |
---|
4193 | } |
---|
4194 | } |
---|
4195 | for (int k=1; k<=nrows(M); k++) |
---|
4196 | { |
---|
4197 | for (l=1; l<=ncols (M); l++) |
---|
4198 | { |
---|
4199 | f=M[k,l]; |
---|
4200 | s=size(f); |
---|
4201 | for (i=1; i<=s; i++) |
---|
4202 | { |
---|
4203 | vnm=leadexp(f); |
---|
4204 | kmin=sum(vnm)+v[l]; |
---|
4205 | rem1[size(rem1)+1]=lead(f); |
---|
4206 | rem2[size(rem2)+1]=kmin; |
---|
4207 | findmax=insert(findmax,kmin); |
---|
4208 | f=f-lead(f); |
---|
4209 | } |
---|
4210 | rempoly[l]=rem1; |
---|
4211 | remk[l]=rem2; |
---|
4212 | rem1=list(); |
---|
4213 | rem2=list(); |
---|
4214 | } |
---|
4215 | if (size(findmax)!=0) |
---|
4216 | { |
---|
4217 | kmin=Max(findmax); |
---|
4218 | } |
---|
4219 | else |
---|
4220 | { |
---|
4221 | kmin=0; |
---|
4222 | } |
---|
4223 | for (l=1; l<=ncols(M); l++) |
---|
4224 | { |
---|
4225 | if (M[k,l]!=0) |
---|
4226 | { |
---|
4227 | M[k,l]=0; |
---|
4228 | for (i=1; i<=size(rempoly[l]);i++) |
---|
4229 | { |
---|
4230 | hexp=kmin-remk[l][i]; |
---|
4231 | maxhexp[size(maxhexp)+1]=hexp; |
---|
4232 | M[k,l]=M[k,l]+h^hexp*rempoly[l][i]; |
---|
4233 | } |
---|
4234 | } |
---|
4235 | } |
---|
4236 | rempoly=list(); |
---|
4237 | remk=list(); |
---|
4238 | findmax=list(); |
---|
4239 | } |
---|
4240 | if (size(maxhexp)!=0) |
---|
4241 | { |
---|
4242 | maxhexp=Max(maxhexp); |
---|
4243 | hexp=maxhexp[1]; |
---|
4244 | } |
---|
4245 | else |
---|
4246 | { |
---|
4247 | hexp=0; |
---|
4248 | } |
---|
4249 | if (size(#)>1) |
---|
4250 | { |
---|
4251 | list forreturn=M,hexp; |
---|
4252 | |
---|
4253 | return(forreturn); |
---|
4254 | } |
---|
4255 | return(M); |
---|
4256 | } |
---|
4257 | |
---|
4258 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4259 | |
---|
4260 | static proc max(int i,int j) |
---|
4261 | { |
---|
4262 | if(i>j){return(i);} |
---|
4263 | return(j); |
---|
4264 | } |
---|
4265 | |
---|
4266 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4267 | |
---|
4268 | static proc nDeg (matrix M,intvec m) |
---|
4269 | {/*we compute an intvec n such that n[i]=max(deg(M[i,j])+m[j]|M[i,j]!=0) (where deg |
---|
4270 | stands for the total degree) if (M[i,j]!=0 for some j) and n[i]=0 else*/ |
---|
4271 | int i; int j; |
---|
4272 | intvec n; |
---|
4273 | list L; |
---|
4274 | for (i=1; i<=nrows(M); i++) |
---|
4275 | { |
---|
4276 | L=list(); |
---|
4277 | for (j=1; j<=ncols(M); j++) |
---|
4278 | { |
---|
4279 | if (M[i,j]!=0) |
---|
4280 | { |
---|
4281 | L=insert(L,deg(M[i,j])+m[j]); |
---|
4282 | } |
---|
4283 | } |
---|
4284 | if (size(L)==0) |
---|
4285 | { |
---|
4286 | n[i]=0; |
---|
4287 | } |
---|
4288 | else |
---|
4289 | { |
---|
4290 | n[i]=Max(L); |
---|
4291 | } |
---|
4292 | } |
---|
4293 | return(n); |
---|
4294 | } |
---|
4295 | |
---|
4296 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4297 | |
---|
4298 | static proc minIntRoot(list L,list #) |
---|
4299 | "USAGE:minIntRoot(L [,M]); L list, M optinonal list |
---|
4300 | ASSUME:L a list of univariate polynomials with rational coefficients @* |
---|
4301 | the variable of the polynomial is s if size(#)==0 (needed for proc |
---|
4302 | MVComplex) and t else (needed for globalBFun) |
---|
4303 | RETURN:-if size(#)==0: int i, where i is an integer root of one of the polynomials |
---|
4304 | and it is minimal with respect to that property@* |
---|
4305 | -if size(#)!=0: list L=(i,j), where i is as above and j is an integer root |
---|
4306 | of one of the polynomials and is maximal with respect to that property (if |
---|
4307 | an integer root exists) or L=list() else |
---|
4308 | " |
---|
4309 | { |
---|
4310 | def B=basering; |
---|
4311 | if (size(#)==0) |
---|
4312 | { |
---|
4313 | ring rnew=0,s,dp; |
---|
4314 | } |
---|
4315 | else |
---|
4316 | { |
---|
4317 | ring rnew=0,t,dp; |
---|
4318 | } |
---|
4319 | list L=imap(B,L); |
---|
4320 | |
---|
4321 | int i; |
---|
4322 | int j; |
---|
4323 | number isint; |
---|
4324 | list possmin; |
---|
4325 | ideal allfac; |
---|
4326 | list allfacs; |
---|
4327 | for (i=1; i<=size(L); i++) |
---|
4328 | { |
---|
4329 | allfac=factorize(L[i],1); |
---|
4330 | for (j=1; j<=ncols(allfac); j++) |
---|
4331 | { |
---|
4332 | allfacs[j]=allfac[j]; |
---|
4333 | } |
---|
4334 | for (j=1; j<=size(allfacs); j++) |
---|
4335 | { |
---|
4336 | if (deg(allfacs[j])==1) |
---|
4337 | { |
---|
4338 | isint=number(subst(allfacs[j],var(1),0)/leadcoef(allfacs[j])); |
---|
4339 | if (isint-int(isint)==0) |
---|
4340 | { |
---|
4341 | possmin[size(possmin)+1]=int(isint); |
---|
4342 | } |
---|
4343 | } |
---|
4344 | } |
---|
4345 | allfacs=list(); |
---|
4346 | } |
---|
4347 | int zerolist; |
---|
4348 | if (size(possmin)!=0) |
---|
4349 | { |
---|
4350 | int miniroot=(-1)*Max(possmin); |
---|
4351 | int maxiroot=(-1)*Min(possmin); |
---|
4352 | } |
---|
4353 | else |
---|
4354 | { |
---|
4355 | zerolist=1; |
---|
4356 | } |
---|
4357 | setring B; |
---|
4358 | if (size(#)==0) |
---|
4359 | { |
---|
4360 | return(miniroot); |
---|
4361 | } |
---|
4362 | else |
---|
4363 | { |
---|
4364 | if (zerolist==0) |
---|
4365 | { |
---|
4366 | return(list(miniroot,maxiroot)); |
---|
4367 | } |
---|
4368 | else |
---|
4369 | { |
---|
4370 | return(list()); |
---|
4371 | } |
---|
4372 | } |
---|
4373 | } |
---|
4374 | |
---|
4375 | |
---|
4376 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4377 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4378 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4379 | /* |
---|
4380 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4381 | FURTHER EXAMPLES FOR TESTING THE PROCEDURES |
---|
4382 | //////////////////////////////////////////////////////////////////////////////////// |
---|
4383 | LIB "derham.lib"; |
---|
4384 | |
---|
4385 | //---------------------------------------- |
---|
4386 | //EXAMPLE 1 |
---|
4387 | //---------------------------------------- |
---|
4388 | ring r=0,(x,y),dp; |
---|
4389 | poly f=y2-x3-2x+3; |
---|
4390 | list L=deRhamCohomology(f); |
---|
4391 | L; |
---|
4392 | kill r; |
---|
4393 | |
---|
4394 | //---------------------------------------- |
---|
4395 | //EXAMPLE 2 |
---|
4396 | //---------------------------------------- |
---|
4397 | ring r=0,(x,y),dp; |
---|
4398 | poly f=y2-x3-x; |
---|
4399 | list L=deRhamCohomology(f); |
---|
4400 | L; |
---|
4401 | kill r; |
---|
4402 | |
---|
4403 | //---------------------------------------- |
---|
4404 | //EXAMPLE 3 |
---|
4405 | //---------------------------------------- |
---|
4406 | ring r=0,(x,y),dp; |
---|
4407 | list C=list(x2-1,(x+1)*y,y*(y2+2x+1)); |
---|
4408 | list L=deRhamCohomology(C); |
---|
4409 | L; |
---|
4410 | kill r; |
---|
4411 | |
---|
4412 | //---------------------------------------- |
---|
4413 | //EXAMPLE 4 |
---|
4414 | //---------------------------------------- |
---|
4415 | ring r=0,(x,y,z),dp; |
---|
4416 | list C=list(x*(x-1),y,z*(z-1),z*(x-1)); |
---|
4417 | list L=deRhamCohomology(C); |
---|
4418 | L; |
---|
4419 | kill r; |
---|
4420 | |
---|
4421 | //---------------------------------------- |
---|
4422 | //EXAMPLE 5 |
---|
4423 | //---------------------------------------- |
---|
4424 | ring r=0,(x,y,z),dp; |
---|
4425 | list C=list(x*y,y*z); |
---|
4426 | list L=deRhamCohomology(C,"Vdres"); |
---|
4427 | L; |
---|
4428 | kill r; |
---|
4429 | |
---|
4430 | //---------------------------------------- |
---|
4431 | //EXAMPLE 6 |
---|
4432 | //---------------------------------------- |
---|
4433 | ring r=0,(x,y,z,u),dp; |
---|
4434 | list C=list(x,y,z,u); |
---|
4435 | list L=deRhamCohomology(C); |
---|
4436 | L; |
---|
4437 | kill r; |
---|
4438 | |
---|
4439 | //---------------------------------------- |
---|
4440 | //EXAMPLE 7 |
---|
4441 | //---------------------------------------- |
---|
4442 | ring r=0,(x,y,z),dp; |
---|
4443 | poly f=x3+y3+z3; |
---|
4444 | list L=deRhamCohomology(f); |
---|
4445 | L; |
---|
4446 | kill r; |
---|
4447 | |
---|
4448 | //---------------------------------------- |
---|
4449 | //EXAMPLE 8 |
---|
4450 | //---------------------------------------- |
---|
4451 | ring r=0,(x,y,z),dp; |
---|
4452 | poly f=x2+y2+z2; |
---|
4453 | list L=deRhamCohomology(f,"Vdres"); |
---|
4454 | L; |
---|
4455 | kill r; |
---|
4456 | |
---|
4457 | //---------------------------------------- |
---|
4458 | //EXAMPLE 9 |
---|
4459 | //---------------------------------------- |
---|
4460 | ring r=0,(x,y,z,u),dp; |
---|
4461 | list C=list(x2+y2+z2,u); |
---|
4462 | list L=deRhamCohomology(C); |
---|
4463 | L; |
---|
4464 | kill r; |
---|
4465 | |
---|
4466 | |
---|
4467 | //---------------------------------------- |
---|
4468 | //EXAMPLE 10 |
---|
4469 | //---------------------------------------- |
---|
4470 | ring r=0,(x,y,z),dp; |
---|
4471 | list C=list((x*(y-1),y2-1)); |
---|
4472 | list L=deRhamCohomology(C); |
---|
4473 | L; |
---|
4474 | kill r; |
---|
4475 | |
---|
4476 | |
---|
4477 | */ |
---|