[0e8a5a] | 1 | /////////////////////////////////////////////////////////////////////////////// |
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[08fa62] | 2 | version="version derham.lib 4.0.0.0 Jun_2013 "; |
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[0e8a5a] | 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 | [W3] Walther, U.: Computing the cup product structure for complements of complex |
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| 25 | affine varieties, J. Pure Appl. Algebra 164, 247-273 (2001) |
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| 26 | |
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| 27 | |
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| 28 | PROCEDURES: |
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| 29 | |
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| 30 | deRhamCohomology(list[,opt]); computes the de Rham cohomology |
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| 31 | MVComplex(list); computes the Mayer-Vietoris complex |
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| 32 | "; |
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| 33 | |
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| 34 | LIB "nctools.lib"; |
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| 35 | LIB "matrix.lib"; |
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| 36 | LIB "qhmoduli.lib"; |
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| 37 | LIB "general.lib"; |
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[08fa62] | 38 | //LIB "dmod.lib"; |
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[0e8a5a] | 39 | LIB "bfun.lib"; |
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| 40 | LIB "dmodapp.lib"; |
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| 41 | LIB "poly.lib"; |
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| 42 | LIB "schreyer.lib"; |
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| 43 | LIB "dmodloc.lib"; |
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| 44 | |
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| 45 | |
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| 46 | //////////////////////////////////////////////////////////////////////////////////// |
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| 47 | |
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| 48 | proc deRhamCohomology(list L,list #) |
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| 49 | "USAGE: deRhamCohomology(L[,choices]); L a list consisting of polynomials, choices |
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| 50 | optional list consisting of one up to three strings @* |
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| 51 | The optional strings may be one of the strings@* |
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| 52 | -'noCE': compute quasi-isomorphic complexes without using Cartan-Eilenberg |
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| 53 | resolutionsq@* |
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| 54 | -'Vdres': compute quasi-isomorphic complexes using Cartan-Eilenberg |
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| 55 | resolutions; the CE resolutions are computed via V__d-homogenization |
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| 56 | and without using Schreyer's method @* |
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| 57 | -'Sres': compute quasi-isomorphic complexes using Cartan-Eilenberg |
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| 58 | resolutions in the homogenized Weyl algebra via Schreyer's method@* |
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| 59 | one of the strings@* |
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| 60 | -'iterativeloc': compute localizations by factorizing the polynomials and |
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| 61 | sucessive localization of the factors @* |
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| 62 | -'no iterativeloc': compute localizations by directly localizing the |
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| 63 | product@* |
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| 64 | and one of the strings |
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| 65 | -'onlybounds': computes bounds for the minimal and maximal interger roots |
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| 66 | of the global b-function |
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| 67 | -'exactroots' computes the minimal and maximal integer root of the global |
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| 68 | b-function |
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| 69 | The default is 'noCE', 'iterativeloc' and 'onlybounds'. |
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| 70 | ASSUME: -The basering must be a polynomial ring over the field of rational numbers@* |
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| 71 | RETURN: list, where the ith entry is the (i-1)st de Rham cohomology group of the |
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| 72 | complement of the complex affine variety given by the polynomials in L |
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| 73 | EXAMPLE:example deRhamCohomology; shows an example |
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| 74 | " |
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| 75 | { |
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| 76 | intvec saveoptions=option(get); |
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| 77 | intvec i1,i2; |
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| 78 | option(none); |
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| 79 | int recursiveloc=1; |
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| 80 | int i,j,nr,nc; |
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| 81 | def R=basering; |
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| 82 | poly islcm, forlcm; |
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| 83 | int n=nvars(R); |
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| 84 | int le=size(L)+n; |
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| 85 | string Syzstring="noCE"; |
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| 86 | int onlybounds=1; |
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| 87 | int diffforms; |
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| 88 | for (i=1; i<=size(#); i++) |
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| 89 | { |
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| 90 | if (#[i]=="Sres") |
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| 91 | { |
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| 92 | Syzstring="Sres"; |
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| 93 | } |
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| 94 | if (#[i]=="Vdres") |
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| 95 | { |
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| 96 | Syzstring="Vdres"; |
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| 97 | } |
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| 98 | if (#[i]=="noiterativeloc") |
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| 99 | { |
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| 100 | recursiveloc=0; |
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| 101 | } |
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| 102 | if (#[i]=="exactroots") |
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| 103 | { |
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| 104 | onlybounds=0; |
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| 105 | } |
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| 106 | if (#[i]=="diffforms") |
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| 107 | { |
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| 108 | diffforms=1; |
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| 109 | } |
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| 110 | } |
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| 111 | for (i=1; i<=size(L); i++) |
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| 112 | { |
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| 113 | if (L[i]==0) |
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| 114 | { |
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| 115 | L=delete(L,i); |
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| 116 | i=i-1; |
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| 117 | } |
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| 118 | } |
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| 119 | if (size(L)==0) |
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| 120 | { |
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| 121 | return (list(0));//////////////////////////////////////////////////////////////////stimmt das jetzt?!?????????????????????????????????? |
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| 122 | } |
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| 123 | for (i=1; i<= size(L); i++) |
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| 124 | { |
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| 125 | if (leadcoef(L[i])-L[i]==0) |
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| 126 | { |
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| 127 | return(list(1)); ///////////////////////////////////////////////////////////////stimmt das jetzt?!???????????????????????????????????? |
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| 128 | } |
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| 129 | } |
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| 130 | if (size(L)==0) |
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| 131 | { |
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| 132 | /*the complement of the variety given by the input is the whole space*/ |
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| 133 | return(list(1)); |
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| 134 | } |
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| 135 | for (i=1; i<=size(L); i++) |
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| 136 | { |
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| 137 | if (typeof(L[i])!="poly") |
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| 138 | { |
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| 139 | print("The input list must consist of polynomials"); |
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| 140 | return(); |
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| 141 | } |
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| 142 | } |
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| 143 | if (size(L)==1 and Syzstring=="noCE") |
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| 144 | { |
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| 145 | Syzstring="Sres"; |
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| 146 | } |
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| 147 | /* 1st step: compute the Mayer-Vietoris Complex and its Fourier transform*/ |
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| 148 | def W=MVComplex(L,recursiveloc);//new ring that contains the MV complex |
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| 149 | setring W; |
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| 150 | list fortoVdstrict=MV; |
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| 151 | if (diffforms==0) |
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| 152 | { |
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| 153 | ideal IFourier=var(n+1); |
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| 154 | for (i=2;i<=n;i++) |
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| 155 | { |
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| 156 | IFourier=IFourier,var(n+i); |
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| 157 | } |
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| 158 | for (i=1; i<=n;i++) |
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| 159 | { |
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| 160 | IFourier=IFourier,-var(i); |
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| 161 | } |
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| 162 | map cFourier=W,IFourier; |
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| 163 | matrix sup; |
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| 164 | for (i=1; i<=size(MV); i++) |
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| 165 | { |
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| 166 | sup=fortoVdstrict[i]; |
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| 167 | /*takes the Fourier transform of the MV complex*/ |
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| 168 | fortoVdstrict[i]=cFourier(sup); |
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| 169 | } |
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| 170 | } |
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| 171 | /* 2nd step: Compute a V_d-strict free complex that is quasi-isomorphic to the |
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| 172 | complex fortoVdstrict |
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| 173 | The 1st entry of the list rem will be the quasi-isomorphic complex, the 2nd |
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| 174 | entry contains the cohomology modules and is needed for the computation of the |
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| 175 | global b-function*/ |
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| 176 | if (Syzstring=="noCE") |
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| 177 | { |
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| 178 | list rem=quasiisomorphicVdComplex(fortoVdstrict,diffforms); |
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| 179 | list quasiiso=rem[3]; |
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| 180 | } |
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| 181 | else |
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| 182 | { |
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| 183 | list rem=toVdStrictFreeComplex(fortoVdstrict,Syzstring,diffforms); |
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| 184 | if (diffforms==1) |
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| 185 | { |
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| 186 | list quasiiso=list(matrix(1,1,1)); |
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| 187 | } |
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| 188 | } |
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| 189 | list newcomplex=rem[1]; |
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| 190 | //////////////////////////////////////////////////////////////////////////////////// |
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| 191 | /* 3rd step: Compute the bounds for the minimal and maximal integer root of the |
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| 192 | global b-function of newcomplex(i.e. compute the lcm of the b-functions of its |
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| 193 | cohomology modules)(if onlybouns=1). Else we compute the minimal and maximal |
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| 194 | integer root. |
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| 195 | |
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| 196 | If we compute only the bounds, we omit additional Groebner basis computations. |
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| 197 | However this leads to a higher-dimensional truncated complex. |
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| 198 | |
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| 199 | Note that the cohomology modules are already contained in rem[2]. |
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| 200 | minmaxk[1] and minmaxk[2] will contain the bounds resp exact roots.*/ |
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| 201 | if (diffforms==1) |
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| 202 | { |
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| 203 | list minmaxk=exactGlobalBFunIntegration(rem[2]); |
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| 204 | } |
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| 205 | else |
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| 206 | { |
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| 207 | if (onlybounds==1) |
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| 208 | { |
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| 209 | list minmaxk=globalBFun(rem[2],Syzstring); |
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| 210 | } |
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| 211 | else |
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| 212 | { |
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| 213 | list minmaxk=exactGlobalBFun(rem[2],Syzstring); |
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| 214 | } |
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| 215 | } |
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| 216 | if (size(minmaxk)==0) |
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| 217 | { |
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| 218 | return (0); |
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| 219 | } |
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| 220 | ///////////////////////////////////////////////////////////////////////////Bis hierhin angepasst |
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| 221 | /*4th step: Truncate the complex D_n/(x_1,...,x_n)\otimes C, (where |
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| 222 | 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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| 223 | m^i=newcomplex[3*i-1], d^i=newcomplex[3*i]), using Thm 5.7 in [W1]: |
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| 224 | The truncated module D_n/(x_1,..,x_n)\otimes C[i] is generated by the set |
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| 225 | (0,...,P_(i_j),0,...), where P_(i_j) is a monomial in C[D(1),...,D(n)] and |
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| 226 | if it is placed in component k it holds that |
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| 227 | minmaxk[1]-m^i[k]<=deg(P_(i_j))<=minmaxk[2]-m^i[k]*/ |
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| 228 | int k,l; |
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| 229 | list truncatedcomplex,shorten,upto; |
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| 230 | for (i=1; i<=size(newcomplex) div 3; i++) |
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| 231 | { |
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| 232 | shorten[3*i-1]=list(); |
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| 233 | for (j=1; j<=size(newcomplex[3*i-1]); j++) |
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| 234 | { |
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| 235 | /*shorten[3*i-1][j][k]=minmaxk[k]-m^i[j]+1 (for k=1,2) if this value is |
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| 236 | positive otherwise we will set it to be list(); |
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| 237 | .- we added +1, because we will use a list, where we put in position l |
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| 238 | polys of degree l+1*/ |
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| 239 | shorten[3*i-1][j]=list(minmaxk[1]-newcomplex[3*i-1][j]+1); |
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| 240 | if (diffforms==1) |
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| 241 | { |
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| 242 | shorten[3*i-1][j][1]=1; |
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| 243 | } |
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| 244 | shorten[3*i-1][j][2]=minmaxk[2]-newcomplex[3*i-1][j]+1; |
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| 245 | upto[size(upto)+1]=shorten[3*i-1][j][2]; |
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| 246 | if (shorten[3*i-1][j][2]<=0) |
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| 247 | { |
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| 248 | shorten[3*i-1][j]=list(); |
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| 249 | } |
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| 250 | else |
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| 251 | { |
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| 252 | if (shorten[3*i-1][j][1]<=0) |
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| 253 | { |
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| 254 | shorten[3*i-1][j][1]=1; |
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| 255 | } |
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| 256 | } |
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| 257 | } |
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| 258 | } |
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| 259 | int iupto=Max(upto);//maximal degree +1 of the polynomials we have to consider |
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| 260 | if (iupto<=0) |
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| 261 | { |
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| 262 | return(list(0)); |
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| 263 | } |
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| 264 | list allpolys; |
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| 265 | /*allpolys[i] will consist list of all monomials in D(1),...,D(n) of degree i-1*/ |
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| 266 | allpolys[1]=list(1); |
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| 267 | list minvar; |
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| 268 | list keepv; |
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| 269 | minvar[1]=list(1); |
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| 270 | for (i=1; i<=iupto-1; i++) |
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| 271 | { |
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| 272 | allpolys[i+1]=list(); |
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| 273 | minvar[i+1]=list(); |
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| 274 | for (k=1; k<=size(allpolys[i]); k++) |
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| 275 | { |
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| 276 | for (j=minvar[i][k]; j<=nvars(W) div 2; j++) |
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| 277 | { |
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| 278 | if (diffforms==0) |
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| 279 | { |
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| 280 | allpolys[i+1][size(allpolys[i+1])+1]=allpolys[i][k]*D(j); |
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| 281 | } |
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| 282 | else |
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| 283 | { |
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| 284 | allpolys[i+1][size(allpolys[i+1])+1]=allpolys[i][k]*x(j); |
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| 285 | } |
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| 286 | minvar[i+1][size(minvar[i+1])+1]=j; |
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| 287 | } |
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| 288 | } |
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| 289 | } |
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| 290 | list keepformatrix,sizetruncom,fortrun,fst; |
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| 291 | int count,stc; |
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| 292 | intvec v,forin; |
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| 293 | matrix subm; |
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| 294 | list keepcount; |
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| 295 | list passendespoly; |
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| 296 | /*now we compute the truncation*/ |
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| 297 | for (i=1; i<=size(newcomplex) div 3; i++) |
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| 298 | { |
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| 299 | /*truncatedcomplex[2*i-1] will contain all the generators for the truncation |
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| 300 | of D_n/(x(1),..,x(n))\otimes C[i]*/ |
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| 301 | truncatedcomplex[2*i-1]=list(); |
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| 302 | sizetruncom[2*i-1]=list(); |
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| 303 | sizetruncom[2*i]=list(); |
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| 304 | passendespoly[i]=list(); |
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| 305 | /*truncatedcomplex[2*i] will be the map trunc(D_n/(x(1),..,x(n))\otimes C[i]) |
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| 306 | ->trunc(D_n/(x(1),..,x(n))\otimes C[i+1])*/ |
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| 307 | truncatedcomplex[2*i]=newcomplex[3*i]; |
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| 308 | v=0;count=0; |
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| 309 | sizetruncom[2*i][1]=0; |
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| 310 | for (j=1; j<=newcomplex[3*i-2]; j++) |
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| 311 | { |
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| 312 | if (size(shorten[3*i-1][j])!=0) |
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| 313 | { |
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| 314 | fortrun=sublist(allpolys,shorten[3*i-1][j][1],shorten[3*i-1][j][2]); |
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| 315 | truncatedcomplex[2*i-1][size(truncatedcomplex[2*i-1])+1]=fortrun[1]; |
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| 316 | for (k=1; k<=size(fortrun[1]); k++) |
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| 317 | { |
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| 318 | for (l=1; l<=size(fortrun[1][k]); l++) |
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| 319 | { |
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| 320 | passendespoly[i][size(passendespoly[i])+1]=list(fortrun[1][k][l][1],j); |
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| 321 | } |
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| 322 | } |
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| 323 | count=count+fortrun[2]; |
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| 324 | fst=list(int(shorten[3*i-1][j][1])-1,int(shorten[3*i-1][j][2])-1); |
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| 325 | sizetruncom[2*i-1][size(sizetruncom[2*i-1])+1]=fst; |
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| 326 | sizetruncom[2*i][size(sizetruncom[2*i])+1]=count; |
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| 327 | if (v!=0) |
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| 328 | { |
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| 329 | v[size(v)+1]=j; |
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| 330 | } |
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| 331 | else |
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| 332 | { |
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| 333 | v[1]=j; |
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| 334 | } |
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| 335 | } |
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| 336 | } |
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| 337 | if (v!=0) |
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| 338 | { |
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| 339 | keepv[i]=v; |
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| 340 | subm=submat(truncatedcomplex[2*i],v,1..ncols(truncatedcomplex[2*i])); |
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| 341 | truncatedcomplex[2*i]=subm; |
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| 342 | if (i!=1) |
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| 343 | { |
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| 344 | i1=1..nrows(truncatedcomplex[2*(i-1)]); |
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| 345 | subm=submat(truncatedcomplex[2*(i-1)],i1,v); |
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| 346 | truncatedcomplex[2*(i-1)]=subm; |
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| 347 | } |
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| 348 | } |
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| 349 | else |
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| 350 | { |
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| 351 | keepv[i]=list(); |
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| 352 | truncatedcomplex[2*i]=matrix(0,1,ncols(truncatedcomplex[2*i])); |
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| 353 | if (i!=1) |
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| 354 | { |
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| 355 | nr=nrows(truncatedcomplex[2*(i-1)]); |
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| 356 | truncatedcomplex[2*(i-1)]=matrix(0,nr,1); |
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| 357 | } |
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| 358 | } |
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| 359 | } |
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| 360 | list keeptruncatedcomplex=truncatedcomplex; |
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| 361 | matrix M; |
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| 362 | int st,pi,pj; |
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| 363 | poly ptc; |
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| 364 | int b,d,ideg,kplus,lplus; |
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| 365 | int z; |
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| 366 | poly form,lform,nform; |
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| 367 | /*computation of the maps*/ |
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| 368 | if (diffforms==1) |
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| 369 | { |
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| 370 | def ConvWeyl=makeConverseWeyl(nvars(basering) div 2); |
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| 371 | setring ConvWeyl; |
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| 372 | poly form,lform,nform; |
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| 373 | poly ptc; |
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| 374 | list truncatedcomplex; |
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| 375 | matrix M; |
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| 376 | ideal I=x(1); |
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| 377 | for (i=2; i<=nvars(basering) div 2; i++) |
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| 378 | { |
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| 379 | I=I,var(nvars(basering) div 2 + i); |
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| 380 | } |
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| 381 | for (i=1; i<=nvars(basering) div 2; i++) |
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| 382 | { |
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| 383 | I=I,var(i); |
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| 384 | } |
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| 385 | map transtc=W,I; |
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| 386 | truncatedcomplex=transtc(truncatedcomplex); |
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| 387 | } |
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| 388 | for (i=1; i<size(truncatedcomplex) div 2; i++) |
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| 389 | { |
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| 390 | nr=max(1,sizetruncom[2*i][size(sizetruncom[2*i])]); |
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| 391 | nc=max(1,sizetruncom[2*i+2][size(sizetruncom[2*i+2])]); |
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| 392 | M=matrix(0,nr,nc); |
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| 393 | for (k=1; k<=size(truncatedcomplex[2*i-1]);k++) |
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| 394 | { |
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| 395 | for (l=1; l<=size(truncatedcomplex[2*(i+1)-1]); l++) |
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| 396 | { |
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| 397 | if (size(sizetruncom[2*i])!=1) |
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| 398 | { |
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| 399 | for (j=1; j<=size(truncatedcomplex[2*i-1][k]); j++) |
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| 400 | { |
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| 401 | for (b=1; b<=size(truncatedcomplex[2*i-1][k][j]); b++) |
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| 402 | { |
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| 403 | form=truncatedcomplex[2*i-1][k][j][b][1]; |
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| 404 | form=form*truncatedcomplex[2*i][k,l]; |
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| 405 | |
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| 406 | |
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| 407 | for (z=1; z<=nvars(basering) div 2; z++)//neu |
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| 408 | {// |
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| 409 | form=subst(form,var(z),0);// |
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| 410 | }// |
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| 411 | |
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| 412 | while (form!=0) |
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| 413 | { |
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| 414 | lform=lead(form); |
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| 415 | v=leadexp(lform); |
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| 416 | v=v[1..n]; |
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| 417 | // if (v==(0:n)) |
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| 418 | //{ |
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| 419 | ideg=deg(lform)-sizetruncom[2*(i+1)-1][l][1]; |
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| 420 | if (ideg>=0) |
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| 421 | { |
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| 422 | nr=ideg+1; |
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| 423 | st=size(truncatedcomplex[2*(i+1)-1][l][nr]); |
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| 424 | for (d=1; d<=st;d++) |
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| 425 | { |
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| 426 | nc=2*(i+1)-1; |
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| 427 | ptc=truncatedcomplex[nc][l][ideg+1][d][1]; |
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| 428 | if (leadmonom(lform)==ptc) |
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| 429 | { |
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| 430 | nr=2*i-1; |
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| 431 | pi=truncatedcomplex[nr][k][j][b][2]; |
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| 432 | pi=pi+sizetruncom[2*i][k]; |
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| 433 | nc=2*(i+1)-1; |
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| 434 | nr=ideg+1; |
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| 435 | pj=truncatedcomplex[nc][l][nr][d][2]; |
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| 436 | pj=pj+sizetruncom[2*(i+1)][l]; |
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| 437 | M[pi,pj]=leadcoef(lform); |
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| 438 | break; |
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| 439 | } |
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| 440 | } |
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| 441 | } |
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| 442 | // } |
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| 443 | |
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| 444 | form=form-lform; |
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| 445 | } |
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| 446 | } |
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| 447 | } |
---|
| 448 | } |
---|
| 449 | } |
---|
| 450 | } |
---|
| 451 | truncatedcomplex[2*i]=M; |
---|
| 452 | truncatedcomplex[2*i-1]=sizetruncom[2*i][size(sizetruncom[2*i])]; |
---|
| 453 | } |
---|
| 454 | truncatedcomplex[2*i-1]=sizetruncom[2*i][size(sizetruncom[2*i])]; |
---|
| 455 | if (truncatedcomplex[2*i-1]!=0) |
---|
| 456 | { |
---|
| 457 | truncatedcomplex[2*i]=matrix(0,truncatedcomplex[2*i-1],1); |
---|
| 458 | } |
---|
| 459 | if (diffforms==1) |
---|
| 460 | { |
---|
| 461 | setring W; |
---|
| 462 | truncatedcomplex=imap(ConvWeyl,truncatedcomplex); |
---|
| 463 | } |
---|
| 464 | setring R; |
---|
| 465 | list truncatedcomplex=imap(W,truncatedcomplex); |
---|
| 466 | /*computes the cohomology of the complex (D^i,d^i) given by truncatedcomplex, |
---|
| 467 | i.e. D^i=C^truncatedcomplex[2*i-1] and d^i=truncatedcomplex[2*i]*/ |
---|
| 468 | if (diffforms==0) |
---|
| 469 | { |
---|
| 470 | list derhamhom=findCohomology(truncatedcomplex,le); |
---|
| 471 | option(set,saveoptions); |
---|
| 472 | return (derhamhom); |
---|
| 473 | } |
---|
| 474 | list outall=findCohomologyDiffForms(truncatedcomplex,le); |
---|
| 475 | setring W; |
---|
| 476 | list dimanddiff=imap(R,outall); |
---|
| 477 | list alldiffforms=dimanddiff[2]; |
---|
| 478 | while(size(alldiffforms)<size(passendespoly)) |
---|
| 479 | { |
---|
| 480 | passendespoly=delete(passendespoly,1); |
---|
| 481 | } |
---|
| 482 | list newdiffforms; |
---|
| 483 | matrix Diff; |
---|
| 484 | for (i=1; i<=size(alldiffforms); i++) |
---|
| 485 | { |
---|
| 486 | newdiffforms[i]=list(); |
---|
| 487 | for (j=1; j<=size(alldiffforms[i]); j++) |
---|
| 488 | { |
---|
| 489 | Diff=matrix(0,1,newcomplex[3*(i+size(newcomplex) div 3 - size(alldiffforms))-2]); |
---|
| 490 | for (k=1; k<=ncols(alldiffforms[i][j]); k++) |
---|
| 491 | { |
---|
| 492 | if (alldiffforms[i][j][1,k]!=0) |
---|
| 493 | { |
---|
| 494 | Diff[1,passendespoly[i][k][2]]=Diff[1,passendespoly[i][k][2]]+alldiffforms[i][j][1,k]*passendespoly[i][k][1]; |
---|
| 495 | } |
---|
| 496 | } |
---|
| 497 | newdiffforms[i][j]=Diff; |
---|
| 498 | } |
---|
| 499 | } |
---|
| 500 | list omegacomplex=makeOmega(nvars(W) div 2); |
---|
| 501 | list newcomplexmod; |
---|
| 502 | for (i=1; i<=size(newcomplex) div 3; i++) |
---|
| 503 | { |
---|
| 504 | newcomplexmod[2*i-1]=newcomplex[3*i-2]; |
---|
| 505 | newcomplexmod[2*i]=newcomplex[3*i]; |
---|
| 506 | } |
---|
| 507 | while (size(dimanddiff[1])<size(newcomplexmod) div 2) |
---|
| 508 | { |
---|
| 509 | newcomplexmod=delete(newcomplexmod,1); |
---|
| 510 | newcomplexmod=delete(newcomplexmod,1); |
---|
| 511 | } |
---|
| 512 | while (size(dimanddiff[1])<size(quasiiso)) |
---|
| 513 | { |
---|
| 514 | quasiiso=delete(quasiiso,1); |
---|
| 515 | } |
---|
| 516 | while (size(dimanddiff[1])>size(generators)) |
---|
| 517 | { |
---|
| 518 | generators=insert(generators,list()); |
---|
| 519 | } |
---|
| 520 | while (size(dimanddiff[1])>size(quasiiso)) |
---|
| 521 | { |
---|
| 522 | quasiiso=insert(quasiiso,list()); |
---|
| 523 | } |
---|
| 524 | int keepsign; |
---|
| 525 | list derhamdiff; |
---|
| 526 | list doublecom=makeDoubleComplex(newcomplexmod,omegacomplex,quasiiso,generators); |
---|
| 527 | matrix diffform; |
---|
| 528 | int stopping; |
---|
| 529 | int p; |
---|
| 530 | matrix convert; |
---|
| 531 | list interim; |
---|
| 532 | list correspondingposition; |
---|
| 533 | list allforms=list(); |
---|
| 534 | for (i=1; i<=size(newdiffforms); i++) |
---|
| 535 | { |
---|
| 536 | derhamdiff[i]=list(); |
---|
| 537 | allforms[i]=list(); |
---|
| 538 | for (j=1; j<=size(newdiffforms[i]); j++) |
---|
| 539 | { |
---|
| 540 | allforms[i][j]=list(); |
---|
| 541 | keepsign=1; |
---|
| 542 | derhamdiff[i][j]=0; |
---|
| 543 | diffform=newdiffforms[i][j];//Zeilenform |
---|
| 544 | correspondingposition=doublecom[i][1];//needed fpr transformation process |
---|
| 545 | interim=transferDiffforms(diffform,correspondingposition); |
---|
| 546 | if (size(interim)!=0) |
---|
| 547 | { |
---|
| 548 | allforms[i][j][size(allforms[i][j])+1]=interim; |
---|
| 549 | } |
---|
| 550 | stopping=0; |
---|
| 551 | p=1; |
---|
| 552 | for (k=i; k<=size(newdiffforms); k++) |
---|
| 553 | { |
---|
| 554 | keepsign=(-1)*keepsign; |
---|
| 555 | if (stopping==0) |
---|
| 556 | { |
---|
| 557 | if (size(doublecom[k][p][2])==0) |
---|
| 558 | { |
---|
| 559 | stopping=1; |
---|
| 560 | } |
---|
| 561 | else |
---|
| 562 | { |
---|
| 563 | if (size(doublecom[k+1][p][3])!=0) |
---|
| 564 | { |
---|
| 565 | diffform=diffform*doublecom[k][p][2];//Spaltenform |
---|
| 566 | if (diffform!=matrix(0,nrows(diffform),ncols(diffform))) |
---|
| 567 | { |
---|
| 568 | diffform=findPreimage(doublecom[k+1][p][3],transpose(diffform));//Zeilenform |
---|
| 569 | correspondingposition=doublecom[k+1][p+1];//needed for transformation process |
---|
| 570 | interim=transferDiffforms(keepsign*diffform,correspondingposition); |
---|
| 571 | if (size(interim)!=0) |
---|
| 572 | { |
---|
| 573 | allforms[i][j][size(allforms[i][j])+1]=interim; |
---|
| 574 | } |
---|
| 575 | p=p+1; |
---|
| 576 | } |
---|
| 577 | else |
---|
| 578 | { |
---|
| 579 | stopping=1; |
---|
| 580 | } |
---|
| 581 | } |
---|
| 582 | else |
---|
| 583 | { |
---|
| 584 | stopping=1; |
---|
| 585 | } |
---|
| 586 | } |
---|
| 587 | } |
---|
| 588 | } |
---|
| 589 | } |
---|
| 590 | } |
---|
| 591 | setring R; |
---|
| 592 | list allforms=fetch(W,allforms); |
---|
| 593 | option(set,saveoptions); |
---|
| 594 | return (allforms); |
---|
| 595 | } |
---|
| 596 | |
---|
| 597 | example |
---|
| 598 | { "EXAMPLE:"; |
---|
| 599 | ring r = 0,(x,y,z),dp; |
---|
| 600 | list L=(xy,xz); |
---|
| 601 | deRhamCohomology(L); |
---|
| 602 | } |
---|
| 603 | |
---|
| 604 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 605 | //COMPUTATION OF THE MAYER-VIETORIS COMPLEX |
---|
| 606 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 607 | |
---|
| 608 | proc MVComplex(list L,list #) |
---|
| 609 | "USAGE:MVComplex(L); L a list of polynomials |
---|
| 610 | ASSUME: -Basering is a polynomial ring with n vwariables and rational coefficients |
---|
| 611 | -L is a list of non-constant polynomials |
---|
| 612 | RETURN: ring W: the nth Weyl algebra @* |
---|
| 613 | W contains a list MV, which represents the Mayer-Vietrois complex (C^i,d^i) of the |
---|
[7fe9f8b] | 614 | polynomials contained in L as follows:@* |
---|
| 615 | the C^i are given by D_n^ncols(C[2*i-1])/im(C[2*i-1]) and the differentials |
---|
| 616 | d^i are given by C[2*i] |
---|
| 617 | EXAMPLE:example MVComplex; shows an example |
---|
| 618 | " |
---|
| 619 | { |
---|
| 620 | /* We follow algorithm 3.2.5 in [R],if #!=0 we use also Remark 3.2.6 in [R] for |
---|
[0e8a5a] | 621 | an additional iterative localization*/ |
---|
[7fe9f8b] | 622 | def R=basering; |
---|
| 623 | int i; |
---|
| 624 | int iterative=1; |
---|
| 625 | if (size(#)!=0) |
---|
[0e8a5a] | 626 | { |
---|
| 627 | iterative=#[1]; |
---|
| 628 | } |
---|
| 629 | for (i=1; i<=size(L); i++) |
---|
| 630 | { |
---|
| 631 | if (L[i]==0) |
---|
| 632 | { |
---|
| 633 | print("localization with respect to 0 not possible"); |
---|
| 634 | return(); |
---|
| 635 | } |
---|
| 636 | if (leadcoef(L[i])-L[i]==0) |
---|
| 637 | { |
---|
| 638 | print("polynomials must be non-constant"); |
---|
| 639 | return(); |
---|
| 640 | } |
---|
| 641 | } |
---|
| 642 | if (iterative==1) |
---|
| 643 | { |
---|
| 644 | /*compute the localizations by factorizing the polynomials and iterative |
---|
| 645 | localization of the factors */ |
---|
| 646 | for (i=1; i<=size(L); i++) |
---|
| 647 | { |
---|
| 648 | L[i]=factorize(L[i],1); |
---|
| 649 | } |
---|
| 650 | } |
---|
| 651 | int r=size(L); |
---|
| 652 | int n=nvars(basering); |
---|
| 653 | int le=size(L)+n; |
---|
| 654 | /*construct the ring Ws*/ |
---|
| 655 | def W=makeWeyl(n); |
---|
| 656 | setring W; |
---|
| 657 | list man=ringlist(W); |
---|
| 658 | if (n==1) |
---|
| 659 | { |
---|
| 660 | man[2][1]="x(1)"; |
---|
| 661 | man[2][2]="D(1)"; |
---|
| 662 | def Wi=ring(man); |
---|
| 663 | setring Wi; |
---|
| 664 | kill W; |
---|
| 665 | def W=Wi; |
---|
| 666 | setring W; |
---|
| 667 | list man=ringlist(W); |
---|
| 668 | } |
---|
| 669 | man[2][size(man[2])+1]="s";; |
---|
| 670 | man[3][3]=man[3][2]; |
---|
| 671 | man[3][2]=list("dp",intvec(1)); |
---|
| 672 | matrix N=UpOneMatrix(size(man[2])); |
---|
| 673 | man[5]=N; |
---|
| 674 | matrix M[1][1]; |
---|
| 675 | man[6]=transpose(concat(transpose(concat(man[6],M)),M)); |
---|
| 676 | def Ws=ring(man); |
---|
| 677 | setring Ws; |
---|
| 678 | int j,k,l,c; |
---|
| 679 | list L=fetch(R,L); |
---|
| 680 | list Cech; |
---|
| 681 | ideal J=var(1+n); |
---|
| 682 | for (i=2; i<=n; i++) |
---|
| 683 | { |
---|
| 684 | J=J,var(i+n); |
---|
| 685 | } |
---|
| 686 | Cech[1]=list(J); |
---|
| 687 | list Theta, remminroots; |
---|
| 688 | Theta[1]=list(list(list(),1,1)); |
---|
| 689 | list rem,findminintroot,diffmaps; |
---|
| 690 | int minroot,st,sk; |
---|
| 691 | intvec k1; |
---|
| 692 | poly fred,forfetch; |
---|
| 693 | matrix subm; |
---|
| 694 | int rmr; |
---|
| 695 | if (iterative==0) |
---|
| 696 | {/*computation of the modules of the MV complex*/ |
---|
| 697 | for (i=1; i<=r; i++) |
---|
| 698 | { |
---|
| 699 | findminintroot=list(); |
---|
| 700 | Cech[i+1]=list(); |
---|
| 701 | Theta[i+1]=list(); |
---|
| 702 | k1=1; |
---|
| 703 | for (j=1; j<=i; j++) |
---|
| 704 | { |
---|
| 705 | k1[size(k1)+1]=size(Theta[j+1]); |
---|
| 706 | for (k=1; k<=k1[j]; k++) |
---|
| 707 | { |
---|
| 708 | Theta[j+1][size(Theta[j+1])+1]=list(Theta[j][k][1]+list(i)); |
---|
| 709 | Theta[j+1][size(Theta[j+1])][2]=Theta[j][k][2]*L[i]; |
---|
| 710 | /*We compute the s-parametric annihilator J(s) and the b-function |
---|
| 711 | of the polynomial L[i] and Cech[i][k] to localize the module |
---|
| 712 | D_n/(D(1),...,D(n))[L[i]^(-1)]\otimes D_n^c/im(Cech[i][k]), |
---|
| 713 | where c=ncols(Cech[i][k]) and the im(Cech[i][k]) is generated by |
---|
| 714 | the rows of the matrix. |
---|
| 715 | If we plug the minimal integer root r(or a smaller integer |
---|
| 716 | value)in J(s), then D_n^ncols(J(s))/im(J(r)) is isomorphic to |
---|
| 717 | the above localization*/ |
---|
| 718 | rem=SannfsIBM(L[i],Cech[j][k]); |
---|
| 719 | Cech[j+1][size(Cech[j+1])+1]=rem[1]; |
---|
| 720 | findminintroot[size(findminintroot)+1]=rem[2]; |
---|
| 721 | } |
---|
| 722 | } |
---|
| 723 | /* we compute the minimal root of all b-functions of L[i] computed above, |
---|
| 724 | because we want to plug in the same root r in all s-parametric |
---|
| 725 | annihilators we computed for L[i] ->this will ensure we can compute |
---|
| 726 | the maps of the MV complex*/ |
---|
[76d26c] | 727 | minroot=minIntRootD(findminintroot); |
---|
[0e8a5a] | 728 | for (j=1; j<=i; j++) |
---|
| 729 | { |
---|
| 730 | for (k=1; k<=k1[j]; k++) |
---|
| 731 | { |
---|
| 732 | sk=size(Cech[j+1])+1-k; |
---|
| 733 | Cech[j+1][size(Cech[j+1])+1-k]=subst(Cech[j+1][sk],s,minroot); |
---|
| 734 | } |
---|
| 735 | } |
---|
| 736 | remminroots[i]=minroot; |
---|
| 737 | } |
---|
| 738 | Cech=delete(Cech,1); |
---|
| 739 | Theta=delete(Theta,1); |
---|
| 740 | list zw; |
---|
| 741 | poly reme; |
---|
| 742 | /*computation of the maps of the MV complex*/ |
---|
| 743 | for (i=1; i<r; i++) |
---|
| 744 | { |
---|
| 745 | diffmaps[i]=matrix(0,size(Cech[i]),size(Cech[i+1])); |
---|
| 746 | for (j=1; j<=size(Cech[i]); j++) |
---|
| 747 | { |
---|
| 748 | for (k=1; k<=size(Cech[i+1]); k++) |
---|
| 749 | { |
---|
| 750 | zw=LMSubset(Theta[i][j][1],Theta[i+1][k][1]); |
---|
| 751 | if (zw[2]!=0) |
---|
| 752 | { |
---|
| 753 | rmr=-remminroots[zw[1]]; |
---|
| 754 | reme=zw[2]*(Theta[i+1][k][2]/Theta[i][j][2])^(rmr); |
---|
| 755 | zw[2]=zw[2]*(Theta[i+1][k][2]/Theta[i][j][2])^(rmr); |
---|
| 756 | diffmaps[i][j,k]=zw[2]; |
---|
| 757 | } |
---|
| 758 | } |
---|
| 759 | } |
---|
| 760 | } |
---|
| 761 | diffmaps[r]=matrix(0,1,1); |
---|
| 762 | } |
---|
| 763 | list generators; |
---|
| 764 | if (iterative==1) |
---|
| 765 | { |
---|
| 766 | for (i=1; i<=r;i++) |
---|
| 767 | { |
---|
| 768 | generators[i]=list();//////////////////////////////////////////////////////////////////// |
---|
| 769 | Cech[i+1]=list(); |
---|
| 770 | Theta[i+1]=list(); |
---|
| 771 | k1=1; |
---|
| 772 | for (c=1; c<=size(L[i]); c++) |
---|
| 773 | { |
---|
| 774 | findminintroot=list(); |
---|
| 775 | for (j=1; j<=i; j++) |
---|
| 776 | { |
---|
| 777 | if (c==1) |
---|
| 778 | { |
---|
| 779 | k1[size(k1)+1]=size(Theta[j+1]); |
---|
| 780 | } |
---|
| 781 | for (k=1; k<=k1[j]; k++) |
---|
| 782 | { |
---|
| 783 | /*We compute the s-parametric annihilator J(s) und the b- |
---|
| 784 | function of the polynomial L[i][c] and Cech[i][k] to |
---|
| 785 | localize the module D_n/(D(1),...,D(n))[L[i][c]^(-1)]\otimes |
---|
| 786 | D_n^c/im(Cech[i][k]), where c=ncols(Cech[i][k]). |
---|
| 787 | If we plug the minimal integer root r(or a smaller integer |
---|
| 788 | value)in J(s), then D_n^ncols(J(s))/im(J(r)) is isomorphic |
---|
| 789 | to the above localization*/ |
---|
| 790 | if (c==1) |
---|
| 791 | { |
---|
| 792 | rmr=size(Theta[j+1])+1; |
---|
| 793 | Theta[j+1][rmr]=list(Theta[j][k][1]+list(i)); |
---|
| 794 | Theta[j+1][size(Theta[j+1])][2]=Theta[j][k][2]*L[i][c]; |
---|
| 795 | rem=SannfsIBM(L[i][c],Cech[j][k]); |
---|
| 796 | Cech[j+1][size(Cech[j+1])+1]=rem[1]; |
---|
| 797 | findminintroot[size(findminintroot)+1]=rem[2]; |
---|
| 798 | } |
---|
| 799 | else |
---|
| 800 | { |
---|
| 801 | st=size(Theta[j+1])-k1[j]+k; |
---|
| 802 | Theta[j+1][st][2]=Theta[j+1][st][2]*L[i][c]; |
---|
| 803 | rem=SannfsIBM(L[i][c],Cech[j+1][size(Cech[j+1])-k1[j]+k]); |
---|
| 804 | Cech[j+1][size(Cech[j+1])-k1[j]+k]=rem[1]; |
---|
| 805 | findminintroot[size(findminintroot)+1]=rem[2]; |
---|
| 806 | } |
---|
| 807 | } |
---|
| 808 | } |
---|
| 809 | /* we compute the minimal root of all b-functions of L[i][c] |
---|
| 810 | computed above,because we want to plug in the same root r in all |
---|
| 811 | s-parametric annihilators we computed for L[i] ->this will |
---|
| 812 | ensure we can compute the maps of the MV complex*/ |
---|
[76d26c] | 813 | minroot=minIntRootD(findminintroot); |
---|
[0e8a5a] | 814 | for (j=1; j<=i; j++) |
---|
| 815 | { |
---|
| 816 | for (k=1; k<=k1[j]; k++) |
---|
| 817 | { |
---|
| 818 | st=size(Cech[j+1])+1-k; |
---|
| 819 | Cech[j+1][st]=subst(Cech[j+1][st],s,minroot); |
---|
| 820 | } |
---|
| 821 | } |
---|
| 822 | if (c==1) |
---|
| 823 | { |
---|
| 824 | remminroots[i]=list(); |
---|
| 825 | } |
---|
| 826 | remminroots[i][c]=minroot; |
---|
| 827 | } |
---|
| 828 | } |
---|
| 829 | Cech=delete(Cech,1); |
---|
| 830 | Theta=delete(Theta,1); |
---|
| 831 | list zw; |
---|
| 832 | poly reme; |
---|
| 833 | /*maps of the MV Complex*/ |
---|
| 834 | for (i=1; i<r; i++) |
---|
| 835 | { |
---|
| 836 | diffmaps[i]=matrix(0,size(Cech[i]),size(Cech[i+1])); |
---|
| 837 | for (j=1; j<=size(Cech[i]); j++) |
---|
| 838 | { |
---|
| 839 | for (k=1; k<=size(Cech[i+1]); k++) |
---|
| 840 | { |
---|
| 841 | zw=LMSubset(Theta[i][j][1],Theta[i+1][k][1]); |
---|
| 842 | if (zw[2]!=0) |
---|
| 843 | { |
---|
| 844 | reme=1; |
---|
| 845 | for (c=1; c<=size(L[zw[1]]);c++) |
---|
| 846 | { |
---|
| 847 | reme=reme*L[zw[1]][c]^(-remminroots[zw[1]][c]); |
---|
| 848 | } |
---|
| 849 | diffmaps[i][j,k]=zw[2]*reme; |
---|
| 850 | } |
---|
| 851 | } |
---|
| 852 | } |
---|
| 853 | } |
---|
| 854 | diffmaps[r]=matrix(0,1,1); |
---|
| 855 | for (i=1; i<=r; i++) |
---|
| 856 | { |
---|
| 857 | for (j=1; j<=size(Theta[i]); j++) |
---|
| 858 | { |
---|
| 859 | generators[i][j]=1; |
---|
| 860 | for (c=1; c<=size(Theta[i][j][1]); c++) |
---|
| 861 | { |
---|
| 862 | for (k=1; k<=size(L[Theta[i][j][1][c]]); k++) |
---|
| 863 | { |
---|
| 864 | generators[i][j]=generators[i][j]*L[Theta[i][j][1][c]][k]^((-1)*remminroots[Theta[i][j][1][c]][k]); |
---|
| 865 | } |
---|
| 866 | } |
---|
| 867 | } |
---|
| 868 | } |
---|
| 869 | } |
---|
| 870 | setring W; |
---|
| 871 | /*map the modules and maps to the Weyl algebra*/ |
---|
| 872 | list diffmaps=imap(Ws,diffmaps); |
---|
| 873 | list Cechmodules=imap(Ws,Cech); |
---|
| 874 | if (iterative==1) |
---|
| 875 | { |
---|
| 876 | list Theta=imap(Ws,Theta); |
---|
| 877 | list generators=imap(Ws,generators); |
---|
| 878 | } |
---|
| 879 | list Cech; |
---|
| 880 | matrix sup; |
---|
| 881 | for (i=1; i<=r; i++) |
---|
| 882 | { |
---|
| 883 | sup=transpose(matrix(Cechmodules[i][1])); |
---|
| 884 | Cech[2*i-1]=sup; |
---|
| 885 | for (j=2; j<=size(Cechmodules[i]); j++) |
---|
| 886 | { |
---|
| 887 | sup=transpose(matrix(Cechmodules[i][j])); |
---|
| 888 | Cech[2*i-1]=dsum(Cech[2*i-1],sup); |
---|
| 889 | } |
---|
| 890 | sup=matrix(diffmaps[i]); |
---|
| 891 | Cech[2*i]=sup; |
---|
| 892 | } |
---|
| 893 | list MV=Cech; |
---|
| 894 | if (iterative==1) |
---|
| 895 | { |
---|
| 896 | export Theta; |
---|
| 897 | export generators; |
---|
| 898 | } |
---|
| 899 | export MV; |
---|
| 900 | |
---|
| 901 | return (W); |
---|
| 902 | } |
---|
| 903 | |
---|
| 904 | example |
---|
| 905 | { "EXAMPLE:"; |
---|
| 906 | ring r = 0,(x,y,z),dp; |
---|
| 907 | list L=xy,xz; |
---|
| 908 | def C=MVComplex(L); |
---|
| 909 | setring C; |
---|
| 910 | MV; |
---|
| 911 | } |
---|
| 912 | |
---|
| 913 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 914 | |
---|
| 915 | static proc SannfsIBM(poly F,ideal myJ) |
---|
| 916 | "USAGE: SannfsIBM(f,J), F poly, J ideal |
---|
| 917 | ASSUME: basering is D_n[s], where D_n is the Weyl algebra and s and extra |
---|
| 918 | commutative variable@* |
---|
| 919 | f is a polynomial in the variables x(1),...,x(n) with rational coefficients |
---|
| 920 | @* |
---|
| 921 | J is holonomic and f-saturated |
---|
| 922 | RETURN AlList of the form (K,g), where K is an ideal and g a univariant polynomial |
---|
| 923 | in the variable s. K is the s-parametric annihilator of F and J and g is |
---|
| 924 | the b-function of F and J. |
---|
| 925 | " |
---|
| 926 | { |
---|
| 927 | /*modified version of the procedure SannfsBM from the library dmod.lib: SannfsBM |
---|
| 928 | computes the s-parametric annihilator for J=(x_1,...,x_n)*/ |
---|
| 929 | /* We use Algorithm 3.1.12 in[R] to compute the s-parametric |
---|
| 930 | annihilator. Then we use the s-parametric annihilator to compute the b-function |
---|
| 931 | via Algorithm 4.7 in [W1].*/ |
---|
| 932 | /* We assume that the basering the the nth Weyl algebra D_n. We create the ring |
---|
| 933 | D_n[s,t], where t*s=s*t-t*/ |
---|
| 934 | def save = basering; |
---|
| 935 | int N = nvars(basering)-1; |
---|
| 936 | int Nnew = N+2; |
---|
| 937 | int i,j; |
---|
| 938 | string s; |
---|
| 939 | list RL = ringlist(basering); |
---|
| 940 | list L, Lord; |
---|
| 941 | list tmp; |
---|
| 942 | intvec iv; |
---|
| 943 | L[1] = RL[1]; |
---|
| 944 | L[4] = RL[4]; |
---|
| 945 | list Name = RL[2]; |
---|
| 946 | Name=delete(Name,size(Name)); |
---|
| 947 | list RName; |
---|
| 948 | RName[1] = "t"; |
---|
| 949 | RName[2] = "s"; |
---|
| 950 | list DName; |
---|
| 951 | for(i=1;i<=N div 2;i++) |
---|
| 952 | { |
---|
| 953 | DName[i] = var(N div 2+i); |
---|
| 954 | Name=delete(Name,N div 2+1); |
---|
| 955 | } |
---|
| 956 | tmp[1] = "t"; |
---|
| 957 | tmp[2] = "s"; |
---|
| 958 | list NName = tmp +Name+DName; |
---|
| 959 | L[2] = NName; |
---|
| 960 | kill NName; |
---|
| 961 | tmp[1] = "lp"; |
---|
| 962 | iv = 1,1; |
---|
| 963 | tmp[2] = iv; |
---|
| 964 | Lord[1] = tmp; |
---|
| 965 | tmp[1] = "dp"; |
---|
| 966 | s = "iv="; |
---|
| 967 | for(i=1;i<=Nnew;i++) |
---|
[7fe9f8b] | 968 | { |
---|
[0e8a5a] | 969 | s = s+"1,"; |
---|
[7fe9f8b] | 970 | } |
---|
[0e8a5a] | 971 | s[size(s)]= ";"; |
---|
| 972 | execute(s); |
---|
| 973 | kill s; |
---|
| 974 | tmp[2] = iv; |
---|
| 975 | Lord[2] = tmp; |
---|
| 976 | tmp[1] = "C"; |
---|
| 977 | iv = 0; |
---|
| 978 | tmp[2] = iv; |
---|
| 979 | Lord[3] = tmp; |
---|
| 980 | tmp = 0; |
---|
| 981 | L[3] = Lord; |
---|
| 982 | def @R@ = ring(L); |
---|
| 983 | setring @R@; |
---|
| 984 | matrix @D[Nnew][Nnew]; |
---|
| 985 | @D[1,2]=t; |
---|
| 986 | for(i=1; i<=N div 2; i++) |
---|
[7fe9f8b] | 987 | { |
---|
[0e8a5a] | 988 | @D[2+i, N div 2+2+i]=1; |
---|
| 989 | } |
---|
| 990 | def @R = nc_algebra(1,@D); |
---|
| 991 | setring @R; |
---|
| 992 | kill @R@; |
---|
| 993 | /*we start with the computation of the s-parametric annihilator*/ |
---|
| 994 | poly F = imap(save,F); |
---|
| 995 | ideal myJ=imap(save,myJ); |
---|
| 996 | for (i=1; i<=N div 2; i++) |
---|
[7fe9f8b] | 997 | { |
---|
[0e8a5a] | 998 | myJ=subst(myJ,D(i),D(i)+diff(F,x(i))*t); |
---|
[7fe9f8b] | 999 | } |
---|
[0e8a5a] | 1000 | ideal I = t*F+s; |
---|
| 1001 | I=I,myJ;//the s-parametric annihilator in D_n[s,t] |
---|
| 1002 | /*we compute the intersection of I and D_n[s]*/ |
---|
| 1003 | ideal J = slimgb(I); |
---|
| 1004 | ideal K = nselect(J,1); |
---|
| 1005 | K = slimgb(K);//the s-parametric annihilator |
---|
| 1006 | /*we use K to compute the b-function*/ |
---|
| 1007 | ideal B=K,F; |
---|
| 1008 | B=slimgb(B); |
---|
| 1009 | vector p=pIntersect(s,B); |
---|
| 1010 | poly f=vec2poly(p,2); |
---|
| 1011 | setring save; |
---|
| 1012 | poly f=imap(@R,f); |
---|
| 1013 | ideal K=imap(@R,K); |
---|
| 1014 | return (list(K,f)); |
---|
| 1015 | } |
---|
| 1016 | |
---|
| 1017 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1018 | //COMPUTATION OF A QUASI-ISOMORPHIC V_D-STRICT FREE COMPLEX |
---|
| 1019 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1020 | |
---|
| 1021 | static proc quasiisomorphicVdComplex(list L,list #) |
---|
| 1022 | "USAGE: quasiisomorphicVdComplex(L[,df]); L a list of the form (M_1,f_1,...,M_s,f_s), |
---|
| 1023 | where the M_i and f_i are matrices |
---|
| 1024 | ASSUME: Basering is the Weyl algebra D_n @* |
---|
| 1025 | (M_1,f_1,...,M_s,f_s) represents a complex 0->D_n^(r_1)/im(M_1)-> |
---|
| 1026 | D_n^(r_2)/im(M_2)->...->D_n^(r_s)->0 with differentials f_i, where im(M_i) |
---|
| 1027 | is generated by the rows of M_i. In particular it hold:@* |
---|
| 1028 | - The M_i are m_i x r_i-matrices and the f_iare r_i x r_(i+1)-matrices @* |
---|
| 1029 | -the image of M_1*f_i is contained in the image of M_(i+1) @* |
---|
| 1030 | d is an integer between 1 and n. If no value for d is given, it is assumed |
---|
| 1031 | to be n @* |
---|
| 1032 | df is an optional int, if df equals 1 a \tilde(V_d)-strict complex |
---|
| 1033 | will be computed (instead of a V_d-strict one) (for a definition see [W3]) |
---|
| 1034 | RETURN: list of the form (L_1,L_2), were L_1 and L_2 are lists @* |
---|
| 1035 | L_1 is of the form (i_(-n-1),g_(-n-1),m_(-n-1),...,i_s,g_s,m_s) such that:@* |
---|
| 1036 | -the i_j are integers, the g_j are i_j x i_(j+1)-matrices, the m_j intvecs |
---|
| 1037 | of size i_j@* |
---|
| 1038 | -D_n^(i_(-n-1))[m_(-n-1)]->...->D_n^(i_s)[m_s]->0 is a V_d-strict complex |
---|
| 1039 | with differentials m_i that is quasi-isomorphic to the complex given by L@* |
---|
| 1040 | L_2 is of the form (H_1,n_1,...,H_s,n_s), where the H_i are matrices and |
---|
| 1041 | the n_i are shift vectors such that:@* |
---|
| 1042 | -coker(H_i) is the ith cohomology group of the complex given by L_1@* |
---|
| 1043 | -the n_i are the shift vectors of the coker(H_i) |
---|
| 1044 | THEORY: We follow Proposition 3.2 and Corollary 3.3 in [W3] |
---|
| 1045 | " |
---|
| 1046 | { |
---|
| 1047 | int tilde; |
---|
| 1048 | if (size(#)!=0) |
---|
[7fe9f8b] | 1049 | { |
---|
[0e8a5a] | 1050 | tilde=#[1]; |
---|
[7fe9f8b] | 1051 | } |
---|
[0e8a5a] | 1052 | def B=basering; |
---|
[08fa62] | 1053 | int n=nvars(B) div 2 + 1;//+1 muesste stimmen! bitte kontrollieren! |
---|
[0e8a5a] | 1054 | int d=nvars(B) div 2; |
---|
| 1055 | int r=size(L) div 2; |
---|
| 1056 | int lonc=n+r; |
---|
| 1057 | int Kiold=0; |
---|
| 1058 | matrix kerold; |
---|
| 1059 | // matrix kernew=out[r][2][2]; |
---|
| 1060 | matrix kernew=diag(1,ncols(L[size(L)-1])); |
---|
| 1061 | module mL; |
---|
| 1062 | int i; |
---|
| 1063 | int k; |
---|
| 1064 | matrix testm; |
---|
| 1065 | int Kinew=nrows(kernew); |
---|
| 1066 | int Jiold=0; |
---|
| 1067 | int Jinew=0; |
---|
| 1068 | matrix Niold; |
---|
| 1069 | matrix Ninew; |
---|
| 1070 | list newcomplex; |
---|
| 1071 | int Aiold=Kinew; |
---|
| 1072 | matrix savediv; |
---|
| 1073 | newcomplex[3*lonc-2]=Kinew; |
---|
| 1074 | newcomplex[3*lonc-1]=intvec(0:Kinew); |
---|
| 1075 | newcomplex[3*lonc]=matrix(0,Kinew,1); |
---|
| 1076 | list quasiiso; |
---|
| 1077 | quasiiso[lonc]=diag(1,Kinew); |
---|
| 1078 | matrix invimage; |
---|
| 1079 | matrix keralpha; |
---|
| 1080 | intvec v; |
---|
| 1081 | int j; |
---|
| 1082 | matrix sc; |
---|
| 1083 | matrix fnc; |
---|
| 1084 | int indk; |
---|
| 1085 | int indj; |
---|
| 1086 | int Aiold; |
---|
| 1087 | list saveres; |
---|
| 1088 | matrix Liplus; |
---|
| 1089 | for (i=r-1; i>=0; i--) |
---|
| 1090 | { |
---|
| 1091 | indk=0; |
---|
| 1092 | indj=0; |
---|
| 1093 | Kiold=Kinew; |
---|
| 1094 | kerold=kernew; |
---|
| 1095 | if (i!=0) |
---|
| 1096 | { |
---|
| 1097 | // kernew=divdr(L[2*i],L[2*i+1],1); |
---|
| 1098 | kernew=divdr(L[2*i],L[2*i+1]); |
---|
| 1099 | mL=slimgb(transpose(L[2*i-1])); |
---|
| 1100 | for (k=1; k<=nrows(kernew); k++) |
---|
| 1101 | { |
---|
| 1102 | testm=reduce(transpose(submat(kernew,k,intvec(1..ncols(kernew)))),mL); |
---|
| 1103 | if (testm==matrix(0,nrows(testm),ncols(testm))) |
---|
| 1104 | { |
---|
| 1105 | kernew=transpose(deletecol(transpose(kernew),k)); |
---|
| 1106 | k=k-1; |
---|
| 1107 | } |
---|
| 1108 | } |
---|
| 1109 | Kinew=nrows(kernew); |
---|
| 1110 | if (kernew==matrix(0,nrows(kernew),ncols(kernew))) |
---|
| 1111 | { |
---|
| 1112 | Kinew=0; |
---|
| 1113 | indk=1; |
---|
| 1114 | } |
---|
| 1115 | } |
---|
| 1116 | else |
---|
| 1117 | { |
---|
| 1118 | Kinew=0; |
---|
| 1119 | indk=1; |
---|
| 1120 | } |
---|
| 1121 | Jiold=Jinew; |
---|
| 1122 | Niold=Ninew; |
---|
| 1123 | keralpha=transpose(syz(transpose(newcomplex[3*(i+n)+3]))); |
---|
| 1124 | if (i!=0) |
---|
| 1125 | { |
---|
| 1126 | invimage=divdr(quasiiso[n+i+1],transpose(concat(transpose(L[2*i]),transpose(L[2*i+1])))); |
---|
| 1127 | Ninew=vdStrictIntersect(keralpha,invimage,newcomplex[3*(n+i+1)-1],tilde);////////////// |
---|
| 1128 | } |
---|
| 1129 | else |
---|
| 1130 | { |
---|
| 1131 | invimage=divdr(quasiiso[n+i+1],L[2*i+1]); |
---|
| 1132 | saveres=vdStrictIntersectPlus(keralpha,invimage,newcomplex[3*(n+i+1)-1],tilde);//////////////////////// |
---|
| 1133 | |
---|
| 1134 | ///////////////////BIS HIERHIN VERALLGEMEINERT//////////////////////////////////////////////////////////////////// |
---|
| 1135 | |
---|
| 1136 | |
---|
| 1137 | Ninew=saveres[1]; |
---|
| 1138 | } |
---|
| 1139 | Jinew=nrows(Ninew); |
---|
| 1140 | if (Ninew==matrix(0,nrows(Ninew),ncols(Ninew))) |
---|
| 1141 | { |
---|
| 1142 | Jinew=0; |
---|
| 1143 | indk=1; |
---|
| 1144 | } |
---|
| 1145 | newcomplex[3*(n+i)-2]=Kinew+Jinew; |
---|
| 1146 | v=0; |
---|
| 1147 | if (indk==0) |
---|
| 1148 | { |
---|
| 1149 | v=(0:Kinew); |
---|
| 1150 | if (indj==0) |
---|
| 1151 | { |
---|
| 1152 | fnc=transpose(concat(transpose(matrix(0,Kinew,Kiold+Jiold)),transpose(Ninew))); |
---|
| 1153 | } |
---|
| 1154 | else |
---|
| 1155 | { |
---|
| 1156 | fnc=matrix(0,Kinew,Kiold+Jiold); |
---|
| 1157 | } |
---|
| 1158 | } |
---|
| 1159 | else |
---|
| 1160 | { |
---|
| 1161 | if (indj==0) |
---|
| 1162 | { |
---|
| 1163 | fnc=Ninew; |
---|
| 1164 | } |
---|
| 1165 | else |
---|
| 1166 | { |
---|
| 1167 | fnc=matrix(0,1,Kiold+Jiold); |
---|
| 1168 | newcomplex[3*(n+i)-2]=1; |
---|
| 1169 | } |
---|
| 1170 | } |
---|
| 1171 | Aiold=Jinew+Kinew; |
---|
| 1172 | if (Aiold==0) |
---|
| 1173 | { |
---|
| 1174 | Aiold=1; |
---|
| 1175 | } |
---|
| 1176 | newcomplex[3*(n+i)]=fnc; |
---|
| 1177 | for (j=1; j<=Jinew; j++) |
---|
| 1178 | { |
---|
| 1179 | if (tilde==0) |
---|
| 1180 | { |
---|
| 1181 | v[Kinew+j]=VdDeg(submat(Ninew,j,(1..ncols(Ninew))),nvars(B) div 2,newcomplex[3*(n+i)+2]); |
---|
| 1182 | } |
---|
| 1183 | else |
---|
| 1184 | { |
---|
| 1185 | v[Kinew+j]=VdDegTilde(submat(Ninew,j,(1..ncols(Ninew))),nvars(B) div 2,newcomplex[3*(n+i)+2]); |
---|
| 1186 | } |
---|
| 1187 | } |
---|
| 1188 | newcomplex[3*(n+i)-1]=v; |
---|
| 1189 | if (i==0) |
---|
| 1190 | { |
---|
| 1191 | quasiiso[n+i]=matrix(0,Jinew,1); |
---|
| 1192 | } |
---|
| 1193 | else |
---|
| 1194 | { |
---|
| 1195 | if (indj==0) |
---|
| 1196 | { |
---|
| 1197 | sc=submat(fnc,intvec(Kinew+1..nrows(fnc)),intvec(1..ncols(fnc)))*quasiiso[n+i+1]; |
---|
| 1198 | Liplus=transpose(concat(transpose(L[2*i]),transpose(L[2*i+1]))); |
---|
| 1199 | sc=matrixLift(Liplus,sc);//stimmt das jetzt |
---|
| 1200 | sc=submat(sc,intvec(1..nrows(sc)),intvec(1..nrows(L[2*i]))); |
---|
| 1201 | if (indk==0) |
---|
| 1202 | { |
---|
| 1203 | //pi=kernew |
---|
| 1204 | quasiiso[n+i]=transpose(concat(transpose(kernew),transpose(sc))); |
---|
| 1205 | } |
---|
| 1206 | else |
---|
| 1207 | { |
---|
| 1208 | quasiiso[n+i]=sc; |
---|
| 1209 | } |
---|
| 1210 | } |
---|
| 1211 | else |
---|
| 1212 | { |
---|
| 1213 | if (indk==0) |
---|
| 1214 | { |
---|
| 1215 | quasiiso[n+i]=kernew; |
---|
| 1216 | } |
---|
| 1217 | else |
---|
| 1218 | { |
---|
| 1219 | quasiiso[n+i]=matrix(0,1,ncols(kernew)); |
---|
| 1220 | } |
---|
| 1221 | } |
---|
| 1222 | } |
---|
| 1223 | } |
---|
| 1224 | for (i=1; i<=n-1; i++) |
---|
[7fe9f8b] | 1225 | { |
---|
[0e8a5a] | 1226 | quasiiso[n-i]=list(); |
---|
| 1227 | if (size(saveres[2][i])!=0) |
---|
| 1228 | { |
---|
| 1229 | newcomplex[3*(n-i)]=saveres[2][i]; |
---|
| 1230 | newcomplex[3*(n-i)-2]=nrows(saveres[2][i]); |
---|
| 1231 | v=0; |
---|
| 1232 | for (j=1; j<=newcomplex[3*(n-i)-2]; j++) |
---|
| 1233 | { |
---|
| 1234 | if (tilde==0) |
---|
| 1235 | { |
---|
| 1236 | v[j]=VdDeg(submat(saveres[2][i],j,(1..ncols(saveres[2][i]))),nvars(B) div 2, newcomplex[3*(n-i)+2]); |
---|
| 1237 | } |
---|
| 1238 | else |
---|
| 1239 | { |
---|
| 1240 | v[j]=VdDegTilde(submat(saveres[2][i],j,(1..ncols(saveres[2][i]))),nvars(B) div 2, newcomplex[3*(n-i)+2]); |
---|
| 1241 | } |
---|
| 1242 | } |
---|
| 1243 | newcomplex[3*(n-i)-1]=v; |
---|
| 1244 | } |
---|
| 1245 | else |
---|
| 1246 | { |
---|
| 1247 | newcomplex[3*(n-i)]=matrix(0,1,1); |
---|
| 1248 | if (newcomplex[3*(n-i)+1]!=0) |
---|
| 1249 | { |
---|
| 1250 | newcomplex[3*(n-i)]=matrix(0,1,newcomplex[3*(n-i)+1]); |
---|
| 1251 | } |
---|
| 1252 | newcomplex[3*(n-i)-2]=int(0); |
---|
| 1253 | newcomplex[3*(n-i)-1]=intvec(0); |
---|
| 1254 | } |
---|
[7fe9f8b] | 1255 | } |
---|
[0e8a5a] | 1256 | list result; |
---|
| 1257 | result[1]=newcomplex; |
---|
| 1258 | result[2]=list(); |
---|
| 1259 | list forsep; |
---|
| 1260 | for (i=1; i<=size(L) div 2+1; i++) |
---|
| 1261 | { |
---|
| 1262 | forsep[2*i]=newcomplex[3*(n+i-1)]; |
---|
| 1263 | forsep[2*i-1]=matrix(0,1,nrows(forsep[2*i])); |
---|
| 1264 | } |
---|
| 1265 | forsep=shortExactPieces(forsep); |
---|
| 1266 | list listofHis; |
---|
| 1267 | matrix forVd; |
---|
| 1268 | for (i=1; i<=size(L) div 2; i++) |
---|
| 1269 | { |
---|
| 1270 | v=0; |
---|
| 1271 | listofHis[i]=list(forsep[i+1][1][5]); |
---|
| 1272 | forVd=forsep[i+1][2][2]; |
---|
| 1273 | for (j=1; j<=nrows(forVd); j++) |
---|
| 1274 | { |
---|
| 1275 | if (tilde==0) |
---|
| 1276 | { |
---|
| 1277 | v[j]=VdDeg(submat(forVd,j,intvec(1..ncols(forVd))),nvars(B) div 2, newcomplex[3*(n+i)-1]); |
---|
| 1278 | } |
---|
| 1279 | else |
---|
| 1280 | { |
---|
| 1281 | v[j]=VdDegTilde(submat(forVd,j,intvec(1..ncols(forVd))),nvars(B) div 2, newcomplex[3*(n+i)-1]); |
---|
| 1282 | } |
---|
| 1283 | } |
---|
| 1284 | listofHis[i][2]=v; |
---|
| 1285 | } |
---|
| 1286 | result[2]=listofHis; |
---|
| 1287 | result[3]=quasiiso; |
---|
| 1288 | return(result); |
---|
| 1289 | } |
---|
| 1290 | |
---|
| 1291 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1292 | |
---|
| 1293 | static proc vdStrictIntersect(matrix M, matrix N, intvec v, int tilde) |
---|
| 1294 | { |
---|
| 1295 | def B=basering; |
---|
| 1296 | option(returnSB);// alternative:erst intersect und dann SB-Berechung mit slimgb |
---|
| 1297 | if (tilde==0) |
---|
| 1298 | { |
---|
| 1299 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2,v); |
---|
| 1300 | } |
---|
| 1301 | else |
---|
| 1302 | { |
---|
| 1303 | def HomWeyl=makeHomogenizedWeylTilde(nvars(B) div 2,v); |
---|
| 1304 | } |
---|
| 1305 | setring HomWeyl; |
---|
| 1306 | matrix M=fetch(B,M); |
---|
| 1307 | matrix N=fetch(B,N); |
---|
| 1308 | M=nHomogenize(M); |
---|
| 1309 | N=nHomogenize(N); |
---|
| 1310 | matrix vdintersection=transpose(intersect(transpose(M),transpose(N))); |
---|
| 1311 | vdintersection=subst(vdintersection,h,1); |
---|
| 1312 | setring B; |
---|
| 1313 | matrix vdintersection=fetch(HomWeyl,vdintersection); |
---|
| 1314 | option(noreturnSB); |
---|
| 1315 | return(vdintersection); |
---|
| 1316 | } |
---|
| 1317 | |
---|
| 1318 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1319 | |
---|
| 1320 | static proc vdStrictIntersectPlus(matrix M, matrix N, intvec v, int tilde) |
---|
| 1321 | { |
---|
| 1322 | def B=basering; |
---|
| 1323 | int n=nvars(B) div 2; |
---|
| 1324 | matrix vdint=transpose(intersect(transpose(M),transpose(N))); |
---|
| 1325 | if (tilde==0) |
---|
| 1326 | { |
---|
| 1327 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2,v); |
---|
| 1328 | } |
---|
| 1329 | else |
---|
| 1330 | { |
---|
| 1331 | def HomWeyl=makeHomogenizedWeylTilde(nvars(B) div 2,v); |
---|
| 1332 | } |
---|
| 1333 | setring HomWeyl; |
---|
| 1334 | matrix vdint=fetch(B,vdint); |
---|
| 1335 | matrix N=fetch(B,N); |
---|
| 1336 | vdint=nHomogenize(vdint); |
---|
| 1337 | intvec i1; |
---|
| 1338 | intvec i2; |
---|
| 1339 | int i; |
---|
| 1340 | int nr; |
---|
| 1341 | int nc; |
---|
| 1342 | def ringofSyz=Sres(transpose(vdint),n);//////////////////////////////////////////////////////////////// |
---|
| 1343 | setring ringofSyz; |
---|
| 1344 | matrix vdint=transpose(matrix(RES[2])); |
---|
| 1345 | vdint=subst(vdint,h,1); |
---|
| 1346 | int logens=ncols(vdint)+1; |
---|
| 1347 | int omitemptylist; |
---|
| 1348 | matrix zerom; |
---|
| 1349 | list rofA; |
---|
[08fa62] | 1350 | for (i=3; i<=n+3; i++)////////////////////////////////////////////////////////////////////////////n und si muessen noch definiert werden |
---|
[0e8a5a] | 1351 | { |
---|
| 1352 | if (size(RES)>=i) |
---|
| 1353 | { |
---|
| 1354 | zerom=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i]))); |
---|
| 1355 | if (RES[i]!=zerom) |
---|
| 1356 | { |
---|
| 1357 | rofA[i-2]=(matrix(RES[i])); |
---|
| 1358 | if (i==3) |
---|
| 1359 | { |
---|
| 1360 | if (nrows(rofA[i-2])-logens+1!=nrows(vdint)) |
---|
| 1361 | { |
---|
| 1362 | //build the resolution |
---|
| 1363 | nr=nrows(vdint)+logens-1; |
---|
| 1364 | nc=ncols(rofA[i-2]); |
---|
| 1365 | rofA[i-2]=matrix(rofA[i-2],nr,nc); |
---|
| 1366 | } |
---|
| 1367 | |
---|
| 1368 | } |
---|
| 1369 | if (i!=3) |
---|
| 1370 | { |
---|
| 1371 | if (nrows(rofA[i-2])-logens+1!=nrows(rofA[i-3])) |
---|
| 1372 | { |
---|
| 1373 | nr=nrows(rofA[i-3])+logens-1; |
---|
| 1374 | nc=ncols(rofA[i-2]); |
---|
| 1375 | rofA[i-2]=matrix(rofA[i-2],nr,nc); |
---|
| 1376 | } |
---|
| 1377 | } |
---|
| 1378 | i1=intvec(logens..nrows(rofA[i-2])); |
---|
| 1379 | i2=intvec(1..ncols(rofA[i-2])); |
---|
| 1380 | rofA[i-2]=transpose(submat(rofA[i-2],i1,i2)); |
---|
| 1381 | logens=logens+ncols(rofA[i-2]); |
---|
| 1382 | rofA[i-2]=subst(rofA[i-2],h,1); |
---|
| 1383 | } |
---|
| 1384 | else |
---|
| 1385 | { |
---|
| 1386 | rofA[i-2]=list(); |
---|
| 1387 | } |
---|
| 1388 | } |
---|
| 1389 | else |
---|
| 1390 | { |
---|
| 1391 | rofA[i-2]=list(); |
---|
| 1392 | } |
---|
| 1393 | } |
---|
| 1394 | if(size(rofA[1])==0) |
---|
[7fe9f8b] | 1395 | { |
---|
[0e8a5a] | 1396 | omitemptylist=1; |
---|
[7fe9f8b] | 1397 | } |
---|
[0e8a5a] | 1398 | setring B; |
---|
| 1399 | vdint=fetch(ringofSyz,vdint); |
---|
| 1400 | if (omitemptylist!=1) |
---|
| 1401 | { |
---|
| 1402 | list rofA=fetch(ringofSyz,rofA); |
---|
| 1403 | } |
---|
| 1404 | kill HomWeyl; |
---|
| 1405 | kill ringofSyz; |
---|
| 1406 | return(list(vdint,rofA)); |
---|
[7fe9f8b] | 1407 | } |
---|
| 1408 | |
---|
| 1409 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1410 | |
---|
| 1411 | static proc toVdStrictFreeComplex(list L,string Syzstring,list #) |
---|
[0e8a5a] | 1412 | "USAGE: toVdStrictFreeComplex(L, Syzstring [,d]); L a list of the form |
---|
| 1413 | (M_1,f_1,...,M_s,f_s), where the M_i and f_i are matrices, Syzstring a |
---|
[7fe9f8b] | 1414 | string, d an optional integer |
---|
| 1415 | ASSUME: Basering is the Weyl algebra D_n @* |
---|
| 1416 | (M_1,f_1,...,M_s,f_s) represents a complex 0->D_n^(r_1)/im(M_1)-> |
---|
[0e8a5a] | 1417 | D_n^(r_2)/im(M_2)->...->D_n^(r_s)->0 with differentials f_i, where im(M_i) |
---|
[7fe9f8b] | 1418 | is generated by the rows of M_i. In particular it hold:@* |
---|
| 1419 | - The M_i are m_i x r_i-matrices and the f_iare r_i x r_(i+1)-matrices @* |
---|
| 1420 | -the image of M_1*f_i is contained in the image of M_(i+1) @* |
---|
[0e8a5a] | 1421 | d is an optional integer which indices in the case size(L)=2, whether a |
---|
| 1422 | V_d-strict or \tilde(V_d)-strict will be computed@* |
---|
[7fe9f8b] | 1423 | Syzstring is either: @* |
---|
[0e8a5a] | 1424 | -'Sres' (computes the resolutions and Groebner bases in the homogenized |
---|
| 1425 | Weyl algebra using Schreyer's method)@* |
---|
[7fe9f8b] | 1426 | or @* |
---|
[0e8a5a] | 1427 | -'Vdres' (computes the resolutions via V_d-homogenization and without |
---|
[7fe9f8b] | 1428 | Schreyer's method)@* |
---|
| 1429 | RETURN: list of the form (L_1,L_2), were L_1 and L_2 are lists @* |
---|
| 1430 | L_1 is of the form (i_(-n-1),g_(-n-1),m_(-n-1),...,i_s,g_s,m_s) such that:@* |
---|
[0e8a5a] | 1431 | -the i_j are integers, the g_j are i_j x i_(j+1)-matrices, the m_j intvecs |
---|
[7fe9f8b] | 1432 | of size i_j@* |
---|
| 1433 | -D_n^(i_(-n-1))[m_(-n-1)]->...->D_n^(i_s)[m_s]->0 is a V_d-strict complex |
---|
| 1434 | with differentials m_i that is quasi-isomorphic to the complex given by L@* |
---|
[0e8a5a] | 1435 | L_2 is of the form (H_1,n_1,...,H_s,n_s), where the H_i are matrices and |
---|
[7fe9f8b] | 1436 | the n_i are shift vectors such that:@* |
---|
[0e8a5a] | 1437 | -coker(H_i) is the ith cohomology group of the complex given by L_1@* |
---|
[7fe9f8b] | 1438 | -the n_i are the shift vectors of the coker(H_i) |
---|
| 1439 | THEORY: We follow Algorithm 3.8 in [W2] |
---|
| 1440 | " |
---|
| 1441 | { |
---|
[0e8a5a] | 1442 | def B=basering; |
---|
[7fe9f8b] | 1443 | int n=nvars(B) div 2+2; |
---|
| 1444 | int d=nvars(B) div 2; |
---|
| 1445 | intvec v; |
---|
| 1446 | list out, outall; |
---|
| 1447 | int i,j,k,indi,nc,nr; |
---|
| 1448 | matrix mem; |
---|
| 1449 | intvec i1,i2; |
---|
[0e8a5a] | 1450 | int tilde; |
---|
[7fe9f8b] | 1451 | if (size(#)!=0) |
---|
| 1452 | { |
---|
[0e8a5a] | 1453 | for (i=1; i<=size(#); i++) |
---|
[7fe9f8b] | 1454 | { |
---|
[0e8a5a] | 1455 | if (typeof(#[i])=="int") |
---|
| 1456 | { |
---|
| 1457 | tilde=#[i]; |
---|
| 1458 | } |
---|
[7fe9f8b] | 1459 | } |
---|
| 1460 | } |
---|
| 1461 | /* If size(L)=2, our complex consists for only one non-trivial module. |
---|
[0e8a5a] | 1462 | Therefore, we just have to compute a V_d-strict resolution of this module.*/ |
---|
| 1463 | if (size(L)==2) |
---|
| 1464 | { |
---|
| 1465 | v=(0:ncols(L[1])); |
---|
| 1466 | out[3*n-1]=v; |
---|
| 1467 | out[3*n-2]=ncols(L[1]); |
---|
| 1468 | out[3*n]=L[2]; |
---|
| 1469 | if (Syzstring=="Vdres") |
---|
[7fe9f8b] | 1470 | { |
---|
[0e8a5a] | 1471 | /*if Syzstring="Vdres", we compute a V_d-strict Groebner basis of L[1] |
---|
| 1472 | using F-homogenization (Prop. 3.9 in [OT]); then we compute the syzygies |
---|
| 1473 | and make them V_d-strict using Prop 3.9[OT] and so on*/ |
---|
| 1474 | out[3*n-3]=VdStrictGB(L[1],d,v); |
---|
| 1475 | for (i=n-1; i>=1; i--) |
---|
| 1476 | { |
---|
| 1477 | out[3*i-2]=nrows(out[3*i]); |
---|
| 1478 | v=0; |
---|
| 1479 | for (j=1; j<=out[3*i-2]; j++) |
---|
| 1480 | { |
---|
| 1481 | mem=submat(out[3*i],j,intvec(1..ncols(out[3*i]))); |
---|
| 1482 | v[j]=VdDeg(mem,d, out[3*i+2]);//next shift vector |
---|
| 1483 | } |
---|
| 1484 | out[3*i-1]=v; |
---|
| 1485 | if (i!=1) |
---|
| 1486 | { |
---|
| 1487 | /*next step in the resolution*/ |
---|
| 1488 | out[3*i-3]=transpose(syz(transpose(out[3*i]))); |
---|
| 1489 | if (out[3*i-3]!=matrix(0,nrows(out[3*i-3]),ncols(out[3*i-3]))) |
---|
| 1490 | { |
---|
| 1491 | /*makes the resolution V_d-strict*/ |
---|
| 1492 | out[3*i-3]=VdStrictGB(out[3*i-3],d,out[3*i-1]); |
---|
| 1493 | } |
---|
| 1494 | else |
---|
| 1495 | { |
---|
| 1496 | /*resolution is already computed*/ |
---|
| 1497 | out[3*i-3]=matrix(0,1,ncols(out[3*i-3])); |
---|
| 1498 | out[3*i-4]=intvec(0); |
---|
| 1499 | out[3*i-5]=int(0); |
---|
| 1500 | for (j=i-2; j>=1; j--) |
---|
| 1501 | { |
---|
| 1502 | out[3*j]=matrix(0,1,1); |
---|
| 1503 | out[3*j-1]=intvec(0); |
---|
| 1504 | out[3*j-2]=int(0); |
---|
| 1505 | } |
---|
| 1506 | break; |
---|
| 1507 | } |
---|
| 1508 | } |
---|
| 1509 | } |
---|
[7fe9f8b] | 1510 | } |
---|
[0e8a5a] | 1511 | else |
---|
[7fe9f8b] | 1512 | { |
---|
[0e8a5a] | 1513 | /*in the case Syzstring!="Vdres" we compute the resolution in the |
---|
| 1514 | homogenized Weyl algebra using Thm 9.10 in[OT]*/ |
---|
| 1515 | if (tilde==0) |
---|
| 1516 | { |
---|
| 1517 | def HomWeyl=makeHomogenizedWeyl(d); |
---|
| 1518 | } |
---|
| 1519 | else |
---|
| 1520 | { |
---|
| 1521 | def HomWeyl=makeHomogenizedWeylTilde(d); |
---|
| 1522 | } |
---|
| 1523 | setring HomWeyl; |
---|
| 1524 | list L=fetch(B,L); |
---|
| 1525 | L[1]=nHomogenize(L[1]); |
---|
| 1526 | list out=fetch(B,out); |
---|
| 1527 | out[3*n-3]=L[1]; |
---|
| 1528 | /*computes a ring with a list RES; RES is a V_d-strict resolution of |
---|
| 1529 | L[1]*/ |
---|
| 1530 | def ringofSyz=Sres(transpose(L[1]),d); |
---|
| 1531 | setring ringofSyz; |
---|
| 1532 | int logens=2; |
---|
| 1533 | matrix mem; |
---|
| 1534 | list out=fetch(HomWeyl,out); |
---|
| 1535 | out[3*n-3]=transpose(matrix(RES[2])); |
---|
| 1536 | out[3*n-3]=subst(out[3*n-3],h,1); |
---|
| 1537 | for (i=n-1; i>=1; i--) |
---|
| 1538 | { |
---|
| 1539 | out[3*i-2]=nrows(out[3*i]); |
---|
| 1540 | v=0; |
---|
| 1541 | for (j=1; j<=out[3*i-2]; j++) |
---|
| 1542 | { |
---|
| 1543 | mem=submat(out[3*i],j,intvec(1..ncols(out[3*i]))); |
---|
| 1544 | if (tilde==0) |
---|
| 1545 | { |
---|
| 1546 | v[j]=VdDeg(mem,d, out[3*i+2]); |
---|
| 1547 | } |
---|
| 1548 | else |
---|
| 1549 | { |
---|
| 1550 | v[j]=VdDegTilde(mem,d, out[3*i+2]); |
---|
| 1551 | } |
---|
| 1552 | } |
---|
| 1553 | out[3*i-1]=v;//shift vector such that the resolution RES is V_d-strict |
---|
| 1554 | if (i!=1) |
---|
| 1555 | { |
---|
| 1556 | indi=0; |
---|
| 1557 | if (size(RES)>=n-i+2) |
---|
| 1558 | { |
---|
| 1559 | nr=nrows(matrix(RES[n-i+2])); |
---|
| 1560 | mem=matrix(0,nr,ncols(matrix(RES[n-i+2]))); |
---|
| 1561 | if (matrix(RES[n-i+2])!=mem) |
---|
| 1562 | { |
---|
| 1563 | indi=1; |
---|
| 1564 | out[3*i-3]=(matrix(RES[n-i+2])); |
---|
| 1565 | if (nrows(out[3*i-3])-logens+1!=nrows(out[3*i])) |
---|
| 1566 | { |
---|
| 1567 | mem=out[3*i-3]; |
---|
| 1568 | out[3*i-3]=matrix(mem,nrows(mem)+logens-1,ncols(mem)); |
---|
| 1569 | } |
---|
| 1570 | mem=out[3*i-3]; |
---|
| 1571 | i1=intvec(logens..nrows(mem)); |
---|
| 1572 | mem=submat(mem,i1,intvec(1..ncols(mem))); |
---|
| 1573 | out[3*i-3]=transpose(mem); |
---|
| 1574 | out[3*i-3]=subst(out[3*i-3],h,1); |
---|
| 1575 | logens=logens+ncols(out[3*i-3]); |
---|
| 1576 | } |
---|
| 1577 | } |
---|
| 1578 | if(indi==0) |
---|
| 1579 | { |
---|
| 1580 | out[3*i-3]=matrix(0,1,nrows(out[3*i])); |
---|
| 1581 | out[3*i-4]=intvec(0); |
---|
| 1582 | out[3*i-5]=int(0); |
---|
| 1583 | for (j=i-2; j>=1; j--) |
---|
| 1584 | { |
---|
| 1585 | out[3*j]=matrix(0,1,1); |
---|
| 1586 | out[3*j-1]=intvec(0); |
---|
| 1587 | out[3*j-2]=int(0); |
---|
| 1588 | } |
---|
| 1589 | break; |
---|
| 1590 | } |
---|
| 1591 | } |
---|
| 1592 | } |
---|
| 1593 | setring B; |
---|
| 1594 | out=fetch(ringofSyz,out);//contains the V_d-strict resolution |
---|
| 1595 | kill ringofSyz; |
---|
| 1596 | } |
---|
| 1597 | outall[1]=out; |
---|
| 1598 | outall[2]=list(list(out[3*n-3],out[3*n-1])); |
---|
| 1599 | return(outall); |
---|
| 1600 | } |
---|
| 1601 | /*case size(L)>2: We compute a quasi-isomorphic free complex following Alg 3.8 in |
---|
| 1602 | [W2]*/ |
---|
| 1603 | /* We denote the complex given by L as (C^i,d^i). |
---|
| 1604 | We start by computing in the proc shortExaxtPieces representations for the |
---|
| 1605 | short exact sequences B^i->Z^i->H^i and Z^i->C^i->B^(i+1), where the B^i, Z^i |
---|
| 1606 | and H^i are coboundaries, cocycles and cohomology groups, respectively.*/ |
---|
| 1607 | out=shortExactPieces(L); |
---|
[7fe9f8b] | 1608 | list rem; |
---|
[0e8a5a] | 1609 | /* shortExactpiecesToVdStrict makes the sequences B^i->Z^i->H^i and |
---|
| 1610 | Z^i->C^i->B^(i+1) V_d-strict*/ |
---|
[7fe9f8b] | 1611 | rem=shortExactPiecesToVdStrict(out,d,Syzstring); |
---|
[0e8a5a] | 1612 | /*VdStrictDoubleComplexes computes V_d-strict resolutions over the seqeunces from |
---|
| 1613 | proc shortExactPiecesToVdstrict*/ |
---|
[7fe9f8b] | 1614 | out=VdStrictDoubleComplexes(rem[1],d,Syzstring); |
---|
| 1615 | for (i=1;i<=size(out); i++) |
---|
[0e8a5a] | 1616 | { |
---|
| 1617 | rem[2][i][1]=out[i][1][5][1]; |
---|
| 1618 | rem[2][i][2]=out[i][1][8][1]; |
---|
| 1619 | } |
---|
| 1620 | /* AssemblingDoubleComplexes puts the resolution of the C^i (from the sequences |
---|
| 1621 | Z^i->C^i->B^(i+1)) together to obtain a Cartan-Eilenberg resolution of |
---|
| 1622 | (C^i,d^i)*/ |
---|
[7fe9f8b] | 1623 | out=assemblingDoubleComplexes(out); |
---|
| 1624 | /*the proc totalComplex takes the total complex of the double complex from the |
---|
[0e8a5a] | 1625 | proc assemblingDoubleComplexes*/ |
---|
[7fe9f8b] | 1626 | out=totalComplex(out); |
---|
| 1627 | outall[1]=out; |
---|
[0e8a5a] | 1628 | outall[2]=rem[2];//contains the cohomology groups and their shift vectors |
---|
[7fe9f8b] | 1629 | return (outall); |
---|
| 1630 | } |
---|
| 1631 | |
---|
| 1632 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1633 | |
---|
| 1634 | |
---|
| 1635 | static proc sublist(list L,int m,int n) |
---|
| 1636 | { |
---|
| 1637 | list out; |
---|
| 1638 | int i; int j; |
---|
| 1639 | int count; |
---|
| 1640 | for (i=m; i<=n; i++) |
---|
| 1641 | { |
---|
[0e8a5a] | 1642 | out[size(out)+1]=list(); |
---|
| 1643 | for (j=1; j<=size(L[i]); j++) |
---|
| 1644 | { |
---|
| 1645 | count=count+1; |
---|
| 1646 | out[size(out)][j]=list(L[i][j],count); |
---|
| 1647 | } |
---|
[7fe9f8b] | 1648 | } |
---|
| 1649 | list o=list(out,count); |
---|
| 1650 | return(o); |
---|
| 1651 | } |
---|
| 1652 | |
---|
| 1653 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1654 | |
---|
[0e8a5a] | 1655 | static proc LMSubset(list L,list M, list #) |
---|
[7fe9f8b] | 1656 | { |
---|
| 1657 | int i; |
---|
| 1658 | int j=1; |
---|
[0e8a5a] | 1659 | if (size(#)==0) |
---|
[7fe9f8b] | 1660 | { |
---|
[0e8a5a] | 1661 | list position=(M[size(M)],(-1)^(size(L))); |
---|
| 1662 | } |
---|
| 1663 | else |
---|
| 1664 | { |
---|
| 1665 | list position=(M[size(M)],1); |
---|
[7fe9f8b] | 1666 | } |
---|
[0e8a5a] | 1667 | for (i=1; i<=size(L); i++) |
---|
| 1668 | { |
---|
| 1669 | if (L[i]!=M[j]) |
---|
| 1670 | { |
---|
| 1671 | if (L[i]!=M[i+1] or j!=i) |
---|
| 1672 | { |
---|
| 1673 | return (L[i],0); |
---|
| 1674 | } |
---|
| 1675 | else |
---|
| 1676 | { |
---|
| 1677 | if (size(#)==0) |
---|
| 1678 | { |
---|
| 1679 | position=(M[i],(-1)^(i-1)); |
---|
| 1680 | } |
---|
| 1681 | else |
---|
| 1682 | { |
---|
| 1683 | position=(M[i],(-1)^(size(L)+1-i)); |
---|
| 1684 | } |
---|
| 1685 | j=j+1; |
---|
| 1686 | } |
---|
| 1687 | } |
---|
| 1688 | j=j+1; |
---|
[7fe9f8b] | 1689 | |
---|
[0e8a5a] | 1690 | } |
---|
[7fe9f8b] | 1691 | return (position); |
---|
| 1692 | } |
---|
| 1693 | |
---|
| 1694 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1695 | |
---|
| 1696 | static proc shortExactPieces(list L) |
---|
| 1697 | { |
---|
| 1698 | /*we follow Section 3.3 in [W2]*/ |
---|
[0e8a5a] | 1699 | /* we assume that L=(M_1,f_1,...,M_s,f_s) defines the complex C=(C^i,d^i) |
---|
| 1700 | as in the procedure toVdstrictcomplex*/ |
---|
[7fe9f8b] | 1701 | matrix Bnew= divdr(L[2],L[3]); |
---|
| 1702 | matrix Bold=Bnew; |
---|
| 1703 | matrix Z=divdr(Bnew,L[1]); |
---|
| 1704 | list bzh,zcb; |
---|
| 1705 | bzh=list(list(),list(),Z,unitmat(ncols(Z)),Z); |
---|
| 1706 | zcb=(Z, Bnew, L[1], unitmat(ncols(L[1])), Bnew); |
---|
| 1707 | list sep; |
---|
| 1708 | /* the list sep will be of size s such that |
---|
[0e8a5a] | 1709 | -sep[i]=(sep[i][1],sep[i][2]) is a list of two lists |
---|
| 1710 | -sep[i][1]=(B^i,f^(BZi),Z^i,f_^(ZHi),H^i) such that coker(B^i)->coker(Z^i) |
---|
| 1711 | ->coker(H^i) represents the short exact seqeuence B^i(C)->Z^i(C)->H^i(C) |
---|
| 1712 | -sep[i][2]=(Z^i,f^(ZCi),C^i,f^(CBi),B^(i+1)) such that coker(Z^i)->coker(C^i)-> |
---|
| 1713 | coker(B^(i+1)) represents the short exact seqeuence Z^i(C)->C^i->B^(i+1)(C)*/ |
---|
[7fe9f8b] | 1714 | sep[1]=list(bzh,zcb); |
---|
| 1715 | int i; |
---|
| 1716 | list out; |
---|
| 1717 | for (i=3; i<=size(L)-2; i=i+2) |
---|
[0e8a5a] | 1718 | { |
---|
| 1719 | /*the proc bzhzcb computes representations for the short exact seqeunces */ |
---|
| 1720 | out=bzhzcb(Bold, L[i-1] , L[i], L[i+1], L[i+2]); |
---|
| 1721 | sep[size(sep)+1]=out[1]; |
---|
| 1722 | Bold=out[2]; |
---|
| 1723 | } |
---|
[7fe9f8b] | 1724 | bzh=(divdr(L[size(L)-2], L[size(L)-1]),L[size(L)-2], L[size(L)-1]); |
---|
| 1725 | bzh[4]=unitmat(ncols(L[size(L)-1])); |
---|
| 1726 | bzh[5]=transpose(concat(transpose(L[size(L)-2]),transpose(L[size(L)-1]))); |
---|
| 1727 | zcb=(L[size(L)-1], unitmat(ncols(L[size(L)-1])), L[size(L)-1],list(),list()); |
---|
| 1728 | sep[size(sep)+1]=list(bzh,zcb); |
---|
| 1729 | return(sep); |
---|
| 1730 | } |
---|
| 1731 | |
---|
| 1732 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1733 | |
---|
| 1734 | static proc bzhzcb (matrix Bold,matrix f0,matrix C1,matrix f1,matrix C2) |
---|
| 1735 | { |
---|
| 1736 | matrix Bnew=divdr(f1,C2); |
---|
| 1737 | matrix Z= divdr(Bnew,C1); |
---|
| 1738 | matrix lift1= matrixLift(Bnew,f0); |
---|
| 1739 | matrix H=transpose(concat(transpose(lift1),transpose(Z))); |
---|
| 1740 | list bzh=(Bold, lift1, Z, unitmat(ncols(Z)),H); |
---|
| 1741 | list zcb=(Z, Bnew, C1, unitmat(ncols(C1)),Bnew); |
---|
| 1742 | list out=(list(bzh, zcb), Bnew); |
---|
| 1743 | return(out); |
---|
| 1744 | } |
---|
| 1745 | |
---|
| 1746 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1747 | |
---|
| 1748 | static proc shortExactPiecesToVdStrict(list C,int d,list #) |
---|
| 1749 | {/* We transform the short exact pieces from procedure shortExactPieces to V_d- |
---|
[0e8a5a] | 1750 | strict short exact sequences. For this, we use Algorithm 3.11 and Lemma 4.2 in |
---|
| 1751 | [W2].*/ |
---|
[7fe9f8b] | 1752 | /* If we compute our Groebner bases in the homogenized Weyl algebra, we already |
---|
[0e8a5a] | 1753 | compute some resolutions it omit additional Groebner basis computations later |
---|
| 1754 | on.*/ |
---|
[7fe9f8b] | 1755 | int s =size(C);int i; int j; |
---|
| 1756 | string Syzstring="Sres"; |
---|
[0e8a5a] | 1757 | intvec v=0:ncols(C[s][1][5]); |
---|
[7fe9f8b] | 1758 | if (size(#)!=0) |
---|
| 1759 | { |
---|
[0e8a5a] | 1760 | for (i=1; i<=size(#); i++) |
---|
| 1761 | { |
---|
| 1762 | if (typeof(#[i])=="string") |
---|
| 1763 | { |
---|
| 1764 | Syzstring=#[i]; |
---|
| 1765 | } |
---|
| 1766 | if (typeof(#[i])=="intvec") |
---|
| 1767 | { |
---|
| 1768 | v=#[i]; |
---|
| 1769 | } |
---|
| 1770 | } |
---|
[7fe9f8b] | 1771 | } |
---|
| 1772 | list out; |
---|
| 1773 | list forout; |
---|
| 1774 | if (Syzstring=="Vdres") |
---|
[0e8a5a] | 1775 | { |
---|
| 1776 | out[s]=list(toVdStrictSequence(C[s][1],d,v, Syzstring,s)); |
---|
| 1777 | } |
---|
[7fe9f8b] | 1778 | else |
---|
[0e8a5a] | 1779 | { |
---|
| 1780 | forout=toVdStrictSequence(C[s][1],d,v, Syzstring,s); |
---|
| 1781 | list resolutionofA=forout[9]; |
---|
| 1782 | list resolutionofC=forout[10]; |
---|
| 1783 | forout=delete(forout,10); |
---|
| 1784 | forout=delete(forout,9); |
---|
| 1785 | out[s]=list(forout); |
---|
| 1786 | for (i=1; i<=size(resolutionofC); i++) |
---|
| 1787 | { |
---|
| 1788 | out[s][1][5][i+1]=resolutionofC[i];//save the resolutions |
---|
| 1789 | out[s][1][1][i+1]=resolutionofA[i]; |
---|
| 1790 | } |
---|
| 1791 | } |
---|
[7fe9f8b] | 1792 | out[s][2]=list(list(out[s][1][3][1])); |
---|
| 1793 | out[s][2][2]=list(unitmat(ncols(out[s][1][3][1]))); |
---|
| 1794 | out[s][2][3]=list(out[s][1][3][1]); |
---|
| 1795 | out[s][2][4]=list(list()); |
---|
| 1796 | out[s][2][5]=list(list()); |
---|
| 1797 | out[s][2][6]=list(out[s][1][7][1]); |
---|
| 1798 | out[s][2][7]=list(out[s][2][6][1]); |
---|
| 1799 | out[s][2][8]=list(list()); |
---|
| 1800 | list resolutionofD; |
---|
| 1801 | list resolutionofF; |
---|
| 1802 | for (i=s-1; i>=2; i--) |
---|
[0e8a5a] | 1803 | { |
---|
| 1804 | C[i][2][5]=out[i+1][1][1][1]; |
---|
| 1805 | forout=toVdStrictSequences(C[i],d,out[i+1][1][6][1],Syzstring,s); |
---|
| 1806 | if (Syzstring=="Sres") |
---|
| 1807 | { |
---|
| 1808 | resolutionofD=forout[3];//save the resolutions |
---|
| 1809 | resolutionofF=forout[4]; |
---|
| 1810 | forout=delete(forout,4); |
---|
| 1811 | forout=delete(forout,3); |
---|
| 1812 | } |
---|
| 1813 | out[i]=forout; |
---|
| 1814 | if(Syzstring=="Sres") |
---|
| 1815 | { |
---|
| 1816 | for (j=2; j<=size(out[i+1][1][1]); j++) |
---|
| 1817 | { |
---|
| 1818 | out[i][2][5][j]=out[i+1][1][1][j]; |
---|
| 1819 | } |
---|
| 1820 | for (j=1; j<=size(resolutionofD);j++) |
---|
| 1821 | { |
---|
| 1822 | out[i][1][1][j+1]=resolutionofD[j]; |
---|
| 1823 | out[i][1][5][j+1]=resolutionofF[j]; |
---|
| 1824 | } |
---|
| 1825 | } |
---|
[7fe9f8b] | 1826 | } |
---|
| 1827 | out[1]=list(list());//initalize our list |
---|
| 1828 | C[1][2][5]=out[2][1][1][1]; |
---|
| 1829 | /*Compute the last V_d-strict seqeunce*/ |
---|
| 1830 | if (Syzstring=="Vdres") |
---|
| 1831 | { |
---|
[0e8a5a] | 1832 | out[1][2]=toVdStrictSequence(C[1][2],d,out[2][1][6][1],Syzstring,s,"J_Agiv"); |
---|
[7fe9f8b] | 1833 | } |
---|
[0e8a5a] | 1834 | else |
---|
[7fe9f8b] | 1835 | { |
---|
[0e8a5a] | 1836 | forout=toVdStrictSequence(C[1][2],d,out[2][1][6][1],Syzstring,s,"J_Agiv"); |
---|
| 1837 | out[1][2]=delete(forout,9); |
---|
| 1838 | list resolutionofA2=forout[9]; |
---|
| 1839 | for (i=1; i<=size(out[2][1][1]); i++) |
---|
| 1840 | { |
---|
| 1841 | /*put the modules for the resolutions in the right spot*/ |
---|
| 1842 | out[1][2][5][i]=out[2][1][1][i]; |
---|
| 1843 | } |
---|
| 1844 | for (i=1; i<=size(resolutionofA2); i++) |
---|
| 1845 | { |
---|
| 1846 | out[1][2][1][i+1]=resolutionofA2[i]; |
---|
| 1847 | } |
---|
[7fe9f8b] | 1848 | } |
---|
| 1849 | out[1][1][3]=list(out[1][2][1][1]); |
---|
| 1850 | out[1][1][5]=list(out[1][2][1][1]); |
---|
| 1851 | out[1][1][4]=list(unitmat(ncols(out[1][1][3][1]))); |
---|
| 1852 | out[1][1][7]=list(out[1][2][6][1]); |
---|
| 1853 | out[1][1][8]=list(out[1][2][6][1]); |
---|
| 1854 | out[1][1][1]=list(list()); |
---|
| 1855 | out[1][1][2]=list(list()); |
---|
| 1856 | out[1][1][6]=list(list()); |
---|
| 1857 | if (Syzstring=="Sres") |
---|
| 1858 | { |
---|
[0e8a5a] | 1859 | for (i=1; i<=size(out[1][2][1]); i++) |
---|
| 1860 | { |
---|
| 1861 | out[1][1][3][i]=out[1][2][1][i]; |
---|
| 1862 | out[1][1][5][i]=out[1][2][1][i]; |
---|
| 1863 | } |
---|
[7fe9f8b] | 1864 | } |
---|
| 1865 | list Hi; |
---|
[0e8a5a] | 1866 | for (i=1; i<=size(out); i++) |
---|
| 1867 | { |
---|
| 1868 | Hi[i]=list(out[i][1][5][1],out[i][1][8][1]); |
---|
| 1869 | } |
---|
[7fe9f8b] | 1870 | list outall; |
---|
| 1871 | outall[1]=out; |
---|
| 1872 | outall[2]=Hi; |
---|
| 1873 | return(outall); |
---|
| 1874 | } |
---|
| 1875 | |
---|
| 1876 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 1877 | |
---|
[0e8a5a] | 1878 | static proc toVdStrictSequence(list C,int n,intvec v,string Syzstring,int si,list #) |
---|
[7fe9f8b] | 1879 | { |
---|
| 1880 | /*this is the Algorithm 3.11 in [W2]*/ |
---|
| 1881 | int omitemptylist; |
---|
| 1882 | int lengthofres=si+n-1; |
---|
| 1883 | int i,j,logens; |
---|
| 1884 | def B=basering; |
---|
| 1885 | matrix bi=slimgb(transpose(C[5])); |
---|
[0e8a5a] | 1886 | /* Computation of a V_d-strict Groebner basis of C[5]: |
---|
| 1887 | -if Syzstring=="Vdres" this is done using the method of weighted homogenization |
---|
| 1888 | (Prop. 3.9 [OT]) |
---|
| 1889 | -else we use the homogenized Weyl algebra for Groebner basis computations |
---|
| 1890 | (Prop 9.9 [OT]), |
---|
| 1891 | in this case we already compute someresolutions (Thm. 9.10 [OT]) to omit |
---|
| 1892 | extra Groebner basis computations later on*/ |
---|
[7fe9f8b] | 1893 | int nr,nc; |
---|
| 1894 | intvec i1,i2; |
---|
| 1895 | if (Syzstring=="Vdres") |
---|
| 1896 | { |
---|
[0e8a5a] | 1897 | if(size(#)==0) |
---|
| 1898 | { |
---|
| 1899 | matrix J_C=VdStrictGB(C[5],n,list(v)); |
---|
| 1900 | } |
---|
| 1901 | else |
---|
| 1902 | { |
---|
| 1903 | matrix J_C=C[5];//C[5] is already a V_d-strict Groebner basis |
---|
| 1904 | } |
---|
[7fe9f8b] | 1905 | } |
---|
| 1906 | else |
---|
| 1907 | { |
---|
[0e8a5a] | 1908 | if (size(#)==0) |
---|
[7fe9f8b] | 1909 | { |
---|
[0e8a5a] | 1910 | matrix MC=C[5]; |
---|
| 1911 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, v); |
---|
| 1912 | setring HomWeyl; |
---|
| 1913 | matrix J_C=fetch(B,MC); |
---|
| 1914 | J_C=nHomogenize(J_C); |
---|
| 1915 | /*computation of V_d-strict resolution of C[5]->needed for proc |
---|
| 1916 | VdstrictDoubleComplexes*/ |
---|
| 1917 | def ringofSyz=Sres(transpose(J_C),lengthofres); |
---|
| 1918 | setring ringofSyz; |
---|
| 1919 | matrix J_C=transpose(matrix(RES[2])); |
---|
| 1920 | J_C=subst(J_C,h,1); |
---|
| 1921 | logens=ncols(J_C)+1; |
---|
| 1922 | matrix zerom; |
---|
| 1923 | list rofC;//will contain resolution of C |
---|
| 1924 | for (i=3; i<=n+si+1; i++) |
---|
[7fe9f8b] | 1925 | { |
---|
[0e8a5a] | 1926 | if (size(RES)>=i) |
---|
| 1927 | { |
---|
| 1928 | zerom=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i]))); |
---|
| 1929 | if (RES[i]!=zerom) |
---|
| 1930 | { |
---|
| 1931 | rofC[i-2]=(matrix(RES[i])); |
---|
[7fe9f8b] | 1932 | |
---|
[0e8a5a] | 1933 | if (i==3) |
---|
| 1934 | { |
---|
| 1935 | if (nrows(rofC[i-2])-logens+1!=nrows(J_C)) |
---|
| 1936 | { |
---|
| 1937 | //build the resolution |
---|
| 1938 | nr=nrows(J_C)+logens-1; |
---|
| 1939 | nc=ncols(rofC[i-2]); |
---|
| 1940 | rofC[i-2]=matrix(rofC[i-2],nr,nc); |
---|
| 1941 | } |
---|
| 1942 | |
---|
| 1943 | } |
---|
| 1944 | if (i!=3) |
---|
| 1945 | { |
---|
| 1946 | if (nrows(rofC[i-2])-logens+1!=nrows(rofC[i-3])) |
---|
| 1947 | { |
---|
| 1948 | nr=nrows(rofC[i-3])+logens-1; |
---|
| 1949 | nc=ncols(rofC[i-2]); |
---|
| 1950 | rofC[i-2]=matrix(rofC[i-2],nr,nc); |
---|
| 1951 | } |
---|
| 1952 | } |
---|
| 1953 | i1=intvec(logens..nrows(rofC[i-2])); |
---|
| 1954 | i2=intvec(1..ncols(rofC[i-2])); |
---|
| 1955 | rofC[i-2]=transpose(submat(rofC[i-2],i1,i2)); |
---|
| 1956 | logens=logens+ncols(rofC[i-2]); |
---|
| 1957 | rofC[i-2]=subst(rofC[i-2],h,1); |
---|
| 1958 | } |
---|
| 1959 | else |
---|
| 1960 | { |
---|
| 1961 | rofC[i-2]=list(); |
---|
| 1962 | } |
---|
| 1963 | } |
---|
| 1964 | else |
---|
| 1965 | { |
---|
| 1966 | rofC[i-2]=list(); |
---|
| 1967 | } |
---|
[7fe9f8b] | 1968 | } |
---|
[0e8a5a] | 1969 | if(size(rofC[1])==0) |
---|
[7fe9f8b] | 1970 | { |
---|
[0e8a5a] | 1971 | omitemptylist=1; |
---|
| 1972 | } |
---|
| 1973 | setring B; |
---|
| 1974 | matrix J_C=fetch(ringofSyz,J_C); |
---|
| 1975 | if (omitemptylist!=1) |
---|
| 1976 | { |
---|
| 1977 | list rofC=fetch(ringofSyz,rofC); |
---|
| 1978 | } |
---|
| 1979 | omitemptylist=0; |
---|
| 1980 | kill HomWeyl; |
---|
| 1981 | kill ringofSyz; |
---|
| 1982 | } |
---|
| 1983 | else |
---|
| 1984 | { |
---|
| 1985 | matrix J_C=C[5];//C[5] is already a V_d-strict Groebner basis |
---|
| 1986 | } |
---|
[7fe9f8b] | 1987 | } |
---|
| 1988 | /* we compute a V_d-strict Groebner basis for C[3]*/ |
---|
| 1989 | matrix J_A=C[1]; |
---|
| 1990 | matrix f_CB=C[4]; |
---|
| 1991 | matrix f_ACB=transpose(concat(transpose(C[2]),transpose(f_CB))); |
---|
| 1992 | matrix J_AC=divdr(f_ACB,C[3]); |
---|
| 1993 | matrix P=matrixLift(J_AC * prodr(ncols(C[1]),ncols(C[5])) ,J_C); |
---|
| 1994 | list storePi; |
---|
| 1995 | matrix Pi[1][ncols(J_AC)]; |
---|
| 1996 | for (i=1; i<=nrows(J_C); i++) |
---|
| 1997 | { |
---|
[0e8a5a] | 1998 | for (j=1; j<=nrows(J_AC);j++) |
---|
| 1999 | { |
---|
| 2000 | Pi=Pi+P[i,j]*submat(J_AC,j,intvec(1..ncols(J_AC))); |
---|
| 2001 | } |
---|
| 2002 | storePi[i]=Pi; |
---|
| 2003 | Pi=0; |
---|
[7fe9f8b] | 2004 | } |
---|
| 2005 | /*we compute the shift vector for C[1]*/ |
---|
| 2006 | intvec m_a; |
---|
| 2007 | list findMin; |
---|
[0e8a5a] | 2008 | int comMin; |
---|
[7fe9f8b] | 2009 | for (i=1; i<=ncols(J_A); i++) |
---|
| 2010 | { |
---|
[0e8a5a] | 2011 | for (j=1; j<=size(storePi);j++) |
---|
| 2012 | { |
---|
| 2013 | if (storePi[j][1,i]!=0) |
---|
| 2014 | { |
---|
| 2015 | comMin=VdDeg(storePi[j]*prodr(ncols(J_A),ncols(C[5])),n,v); |
---|
| 2016 | comMin=comMin-VdDeg(storePi[j][1,i],n,intvec(0)); |
---|
| 2017 | findMin[size(findMin)+1]=comMin; |
---|
| 2018 | } |
---|
| 2019 | } |
---|
| 2020 | if (size(findMin)!=0) |
---|
| 2021 | { |
---|
| 2022 | m_a[i]=Min(findMin); |
---|
| 2023 | findMin=list(); |
---|
| 2024 | } |
---|
| 2025 | else |
---|
| 2026 | { |
---|
| 2027 | m_a[i]=0; |
---|
| 2028 | } |
---|
[7fe9f8b] | 2029 | } |
---|
| 2030 | matrix zero[ncols(J_A)][ncols(J_C)]; |
---|
| 2031 | matrix g_AB=concat(unitmat(ncols(J_A)),zero); |
---|
| 2032 | matrix g_BC= transpose(concat(transpose(zero),transpose(unitmat(ncols(J_C))))); |
---|
| 2033 | intvec m_b=m_a,v; |
---|
[0e8a5a] | 2034 | /* computation of a V_d-strict Groebner basis of C[1] (and resolution if |
---|
| 2035 | Syzstring=='Vdres') */ |
---|
[7fe9f8b] | 2036 | if (Syzstring=="Vdres") |
---|
[0e8a5a] | 2037 | { |
---|
| 2038 | J_A=VdStrictGB(J_A,n,m_a); |
---|
| 2039 | } |
---|
[7fe9f8b] | 2040 | else |
---|
| 2041 | { |
---|
[0e8a5a] | 2042 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, m_a); |
---|
| 2043 | setring HomWeyl; |
---|
| 2044 | matrix J_A=fetch(B,J_A); |
---|
| 2045 | J_A=nHomogenize(J_A); |
---|
| 2046 | def ringofSyz=Sres(transpose(J_A),lengthofres); |
---|
| 2047 | setring ringofSyz; |
---|
| 2048 | matrix J_A=transpose(matrix(RES[2])); |
---|
| 2049 | matrix zerom; |
---|
| 2050 | J_A=subst(J_A,h,1); |
---|
| 2051 | logens=ncols(J_A)+1; |
---|
| 2052 | list rofA; |
---|
| 2053 | for (i=3; i<=n+si+1; i++) |
---|
[7fe9f8b] | 2054 | { |
---|
[0e8a5a] | 2055 | if (size(RES)>=i) |
---|
[7fe9f8b] | 2056 | { |
---|
[0e8a5a] | 2057 | zerom=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i]))); |
---|
| 2058 | if (RES[i]!=zerom) |
---|
| 2059 | { |
---|
| 2060 | rofA[i-2]=matrix(RES[i]);// resolution for C[1] |
---|
| 2061 | if (i==3) |
---|
| 2062 | { |
---|
| 2063 | if (nrows(rofA[i-2])-logens+1!=nrows(J_A)) |
---|
| 2064 | { |
---|
| 2065 | nr=nrows(J_A)+logens-1; |
---|
| 2066 | nc=ncols(rofA[i-2]); |
---|
| 2067 | rofA[i-2]=matrix(rofA[i-2],nr,nc); |
---|
| 2068 | } |
---|
| 2069 | } |
---|
| 2070 | if (i!=3) |
---|
| 2071 | { |
---|
| 2072 | if (nrows(rofA[i-2])-logens+1!=nrows(rofA[i-3])) |
---|
| 2073 | { |
---|
| 2074 | nr=nrows(rofA[i-3])+logens-1; |
---|
| 2075 | nc=ncols(rofA[i-2]); |
---|
| 2076 | rofA[i-2]=matrix(rofA[i-2],nr,nc); |
---|
| 2077 | } |
---|
| 2078 | } |
---|
| 2079 | i1=intvec(logens..nrows(rofA[i-2])); |
---|
| 2080 | i2=intvec(1..ncols(rofA[i-2])); |
---|
| 2081 | rofA[i-2]=transpose(submat(rofA[i-2],i1,i2)); |
---|
| 2082 | logens=logens+ncols(rofA[i-2]); |
---|
| 2083 | rofA[i-2]=subst(rofA[i-2],h,1); |
---|
| 2084 | } |
---|
| 2085 | else |
---|
| 2086 | { |
---|
| 2087 | rofA[i-2]=list(); |
---|
| 2088 | } |
---|
[7fe9f8b] | 2089 | } |
---|
[0e8a5a] | 2090 | else |
---|
[7fe9f8b] | 2091 | { |
---|
[0e8a5a] | 2092 | rofA[i-2]=list(); |
---|
[7fe9f8b] | 2093 | } |
---|
| 2094 | } |
---|
[0e8a5a] | 2095 | if(size(rofA[1])==0) |
---|
[7fe9f8b] | 2096 | { |
---|
[0e8a5a] | 2097 | omitemptylist=1; |
---|
[7fe9f8b] | 2098 | } |
---|
[0e8a5a] | 2099 | setring B; |
---|
| 2100 | J_A=fetch(ringofSyz,J_A); |
---|
| 2101 | if (omitemptylist!=1) |
---|
| 2102 | { |
---|
| 2103 | list rofA=fetch(ringofSyz,rofA); |
---|
| 2104 | } |
---|
| 2105 | omitemptylist=0; |
---|
| 2106 | kill HomWeyl; |
---|
| 2107 | kill ringofSyz; |
---|
[7fe9f8b] | 2108 | } |
---|
| 2109 | J_AC=transpose(storePi[1]); |
---|
| 2110 | for (i=2; i<= size(storePi); i++) |
---|
[0e8a5a] | 2111 | { |
---|
| 2112 | J_AC=concat(J_AC, transpose(storePi[i])); |
---|
| 2113 | } |
---|
[7fe9f8b] | 2114 | J_AC=transpose(concat(transpose(matrix(J_A,nrows(J_A),nrows(J_AC))),J_AC)); |
---|
| 2115 | list Vdstrict=(list(J_A),list(g_AB),list(J_AC),list(g_BC),list(J_C),list(m_a)); |
---|
| 2116 | Vdstrict[7]=list(m_b); |
---|
[0e8a5a] | 2117 | Vdstrict[8]=list(v); |
---|
[7fe9f8b] | 2118 | if(Syzstring=="Sres") |
---|
| 2119 | { |
---|
[0e8a5a] | 2120 | Vdstrict[9]=rofA; |
---|
| 2121 | if(size(#)==0) |
---|
| 2122 | { |
---|
| 2123 | Vdstrict[10]=rofC; |
---|
| 2124 | } |
---|
[7fe9f8b] | 2125 | } |
---|
| 2126 | return (Vdstrict); |
---|
| 2127 | } |
---|
| 2128 | |
---|
| 2129 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 2130 | |
---|
[0e8a5a] | 2131 | static proc toVdStrictSequences (list L,int d,intvec v,string Syzstring,int sizeL) |
---|
[7fe9f8b] | 2132 | { |
---|
[0e8a5a] | 2133 | /* this is Argorithm 3.11 combined with Lemma 4.2 in [W2] for two short exact |
---|
| 2134 | pieces. |
---|
| 2135 | We asume that we are given two sequences of the form coker(L[i][1])-> |
---|
| 2136 | coker(L[i][3])->coker(L[i][5]) with differentials L[i][2] and L[i][4] such |
---|
| 2137 | that L[1][3]=L[2][1].We are going to transform them to V_d-strict sequences |
---|
| 2138 | J_D->J_A->J_F and J_A->J_B->J_C*/ |
---|
[7fe9f8b] | 2139 | int omitemptylist; |
---|
| 2140 | int lengthofres=sizeL+d-1; |
---|
| 2141 | int logens; |
---|
| 2142 | def B=basering; |
---|
| 2143 | matrix J_F=L[1][5]; |
---|
| 2144 | matrix J_D=L[1][1]; |
---|
| 2145 | matrix f_FA=L[1][4]; |
---|
[0e8a5a] | 2146 | /*We find new presentations coker(J_DF) and coker(J_DFC) for L[1][4]=L[2][1] |
---|
| 2147 | and L[2][4],resp. such that ncols(L[i][1])+ncols(L[i][5])=ncols(L[i][3]) */ |
---|
[7fe9f8b] | 2148 | matrix f_DFA=transpose(concat(transpose(L[1][2]),transpose(f_FA))); |
---|
| 2149 | matrix J_DF=divdr(f_DFA,L[1][3]);//coker(J_DF) is isomorphic to coker(L[2][1]); |
---|
| 2150 | matrix J_C=L[2][5]; |
---|
| 2151 | matrix f_CB=L[2][4]; |
---|
| 2152 | matrix f_DFCB=transpose(concat(transpose(f_DFA*L[2][2]),transpose(f_CB))); |
---|
| 2153 | matrix J_DFC=divdr(f_DFCB,L[2][3]);//coker(J_DFC) are coker(L[2][3)]) isomorphic |
---|
[0e8a5a] | 2154 | /* find a shift vector on the range of J_F such that the first sequence is |
---|
| 2155 | exact*/ |
---|
[7fe9f8b] | 2156 | matrix P=matrixLift(J_DFC*prodr(ncols(J_DF),ncols(L[2][5])),J_C); |
---|
| 2157 | list storePi; |
---|
| 2158 | matrix Pi[1][ncols(J_DFC)]; |
---|
| 2159 | int i; int j; |
---|
| 2160 | for (i=1; i<=nrows(J_C); i++) |
---|
| 2161 | { |
---|
[0e8a5a] | 2162 | for (j=1; j<=nrows(J_DFC);j++) |
---|
| 2163 | { |
---|
| 2164 | Pi=Pi+P[i,j]*submat(J_DFC,j,intvec(1..ncols(J_DFC))); |
---|
| 2165 | } |
---|
| 2166 | storePi[i]=Pi; |
---|
| 2167 | Pi=0; |
---|
[7fe9f8b] | 2168 | } |
---|
| 2169 | intvec m_a; |
---|
| 2170 | list findMin; |
---|
| 2171 | list noMin; |
---|
| 2172 | int comMin; |
---|
| 2173 | int nr,nc; |
---|
| 2174 | intvec i1,i2; |
---|
| 2175 | for (i=1; i<=ncols(J_DF); i++) |
---|
| 2176 | { |
---|
[0e8a5a] | 2177 | for (j=1; j<=size(storePi);j++) |
---|
| 2178 | { |
---|
| 2179 | if (storePi[j][1,i]!=0) |
---|
| 2180 | { |
---|
| 2181 | comMin=VdDeg(storePi[j]*prodr(ncols(J_DF),ncols(J_C)),d,v); |
---|
| 2182 | comMin=comMin-VdDeg(storePi[j][1,i],d,intvec(0)); |
---|
| 2183 | findMin[size(findMin)+1]=comMin; |
---|
| 2184 | } |
---|
| 2185 | } |
---|
| 2186 | if (size(findMin)!=0) |
---|
| 2187 | { |
---|
| 2188 | m_a[i]=Min(findMin);// shift vector for L[2][1] |
---|
| 2189 | findMin=list(); |
---|
| 2190 | noMin[i]=0; |
---|
| 2191 | } |
---|
| 2192 | else |
---|
| 2193 | { |
---|
| 2194 | noMin[i]=1; |
---|
| 2195 | } |
---|
[7fe9f8b] | 2196 | } |
---|
[0e8a5a] | 2197 | if (size(m_a) < ncols(J_DF)) |
---|
[7fe9f8b] | 2198 | { |
---|
[0e8a5a] | 2199 | m_a[ncols(J_DF)]=0; |
---|
[7fe9f8b] | 2200 | } |
---|
| 2201 | intvec m_f=m_a[ncols(J_D)+1..size(m_a)]; |
---|
[0e8a5a] | 2202 | /* Computation of a V_d-strict Groebner basis of J_F=L[1][5]: |
---|
| 2203 | if Syzstring=="Vdres" this is done using the method of weighted homogenization |
---|
| 2204 | (Prop. 3.9 [OT]) |
---|
| 2205 | else we use the homogenized Weyl algerba for Groebner basis computations |
---|
| 2206 | (Prop 9.9 [OT]), in this case we already compute resolutions |
---|
| 2207 | (Thm. 9.10 in [OT]) to omit extra Groebner basis computations later on*/ |
---|
[7fe9f8b] | 2208 | if (Syzstring=="Vdres") |
---|
[0e8a5a] | 2209 | { |
---|
| 2210 | J_F=VdStrictGB(J_F,d,m_f); |
---|
| 2211 | } |
---|
[7fe9f8b] | 2212 | else |
---|
| 2213 | { |
---|
[0e8a5a] | 2214 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, m_f); |
---|
| 2215 | setring HomWeyl; |
---|
| 2216 | matrix J_F=fetch(B,J_F); |
---|
| 2217 | J_F=nHomogenize(J_F); |
---|
| 2218 | def ringofSyz=Sres(transpose(J_F),lengthofres); |
---|
| 2219 | setring ringofSyz; |
---|
| 2220 | matrix J_F=transpose(matrix(RES[2])); |
---|
| 2221 | J_F=subst(J_F,h,1); |
---|
| 2222 | logens=ncols(J_F)+1; |
---|
| 2223 | list rofF; |
---|
| 2224 | for (i=3; i<=d+sizeL+1; i++) |
---|
[7fe9f8b] | 2225 | { |
---|
[0e8a5a] | 2226 | if (size(RES)>=i) |
---|
[7fe9f8b] | 2227 | { |
---|
[0e8a5a] | 2228 | if (RES[i]!=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i])))) |
---|
| 2229 | { |
---|
| 2230 | rofF[i-2]=(matrix(RES[i]));// resolution for J_F |
---|
| 2231 | if (i==3) |
---|
| 2232 | { |
---|
| 2233 | if (nrows(rofF[i-2])-logens+1!=nrows(J_F)) |
---|
| 2234 | { |
---|
| 2235 | nr=nrows(J_F)+logens-1; |
---|
| 2236 | nc=ncols(rofF[i-2]); |
---|
| 2237 | rofF[i-2]=matrix(rofF[i-2],nr,nc); |
---|
| 2238 | } |
---|
| 2239 | } |
---|
| 2240 | if (i!=3) |
---|
| 2241 | { |
---|
| 2242 | if (nrows(rofF[i-2])-logens+1!=nrows(rofF[i-3])) |
---|
| 2243 | { |
---|
| 2244 | nr=nrows(rofF[i-3])+logens-1; |
---|
| 2245 | rofF[i-2]=matrix(rofF[i-2],nr,ncols(rofF[i-2])); |
---|
| 2246 | } |
---|
| 2247 | } |
---|
| 2248 | i1=intvec(logens..nrows(rofF[i-2])); |
---|
| 2249 | i2=intvec(1..ncols(rofF[i-2])); |
---|
| 2250 | rofF[i-2]=transpose(submat(rofF[i-2],i1,i2)); |
---|
| 2251 | logens=logens+ncols(rofF[i-2]); |
---|
| 2252 | rofF[i-2]=subst(rofF[i-2],h,1); |
---|
| 2253 | } |
---|
| 2254 | else |
---|
| 2255 | { |
---|
| 2256 | rofF[i-2]=list(); |
---|
| 2257 | } |
---|
[7fe9f8b] | 2258 | } |
---|
[0e8a5a] | 2259 | else |
---|
[7fe9f8b] | 2260 | { |
---|
[0e8a5a] | 2261 | rofF[i-2]=list(); |
---|
[7fe9f8b] | 2262 | } |
---|
| 2263 | } |
---|
[0e8a5a] | 2264 | if(size(rofF[1])==0) |
---|
[7fe9f8b] | 2265 | { |
---|
[0e8a5a] | 2266 | omitemptylist=1; |
---|
[7fe9f8b] | 2267 | } |
---|
[0e8a5a] | 2268 | setring B; |
---|
| 2269 | J_F=fetch(ringofSyz,J_F); |
---|
| 2270 | if (omitemptylist!=1) |
---|
| 2271 | { |
---|
| 2272 | list rofF=fetch(ringofSyz,rofF); |
---|
| 2273 | } |
---|
| 2274 | omitemptylist=0; |
---|
| 2275 | kill HomWeyl; |
---|
| 2276 | kill ringofSyz; |
---|
[7fe9f8b] | 2277 | } |
---|
| 2278 | /*find shift vectors on the range of J_D*/ |
---|
| 2279 | P=matrixLift(J_DF * prodr(ncols(L[1][1]),ncols(L[1][5])) ,J_F); |
---|
| 2280 | list storePinew; |
---|
| 2281 | matrix Pidf[1][ncols(J_DF)]; |
---|
| 2282 | for (i=1; i<=nrows(J_F); i++) |
---|
| 2283 | { |
---|
[0e8a5a] | 2284 | for (j=1; j<=nrows(J_DF);j++) |
---|
| 2285 | { |
---|
| 2286 | Pidf=Pidf+P[i,j]*submat(J_DF,j,intvec(1..ncols(J_DF))); |
---|
| 2287 | } |
---|
| 2288 | storePinew[i]=Pidf; |
---|
| 2289 | Pidf=0; |
---|
[7fe9f8b] | 2290 | } |
---|
| 2291 | intvec m_d; |
---|
| 2292 | for (i=1; i<=ncols(J_D); i++) |
---|
[0e8a5a] | 2293 | { |
---|
| 2294 | for (j=1; j<=size(storePinew);j++) |
---|
| 2295 | { |
---|
| 2296 | if (storePinew[j][1,i]!=0) |
---|
| 2297 | { |
---|
| 2298 | comMin=VdDeg(storePinew[j]*prodr(ncols(J_D),ncols(L[1][5])),d,m_f); |
---|
| 2299 | comMin=comMin-VdDeg(storePinew[j][1,i],d,intvec(0)); |
---|
| 2300 | findMin[size(findMin)+1]=comMin; |
---|
| 2301 | } |
---|
| 2302 | } |
---|
| 2303 | if (size(findMin)!=0) |
---|
| 2304 | { |
---|
| 2305 | if (noMin[i]==0) |
---|
| 2306 | { |
---|
| 2307 | m_d[i]=Min(insert(findMin,m_a[i])); |
---|
| 2308 | m_a[i]=m_d[i]; |
---|
| 2309 | } |
---|
| 2310 | else |
---|
| 2311 | { |
---|
| 2312 | m_d[i]=Min(findMin); |
---|
| 2313 | m_a[i]=m_d[i]; |
---|
| 2314 | } |
---|
| 2315 | } |
---|
[7fe9f8b] | 2316 | else |
---|
[0e8a5a] | 2317 | { |
---|
| 2318 | m_d[i]=m_a[i]; |
---|
| 2319 | } |
---|
| 2320 | findMin=list(); |
---|
[7fe9f8b] | 2321 | } |
---|
[0e8a5a] | 2322 | /* compute a V_d-strict Groebner basis (and resolution of J_D if |
---|
| 2323 | Syzstring!='Vdres') for J_D*/ |
---|
| 2324 | if (Syzstring=="Vdres") |
---|
[7fe9f8b] | 2325 | { |
---|
[0e8a5a] | 2326 | J_D=VdStrictGB(J_D,d,m_d); |
---|
[7fe9f8b] | 2327 | } |
---|
| 2328 | else |
---|
| 2329 | { |
---|
[0e8a5a] | 2330 | def HomWeyl=makeHomogenizedWeyl(nvars(B) div 2, m_d); |
---|
| 2331 | setring HomWeyl; |
---|
| 2332 | matrix J_D=fetch(B,J_D); |
---|
| 2333 | J_D=nHomogenize(J_D); |
---|
| 2334 | def ringofSyz=Sres(transpose(J_D),lengthofres); |
---|
| 2335 | setring ringofSyz; |
---|
| 2336 | matrix J_D=transpose(matrix(RES[2])); |
---|
| 2337 | J_D=subst(J_D,h,1); |
---|
| 2338 | logens=ncols(J_D)+1; |
---|
| 2339 | list rofD; |
---|
| 2340 | for (i=3; i<=d+sizeL+1; i++) |
---|
[7fe9f8b] | 2341 | { |
---|
[0e8a5a] | 2342 | if (size(RES)>=i) |
---|
[7fe9f8b] | 2343 | { |
---|
[0e8a5a] | 2344 | if (RES[i]!=matrix(0,nrows(matrix(RES[i])),ncols(matrix(RES[i])))) |
---|
| 2345 | { |
---|
| 2346 | rofD[i-2]=(matrix(RES[i]));// resolution for J_D |
---|
| 2347 | if (i==3) |
---|
| 2348 | { |
---|
| 2349 | if (nrows(rofD[i-2])-logens+1!=nrows(J_D)) |
---|
| 2350 | { |
---|
| 2351 | nr=nrows(J_D)+logens-1; |
---|
| 2352 | rofD[i-2]=matrix(rofD[i-2],nr,ncols(rofD[i-2])); |
---|
| 2353 | } |
---|
| 2354 | } |
---|
| 2355 | if (i!=3) |
---|
| 2356 | { |
---|
| 2357 | if (nrows(rofD[i-2])-logens+1!=nrows(rofD[i-3])) |
---|
| 2358 | { |
---|
| 2359 | nr=nrows(rofD[i-3])+logens-1; |
---|
| 2360 | rofD[i-2]=matrix(rofD[i-2],nr,ncols(rofD[i-2])); |
---|
| 2361 | } |
---|
| 2362 | } |
---|
| 2363 | i1=intvec(logens..nrows(rofD[i-2])); |
---|
| 2364 | i2=intvec(1..ncols(rofD[i-2])); |
---|
| 2365 | rofD[i-2]=transpose(submat(rofD[i-2],i1,i2)); |
---|
| 2366 | logens=logens+ncols(rofD[i-2]); |
---|
| 2367 | rofD[i-2]=subst(rofD[i-2],h,1); |
---|
| 2368 | } |
---|
| 2369 | else |
---|
| 2370 | { |
---|
| 2371 | rofD[i-2]=list(); |
---|
| 2372 | } |
---|
[7fe9f8b] | 2373 | } |
---|
[0e8a5a] | 2374 | else |
---|
[7fe9f8b] | 2375 | { |
---|
[0e8a5a] | 2376 | rofD[i-2]=list(); |
---|
[7fe9f8b] | 2377 | } |
---|
| 2378 | } |
---|
[0e8a5a] | 2379 | if(size(rofD[1])==0) |
---|
[7fe9f8b] | 2380 | { |
---|
[0e8a5a] | 2381 | omitemptylist=1; |
---|
[7fe9f8b] | 2382 | } |
---|
[0e8a5a] | 2383 | setring B; |
---|
| 2384 | J_D=fetch(ringofSyz,J_D); |
---|
| 2385 | if (omitemptylist!=1) |
---|
| 2386 | { |
---|
| 2387 | list rofD=fetch(ringofSyz,rofD); |
---|
| 2388 | } |
---|
| 2389 | omitemptylist=0; |
---|
| 2390 | kill HomWeyl; |
---|
| 2391 | kill ringofSyz; |
---|
[7fe9f8b] | 2392 | } |
---|
[0e8a5a] | 2393 | /* compute new matrices for J_A and J_B such that their rows form a V_d-strict |
---|
| 2394 | Groebner basis and nrows(J_A)=nrows(J_D)+nrows(J_F) and |
---|
| 2395 | nrows(J_B)=nrows(J_A)+nrows(J_C)*/ |
---|
[7fe9f8b] | 2396 | J_DF=transpose(storePinew[1]); |
---|
| 2397 | for (i=2; i<=nrows(J_F); i++) |
---|
[0e8a5a] | 2398 | { |
---|
| 2399 | J_DF=concat(J_DF,transpose(storePinew[i])); |
---|
| 2400 | } |
---|
[7fe9f8b] | 2401 | J_DF=transpose(concat(transpose(matrix(J_D,nrows(J_D),nrows(J_DF))),J_DF)); |
---|
| 2402 | J_DFC=transpose(storePi[1]); |
---|
| 2403 | for (i=2; i<=nrows(J_C); i++) |
---|
[0e8a5a] | 2404 | { |
---|
| 2405 | J_DFC=concat(J_DFC,transpose(storePi[i])); |
---|
| 2406 | } |
---|
[7fe9f8b] | 2407 | J_DFC=transpose(concat(transpose(matrix(J_DF,nrows(J_DF),nrows(J_DFC))),J_DFC)); |
---|
| 2408 | intvec m_b=m_a,v; |
---|
| 2409 | matrix zero[ncols(J_D)][ncols(J_F)]; |
---|
| 2410 | matrix g_DA=concat(unitmat(ncols(J_D)),zero); |
---|
| 2411 | matrix g_AF=transpose(concat(transpose(zero),unitmat(ncols(J_F)))); |
---|
| 2412 | matrix zero1[ncols(J_DF)][ncols(J_C)]; |
---|
| 2413 | matrix g_AB=concat(unitmat(ncols(J_DF)),zero1); |
---|
| 2414 | matrix g_BC=transpose(concat(transpose(zero1),unitmat(ncols(J_C)))); |
---|
[0e8a5a] | 2415 | list out; |
---|
[7fe9f8b] | 2416 | out[1]=list(list(J_D),list(g_DA),list(J_DF),list(g_AF),list(J_F)); |
---|
| 2417 | out[1]=out[1]+list(list(m_d),list(m_a),list(m_f)); |
---|
| 2418 | out[2]=list(list(J_DF),list(g_AB),list(J_DFC),list(g_BC),list(J_C)); |
---|
| 2419 | out[2]=out[2]+list(list(m_a),list(m_b),list(v)); |
---|
| 2420 | if (Syzstring=="Sres") |
---|
[0e8a5a] | 2421 | { |
---|
| 2422 | out[3]=rofD; |
---|
| 2423 | out[4]=rofF; |
---|
| 2424 | } |
---|
[7fe9f8b] | 2425 | return(out); |
---|
| 2426 | } |
---|
| 2427 | |
---|
| 2428 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 2429 | |
---|
| 2430 | static proc VdStrictDoubleComplexes(list L,int d,string Syzstring) |
---|
| 2431 | { |
---|
[0e8a5a] | 2432 | /* We compute V_d-strict resolutions over the V_d-strict short exact pieces from |
---|
| 2433 | the procedure shortExactPiecesToVdStrict. |
---|
| 2434 | We use Algorithms 3.14 and 3.15 in [W2]*/ |
---|
[7fe9f8b] | 2435 | int i,k,c,j,l,totaldeg,comparedegs,SBcom,verk; |
---|
[0e8a5a] | 2436 | intvec fordegs; |
---|
[7fe9f8b] | 2437 | intvec n_b,i1,i2; |
---|
| 2438 | matrix rem,forML,subm,zerom,unitm,subm2; |
---|
| 2439 | matrix J_B; |
---|
| 2440 | list store; |
---|
| 2441 | int t=size(L)+d; |
---|
| 2442 | int vd1,vd2,nr,nc; |
---|
| 2443 | def B=basering; |
---|
| 2444 | int n=nvars(B) div 2; |
---|
| 2445 | intvec v; |
---|
| 2446 | list forhW; |
---|
| 2447 | if (Syzstring=="Sres") |
---|
| 2448 | { |
---|
[0e8a5a] | 2449 | /*we already computed some of the resolutions in the procedure |
---|
| 2450 | shortExactPiecesToVdStrict*/ |
---|
| 2451 | matrix Pold,Pnew,Picombined; intvec containsndeg; matrix Pinew; |
---|
| 2452 | for (k=1; k<=(size(L)+d-1); k++) |
---|
| 2453 | { |
---|
| 2454 | L[1][1][1][k+1]=list(); |
---|
| 2455 | L[1][1][2][k+1]=list(); |
---|
| 2456 | L[1][1][6][k+1]=list(); |
---|
| 2457 | } |
---|
| 2458 | L[1][1][6][size(L)+d+1]=list(); |
---|
| 2459 | matrix mem; |
---|
| 2460 | for (i=2; i<=d+size(L)+1; i++) |
---|
| 2461 | {; |
---|
| 2462 | v=0; |
---|
| 2463 | if(size(L[1][1][3][i-1])!=0) |
---|
| 2464 | { |
---|
| 2465 | if(i!=d+size(L)+1) |
---|
| 2466 | { |
---|
| 2467 | /*horizontal differential*/ |
---|
| 2468 | L[1][1][4][i-1]=unitmat(nrows(L[1][1][3][i-1])); |
---|
| 2469 | } |
---|
| 2470 | for (j=1; j<=nrows(L[1][1][3][i-1]); j++) |
---|
| 2471 | { |
---|
| 2472 | mem=submat(L[1][1][3][i-1],j,intvec(1..ncols(L[1][1][3][i-1]))); |
---|
| 2473 | v[j]=VdDeg(mem,d,L[1][1][7][i-1]); |
---|
| 2474 | } |
---|
| 2475 | L[1][1][7][i]=v;//new shift vector |
---|
| 2476 | L[1][1][8][i]=v; |
---|
| 2477 | L[1][2][6][i]=v; |
---|
| 2478 | } |
---|
| 2479 | else |
---|
| 2480 | { |
---|
| 2481 | if (i!=d+size(L)+1) |
---|
| 2482 | { |
---|
| 2483 | L[1][1][4][i-1]=list(); |
---|
| 2484 | } |
---|
| 2485 | L[1][1][7][i]=list(); |
---|
| 2486 | L[1][1][8][i]=list(); |
---|
| 2487 | L[1][2][6][i]=list(); |
---|
| 2488 | } |
---|
| 2489 | } |
---|
| 2490 | if (size(L[1][1][3][d+size(L)])!=0) |
---|
| 2491 | { |
---|
| 2492 | /*horizontal differential*/ |
---|
| 2493 | L[1][1][4][d+size(L)]=unitmat(nrows(L[1][1][3][d+size(L)])); |
---|
| 2494 | } |
---|
| 2495 | else |
---|
| 2496 | { |
---|
| 2497 | L[1][1][4][d+size(L)]=list(); |
---|
| 2498 | } |
---|
| 2499 | for (k=1; k<size(L); k++) |
---|
| 2500 | { |
---|
| 2501 | /* We build a V_d-strict resolution for coker(L[k][2][1][1])-> |
---|
| 2502 | coker(L[k][2][3][1])->coker(L[k][2][5][1]) using the resolution |
---|
| 2503 | obtained for coker(L[k][1][3][1]). |
---|
| 2504 | L[k][2][i][j] will be the jth module in the resolution of L[k][2][i][1] |
---|
| 2505 | for i=1,3,5. |
---|
| 2506 | L[k][2][i+5][j] will be the jth shift vector in the resolution of |
---|
| 2507 | L[k][2][i][1](this holds also for the case Syzstring=="Vdres")*/ |
---|
| 2508 | for (i=2; i<=d+size(L); i++) |
---|
| 2509 | { |
---|
| 2510 | v=0; |
---|
| 2511 | if (size(L[k][2][5][i-1])!=0) |
---|
| 2512 | { |
---|
| 2513 | for (j=1; j<=nrows(L[k][2][5][i-1]); j++) |
---|
| 2514 | { |
---|
| 2515 | i1=intvec(1..ncols(L[k][2][5][i-1])); |
---|
| 2516 | mem=submat(L[k][2][5][i-1],j,i1); |
---|
| 2517 | v[j]=VdDeg(mem,d,L[k][2][8][i-1]); |
---|
| 2518 | } |
---|
| 2519 | /*next shift vector in th resolution of coker(L[k][2][5][1])*/ |
---|
| 2520 | L[k][2][8][i]=v; |
---|
| 2521 | } |
---|
[7fe9f8b] | 2522 | else |
---|
| 2523 | { |
---|
[0e8a5a] | 2524 | L[k][2][8][i]=list(); |
---|
[7fe9f8b] | 2525 | } |
---|
[0e8a5a] | 2526 | /* we build step by step a resolution for coker(L[k][2][5][1]) using |
---|
| 2527 | the resolutions of coker(L[k][2][1][1]) and coker(L[k][2][5][1])*/ |
---|
| 2528 | if (size(L[k][2][5][i])!=0) |
---|
[7fe9f8b] | 2529 | { |
---|
[0e8a5a] | 2530 | if (size(L[k][2][1][i])!=0 or size(L[k][2][1][i-1])!=0) |
---|
| 2531 | { |
---|
| 2532 | L[k][2][3][i]=transpose(syz(transpose(L[k][2][3][i-1]))); |
---|
| 2533 | nr= nrows(L[k][2][1][i-1]); |
---|
| 2534 | nc=ncols(L[k][2][5][i]); |
---|
| 2535 | Pold=matrixLift(L[k][2][3][i]*prodr(nr,nc), L[k][2][5][i]); |
---|
| 2536 | matrix Pi[1][ncols(L[k][2][3][i])]; |
---|
| 2537 | for (l=1; l<=nrows(L[k][2][5][i]); l++) |
---|
| 2538 | { |
---|
| 2539 | for (j=1; j<=nrows(L[k][2][3][i]); j++) |
---|
| 2540 | { |
---|
| 2541 | i2=intvec(1..ncols(L[k][2][3][i])); |
---|
| 2542 | Pi=Pi+Pold[l,j]*submat(L[k][2][3][i],j,i2); |
---|
| 2543 | } |
---|
| 2544 | if (l==1) |
---|
| 2545 | { |
---|
| 2546 | Picombined=transpose(Pi); |
---|
| 2547 | } |
---|
| 2548 | else |
---|
| 2549 | { |
---|
| 2550 | Picombined=concat(Picombined,transpose(Pi)); |
---|
| 2551 | } |
---|
| 2552 | Pi=0; |
---|
| 2553 | } |
---|
| 2554 | kill Pi; |
---|
| 2555 | Picombined=transpose(Picombined); |
---|
| 2556 | if (size(L[k][2][1][i])!=0) |
---|
| 2557 | { |
---|
| 2558 | if (i==2) |
---|
| 2559 | { |
---|
| 2560 | containsndeg=(0:ncols(L[k][2][1][1])); |
---|
| 2561 | } |
---|
| 2562 | containsndeg=nDeg(L[k][2][1][i-1],containsndeg); |
---|
| 2563 | forhW=list(L[k][2][6][i],containsndeg); |
---|
| 2564 | def HomWeyl=makeHomogenizedWeyl(n,forhW); |
---|
| 2565 | setring HomWeyl; |
---|
| 2566 | list L=fetch(B,L); |
---|
| 2567 | matrix M=L[k][2][1][i]; |
---|
| 2568 | module Mmod; |
---|
| 2569 | list forM=nHomogenize(M,containsndeg,1); |
---|
| 2570 | M=forM[1]; |
---|
| 2571 | totaldeg=forM[2]; |
---|
| 2572 | kill forM; |
---|
| 2573 | matrix Maorig=fetch(B,Picombined); |
---|
| 2574 | matrix Ma=submat(Maorig,(1..nrows(Maorig)),(1..ncols(M))); |
---|
| 2575 | matrix mem,subm,zerom; |
---|
| 2576 | matrix Pinew; |
---|
| 2577 | M=transpose(M); |
---|
| 2578 | SBcom=0; |
---|
| 2579 | for (l=1; l<=nrows(Ma); l++) |
---|
| 2580 | { |
---|
| 2581 | zerom=matrix(0,1,(ncols(Maorig)-ncols(Ma))); |
---|
| 2582 | i1=(ncols(Ma)+1..ncols(Maorig)); |
---|
| 2583 | if (submat(Maorig,l,i1)==zerom) |
---|
| 2584 | { |
---|
| 2585 | for (cc=1; cc<=ncols(Ma); cc++) |
---|
| 2586 | { |
---|
| 2587 | Maorig[l,cc]=0; |
---|
| 2588 | } |
---|
| 2589 | } |
---|
| 2590 | i2=(ncols(Ma)+1..ncols(Maorig)); |
---|
| 2591 | i1=(1..ncols(Ma)); |
---|
| 2592 | if (VdDeg(submat(Maorig,l,i1),d,L[k][2][6][i])> |
---|
| 2593 | VdDeg(submat(Maorig,l,i2),d,L[k][2][8][i]) and |
---|
| 2594 | submat(Maorig,l,i1)!=matrix(0,1,ncols(Ma))) |
---|
| 2595 | { |
---|
| 2596 | /*V_d-Grad is to big--> we make it smaller using |
---|
| 2597 | Vdnormal form computations*/ |
---|
| 2598 | if (SBcom==0) |
---|
| 2599 | { |
---|
| 2600 | Mmod=slimgb(M); |
---|
| 2601 | M=Mmod; |
---|
| 2602 | SBcom=1; |
---|
| 2603 | } |
---|
| 2604 | //print("Reduzierung des V_d-Grades(Stelle1)"); |
---|
| 2605 | i2=(ncols(Ma)+1..ncols(Maorig)); |
---|
| 2606 | vd1=VdDeg(submat(Maorig,l,i2),d,L[k][2][8][i]); |
---|
| 2607 | mem=submat(Ma,l,(1..ncols(Ma))); |
---|
| 2608 | mem=nHomogenize(mem,containsndeg); |
---|
| 2609 | mem=h^totaldeg*mem; |
---|
| 2610 | mem=transpose(mem); |
---|
| 2611 | mem=reduce(mem,Mod);////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
| 2612 | matrix jt=transpose(subst(mem,h,1)); |
---|
| 2613 | setring B; |
---|
| 2614 | matrix jt=fetch(HomWeyl,jt); |
---|
| 2615 | matrix need=fetch(HomWeyl,Maorig); |
---|
| 2616 | need=submat(need,l,(1..ncols(need))); |
---|
| 2617 | i1=L[k][2][6][i]; |
---|
| 2618 | i2=L[k][2][8][i]; |
---|
| 2619 | jt=VdNormalForm(need,L[k][2][1][i],d,i1,i2); |
---|
| 2620 | setring HomWeyl; |
---|
| 2621 | mem=fetch(B,jt); |
---|
| 2622 | mem=transpose(mem); |
---|
| 2623 | if (l==1) |
---|
| 2624 | { |
---|
| 2625 | Pinew=mem; |
---|
| 2626 | } |
---|
| 2627 | else |
---|
| 2628 | { |
---|
| 2629 | Pinew=concat(Pinew,mem); |
---|
| 2630 | } |
---|
| 2631 | vd2=VdDeg(transpose(mem),d,L[k][2][6][i]); |
---|
| 2632 | if (vd2>vd1 and mem!=matrix(0,nrows(mem),ncols(mem))) |
---|
| 2633 | {//should not happen!! |
---|
| 2634 | //print("Reduzierung fehlgeschlagen!!(Stelle1)"); |
---|
| 2635 | } |
---|
| 2636 | } |
---|
| 2637 | else |
---|
| 2638 | { |
---|
| 2639 | if (l==1) |
---|
| 2640 | { |
---|
| 2641 | Pinew=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
| 2642 | } |
---|
| 2643 | else |
---|
| 2644 | { |
---|
| 2645 | subm=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
| 2646 | Pinew=concat(Pinew,subm); |
---|
| 2647 | } |
---|
| 2648 | } |
---|
| 2649 | } |
---|
| 2650 | Pinew=subst(Pinew,h,1); |
---|
| 2651 | Pinew=transpose(Pinew); |
---|
| 2652 | setring B; |
---|
| 2653 | Pinew=fetch(HomWeyl,Pinew); |
---|
| 2654 | kill HomWeyl; |
---|
| 2655 | L[k][2][3][i]=concat(Pinew,L[k][2][5][i]); |
---|
| 2656 | subm=transpose(L[k][2][3][i]); |
---|
| 2657 | subm=concat(transpose(L[k][2][1][i]),subm); |
---|
| 2658 | L[k][2][3][i]=transpose(subm); |
---|
| 2659 | } |
---|
| 2660 | else |
---|
| 2661 | { |
---|
| 2662 | L[k][2][3][i]=Picombined; |
---|
| 2663 | } |
---|
| 2664 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
| 2665 | nr=nrows(L[k][2][1][i-1]); |
---|
| 2666 | nc=ncols(L[k][2][5][i]); |
---|
| 2667 | L[k][2][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
| 2668 | L[k][2][4][i]=prodr(nrows(L[k][2][1][i-1]),nc); |
---|
| 2669 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
| 2670 | L[k][2][7][i]=v; |
---|
| 2671 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
| 2672 | } |
---|
[7fe9f8b] | 2673 | else |
---|
[0e8a5a] | 2674 | { |
---|
| 2675 | L[k][2][3][i]=L[k][2][5][i]; |
---|
| 2676 | L[k][2][2][i]=list(); |
---|
| 2677 | L[k][2][7][i]=L[k][2][8][i]; |
---|
| 2678 | L[k][2][4][i]=unitmat(nrows(L[k][2][5][i-1])); |
---|
| 2679 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
| 2680 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
| 2681 | } |
---|
[7fe9f8b] | 2682 | } |
---|
[0e8a5a] | 2683 | else |
---|
[7fe9f8b] | 2684 | { |
---|
[0e8a5a] | 2685 | if (size(L[k][2][1][i])!=0) |
---|
| 2686 | { |
---|
| 2687 | if (size(L[k][2][5][i-1])!=0) |
---|
| 2688 | { |
---|
| 2689 | nr=nrows(L[k][2][5][i-1]); |
---|
| 2690 | L[k][2][3][i]=concat(L[k][2][1][i],matrix(0,1,nr)); |
---|
| 2691 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
| 2692 | L[k][2][7][i]=v; |
---|
| 2693 | nc=nrows(L[k][2][1][i-1]); |
---|
| 2694 | L[k][2][2][i]=concat(unitmat(nc),matrix(0,nc,nr)); |
---|
| 2695 | L[k][2][4][i]=prodr(nrows(L[k][2][1][i-1]),nr); |
---|
| 2696 | } |
---|
| 2697 | else |
---|
| 2698 | { |
---|
| 2699 | L[k][2][3][i]=L[k][2][1][i]; |
---|
| 2700 | L[k][2][7][i]=L[k][2][6][i]; |
---|
| 2701 | L[k][2][2][i]=unitmat(nrows(L[k][2][1][i-1])); |
---|
| 2702 | L[k][2][4][i]=list(); |
---|
| 2703 | } |
---|
| 2704 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
| 2705 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
| 2706 | } |
---|
[7fe9f8b] | 2707 | else |
---|
[0e8a5a] | 2708 | { |
---|
| 2709 | L[k][2][3][i]=list(); |
---|
| 2710 | if (size(L[k][2][6][i])!=0) |
---|
| 2711 | { |
---|
| 2712 | if (size(L[k][2][8][i])!=0) |
---|
| 2713 | { |
---|
| 2714 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
| 2715 | L[k][2][7][i]=v; |
---|
| 2716 | nr=nrows(L[k][2][1][i-1]); |
---|
| 2717 | nc=nrows(L[k][2][5][i-1]); |
---|
| 2718 | L[k][2][2][i]=concat(unitmat(nc),matrix(0,nr,nc)); |
---|
| 2719 | L[k][2][4][i]=prodr(nr,nrows(L[k][2][5][i-1])); |
---|
| 2720 | } |
---|
| 2721 | else |
---|
| 2722 | { |
---|
| 2723 | L[k][2][7][i]=L[k][2][6][i]; |
---|
| 2724 | L[k][2][2][i]=unitmat(nrows(L[k][2][1][i-1])); |
---|
| 2725 | L[k][2][4][i]=list(); |
---|
| 2726 | } |
---|
| 2727 | } |
---|
| 2728 | else |
---|
| 2729 | { |
---|
| 2730 | if (size(L[k][2][8][i])!=0) |
---|
| 2731 | { |
---|
| 2732 | L[k][2][7][i]=L[k][2][8][i]; |
---|
| 2733 | L[k][2][2][i]=list(); |
---|
| 2734 | L[k][2][4][i]=unitmat(nrows(L[k][2][5][i-1])); |
---|
| 2735 | } |
---|
| 2736 | else |
---|
| 2737 | { |
---|
| 2738 | L[k][2][7][i]=list(); |
---|
| 2739 | L[k][2][2][i]=list(); |
---|
| 2740 | L[k][2][4][i]=list(); |
---|
| 2741 | } |
---|
| 2742 | } |
---|
| 2743 | L[k+1][1][1][i]=L[k][2][5][i]; |
---|
| 2744 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
| 2745 | } |
---|
[7fe9f8b] | 2746 | } |
---|
| 2747 | } |
---|
[0e8a5a] | 2748 | i=d+size(L)+1; |
---|
| 2749 | v=0; |
---|
| 2750 | if (size(L[k][2][5][i-1])!=0) |
---|
[7fe9f8b] | 2751 | { |
---|
[0e8a5a] | 2752 | for (j=1; j<=nrows(L[k][2][5][i-1]); j++) |
---|
| 2753 | { |
---|
| 2754 | mem=submat(L[k][2][5][i-1],j,intvec(1..ncols(L[k][2][5][i-1]))); |
---|
| 2755 | v[j]=VdDeg(mem,d,L[k][2][8][i-1]); |
---|
| 2756 | } |
---|
| 2757 | L[k][2][8][i]=v; |
---|
| 2758 | if (size(L[k][2][6][i])!=0) |
---|
| 2759 | { |
---|
| 2760 | v=L[k][2][6][i],L[k][2][8][i]; |
---|
| 2761 | L[k][2][7][i]=v; |
---|
| 2762 | } |
---|
[7fe9f8b] | 2763 | else |
---|
[0e8a5a] | 2764 | { |
---|
| 2765 | L[k][2][7][i]=L[k][2][8][i]; |
---|
| 2766 | } |
---|
[7fe9f8b] | 2767 | } |
---|
[0e8a5a] | 2768 | else |
---|
[7fe9f8b] | 2769 | { |
---|
[0e8a5a] | 2770 | L[k][2][8][i]=list(); |
---|
| 2771 | L[k][2][7][i]=L[k][2][6][i]; |
---|
| 2772 | } |
---|
| 2773 | L[k+1][1][6][i]=L[k][2][8][i]; |
---|
| 2774 | /* now we build V_d-strict resolutions for the sequences |
---|
| 2775 | coker(L[k+1][1][1][1])->coker(L[k+1][1][3][1])->coker(L[k+1][1][5][i]) |
---|
| 2776 | using the resolutions for coker(L[k][2][5][1]) we just obtained |
---|
| 2777 | (works exactly the same as above)*/ |
---|
| 2778 | for (i=2; i<=d+size(L); i++) |
---|
| 2779 | { |
---|
| 2780 | v=0; |
---|
| 2781 | if (size(L[k+1][1][5][i-1])!=0) |
---|
| 2782 | { |
---|
| 2783 | for (j=1; j<=nrows(L[k+1][1][5][i-1]); j++) |
---|
| 2784 | { |
---|
| 2785 | i1=intvec(1..ncols(L[k+1][1][5][i-1])); |
---|
| 2786 | mem=submat(L[k+1][1][5][i-1],j,i1); |
---|
| 2787 | v[j]=VdDeg(mem,d,L[k+1][1][8][i-1]); |
---|
| 2788 | } |
---|
| 2789 | L[k+1][1][8][i]=v; |
---|
| 2790 | } |
---|
[7fe9f8b] | 2791 | else |
---|
| 2792 | { |
---|
[0e8a5a] | 2793 | L[k+1][1][8][i]=list(); |
---|
[7fe9f8b] | 2794 | } |
---|
[0e8a5a] | 2795 | if (size(L[k+1][1][5][i])!=0) |
---|
[7fe9f8b] | 2796 | { |
---|
[0e8a5a] | 2797 | if (size(L[k+1][1][1][i])!=0 or size(L[k+1][1][1][i-1])!=0) |
---|
| 2798 | { |
---|
| 2799 | L[k+1][1][3][i]=transpose(syz(transpose(L[k+1][1][3][i-1]))); |
---|
| 2800 | nr=nrows(L[k+1][1][1][i-1]); |
---|
| 2801 | nc=ncols(L[k+1][1][5][i]); |
---|
| 2802 | Pold=matrixLift(L[k+1][1][3][i]*prodr(nr,nc),L[k+1][1][5][i]); |
---|
| 2803 | matrix Pi[1][ncols(L[k+1][1][3][i])]; |
---|
| 2804 | for (l=1; l<=nrows(L[k+1][1][5][i]); l++) |
---|
| 2805 | { |
---|
| 2806 | for (j=1; j<=nrows(L[k+1][1][3][i]); j++) |
---|
| 2807 | { |
---|
| 2808 | i2=intvec(1..ncols(L[k+1][1][3][i])); |
---|
| 2809 | Pi=Pi+Pold[l,j]*submat(L[k+1][1][3][i],j,i2); |
---|
| 2810 | } |
---|
| 2811 | if (l==1) |
---|
| 2812 | { |
---|
| 2813 | Picombined=transpose(Pi); |
---|
| 2814 | } |
---|
| 2815 | else |
---|
| 2816 | { |
---|
| 2817 | Picombined=concat(Picombined,transpose(Pi)); |
---|
| 2818 | } |
---|
| 2819 | Pi=0; |
---|
| 2820 | } |
---|
| 2821 | kill Pi; |
---|
| 2822 | Picombined=transpose(Picombined); |
---|
| 2823 | if(size(L[k+1][1][1][i])!=0) |
---|
| 2824 | { |
---|
| 2825 | if (i==2) |
---|
| 2826 | { |
---|
| 2827 | containsndeg=(0:ncols(L[k+1][1][1][i-1])); |
---|
| 2828 | } |
---|
| 2829 | containsndeg=nDeg(L[k+1][1][1][i-1],containsndeg); |
---|
| 2830 | forhW=list(L[k+1][1][6][i], containsndeg); |
---|
| 2831 | def HomWeyl=makeHomogenizedWeyl(n,forhW); |
---|
| 2832 | setring HomWeyl; |
---|
| 2833 | list L=fetch(B,L); |
---|
| 2834 | matrix M=L[k+1][1][1][i]; |
---|
| 2835 | module Mmod; |
---|
| 2836 | list forM=nHomogenize(M,containsndeg,1); |
---|
| 2837 | M=forM[1]; |
---|
| 2838 | totaldeg=forM[2]; |
---|
| 2839 | kill forM; |
---|
| 2840 | matrix Maorig=fetch(B,Picombined); |
---|
| 2841 | matrix Ma=submat(Maorig,(1..nrows(Maorig)),(1..ncols(M))); |
---|
| 2842 | Ma=nHomogenize(Ma,containsndeg); |
---|
| 2843 | matrix mem,subm,zerom,subm2; |
---|
| 2844 | matrix Pinew; |
---|
| 2845 | M=transpose(M); |
---|
| 2846 | SBcom=0; |
---|
| 2847 | for (l=1; l<=nrows(Ma); l++) |
---|
| 2848 | { |
---|
| 2849 | i2=(ncols(Ma)+1..ncols(Maorig)); |
---|
| 2850 | nc=ncols(Maorig)-ncols(Ma); |
---|
| 2851 | if (submat(Maorig,l,i2)==matrix(0,1,nc)) |
---|
| 2852 | { |
---|
| 2853 | for (cc=1; cc<=ncols(Ma); cc++) |
---|
| 2854 | { |
---|
| 2855 | Maorig[l,cc]=0; |
---|
| 2856 | } |
---|
| 2857 | } |
---|
| 2858 | i1=(1..ncols(Ma)); |
---|
| 2859 | i2=L[k+1][1][8][i]; |
---|
| 2860 | subm=submat(Maorig,l,i1); |
---|
| 2861 | subm2=submat(Maorig,l,(ncols(Ma)+1..ncols(Maorig))); |
---|
| 2862 | if (VdDeg(subm,d,L[k+1][1][6][i])>VdDeg(subm2,d,i2) |
---|
| 2863 | and subm!=matrix(0,1,ncols(Ma))) |
---|
| 2864 | { |
---|
| 2865 | //print("Reduzierung des Vd-Grades (Stelle2)"); |
---|
| 2866 | if (SBcom==0) |
---|
| 2867 | { |
---|
| 2868 | Mmod=slimgb(M); |
---|
| 2869 | M=Mmod; |
---|
| 2870 | SBcom=1; |
---|
| 2871 | } |
---|
| 2872 | vd1=VdDeg(subm2,d,L[k+1][1][8][i]); |
---|
| 2873 | mem=submat(Ma,l,(1..ncols(Ma))); |
---|
| 2874 | mem=nHomogenize(mem,containsndeg); |
---|
| 2875 | mem=h^totaldeg*mem; |
---|
| 2876 | mem=transpose(mem); |
---|
| 2877 | mem=reduce(mem,Mmod); |
---|
| 2878 | if (l==1) |
---|
| 2879 | { |
---|
| 2880 | Pinew=mem; |
---|
| 2881 | } |
---|
| 2882 | else |
---|
| 2883 | { |
---|
| 2884 | Pinew=concat(Pinew,mem); |
---|
| 2885 | } |
---|
| 2886 | vd2=VdDeg(transpose(mem),d,L[k+1][1][6][i]); |
---|
| 2887 | if (vd2>vd1 and mem!=matrix(0,nrows(mem),ncols(mem))) |
---|
| 2888 | {//should not happen |
---|
| 2889 | //print("Reduzierung fehlgeschlagen!!!!(Stelle2)"); |
---|
| 2890 | } |
---|
| 2891 | } |
---|
| 2892 | else |
---|
| 2893 | { |
---|
| 2894 | if (l==1) |
---|
| 2895 | { |
---|
| 2896 | Pinew=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
| 2897 | } |
---|
| 2898 | else |
---|
| 2899 | { |
---|
| 2900 | subm=transpose(submat(Ma,l,(1..ncols(Ma)))); |
---|
| 2901 | Pinew=concat(Pinew,subm); |
---|
| 2902 | } |
---|
| 2903 | } |
---|
| 2904 | } |
---|
| 2905 | Pinew=subst(Pinew,h,1); |
---|
| 2906 | Pinew=transpose(Pinew); |
---|
| 2907 | setring B; |
---|
| 2908 | Pinew=fetch(HomWeyl,Pinew); |
---|
| 2909 | kill HomWeyl; |
---|
| 2910 | L[k+1][1][3][i]=concat(Pinew,L[k+1][1][5][i]); |
---|
| 2911 | subm=transpose(L[k+1][1][1][i]); |
---|
| 2912 | subm2=transpose(L[k+1][1][3][i]); |
---|
| 2913 | L[k+1][1][3][i]=transpose(concat(subm,subm2)); |
---|
| 2914 | } |
---|
| 2915 | else |
---|
| 2916 | { |
---|
| 2917 | L[k+1][1][3][i]=Picombined; |
---|
| 2918 | } |
---|
| 2919 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
| 2920 | nr=nrows(L[k+1][1][1][i-1]); |
---|
| 2921 | nc=ncols(L[k+1][1][5][i]); |
---|
| 2922 | L[k+1][1][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
| 2923 | L[k+1][1][4][i]=prodr(nr,nc); |
---|
| 2924 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
| 2925 | L[k+1][1][7][i]=v; |
---|
| 2926 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
| 2927 | } |
---|
[7fe9f8b] | 2928 | else |
---|
[0e8a5a] | 2929 | { |
---|
| 2930 | L[k+1][1][3][i]=L[k+1][1][5][i]; |
---|
| 2931 | L[k+1][1][2][i]=list(); |
---|
| 2932 | L[k+1][1][4][i]=unitmat(nrows(L[k+1][1][5][i-1])); |
---|
| 2933 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
| 2934 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
| 2935 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
| 2936 | } |
---|
[7fe9f8b] | 2937 | } |
---|
[0e8a5a] | 2938 | else |
---|
[7fe9f8b] | 2939 | { |
---|
[0e8a5a] | 2940 | if (size(L[k+1][1][1][i])!=0) |
---|
| 2941 | { |
---|
| 2942 | if (size(L[k+1][1][5][i-1])!=0) |
---|
| 2943 | { |
---|
| 2944 | zerom=matrix(0,1,nrows(L[k+1][1][5][i-1])); |
---|
| 2945 | L[k+1][1][3][i]=concat(L[k+1][1][1][i],zerom); |
---|
| 2946 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
| 2947 | L[k+1][1][7][i]=v; |
---|
| 2948 | nr=nrows(L[k+1][1][1][i-1]); |
---|
| 2949 | nc=nrows(L[k+1][1][5][i-1]); |
---|
| 2950 | L[k+1][1][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
| 2951 | L[k+1][1][4][i]=prodr(nr,nc); |
---|
| 2952 | } |
---|
| 2953 | else |
---|
| 2954 | { |
---|
| 2955 | L[k+1][1][3][i]=L[k+1][1][1][i]; |
---|
| 2956 | L[k+1][1][7][i]=L[k+1][1][6][i]; |
---|
| 2957 | L[k+1][1][2][i]=unitmat(nrows(L[k+1][1][1][i-1])); |
---|
| 2958 | L[k+1][1][4][i]=list(); |
---|
| 2959 | } |
---|
| 2960 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
| 2961 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
| 2962 | } |
---|
[7fe9f8b] | 2963 | else |
---|
[0e8a5a] | 2964 | { |
---|
| 2965 | L[k+1][1][3][i]=list(); |
---|
| 2966 | if (size(L[k+1][1][6][i])!=0) |
---|
| 2967 | { |
---|
| 2968 | if (size(L[k+1][1][8][i])!=0) |
---|
| 2969 | { |
---|
| 2970 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
| 2971 | L[k+1][1][7][i]=v; |
---|
| 2972 | nr=nrows(L[k+1][1][1][i-1]); |
---|
| 2973 | nc=nrows(L[k+1][1][5][i-1]); |
---|
| 2974 | L[k+1][1][2][i]=concat(unitmat(nr),matrix(0,nr,nc)); |
---|
| 2975 | L[k+1][1][4][i]=prodr(nr,nrows(L[k+1][1][5][i-1])); |
---|
| 2976 | } |
---|
| 2977 | else |
---|
| 2978 | { |
---|
| 2979 | L[k+1][1][7][i]=L[k+1][1][6][i]; |
---|
| 2980 | L[k+1][1][2][i]=unitmat(nrows(L[k+1][1][1][i-1])); |
---|
| 2981 | L[k+1][1][4][i]=list(); |
---|
| 2982 | } |
---|
| 2983 | } |
---|
| 2984 | else |
---|
| 2985 | { |
---|
| 2986 | if (size(L[k+1][1][8][i])!=0) |
---|
| 2987 | { |
---|
| 2988 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
| 2989 | L[k+1][1][2][i]=list(); |
---|
| 2990 | L[k+1][1][4][i]=unitmat(nrows(L[k+1][1][5][i-1])); |
---|
| 2991 | } |
---|
| 2992 | else |
---|
| 2993 | { |
---|
| 2994 | L[k+1][1][7][i]=list(); |
---|
| 2995 | L[k+1][1][2][i]=list(); |
---|
| 2996 | L[k+1][1][4][i]=list(); |
---|
| 2997 | } |
---|
| 2998 | } |
---|
| 2999 | |
---|
| 3000 | L[k+1][2][1][i]=L[k+1][1][3][i]; |
---|
| 3001 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
| 3002 | } |
---|
[7fe9f8b] | 3003 | } |
---|
[0e8a5a] | 3004 | } |
---|
| 3005 | i=size(L)+d+1; |
---|
| 3006 | v=0; |
---|
| 3007 | if (size(L[k+1][1][5][i-1])!=0) |
---|
[7fe9f8b] | 3008 | { |
---|
[0e8a5a] | 3009 | for (j=1; j<=nrows(L[k+1][1][5][i-1]); j++) |
---|
| 3010 | { |
---|
| 3011 | i1=intvec(1..ncols(L[k+1][1][5][i-1])); |
---|
| 3012 | mem=submat(L[k+1][1][5][i-1],j,i1); |
---|
| 3013 | v[j]=VdDeg(mem,d,L[k+1][1][8][i-1]); |
---|
| 3014 | } |
---|
| 3015 | L[k+1][1][8][i]=v; |
---|
| 3016 | if (size(L[k+1][1][6][i])!=0) |
---|
| 3017 | { |
---|
| 3018 | v=L[k+1][1][6][i],L[k+1][1][8][i]; |
---|
| 3019 | L[k+1][1][7][i]=v; |
---|
| 3020 | } |
---|
[7fe9f8b] | 3021 | else |
---|
[0e8a5a] | 3022 | { |
---|
| 3023 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
| 3024 | } |
---|
[7fe9f8b] | 3025 | } |
---|
[0e8a5a] | 3026 | else |
---|
[7fe9f8b] | 3027 | { |
---|
[0e8a5a] | 3028 | L[k+1][1][8][i]=list(); |
---|
| 3029 | L[k+1][1][7][i]=L[k+1][1][8][i]; |
---|
[7fe9f8b] | 3030 | } |
---|
[0e8a5a] | 3031 | L[k+1][2][6][i]=L[k+1][1][7][i]; |
---|
[7fe9f8b] | 3032 | } |
---|
[0e8a5a] | 3033 | for (k=1; k<=(size(L)+d); k++) |
---|
[7fe9f8b] | 3034 | { |
---|
[0e8a5a] | 3035 | L[size(L)][2][5][k]=list(); |
---|
| 3036 | L[size(L)][2][4][k]=list(); |
---|
| 3037 | L[size(L)][2][8][k]=list(); |
---|
| 3038 | L[size(L)][2][3][k]=L[size(L)][2][1][k]; |
---|
| 3039 | L[size(L)][2][7][k]=L[size(L)][2][6][k]; |
---|
[7fe9f8b] | 3040 | } |
---|
[0e8a5a] | 3041 | L[size(L)][2][7][size(L)+d+1]=L[size(L)][2][6][size(L)+d+1]; |
---|
| 3042 | L[size(L)][2][8][size(L)+d+1]=list(); |
---|
| 3043 | /* building the resolution of the last short exact piece*/ |
---|
| 3044 | for (i=2; i<=d+size(L); i++) |
---|
[7fe9f8b] | 3045 | { |
---|
[0e8a5a] | 3046 | v=0; |
---|
| 3047 | if(size(L[size(L)][2][1][i-1])!=0) |
---|
| 3048 | { |
---|
| 3049 | L[size(L)][2][2][i]=unitmat(nrows(L[size(L)][2][1][i-1])); |
---|
| 3050 | } |
---|
| 3051 | else |
---|
| 3052 | { |
---|
| 3053 | L[size(L)][2][2][i-1]=list(); |
---|
| 3054 | } |
---|
[7fe9f8b] | 3055 | } |
---|
[0e8a5a] | 3056 | return(L); |
---|
[7fe9f8b] | 3057 | } |
---|
| 3058 | /*case Syzstring=="Vdres"*/ |
---|
| 3059 | list forVd; |
---|
| 3060 | for (k=1; k<=(size(L)+d); k++)//????? |
---|
| 3061 | { |
---|
[0e8a5a] | 3062 | /* we compute a V_d-strict resolution for the first short exact piece*/ |
---|
| 3063 | L[1][1][1][k+1]=list(); |
---|
| 3064 | L[1][1][2][k+1]=list(); |
---|
| 3065 | L[1][1][6][k+1]=list(); |
---|
| 3066 | if (size(L[1][1][3][k])!=0) |
---|
| 3067 | { |
---|
| 3068 | for (i=1; i<=nrows(L[1][1][3][k]); i++) |
---|
| 3069 | { |
---|
| 3070 | rem=submat(L[1][1][3][k],i,(1..ncols(L[1][1][3][k]))); |
---|
| 3071 | n_b[i]=VdDeg(rem,d,L[1][1][7][k]); |
---|
| 3072 | } |
---|
| 3073 | J_B=transpose(syz(transpose(L[1][1][3][k]))); |
---|
| 3074 | L[1][1][7][k+1]=n_b; |
---|
| 3075 | L[1][1][8][k+1]=n_b; |
---|
| 3076 | L[1][1][4][k+1]=unitmat(nrows(L[1][1][3][k])); |
---|
| 3077 | if (J_B!=matrix(0,nrows(J_B),ncols(J_B))) |
---|
| 3078 | { |
---|
| 3079 | J_B=VdStrictGB(J_B,d,n_b); |
---|
| 3080 | L[1][1][3][k+1]=J_B; |
---|
| 3081 | L[1][1][5][k+1]=J_B; |
---|
| 3082 | } |
---|
| 3083 | else |
---|
| 3084 | { |
---|
| 3085 | L[1][1][3][k+1]=list(); |
---|
| 3086 | L[1][1][5][k+1]=list(); |
---|
| 3087 | } |
---|
| 3088 | n_b=0; |
---|
| 3089 | } |
---|
[7fe9f8b] | 3090 | else |
---|
| 3091 | { |
---|
[0e8a5a] | 3092 | L[1][1][3][k+1]=list(); |
---|
| 3093 | L[1][1][5][k+1]=list(); |
---|
| 3094 | L[1][1][7][k+1]=list(); |
---|
| 3095 | L[1][1][8][k+1]=list(); |
---|
| 3096 | L[1][1][4][k+1]=list(); |
---|
[7fe9f8b] | 3097 | } |
---|
[0e8a5a] | 3098 | /* we compute step by step V_d-strict resolutions over |
---|
| 3099 | coker(L[i][2][1][1])->coker(L[i][2][3][1])->coker(L[i][2][1][5]) |
---|
| 3100 | and coker(L[i+1][1][1][1])->coker(L[i+1][1][3][1])->coker(L[i+1][1][1][5]) |
---|
| 3101 | using the already computed resolutions for coker(L[i][2][1][1])= |
---|
| 3102 | coker(L[i][1][3][1]) and coker(L[i+1][1][1][1])=coker(L[i][2][5][1])*/ |
---|
| 3103 | for (i=1; i<size(L); i++) |
---|
[7fe9f8b] | 3104 | { |
---|
[0e8a5a] | 3105 | forVd[1]=L[i][2][1][k]; |
---|
| 3106 | forVd[2]=L[i][2][2][k]; |
---|
| 3107 | forVd[3]=L[i][2][3][k]; |
---|
| 3108 | forVd[4]=L[i][2][4][k]; |
---|
| 3109 | forVd[5]=L[i][2][5][k]; |
---|
| 3110 | forVd[6]=L[i][2][6][k]; |
---|
| 3111 | forVd[7]=L[i][2][7][k]; |
---|
| 3112 | forVd[8]=L[i][2][8][k]; |
---|
| 3113 | store=toVdStrict2x3Complex(forVd,d,L[i][1][3][k+1],L[i][1][7][k+1]); |
---|
| 3114 | for (j=1; j<=8; j++) |
---|
| 3115 | { |
---|
| 3116 | L[i][2][j][k+1]=store[j]; |
---|
| 3117 | } |
---|
| 3118 | forVd[1]=L[i+1][1][1][k]; |
---|
| 3119 | forVd[2]=L[i+1][1][2][k]; |
---|
| 3120 | forVd[3]=L[i+1][1][3][k]; |
---|
| 3121 | forVd[4]=L[i+1][1][4][k]; |
---|
| 3122 | forVd[5]=L[i+1][1][5][k]; |
---|
| 3123 | forVd[6]=L[i+1][1][6][k]; |
---|
| 3124 | forVd[7]=L[i+1][1][7][k]; |
---|
| 3125 | forVd[8]=L[i+1][1][8][k]; |
---|
| 3126 | store=toVdStrict2x3Complex(forVd,d,L[i][2][5][k+1],L[i][2][8][k+1]); |
---|
| 3127 | for (j=1; j<=8; j++) |
---|
| 3128 | { |
---|
| 3129 | L[i+1][1][j][k+1]=store[j]; |
---|
| 3130 | } |
---|
[7fe9f8b] | 3131 | } |
---|
[0e8a5a] | 3132 | if (size(L[size(L)][1][7][k+1])!=0) |
---|
[7fe9f8b] | 3133 | { |
---|
[0e8a5a] | 3134 | L[size(L)][2][4][k+1]=list(); |
---|
| 3135 | L[size(L)][2][5][k+1]=list(); |
---|
| 3136 | L[size(L)][2][6][k+1]=L[size(L)][1][7][k+1]; |
---|
| 3137 | L[size(L)][2][7][k+1]=L[size(L)][1][7][k+1]; |
---|
| 3138 | L[size(L)][2][8][k+1]=list(); |
---|
| 3139 | L[size(L)][2][2][k+1]=unitmat(size(L[size(L)][1][7][k+1])); |
---|
| 3140 | if (size(L[size(L)][1][3][k+1])!=0) |
---|
| 3141 | { |
---|
| 3142 | L[size(L)][2][1][k+1]=L[size(L)][1][3][k+1]; |
---|
| 3143 | L[size(L)][2][3][k+1]=L[size(L)][1][3][k+1]; |
---|
| 3144 | } |
---|
| 3145 | else |
---|
| 3146 | { |
---|
| 3147 | L[size(L)][2][1][k+1]=list(); |
---|
| 3148 | L[size(L)][2][3][k+1]=list(); |
---|
| 3149 | } |
---|
[7fe9f8b] | 3150 | } |
---|
| 3151 | else |
---|
| 3152 | { |
---|
[0e8a5a] | 3153 | for (j=1; j<=8; j++) |
---|
| 3154 | { |
---|
| 3155 | L[size(L)][2][j][k+1]=list(); |
---|
| 3156 | } |
---|
[7fe9f8b] | 3157 | } |
---|
[0e8a5a] | 3158 | } |
---|
| 3159 | k=t; |
---|
| 3160 | intvec n_c; |
---|
| 3161 | intvec vn_b; |
---|
| 3162 | list N_b; |
---|
| 3163 | int n; |
---|
| 3164 | /*computation of the shift vectors*/ |
---|
| 3165 | for (i=1; i<=size(L); i++) |
---|
| 3166 | { |
---|
| 3167 | for (n=1; n<=2; n++) |
---|
[7fe9f8b] | 3168 | { |
---|
[0e8a5a] | 3169 | if (i==1 and n==1) |
---|
| 3170 | { |
---|
| 3171 | L[i][n][6][k+1]=list(); |
---|
| 3172 | } |
---|
| 3173 | else |
---|
| 3174 | { |
---|
| 3175 | if (n==1) |
---|
| 3176 | { |
---|
| 3177 | L[i][1][6][k+1]=L[i-1][2][8][k+1]; |
---|
| 3178 | } |
---|
| 3179 | else |
---|
| 3180 | { |
---|
| 3181 | L[i][2][6][k+1]=L[i][1][7][k+1]; |
---|
| 3182 | } |
---|
| 3183 | } |
---|
| 3184 | N_b[1]=L[i][n][6][k+1]; |
---|
| 3185 | if (size(L[i][n][5][k])!=0) |
---|
| 3186 | { |
---|
| 3187 | for (j=1; j<=nrows(L[i][n][5][k]); j++) |
---|
| 3188 | { |
---|
| 3189 | rem=submat(L[i][n][5][k],j,(1..ncols(L[i][n][5][k]))); |
---|
| 3190 | n_c[j]=VdDeg(rem,d,L[i][n][8][k]); |
---|
| 3191 | } |
---|
| 3192 | L[i][n][8][k+1]=n_c; |
---|
| 3193 | } |
---|
| 3194 | else |
---|
| 3195 | { |
---|
| 3196 | L[i][n][8][k+1]=list(); |
---|
| 3197 | } |
---|
| 3198 | N_b[2]=L[i][n][8][k+1]; |
---|
| 3199 | n_c=0; |
---|
| 3200 | if (size(N_b[1])!=0) |
---|
| 3201 | { |
---|
| 3202 | vn_b=N_b[1]; |
---|
| 3203 | if (size(N_b[2])!=0) |
---|
| 3204 | { |
---|
| 3205 | vn_b=vn_b,N_b[2]; |
---|
| 3206 | } |
---|
| 3207 | L[i][n][7][k+1]=vn_b; |
---|
| 3208 | } |
---|
| 3209 | else |
---|
| 3210 | { |
---|
| 3211 | if (size(N_b[2])!=0) |
---|
| 3212 | { |
---|
| 3213 | L[i][n][7][k+1]=N_b[2]; |
---|
| 3214 | } |
---|
| 3215 | else |
---|
| 3216 | { |
---|
| 3217 | L[i][n][7][k+1]=list(); |
---|
| 3218 | } |
---|
| 3219 | } |
---|
[7fe9f8b] | 3220 | } |
---|
| 3221 | } |
---|
| 3222 | return(L); |
---|
| 3223 | } |
---|
| 3224 | |
---|
[0e8a5a] | 3225 | //////////////////////////////////////////////////////////////////////////////////// |
---|
[7fe9f8b] | 3226 | |
---|
| 3227 | static proc toVdStrict2x3Complex(list L,int d,list #) |
---|
| 3228 | { |
---|
[0e8a5a] | 3229 | /* We build a one-step free resolution over a V_d-strict short exact piece |
---|
| 3230 | (Algorithm 3.14 in [W2]). |
---|
| 3231 | This procedure is called from the procedure VdStrictDoubleComplexes |
---|
| 3232 | if Syzstring=='Vdres'*/ |
---|
| 3233 | matrix rem; |
---|
[7fe9f8b] | 3234 | int i,j,cc; |
---|
[0e8a5a] | 3235 | int nr; |
---|
[7fe9f8b] | 3236 | list J_A=list(list()); |
---|
| 3237 | list J_B=list(list()); |
---|
| 3238 | list J_C=list(list()); |
---|
| 3239 | list g_AB=list(list()); |
---|
| 3240 | list g_BC=list(list()); |
---|
| 3241 | list n_a=list(list()); |
---|
| 3242 | list n_b=list(list()); |
---|
| 3243 | list n_c=list(list()); |
---|
| 3244 | intvec n_b1; |
---|
| 3245 | matrix fromnf; |
---|
| 3246 | intvec i1,i2; |
---|
| 3247 | /* compute a one step V_d-strict resolution for L[5]*/ |
---|
| 3248 | if (size(L[5])!=0) |
---|
| 3249 | { |
---|
[0e8a5a] | 3250 | intvec n_c1; |
---|
| 3251 | for (i=1; i<=nrows(L[5]); i++) |
---|
| 3252 | { |
---|
| 3253 | rem=submat(L[5],i,intvec(1..ncols(L[5]))); |
---|
| 3254 | n_c1[i]=VdDeg(rem,d, L[8]);//new shift vector |
---|
| 3255 | } |
---|
| 3256 | n_c[1]=n_c1; |
---|
| 3257 | J_C[1]=transpose(syz(transpose(L[5]))); |
---|
| 3258 | if (J_C[1]!=matrix(0,nrows(J_C[1]),ncols(J_C[1]))) |
---|
| 3259 | { |
---|
| 3260 | J_C[1]=VdStrictGB(J_C[1],d,n_c1); |
---|
| 3261 | if (size(#[2])!=0)// new shift vector for the resolution of L[1] |
---|
| 3262 | { |
---|
| 3263 | n_a[1]=#[2]; |
---|
| 3264 | n_b1=n_a[1],n_c[1]; |
---|
| 3265 | n_b[1]=n_b1; |
---|
| 3266 | matrix zero[nrows(L[1])][nrows(L[5])]; |
---|
| 3267 | g_AB=concat(unitmat(nrows(L[1])),matrix(0,nrows(L[1]),nrows(L[5]))); |
---|
| 3268 | if (size(#[1])!=0) |
---|
| 3269 | { |
---|
| 3270 | J_A=#[1];// one step V_d-strict resolution for L[1] |
---|
| 3271 | /* use resolutions of L[1] and L[5] to build a resolution for |
---|
| 3272 | L[3]*/ |
---|
| 3273 | J_B[1]=transpose(matrix(syz(transpose(L[3])))); |
---|
| 3274 | matrix P=matrixLift(J_B[1]*prodr(nrows(L[1]),nrows(L[5])),J_C[1]); |
---|
| 3275 | matrix Pi[1][ncols(J_B[1])]; |
---|
| 3276 | matrix Picombined; |
---|
| 3277 | for (i=1; i<=nrows(J_C[1]); i++) |
---|
| 3278 | { |
---|
| 3279 | for (j=1; j<=nrows(J_B[1]);j++) |
---|
| 3280 | { |
---|
| 3281 | Pi=Pi+P[i,j]*submat(J_B[1],j,intvec(1..ncols(J_B[1]))); |
---|
| 3282 | } |
---|
| 3283 | if (i==1) |
---|
| 3284 | { |
---|
| 3285 | Picombined=transpose(Pi); |
---|
| 3286 | } |
---|
| 3287 | else |
---|
| 3288 | { |
---|
| 3289 | Picombined=concat(Picombined,transpose(Pi)); |
---|
| 3290 | } |
---|
| 3291 | Pi=0; |
---|
| 3292 | } |
---|
| 3293 | Picombined=transpose(Picombined); |
---|
| 3294 | fromnf=VdNormalForm(Picombined,J_A[1],d,n_a[1],n_c[1]); |
---|
| 3295 | i1=intvec(1..nrows(Picombined)); |
---|
| 3296 | i2=intvec((ncols(J_A[1])+1)..ncols(Picombined)); |
---|
| 3297 | Picombined=concat(fromnf,submat(Picombined,i1,i2)); |
---|
| 3298 | J_B[1]=transpose(matrix(J_A[1],nrows(J_A[1]),ncols(J_B[1]))); |
---|
| 3299 | J_B[1]=transpose(concat(J_B[1],transpose(Picombined))); |
---|
| 3300 | g_BC=transpose(concat(transpose(zero),unitmat(nrows(L[5])))); |
---|
| 3301 | } |
---|
| 3302 | else//L[1] is already a resolution |
---|
| 3303 | { |
---|
| 3304 | //compute a resolution for L[3] |
---|
| 3305 | J_B=transpose(matrix(syz(transpose(L[3])))); |
---|
| 3306 | matrix P=matrixLift(J_B[1]*prodr(nrows(L[1]),nrows(L[5])),J_C[1]); |
---|
| 3307 | matrix Pi[1][ncols(J_B[1])]; |
---|
| 3308 | matrix Picombined; |
---|
| 3309 | for (i=1; i<=nrows(J_C[1]); i++) |
---|
| 3310 | { |
---|
| 3311 | for (j=1; j<=nrows(J_B[1]);j++) |
---|
| 3312 | { |
---|
| 3313 | Pi=Pi+P[i,j]*submat(J_B[1],j,intvec(1..ncols(J_B[1]))); |
---|
| 3314 | } |
---|
| 3315 | if (i==1) |
---|
| 3316 | { |
---|
| 3317 | Picombined=transpose(Pi); |
---|
| 3318 | } |
---|
| 3319 | else |
---|
| 3320 | { |
---|
| 3321 | Picombined=concat(Picombined,transpose(Pi)); |
---|
| 3322 | } |
---|
| 3323 | Pi=0; |
---|
| 3324 | } |
---|
| 3325 | Picombined=transpose(Picombined); |
---|
| 3326 | J_B[1]=Picombined; |
---|
| 3327 | g_BC=transpose(concat(transpose(zero),unitmat(nrows(L[5])))); |
---|
| 3328 | } |
---|
| 3329 | } |
---|
| 3330 | else |
---|
| 3331 | { |
---|
| 3332 | n_b=n_c[1]; |
---|
| 3333 | J_B[1]=J_C[1]; |
---|
| 3334 | g_BC=unitmat(ncols(J_C[1])); |
---|
| 3335 | } |
---|
| 3336 | } |
---|
[7fe9f8b] | 3337 | else |
---|
[0e8a5a] | 3338 | { |
---|
| 3339 | J_C=list(list());// L[5] is already a resolution |
---|
| 3340 | if (size(#[2])!=0) |
---|
| 3341 | { |
---|
| 3342 | matrix zero[nrows(L[1])][nrows(L[5])]; |
---|
| 3343 | g_BC=transpose(concat(transpose(zero),unitmat(nrows(L[5])))); |
---|
| 3344 | n_a[1]=#[2]; |
---|
| 3345 | n_b1=n_a[1],n_c[1]; |
---|
| 3346 | n_b[1]=n_b1; |
---|
| 3347 | g_AB=concat(unitmat(nrows(L[1])),matrix(0,nrows(L[1]),nrows(L[5]))); |
---|
| 3348 | if (size(#[1])!=0) |
---|
| 3349 | { |
---|
| 3350 | J_A=#[1]; |
---|
| 3351 | /*resolution of L[3]*/ |
---|
| 3352 | nr=nrows(J_A[1]); |
---|
| 3353 | J_B=concat(J_A[1],matrix(0,nr,nrows(L[3])-nrows(L[1]))); |
---|
| 3354 | } |
---|
| 3355 | } |
---|
| 3356 | else |
---|
| 3357 | { |
---|
| 3358 | n_b=n_c[1]; |
---|
| 3359 | g_BC=unitmat(ncols(L[5])); |
---|
| 3360 | } |
---|
| 3361 | } |
---|
[7fe9f8b] | 3362 | } |
---|
| 3363 | else// L[5]=list(); |
---|
| 3364 | { |
---|
[0e8a5a] | 3365 | if (size(#[2])!=0) |
---|
| 3366 | { |
---|
| 3367 | n_a[1]=#[2]; |
---|
| 3368 | n_b=n_a[1]; |
---|
| 3369 | g_AB=unitmat(size(n_b[1])); |
---|
| 3370 | if (size(#[1])!=0) |
---|
| 3371 | { |
---|
| 3372 | J_A=#[1]; |
---|
| 3373 | J_B[1]=J_A[1];// resolution of L[3] equals that of L[1] |
---|
| 3374 | } |
---|
| 3375 | } |
---|
[7fe9f8b] | 3376 | } |
---|
| 3377 | 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]); |
---|
| 3378 | return (out); |
---|
| 3379 | } |
---|
| 3380 | |
---|
| 3381 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 3382 | |
---|
| 3383 | static proc assemblingDoubleComplexes(list L) |
---|
| 3384 | { |
---|
[0e8a5a] | 3385 | /* The input is the output of VdStrictDoubleComplexes, we assemble the |
---|
| 3386 | resolutions of the L[i][2][3][1] to obtain a V_d-strict free Cartan-Eilenberg |
---|
| 3387 | resolution with modules P^i_j (1<=i<=size(L), j>=0) for the seqeunce |
---|
| 3388 | coker(L[1][2][3][1])->...->coker(L[size(L)][2][3][1])*/ |
---|
[7fe9f8b] | 3389 | list out; |
---|
| 3390 | int i,j,k,l,oldj,newj,nr,nc; |
---|
| 3391 | for (i=1; i<=size(L); i++) |
---|
[0e8a5a] | 3392 | { |
---|
| 3393 | out[i]=list(list()); |
---|
| 3394 | out[i][1][1]=ncols(L[i][2][3][1]);//rank of module P^i_0 |
---|
| 3395 | if (size(L[i][2][5][1])!=0) |
---|
| 3396 | { |
---|
| 3397 | /*horizontal differential P^i_0->P^(i+1)_0*/ |
---|
| 3398 | nc=ncols(L[i][2][5][1]); |
---|
| 3399 | out[i][1][4]=prodr(ncols(L[i][2][3][1])-ncols(L[i][2][5][1]),nc); |
---|
| 3400 | } |
---|
[7fe9f8b] | 3401 | else |
---|
[0e8a5a] | 3402 | { |
---|
| 3403 | /*horizontal differential P^i_0->0*/ |
---|
| 3404 | out[i][1][4]=matrix(0,ncols(L[i][2][3][1]),1); |
---|
| 3405 | } |
---|
| 3406 | oldj=newj; |
---|
| 3407 | for (j=1; j<=size(L[i][2][3]);j++) |
---|
| 3408 | { |
---|
| 3409 | out[i][j][2]=L[i][2][7][j];//shift vector of P^i_{j-1} |
---|
| 3410 | if (size(L[i][2][3][j])==0) |
---|
| 3411 | { |
---|
| 3412 | newj =j; |
---|
| 3413 | break; |
---|
| 3414 | } |
---|
| 3415 | out[i][j+1]=list(); |
---|
| 3416 | out[i][j+1][1]=nrows(L[i][2][3][j]);//rank of the module P^i_j |
---|
| 3417 | out[i][j+1][3]=L[i][2][3][j];//vertical differential P^i_j->P^(i+1)_j |
---|
| 3418 | if (size(L[i][2][5][j])!=0) |
---|
| 3419 | { |
---|
| 3420 | //horizonal differential P^i_j->P^(i-1)_j |
---|
| 3421 | nr=nrows(L[i][2][3][j])-nrows(L[i][2][5][j]); |
---|
| 3422 | out[i][j+1][4]=(-1)^j*prodr(nr,nrows(L[i][2][5][j])); |
---|
| 3423 | } |
---|
| 3424 | else |
---|
| 3425 | { |
---|
| 3426 | /*horizontal differential P^i_j->P^(i-1)_j*/ |
---|
| 3427 | out[i][j+1][4]=matrix(0,nrows(L[i][2][3][j]),1); |
---|
| 3428 | } |
---|
| 3429 | if(j==size(L[i][2][3])) |
---|
| 3430 | { |
---|
| 3431 | out[i][j+1][2]=L[i][2][7][j+1];//shift vector of P^i_j |
---|
| 3432 | newj=j+1; |
---|
| 3433 | } |
---|
| 3434 | } |
---|
| 3435 | if (i>1) |
---|
| 3436 | { |
---|
| 3437 | |
---|
| 3438 | for (k=1; k<=Min(list(oldj,newj)); k++) |
---|
| 3439 | { |
---|
| 3440 | /*horizonal differential P^(i-1)_(k-1)->P^i_(k-1)*/ |
---|
| 3441 | nr=nrows(out[i-1][k][4]); |
---|
| 3442 | out[i-1][k][4]=matrix(out[i-1][k][4],nr,out[i][k][1]); |
---|
| 3443 | } |
---|
| 3444 | for (k=newj+1; k<=oldj; k++) |
---|
| 3445 | { |
---|
| 3446 | /*no differential needed*/ |
---|
| 3447 | out[i-1][k]=delete(out[i-1][k],4); |
---|
| 3448 | } |
---|
| 3449 | } |
---|
[7fe9f8b] | 3450 | } |
---|
| 3451 | return (out); |
---|
| 3452 | } |
---|
| 3453 | |
---|
[0e8a5a] | 3454 | //////////////////////////////////////////////////////////////////////////////////// |
---|
[7fe9f8b] | 3455 | |
---|
| 3456 | static proc totalComplex(list L); |
---|
| 3457 | { |
---|
| 3458 | /* Input is the output of assemblingDoubleComplexes. |
---|
[0e8a5a] | 3459 | We obtain a complex C^1[m^1]->...->C^(r)[m^r] with differentials d^i and |
---|
| 3460 | shift vectors m^i (where C^r is placed in degree size(L)-1). |
---|
| 3461 | This complex is dercribed in the list out as follows: |
---|
| 3462 | rank(C^i)=out[3*i-2]; m_i=out[3*i-1] and d^i=out[3*i]*/ |
---|
[7fe9f8b] | 3463 | list out;intvec rem1;intvec v; list remsize; int emp; |
---|
| 3464 | int i; int j; int c; int d; matrix M; int k; int l; |
---|
| 3465 | int n=nvars(basering) div 2; |
---|
| 3466 | list K; |
---|
| 3467 | for (i=1; i<=n+1; i++) |
---|
| 3468 | { |
---|
[0e8a5a] | 3469 | K[i]=list(); |
---|
[7fe9f8b] | 3470 | } |
---|
[0e8a5a] | 3471 | L=K+L; |
---|
| 3472 | for (i=1; i<=size(L); i++) |
---|
[7fe9f8b] | 3473 | { |
---|
[0e8a5a] | 3474 | emp=0; |
---|
| 3475 | if (size(L[i])!=0) |
---|
[7fe9f8b] | 3476 | { |
---|
[0e8a5a] | 3477 | out[3*i-2]=L[i][1][1]; |
---|
| 3478 | v=L[i][1][1]; |
---|
| 3479 | rem1=L[i][1][2]; |
---|
[7fe9f8b] | 3480 | emp=1; |
---|
| 3481 | } |
---|
[0e8a5a] | 3482 | else |
---|
[7fe9f8b] | 3483 | { |
---|
[0e8a5a] | 3484 | out[3*i-2]=0; |
---|
| 3485 | v=0; |
---|
[7fe9f8b] | 3486 | } |
---|
[0e8a5a] | 3487 | for (j=i+1; j<=size(L); j++) |
---|
| 3488 | { |
---|
| 3489 | if (size(L[j])>=j-i+1) |
---|
| 3490 | { |
---|
| 3491 | out[3*i-2]=out[3*i-2]+L[j][j-i+1][1]; |
---|
| 3492 | if (emp==0) |
---|
| 3493 | { |
---|
| 3494 | rem1=L[j][j-i+1][2]; |
---|
| 3495 | emp=1; |
---|
| 3496 | } |
---|
| 3497 | else |
---|
| 3498 | { |
---|
| 3499 | rem1=rem1,L[j][j-i+1][2]; |
---|
| 3500 | } |
---|
| 3501 | v[size(v)+1]=L[j][j-i+1][1]; |
---|
| 3502 | } |
---|
| 3503 | else |
---|
| 3504 | { |
---|
| 3505 | v[size(v)+1]=0; |
---|
| 3506 | } |
---|
| 3507 | } |
---|
| 3508 | out[3*i-1]=rem1; |
---|
| 3509 | v[size(v)+1]=0; |
---|
| 3510 | remsize[i]=v; |
---|
[7fe9f8b] | 3511 | } |
---|
| 3512 | int o1; |
---|
| 3513 | int o2; |
---|
| 3514 | for (i=1; i<=size(L)-1; i++) |
---|
| 3515 | { |
---|
[0e8a5a] | 3516 | o1=1; |
---|
| 3517 | o2=1; |
---|
| 3518 | if (size(out[3*i-2])!=0) |
---|
| 3519 | { |
---|
| 3520 | o1=out[3*i-2]; |
---|
| 3521 | } |
---|
| 3522 | if (size(out[3*i+1])!=0) |
---|
[7fe9f8b] | 3523 | { |
---|
[0e8a5a] | 3524 | o2=out[3*i+1]; |
---|
[7fe9f8b] | 3525 | } |
---|
[0e8a5a] | 3526 | M=matrix(0,o1,o2); |
---|
| 3527 | if (size(L[i])!=0) |
---|
[7fe9f8b] | 3528 | { |
---|
[0e8a5a] | 3529 | if (size(L[i][1][4])!=0) |
---|
[7fe9f8b] | 3530 | { |
---|
[0e8a5a] | 3531 | M=matrix(L[i][1][4],o1,o2); |
---|
[7fe9f8b] | 3532 | } |
---|
[0e8a5a] | 3533 | } |
---|
| 3534 | c=remsize[i][1]; |
---|
| 3535 | for (j=i+1; j<=size(L); j++) |
---|
| 3536 | { |
---|
| 3537 | if (remsize[i][j-i+1]!=0) |
---|
| 3538 | { |
---|
| 3539 | for (k=c+1; k<=c+remsize[i][j-i+1]; k++) |
---|
| 3540 | { |
---|
| 3541 | for (l=d+1; l<=d+remsize[i+1][j-i];l++) |
---|
| 3542 | { |
---|
| 3543 | M[k,l]=L[j][j-i+1][3][(k-c),(l-d)]; |
---|
| 3544 | } |
---|
| 3545 | } |
---|
| 3546 | d=d+remsize[i+1][j-i]; |
---|
| 3547 | if (remsize[i+1][j-i+1]!=0) |
---|
| 3548 | { |
---|
| 3549 | for (k=c+1; k<=c+remsize[i][j-i+1]; k++) |
---|
| 3550 | { |
---|
| 3551 | for (l=d+1; l<=d+remsize[i+1][j-i+1];l++) |
---|
| 3552 | { |
---|
| 3553 | M[k,l]=L[j][j-i+1][4][k-c,l-d]; |
---|
| 3554 | } |
---|
| 3555 | } |
---|
| 3556 | c=c+remsize[i][j-i+1]; |
---|
| 3557 | } |
---|
| 3558 | } |
---|
| 3559 | else |
---|
| 3560 | { |
---|
| 3561 | d=d+remsize[i+1][j-i]; |
---|
| 3562 | } |
---|
| 3563 | } |
---|
| 3564 | out[3*i]=M; |
---|
| 3565 | d=0; c=0; |
---|
[7fe9f8b] | 3566 | } |
---|
| 3567 | out[3*size(L)]=matrix(0,out[3*size(L)-2],1); |
---|
| 3568 | return (out); |
---|
| 3569 | |
---|
| 3570 | } |
---|
| 3571 | |
---|
[0e8a5a] | 3572 | //////////////////////////////////////////////////////////////////////////////////// |
---|
[7fe9f8b] | 3573 | //COMPUTATION OF THE BLOBAL B-FUNCTION |
---|
[0e8a5a] | 3574 | //////////////////////////////////////////////////////////////////////////////////// |
---|
[7fe9f8b] | 3575 | |
---|
| 3576 | static proc globalBFun(list L,list #) |
---|
| 3577 | { |
---|
[0e8a5a] | 3578 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
| 3579 | 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 |
---|
| 3580 | intvec of size n_i. |
---|
| 3581 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
| 3582 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
| 3583 | 6.1.1 in [R]) by combining Algorithm 6.1.6 in [R] and the method of principal |
---|
| 3584 | intersection (cf. Remark 6.1.7 in [R] 2012). |
---|
| 3585 | This works ONLY IF ALL B-FUNCTIONS ARE NON-ZERO, but this is the case since this |
---|
| 3586 | proc is only called from the procedure deRhamCohomology and the input comes |
---|
| 3587 | originally from the procedure toVdstrictFreeComplex*/ |
---|
[7fe9f8b] | 3588 | if (size(#)==0)//# may contain the Syzstring |
---|
[0e8a5a] | 3589 | { |
---|
| 3590 | string Syzstring="Sres"; |
---|
| 3591 | } |
---|
[7fe9f8b] | 3592 | else |
---|
[0e8a5a] | 3593 | { |
---|
| 3594 | string Syzstring=#[1]; |
---|
| 3595 | } |
---|
[7fe9f8b] | 3596 | int i,j; |
---|
| 3597 | def W=basering; |
---|
| 3598 | int n=nvars(W) div 2; |
---|
| 3599 | list G0; |
---|
| 3600 | ideal I; |
---|
| 3601 | for (j=1; j<=size(L); j++) |
---|
| 3602 | { |
---|
[0e8a5a] | 3603 | G0[j]=list(); |
---|
| 3604 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
| 3605 | { |
---|
| 3606 | G0[j][i]=I; |
---|
| 3607 | } |
---|
[7fe9f8b] | 3608 | } |
---|
| 3609 | list out; |
---|
| 3610 | ideal I; poly f; |
---|
| 3611 | intvec i1; |
---|
| 3612 | for (j=1; j<=size(L); j++) |
---|
| 3613 | { |
---|
[0e8a5a] | 3614 | /*if the shift vector L[j][2] is non-zero we have to compute a V_d-strict |
---|
| 3615 | Groebner basis of L[j][1] with respect to the zero shift; otherwise L[i][1] |
---|
| 3616 | is already a V_d-strict Groebner basis, because it was obtained by the |
---|
| 3617 | procedure toVdStrictFreeComplex*/ |
---|
| 3618 | if (L[j][2]!=intvec(0:size(L[j][2])) or Syzstring=="noCE") |
---|
| 3619 | { |
---|
| 3620 | if (Syzstring=="Vdres") |
---|
| 3621 | { |
---|
| 3622 | L[j][1]=VdStrictGB(L[j][1],n); |
---|
| 3623 | } |
---|
| 3624 | else |
---|
| 3625 | { |
---|
| 3626 | def HomWeyl=makeHomogenizedWeyl(n); |
---|
| 3627 | setring HomWeyl; |
---|
| 3628 | list L=fetch(W,L); |
---|
| 3629 | L[j][1]=nHomogenize(L[j][1]); |
---|
| 3630 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
| 3631 | L[j][1]=subst(L[j][1],h,1); |
---|
| 3632 | setring W; |
---|
| 3633 | L=fetch(HomWeyl,L); |
---|
| 3634 | kill HomWeyl; |
---|
| 3635 | } |
---|
| 3636 | } |
---|
| 3637 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
| 3638 | { |
---|
| 3639 | G0[j][i]=I; |
---|
| 3640 | } |
---|
| 3641 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
| 3642 | { |
---|
| 3643 | /*computes the terms of maximal V_d-degree of the biggest non-zero |
---|
| 3644 | component of submat(L[j][1],i,(1..ncols(L[j][1])))*/ |
---|
| 3645 | i1=(1..ncols(L[j][1])); |
---|
| 3646 | out=VdDeg(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1); |
---|
| 3647 | // f=L[j][1][i,out[2]]; |
---|
| 3648 | G0[j][out[2]]=G0[j][out[2]],out[1]; |
---|
| 3649 | G0[j][out[2]]=compress(G0[j][out[2]]); |
---|
| 3650 | } |
---|
[7fe9f8b] | 3651 | } |
---|
| 3652 | list save; |
---|
| 3653 | int l; |
---|
| 3654 | list weights; |
---|
[0e8a5a] | 3655 | /*bFctIdealModified computes the intersection of G0[j][i] and |
---|
| 3656 | x(1)D(1)+...+x(n)D(n) using the method of principal intersection*/ |
---|
[7fe9f8b] | 3657 | for (j=1; j<=size(G0); j++) |
---|
| 3658 | { |
---|
[0e8a5a] | 3659 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3660 | { |
---|
| 3661 | G0[j][i]=bFctIdealModified(G0[j][i]); |
---|
| 3662 | } |
---|
| 3663 | for (i=1; i<=size(G0[j]); i++) |
---|
[7fe9f8b] | 3664 | { |
---|
[0e8a5a] | 3665 | weights=list(); |
---|
| 3666 | if (size(G0[j][i])!=0) |
---|
| 3667 | { |
---|
| 3668 | for (l=i; l<=size(G0[j]); l++) |
---|
| 3669 | { |
---|
| 3670 | weights[size(weights)+1]=L[j][2][l]; |
---|
| 3671 | } |
---|
| 3672 | G0[j][i]=list(G0[j][i][1]+Min(weights),G0[j][i][2]+Max(weights)); |
---|
| 3673 | } |
---|
[7fe9f8b] | 3674 | } |
---|
| 3675 | } |
---|
| 3676 | list allmin; |
---|
| 3677 | list allmax; |
---|
| 3678 | for (j=1; j<=size(G0); j++) |
---|
| 3679 | { |
---|
[0e8a5a] | 3680 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3681 | { |
---|
| 3682 | if (size(G0[j][i])!=0) |
---|
| 3683 | { |
---|
| 3684 | allmin[size(allmin)+1]=G0[j][i][1]; |
---|
| 3685 | allmax[size(allmax)+1]=G0[j][i][2]; |
---|
| 3686 | } |
---|
| 3687 | } |
---|
[7fe9f8b] | 3688 | } |
---|
| 3689 | list minmax=list(Min(allmin),Max(allmax)); |
---|
| 3690 | return(minmax); |
---|
| 3691 | } |
---|
| 3692 | |
---|
[0e8a5a] | 3693 | //////////////////////////////////////////////////////////////////////////////////// |
---|
[7fe9f8b] | 3694 | |
---|
| 3695 | static proc exactGlobalBFun(list L,list #) |
---|
| 3696 | { |
---|
[0e8a5a] | 3697 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
| 3698 | 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 |
---|
| 3699 | intvec of size n_i. |
---|
| 3700 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
| 3701 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
| 3702 | 6.1.1 in [R]) by combining Algorithm 6.1.6 in [R] and the method of principal |
---|
| 3703 | intersection (cf. Remark 6.1.7 in [R] 2012). |
---|
| 3704 | This works ONLY IF ALL B-FUNCTIONS ARE NON-ZERO, but this is the case since this |
---|
| 3705 | proc is only called from the procedure deRhamCohomology and the input comes |
---|
| 3706 | originally from the procedure toVdstrictFreeComplex*/ |
---|
[7fe9f8b] | 3707 | if (size(#)==0)//# may contain the Syzstring |
---|
[0e8a5a] | 3708 | { |
---|
| 3709 | string Syzstring="Sres"; |
---|
| 3710 | } |
---|
[7fe9f8b] | 3711 | else |
---|
[0e8a5a] | 3712 | { |
---|
| 3713 | string Syzstring=#[1]; |
---|
| 3714 | } |
---|
[7fe9f8b] | 3715 | int i,j,k; |
---|
| 3716 | def W=basering; |
---|
| 3717 | int n=nvars(W) div 2; |
---|
| 3718 | list G0; |
---|
| 3719 | ideal I; |
---|
| 3720 | for (j=1; j<=size(L); j++) |
---|
| 3721 | { |
---|
[0e8a5a] | 3722 | G0[j]=list(); |
---|
| 3723 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
| 3724 | { |
---|
| 3725 | G0[j][i]=I; |
---|
| 3726 | } |
---|
[7fe9f8b] | 3727 | } |
---|
| 3728 | list out; |
---|
| 3729 | matrix M; |
---|
| 3730 | ideal I; poly f; |
---|
| 3731 | intvec i1; |
---|
| 3732 | for (j=1; j<=size(L); j++) |
---|
[0e8a5a] | 3733 | { |
---|
| 3734 | M=L[j][1]; |
---|
| 3735 | /*if the shift vector L[j][2] is non-zero we have to compute a V_d-strict |
---|
| 3736 | Groebner basis of L[j][1] with respect to the zero shift; otherwise L[i][1] |
---|
| 3737 | is already a V_d-strict Groebner basis, because it was obtained by the |
---|
| 3738 | procedure toVdStrictFreeComplex*/ |
---|
| 3739 | for (k=1; k<=ncols(L[j][1]); k++) |
---|
[7fe9f8b] | 3740 | { |
---|
[0e8a5a] | 3741 | L[j][1]=permcol(M,1,k); |
---|
| 3742 | if (Syzstring=="Vdres") |
---|
| 3743 | { |
---|
| 3744 | L[j][1]=VdStrictGB(L[j][1],n); |
---|
| 3745 | } |
---|
| 3746 | else |
---|
| 3747 | { |
---|
| 3748 | def HomWeyl=makeHomogenizedWeyl(n); |
---|
| 3749 | setring HomWeyl; |
---|
| 3750 | list L=fetch(W,L); |
---|
| 3751 | L[j][1]=nHomogenize(L[j][1]); |
---|
| 3752 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
| 3753 | L[j][1]=subst(L[j][1],h,1); |
---|
| 3754 | setring W; |
---|
| 3755 | L=fetch(HomWeyl,L); |
---|
| 3756 | kill HomWeyl; |
---|
| 3757 | } |
---|
| 3758 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
| 3759 | { |
---|
| 3760 | /*computes the terms of maximal V_d-degree of the biggest non-zero |
---|
| 3761 | component of submat(L[j][1],i,(1..ncols(L[j][1])))*/ |
---|
| 3762 | i1=(1..ncols(L[j][1])); |
---|
| 3763 | out=VdDeg(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1); |
---|
| 3764 | if (out[2]==1) |
---|
| 3765 | { |
---|
| 3766 | G0[j][k]=G0[j][k],out[1]; |
---|
| 3767 | G0[j][k]=compress(G0[j][k]); |
---|
| 3768 | } |
---|
| 3769 | } |
---|
[7fe9f8b] | 3770 | } |
---|
| 3771 | } |
---|
| 3772 | list save; |
---|
| 3773 | int l; |
---|
| 3774 | list weights; |
---|
[0e8a5a] | 3775 | /*bFctIdealModified computes the intersection of G0[j][i] and |
---|
| 3776 | x(1)D(1)+...+x(n)D(n) using the method of principal intersection*/ |
---|
| 3777 | for (j=1; j<=size(G0); j++) |
---|
| 3778 | { |
---|
| 3779 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3780 | { |
---|
| 3781 | G0[j][i]=bFctIdealModified(G0[j][i]); |
---|
| 3782 | } |
---|
| 3783 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3784 | { |
---|
| 3785 | if (size(G0[j][i])!=0) |
---|
| 3786 | { |
---|
| 3787 | G0[j][i]=list(G0[j][i][1]+L[j][2][i],G0[j][i][2]+L[j][2][i]); |
---|
| 3788 | } |
---|
| 3789 | } |
---|
| 3790 | } |
---|
| 3791 | list allmin; |
---|
| 3792 | list allmax; |
---|
[7fe9f8b] | 3793 | for (j=1; j<=size(G0); j++) |
---|
| 3794 | { |
---|
[0e8a5a] | 3795 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3796 | { |
---|
| 3797 | if (size(G0[j][i])!=0) |
---|
| 3798 | { |
---|
| 3799 | allmin[size(allmin)+1]=G0[j][i][1]; |
---|
| 3800 | allmax[size(allmax)+1]=G0[j][i][2]; |
---|
| 3801 | } |
---|
| 3802 | } |
---|
| 3803 | } |
---|
| 3804 | list minmax=list(Min(allmin),Max(allmax)); |
---|
| 3805 | return(minmax); |
---|
| 3806 | } |
---|
| 3807 | |
---|
| 3808 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 3809 | |
---|
| 3810 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 3811 | |
---|
| 3812 | static proc exactGlobalBFunIntegration(list L,list #) |
---|
| 3813 | { |
---|
| 3814 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
| 3815 | 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 |
---|
| 3816 | intvec of size n_i. |
---|
| 3817 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
| 3818 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
| 3819 | 6.1.1 in [R]) by combining Algorithm 6.1.6 in [R] and the method of principal |
---|
| 3820 | intersection (cf. Remark 6.1.7 in [R] 2012). |
---|
| 3821 | This works ONLY IF ALL B-FUNCTIONS ARE NON-ZERO, but this is the case since this |
---|
| 3822 | proc is only called from the procedure deRhamCohomology and the input comes |
---|
| 3823 | originally from the procedure toVdstrictFreeComplex*/ |
---|
| 3824 | string Syzstring="Sres"; |
---|
| 3825 | int i,j,k; |
---|
| 3826 | def W=basering; |
---|
| 3827 | int n=nvars(W) div 2; |
---|
| 3828 | // def C=makeConverseWeyl(n); |
---|
| 3829 | // setring C; |
---|
| 3830 | // ideal Jn=x(1); |
---|
| 3831 | // for (i=2; i<=nvars(basering) div 2; i++) |
---|
| 3832 | // { |
---|
| 3833 | // Jn=Jn,var(nvars(basering) div 2 + i); |
---|
| 3834 | // } |
---|
| 3835 | // for (i=1; i<=nvars(basering) div 2; i++) |
---|
| 3836 | // { |
---|
| 3837 | // Jn=Jn,var(i); |
---|
| 3838 | // } |
---|
| 3839 | // map transtc=W,Jn; |
---|
| 3840 | // list L=transtc(L); |
---|
| 3841 | list G0; |
---|
| 3842 | ideal I; |
---|
| 3843 | for (j=1; j<=size(L); j++) |
---|
[7fe9f8b] | 3844 | { |
---|
[0e8a5a] | 3845 | G0[j]=list(); |
---|
| 3846 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
| 3847 | { |
---|
| 3848 | G0[j][i]=I; |
---|
| 3849 | } |
---|
| 3850 | } |
---|
| 3851 | list out; |
---|
| 3852 | matrix M; |
---|
| 3853 | ideal I; |
---|
| 3854 | poly f; |
---|
| 3855 | intvec i1; |
---|
| 3856 | for (j=1; j<=size(L); j++) |
---|
| 3857 | { |
---|
| 3858 | M=L[j][1]; |
---|
| 3859 | /*if the shift vector L[j][2] is non-zero we have to compute a V_d-strict |
---|
| 3860 | Groebner basis of L[j][1] with respect to the zero shift; otherwise L[i][1] |
---|
| 3861 | is already a V_d-strict Groebner basis, because it was obtained by the |
---|
| 3862 | procedure toVdStrictFreeComplex*/ |
---|
| 3863 | for (k=1; k<=ncols(L[j][1]); k++) |
---|
| 3864 | { |
---|
| 3865 | L[j][1]=permcol(M,1,k); |
---|
| 3866 | def HomWeyl=makeHomogenizedWeylTilde(n); |
---|
| 3867 | setring HomWeyl; |
---|
| 3868 | list L=fetch(W,L); |
---|
| 3869 | L[j][1]=nHomogenize(L[j][1]); |
---|
| 3870 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
| 3871 | L[j][1]=subst(L[j][1],h,1); |
---|
| 3872 | setring W; |
---|
| 3873 | L=fetch(HomWeyl,L); |
---|
| 3874 | kill HomWeyl; |
---|
| 3875 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
| 3876 | { |
---|
| 3877 | /*computes the terms of maximal V_d-degree of the biggest non-zero |
---|
| 3878 | component of submat(L[j][1],i,(1..ncols(L[j][1])))*/ |
---|
| 3879 | i1=(1..ncols(L[j][1])); |
---|
[08fa62] | 3880 | out=VdDegTilde(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1);//hier koennte es evtl noch einen Fehler geben!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
---|
[0e8a5a] | 3881 | f=L[j][1][i,out[2]]; |
---|
| 3882 | if (out[2]==1) |
---|
| 3883 | { |
---|
| 3884 | G0[j][k]=G0[j][k],out[1]; |
---|
| 3885 | G0[j][k]=compress(G0[j][k]); |
---|
| 3886 | } |
---|
| 3887 | } |
---|
| 3888 | } |
---|
| 3889 | } |
---|
| 3890 | list save; |
---|
| 3891 | int l; |
---|
| 3892 | list weights; |
---|
| 3893 | /*bFctIdealModified computes the intersection of G0[j][i] and |
---|
| 3894 | x(1)D(1)+...+x(n)D(n) using the method of principal intersection*/ |
---|
| 3895 | for (j=1; j<=size(G0); j++) |
---|
| 3896 | { |
---|
| 3897 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3898 | { |
---|
| 3899 | G0[j][i]=bFctIdealModified(G0[j][i],1); |
---|
| 3900 | } |
---|
| 3901 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3902 | { |
---|
| 3903 | if (size(G0[j][i])!=0) |
---|
| 3904 | { |
---|
| 3905 | G0[j][i]=list(G0[j][i][1]+L[j][2][i],G0[j][i][2]+L[j][2][i]); |
---|
| 3906 | } |
---|
| 3907 | } |
---|
[7fe9f8b] | 3908 | } |
---|
| 3909 | list allmin; |
---|
| 3910 | list allmax; |
---|
| 3911 | for (j=1; j<=size(G0); j++) |
---|
| 3912 | { |
---|
[0e8a5a] | 3913 | for (i=1; i<=size(G0[j]); i++) |
---|
| 3914 | { |
---|
| 3915 | if (size(G0[j][i])!=0) |
---|
| 3916 | { |
---|
| 3917 | allmin[size(allmin)+1]=G0[j][i][1]; |
---|
| 3918 | allmax[size(allmax)+1]=G0[j][i][2]; |
---|
| 3919 | } |
---|
| 3920 | } |
---|
[7fe9f8b] | 3921 | } |
---|
| 3922 | list minmax=list(Min(allmin),Max(allmax)); |
---|
| 3923 | return(minmax); |
---|
| 3924 | } |
---|
| 3925 | |
---|
| 3926 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 3927 | |
---|
[0e8a5a] | 3928 | static proc bFctIdealModified (ideal I, list #) |
---|
[7fe9f8b] | 3929 | {/*modified version of the procedure bfunIdeal from bfun.lib*/ |
---|
[0e8a5a] | 3930 | int tilde; |
---|
| 3931 | if (size(#)!=0) |
---|
| 3932 | { |
---|
| 3933 | tilde=#[1]; |
---|
| 3934 | } |
---|
[7fe9f8b] | 3935 | def B= basering; |
---|
| 3936 | int n = nvars(B) div 2; |
---|
| 3937 | intvec w=(1:n); |
---|
[0e8a5a] | 3938 | // if (tilde==0) |
---|
| 3939 | // { |
---|
| 3940 | I= initialIdealW(I,-w,w); |
---|
| 3941 | // } |
---|
| 3942 | // else |
---|
| 3943 | // { |
---|
| 3944 | // I= initialIdealW(I,w,-w); |
---|
| 3945 | // } |
---|
[7fe9f8b] | 3946 | poly s; int i; |
---|
[0e8a5a] | 3947 | if (tilde==0) |
---|
| 3948 | { |
---|
| 3949 | for (i=1; i<=n; i++) |
---|
| 3950 | { |
---|
| 3951 | s=s+x(i)*D(i); |
---|
| 3952 | } |
---|
| 3953 | } |
---|
| 3954 | else |
---|
| 3955 | { |
---|
| 3956 | for (i=1; i<=n; i++) |
---|
| 3957 | { |
---|
| 3958 | s=s-D(i)*x(i); |
---|
| 3959 | } |
---|
| 3960 | } |
---|
[7fe9f8b] | 3961 | /*pIntersect computes the intersection on s and I*/ |
---|
| 3962 | vector b = pIntersect(s,I); |
---|
| 3963 | list RL = ringlist(B); RL = RL[1..4]; |
---|
| 3964 | RL[2] = list(safeVarName("s")); |
---|
| 3965 | RL[3] = list(list("dp",intvec(1)),list("C",intvec(0))); |
---|
| 3966 | def @S = ring(RL); setring @S; |
---|
| 3967 | vector b = imap(B,b); |
---|
| 3968 | poly bs = vec2poly(b); |
---|
| 3969 | ring r=0,s,dp; |
---|
| 3970 | poly bs=imap(@S,bs); |
---|
| 3971 | /*find minimal and maximal integer root*/ |
---|
| 3972 | ideal allfac=factorize(bs,1); |
---|
| 3973 | list allfacs; |
---|
| 3974 | for (i=1; i<=ncols(allfac); i++) |
---|
[0e8a5a] | 3975 | { |
---|
| 3976 | allfacs[i]=allfac[i]; |
---|
| 3977 | } |
---|
[7fe9f8b] | 3978 | number testzero; |
---|
| 3979 | list zeros; |
---|
| 3980 | for (i=1; i<=size(allfacs); i++) |
---|
| 3981 | { |
---|
[0e8a5a] | 3982 | if (deg(allfacs[i])==1) |
---|
| 3983 | { |
---|
| 3984 | testzero=number(subst(allfacs[i],s,0))/leadcoef(allfacs[i]); |
---|
| 3985 | if (testzero-int(testzero)==0) |
---|
| 3986 | { |
---|
| 3987 | zeros[size(zeros)+1]=int(-1)*int(testzero); |
---|
| 3988 | } |
---|
| 3989 | } |
---|
[7fe9f8b] | 3990 | } |
---|
| 3991 | if (size(zeros)!=0) |
---|
[0e8a5a] | 3992 | { |
---|
| 3993 | list minmax=(Min(zeros),Max(zeros)); |
---|
| 3994 | } |
---|
[7fe9f8b] | 3995 | else |
---|
[0e8a5a] | 3996 | { |
---|
| 3997 | list minmax=list(); |
---|
| 3998 | } |
---|
[7fe9f8b] | 3999 | setring B; |
---|
| 4000 | return(minmax); |
---|
| 4001 | } |
---|
| 4002 | |
---|
| 4003 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4004 | |
---|
| 4005 | static proc safeVarName (string s) |
---|
| 4006 | {/* from the library "bfun.lib"*/ |
---|
| 4007 | string S = "," + charstr(basering) + "," + varstr(basering) + ","; |
---|
| 4008 | s = "," + s + ","; |
---|
| 4009 | while (find(S,s) <> 0) |
---|
| 4010 | { |
---|
| 4011 | s[1] = "@"; |
---|
| 4012 | s = "," + s; |
---|
| 4013 | } |
---|
| 4014 | s = s[2..size(s)-1]; |
---|
| 4015 | return(s) |
---|
| 4016 | } |
---|
| 4017 | |
---|
| 4018 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4019 | |
---|
| 4020 | static proc globalBFunOT(list L,list #) |
---|
| 4021 | { |
---|
[0e8a5a] | 4022 | /*this proc is currently not used since globalBFun computes the same output and is |
---|
| 4023 | faster, however globalBFun works only for non-zero b-functions!*/ |
---|
| 4024 | /*We assume that the basering is the nth Weyl algebra and that L=(L[1],...,L[s]), |
---|
| 4025 | 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 |
---|
| 4026 | intvec of size n_i. |
---|
| 4027 | We compute bounds for the minimal and maximal integer roots of the b-functions |
---|
| 4028 | of coker(L[i][1])[L[i][2]], where L[i][2] is the shift vector (cf. Def. |
---|
| 4029 | 6.1.1 in [R]) using Algorithm 6.1.6 in [R].*/ |
---|
[7fe9f8b] | 4030 | if (size(#)==0) |
---|
[0e8a5a] | 4031 | { |
---|
| 4032 | string Syzstring="Sres"; |
---|
| 4033 | } |
---|
[7fe9f8b] | 4034 | else |
---|
[0e8a5a] | 4035 | { |
---|
| 4036 | string Syzstring=#[1]; |
---|
| 4037 | } |
---|
[7fe9f8b] | 4038 | int i; int j; |
---|
| 4039 | def W=basering; |
---|
| 4040 | int n=nvars(W) div 2; |
---|
| 4041 | list G0; |
---|
| 4042 | ideal I; |
---|
| 4043 | intvec i1; |
---|
| 4044 | for (j=1; j<=size(L); j++) |
---|
| 4045 | { |
---|
[0e8a5a] | 4046 | G0[j]=list(); |
---|
| 4047 | for (i=1; i<=ncols(L[j][1]); i++) |
---|
| 4048 | { |
---|
| 4049 | G0[j][i]=I; |
---|
| 4050 | } |
---|
[7fe9f8b] | 4051 | } |
---|
| 4052 | list out; |
---|
| 4053 | for (j=1; j<=size(L); j++) |
---|
| 4054 | { |
---|
[0e8a5a] | 4055 | if (L[j][2]!=intvec(0:size(L[j][2]))) |
---|
| 4056 | { |
---|
| 4057 | if (Syzstring=="Vdres") |
---|
| 4058 | { |
---|
| 4059 | L[j][1]=VdStrictGB(L[j][1],n); |
---|
| 4060 | } |
---|
| 4061 | else |
---|
| 4062 | { |
---|
| 4063 | def HomWeyl=makeHomogenizedWeyl(n); |
---|
| 4064 | setring HomWeyl; |
---|
| 4065 | list L=fetch(W,L); |
---|
| 4066 | L[j][1]=nHomogenize(L[j][1]); |
---|
| 4067 | L[j][1]=transpose(matrix(slimgb(transpose(L[j][1])))); |
---|
| 4068 | L[j][1]=subst(L[j][1],h,1); |
---|
| 4069 | setring W; |
---|
| 4070 | L=fetch(HomWeyl,L); |
---|
| 4071 | kill HomWeyl; |
---|
| 4072 | } |
---|
| 4073 | } |
---|
| 4074 | for (i=1; i<=nrows(L[j][1]); i++) |
---|
| 4075 | { |
---|
| 4076 | i1=(1..ncols(L[j][1])); |
---|
| 4077 | out=VdDeg(submat(L[j][1],i,i1),n,intvec(0:size(L[j][2])),1); |
---|
| 4078 | G0[j][out[2]][size(G0[j][out[2]])+1]=(out[1]); |
---|
| 4079 | } |
---|
| 4080 | } |
---|
[7fe9f8b] | 4081 | list Data=ringlist(W); |
---|
| 4082 | for (i=1; i<=n; i++) |
---|
[0e8a5a] | 4083 | { |
---|
| 4084 | Data[2][2*n+i]=Data[2][i]; |
---|
| 4085 | Data[2][3*n+i]=Data[2][n+i]; |
---|
| 4086 | Data[2][i]="v("+string(i)+")"; |
---|
| 4087 | Data[2][n+i]="w("+string(i)+")"; |
---|
| 4088 | } |
---|
[7fe9f8b] | 4089 | Data[3][1][1]="M"; |
---|
| 4090 | intvec mord=(0:16*n^2); |
---|
| 4091 | mord[1..2*n]=(1:2*n); |
---|
| 4092 | mord[6*n+1..8*n]=(1:2*n); |
---|
| 4093 | for (i=0; i<=2*n-2; i++) |
---|
[0e8a5a] | 4094 | { |
---|
| 4095 | mord[(3+i)*4*n-i]=-1; |
---|
| 4096 | mord[(2*n+2+i)*4*n-2*n-i]=-1; |
---|
| 4097 | } |
---|
[7fe9f8b] | 4098 | Data[3][1][2]=mord; |
---|
| 4099 | matrix Ones=UpOneMatrix(4*n); |
---|
| 4100 | Data[5]=Ones; |
---|
| 4101 | matrix con[2*n][2*n]; |
---|
| 4102 | Data[6]=transpose(concat(con,transpose(concat(con,Data[6])))); |
---|
| 4103 | def Wuv=ring(Data); |
---|
| 4104 | setring Wuv; |
---|
| 4105 | list G0=imap(W,G0); list G3; poly lterm;intvec lexp; |
---|
[0e8a5a] | 4106 | list G1,G2,LL; |
---|
| 4107 | intvec e,f; |
---|
| 4108 | int kapp,k,l; |
---|
| 4109 | poly h; |
---|
[7fe9f8b] | 4110 | ideal I; |
---|
| 4111 | for (l=1; l<=size(G0); l++) |
---|
[0e8a5a] | 4112 | { |
---|
| 4113 | G1[l]=list(); G2[l]=list(); G3[l]=list(); |
---|
| 4114 | for (i=1; i<=size(G0[l]); i++) |
---|
| 4115 | { |
---|
| 4116 | for (j=1; j<=ncols(G0[l][i]);j++) |
---|
| 4117 | { |
---|
| 4118 | G0[l][i][j]=mHom(G0[l][i][j]); |
---|
| 4119 | } |
---|
| 4120 | for (j=1; j<=nvars(Wuv) div 4; j++) |
---|
| 4121 | { |
---|
| 4122 | G0[l][i][size(G0[l][i])+1]=1-v(j)*w(j); |
---|
| 4123 | } |
---|
| 4124 | G1[l][i]=slimgb(G0[l][i]); |
---|
| 4125 | G2[l][i]=I; |
---|
| 4126 | G3[l][i]=list(); |
---|
| 4127 | for (j=1; j<=ncols(G1[l][i]); j++) |
---|
| 4128 | { |
---|
| 4129 | e=leadexp(G1[l][i][j]); |
---|
| 4130 | f=e[1..2*n]; |
---|
| 4131 | if (f==intvec(0:(2*n))) |
---|
| 4132 | { |
---|
| 4133 | for (k=1; k<=n; k++) |
---|
| 4134 | { |
---|
| 4135 | kapp=-e[2*n+k]+e[3*n+k]; |
---|
| 4136 | if (kapp>0) |
---|
| 4137 | { |
---|
| 4138 | G1[l][i][j]=(x(k)^kapp)*G1[l][i][j]; |
---|
| 4139 | } |
---|
| 4140 | if (kapp<0) |
---|
| 4141 | { |
---|
| 4142 | G1[l][i][j]=(D(k)^(-kapp))*G1[l][i][j]; |
---|
| 4143 | } |
---|
| 4144 | } |
---|
| 4145 | G2[l][i][size(G2[l][i])+1]=G1[l][i][j]; |
---|
| 4146 | G3[l][i][size(G3[l][i])+1]=list(); |
---|
| 4147 | while (G1[l][i][j]!=0) |
---|
| 4148 | { |
---|
| 4149 | lterm=lead(G1[l][i][j]); |
---|
| 4150 | G1[l][i][j]=G1[l][i][j]-lterm; |
---|
| 4151 | lexp=leadexp(lterm); |
---|
| 4152 | lexp=lexp[2*n+1..3*n]; |
---|
| 4153 | LL=list(lexp,leadcoef(lterm)); |
---|
| 4154 | G3[l][i][size(G3[l][i])][size(G3[l][i][size(G3[l][i])])+1]=LL; |
---|
| 4155 | } |
---|
| 4156 | } |
---|
| 4157 | } |
---|
| 4158 | } |
---|
[7fe9f8b] | 4159 | } |
---|
| 4160 | ring r=0,(s(1..n)),dp; |
---|
| 4161 | ideal I; |
---|
| 4162 | map G3forr=Wuv,I; |
---|
| 4163 | list G3=G3forr(G3); |
---|
| 4164 | poly fs,gs; |
---|
| 4165 | int a; |
---|
| 4166 | list G4; |
---|
| 4167 | for (l=1; l<=size(G3); l++) |
---|
| 4168 | { |
---|
[0e8a5a] | 4169 | G4[l]=list(); |
---|
| 4170 | for (i=1; i<=size(G3[l]);i++) |
---|
[7fe9f8b] | 4171 | { |
---|
[0e8a5a] | 4172 | G4[l][i]=I; |
---|
| 4173 | |
---|
| 4174 | for (j=1; j<=size(G3[l][i]); j++) |
---|
[7fe9f8b] | 4175 | { |
---|
[0e8a5a] | 4176 | fs=0; |
---|
| 4177 | for (k=1; k<=size(G3[l][i][j]); k++) |
---|
| 4178 | { |
---|
| 4179 | gs=1; |
---|
| 4180 | for (a=1; a<=n; a++) |
---|
| 4181 | { |
---|
| 4182 | if (G3[l][i][j][k][1][a]!=0) |
---|
| 4183 | { |
---|
| 4184 | gs=gs*permuteVar(list(G3[l][i][j][k][1][a]),a); |
---|
| 4185 | } |
---|
| 4186 | } |
---|
| 4187 | gs=gs*G3[l][i][j][k][2]; |
---|
| 4188 | fs=fs+gs; |
---|
| 4189 | } |
---|
| 4190 | G4[l][i]=G4[l][i],fs; |
---|
[7fe9f8b] | 4191 | } |
---|
| 4192 | } |
---|
| 4193 | } |
---|
| 4194 | if (n==1) |
---|
| 4195 | { |
---|
[0e8a5a] | 4196 | ring rnew=0,t,dp; |
---|
[7fe9f8b] | 4197 | } |
---|
[0e8a5a] | 4198 | else |
---|
[7fe9f8b] | 4199 | { |
---|
[0e8a5a] | 4200 | ring rnew=0,(t,s(2..n)),dp; |
---|
[7fe9f8b] | 4201 | } |
---|
[0e8a5a] | 4202 | ideal Iformap; |
---|
| 4203 | Iformap[1]=t; |
---|
| 4204 | poly forel=1; |
---|
| 4205 | for (i=2; i<=n; i++) |
---|
| 4206 | { |
---|
| 4207 | Iformap[1]=Iformap[1]-s(i); |
---|
| 4208 | Iformap[i]=s(i); |
---|
| 4209 | forel=forel*s(i); |
---|
| 4210 | } |
---|
| 4211 | map rtornew=r,Iformap; |
---|
| 4212 | list G4=rtornew(G4); |
---|
| 4213 | list getintvecs=fetch(W,L); |
---|
| 4214 | ideal J; |
---|
| 4215 | option(redSB); |
---|
| 4216 | for (l=1; l<=size(G4); l++) |
---|
| 4217 | { |
---|
| 4218 | J=1; |
---|
| 4219 | for (i=1; i<=size(G4[l]); i++) |
---|
| 4220 | { |
---|
| 4221 | G4[l][i]=eliminate(G4[l][i],forel); |
---|
| 4222 | J=intersect(J,G4[l][i]); |
---|
| 4223 | } |
---|
| 4224 | G4[l]=poly(std(J)[1]); |
---|
| 4225 | } |
---|
| 4226 | list minmax; |
---|
| 4227 | list mini=list(); |
---|
| 4228 | list maxi=list(); |
---|
| 4229 | list L=fetch(W,L); |
---|
| 4230 | for (i=1; i<=size(G4); i++) |
---|
| 4231 | { |
---|
[76d26c] | 4232 | minmax[i]=minIntRootD(G4[i],1); |
---|
[0e8a5a] | 4233 | if (size(minmax[i])!=0) |
---|
| 4234 | { |
---|
| 4235 | mini=insert(mini,minmax[i][1]+Min(L[i][2])); |
---|
| 4236 | maxi=insert(maxi,minmax[i][2]+Max(L[i][2])); |
---|
| 4237 | } |
---|
| 4238 | } |
---|
| 4239 | mini=Min(mini); |
---|
| 4240 | maxi=Max(maxi); |
---|
| 4241 | minmax=list(mini[1],maxi[1]); |
---|
| 4242 | option(none); |
---|
[7fe9f8b] | 4243 | return(minmax); |
---|
| 4244 | } |
---|
| 4245 | |
---|
| 4246 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4247 | //COMPUTATION OF THE COHOMOLOGY |
---|
| 4248 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4249 | |
---|
| 4250 | static proc findCohomology(list L,int le) |
---|
| 4251 | { |
---|
[0e8a5a] | 4252 | /*computes the cohomology of the complex (D^i,d^i) given by D^i=C^L[2*i-1] and |
---|
[7fe9f8b] | 4253 | d^i=L[2*i]*/ |
---|
| 4254 | def R=basering; |
---|
| 4255 | ring r=0,(x),dp; |
---|
| 4256 | list L=imap(R,L); |
---|
| 4257 | list out; |
---|
| 4258 | int i, ker, im; |
---|
| 4259 | matrix S; |
---|
| 4260 | option(returnSB); |
---|
| 4261 | option(redSB); |
---|
| 4262 | for (i=2; i<=size(L); i=i+2) |
---|
| 4263 | { |
---|
[0e8a5a] | 4264 | if (L[i-1]==0) |
---|
| 4265 | { |
---|
| 4266 | out[i div 2]=0; |
---|
| 4267 | im=0; |
---|
| 4268 | } |
---|
| 4269 | else |
---|
| 4270 | { |
---|
| 4271 | S=matrix(syz(transpose(L[i]))); |
---|
| 4272 | if (S!=matrix(0,nrows(S),ncols(S))) |
---|
| 4273 | { |
---|
| 4274 | ker=ncols(S); |
---|
| 4275 | out[i div 2]=ker-im; |
---|
| 4276 | im=L[i-1]-ker; |
---|
| 4277 | } |
---|
| 4278 | else |
---|
| 4279 | { |
---|
[08fa62] | 4280 | out[i div 2]=0;////achtung geaendert??????????????????????????????????????????????????!!!!!!!!!!!!!!!!!!!!!!!!!war mal out[i-1] |
---|
[0e8a5a] | 4281 | im=L[i-1]; |
---|
| 4282 | } |
---|
| 4283 | } |
---|
| 4284 | } |
---|
| 4285 | option(none); |
---|
| 4286 | while (size(out)>le) |
---|
| 4287 | { |
---|
| 4288 | out=delete(out,1); |
---|
[7fe9f8b] | 4289 | } |
---|
[0e8a5a] | 4290 | setring R; |
---|
| 4291 | return(out); |
---|
| 4292 | } |
---|
| 4293 | |
---|
| 4294 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4295 | |
---|
| 4296 | |
---|
| 4297 | static proc findCohomologyDiffForms(list L,int le) |
---|
| 4298 | { |
---|
| 4299 | /*computes the cohomology of the complex (D^i,d^i) given by D^i=C^L[2*i-1] and |
---|
| 4300 | d^i=L[2*i]*/ |
---|
| 4301 | def R=basering; |
---|
| 4302 | list outdiffforms=list(var(1)); |
---|
| 4303 | ring r=0,(x),dp; |
---|
| 4304 | list L=imap(R,L); |
---|
| 4305 | list out; |
---|
| 4306 | list outdiffforms; |
---|
| 4307 | int i, ker, im, j; |
---|
| 4308 | matrix S; |
---|
| 4309 | matrix concreteimage=matrix(0); |
---|
| 4310 | module concreteimagemod=concreteimage; |
---|
| 4311 | option(returnSB); |
---|
| 4312 | option(redSB); |
---|
| 4313 | matrix redS; |
---|
| 4314 | for (i=2; i<=size(L); i=i+2) |
---|
[7fe9f8b] | 4315 | { |
---|
[0e8a5a] | 4316 | if (L[i-1]==0) |
---|
| 4317 | { |
---|
| 4318 | out[i div 2]=0; |
---|
| 4319 | im=0; |
---|
| 4320 | concreteimage=matrix(0); |
---|
| 4321 | concreteimagemod=concreteimage; |
---|
| 4322 | outdiffforms[i div 2]=list(); |
---|
| 4323 | } |
---|
[7fe9f8b] | 4324 | else |
---|
[0e8a5a] | 4325 | { |
---|
| 4326 | S=matrix(transpose(syz(transpose(L[i])))); |
---|
| 4327 | if (S!=matrix(0,nrows(S),ncols(S))) |
---|
| 4328 | { |
---|
| 4329 | ker=nrows(S); |
---|
| 4330 | out[i div 2]=ker-im; |
---|
| 4331 | if(out[i div 2]==0) |
---|
| 4332 | { |
---|
| 4333 | outdiffforms[i div 2]=list(); |
---|
| 4334 | } |
---|
| 4335 | else |
---|
| 4336 | { |
---|
| 4337 | outdiffforms[i div 2]=list(); |
---|
| 4338 | if (concreteimage==matrix(0)) |
---|
| 4339 | { |
---|
| 4340 | for (j=1; j<=nrows(S); j++) |
---|
| 4341 | { |
---|
| 4342 | outdiffforms[ i div 2][j]=submat(S,j,intvec(1..ncols(S))); |
---|
| 4343 | } |
---|
| 4344 | } |
---|
| 4345 | else |
---|
| 4346 | { |
---|
| 4347 | redS=transpose(std(reduce(transpose(S),concreteimagemod))); |
---|
| 4348 | for (j=1; j<=nrows(redS); j++) |
---|
| 4349 | { |
---|
| 4350 | if (submat(redS,j, intvec(1..ncols(redS)))!=matrix(0,1,ncols(redS))) |
---|
| 4351 | { |
---|
| 4352 | outdiffforms[i div 2][size(outdiffforms[i div 2])+1]=submat(redS,j, intvec(1..ncols(redS))); |
---|
| 4353 | } |
---|
| 4354 | } |
---|
| 4355 | } |
---|
| 4356 | } |
---|
| 4357 | im=L[i-1]-ker; |
---|
| 4358 | concreteimagemod=std(transpose(L[i])); |
---|
| 4359 | concreteimage=concreteimagemod; |
---|
| 4360 | concreteimage=transpose(concreteimage); |
---|
| 4361 | |
---|
| 4362 | |
---|
| 4363 | //concreteimage=transpose(std(transpose(L[i])));//Achtung:hier wieder das Problem mit no Standard basis!!!!!!!!!!!!! |
---|
| 4364 | } |
---|
| 4365 | else |
---|
| 4366 | { |
---|
| 4367 | out[i div 2]=0; |
---|
| 4368 | outdiffforms[i div 2]=0; |
---|
| 4369 | im=L[i-1]; |
---|
| 4370 | concreteimagemod=std(transpose(L[i])); |
---|
| 4371 | concreteimage=concreteimagemod; |
---|
| 4372 | concreteimage=transpose(concreteimage); |
---|
| 4373 | //concreteimage=transpose(std(transpose(L[i]))); |
---|
| 4374 | } |
---|
| 4375 | } |
---|
[7fe9f8b] | 4376 | } |
---|
| 4377 | option(none); |
---|
| 4378 | while (size(out)>le) |
---|
[0e8a5a] | 4379 | { |
---|
| 4380 | out=delete(out,1); |
---|
| 4381 | outdiffforms=delete(outdiffforms,1); |
---|
| 4382 | } |
---|
[7fe9f8b] | 4383 | setring R; |
---|
[0e8a5a] | 4384 | outdiffforms=imap(r,outdiffforms); |
---|
| 4385 | list outall=list(out,outdiffforms); |
---|
| 4386 | option(noredSB); |
---|
| 4387 | option(noreturnSB); |
---|
| 4388 | return(outall); |
---|
[7fe9f8b] | 4389 | } |
---|
| 4390 | |
---|
[0e8a5a] | 4391 | |
---|
| 4392 | |
---|
[7fe9f8b] | 4393 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4394 | //AUXILIARY PROCEDURES |
---|
| 4395 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4396 | |
---|
[0e8a5a] | 4397 | static proc findPreimage(matrix m, matrix n) |
---|
[7fe9f8b] | 4398 | { |
---|
[0e8a5a] | 4399 | def W=basering;//input wird in spaltenform angenommen, output in zeilenform |
---|
| 4400 | list rl=ringlist(W); |
---|
| 4401 | list rlnew=rl; |
---|
| 4402 | rlnew[3][1]=rl[3][2]; |
---|
| 4403 | rlnew[3][2]=rl[3][1]; |
---|
| 4404 | def Wnew=ring(rlnew); |
---|
| 4405 | setring Wnew; |
---|
| 4406 | matrix m=imap(W,m); |
---|
| 4407 | matrix n=imap(W,n); |
---|
| 4408 | def Opp=opposite(Wnew); |
---|
| 4409 | setring Opp; |
---|
| 4410 | matrix m=oppose(Wnew,m); |
---|
| 4411 | matrix n=oppose(Wnew,n); |
---|
| 4412 | option(redSB); |
---|
| 4413 | //matrix m=imap(W,m); |
---|
| 4414 | // matrix n=imap(W,n); |
---|
| 4415 | int i; |
---|
| 4416 | matrix preim; |
---|
| 4417 | if (n!=matrix(0,nrows(n),ncols(n))) |
---|
| 4418 | { |
---|
| 4419 | matrix con=concat(m,n); |
---|
| 4420 | matrix s=syz(con); |
---|
| 4421 | for (i=1; i<=ncols(s); i++) |
---|
| 4422 | { |
---|
| 4423 | if (s[nrows(s),i]==1) |
---|
| 4424 | { |
---|
| 4425 | preim=(-1)*submat(s,1..ncols(m),i); |
---|
| 4426 | break; |
---|
| 4427 | } |
---|
| 4428 | } |
---|
| 4429 | } |
---|
| 4430 | else |
---|
| 4431 | { |
---|
| 4432 | matrix s=syz(m); |
---|
| 4433 | preim=submat(s,1..ncols(m),1); |
---|
| 4434 | } |
---|
| 4435 | option(noredSB); |
---|
| 4436 | setring Wnew; |
---|
| 4437 | matrix preim=oppose(Opp,preim); |
---|
| 4438 | setring W; |
---|
| 4439 | matrix preim=imap(Wnew,preim); |
---|
| 4440 | return(transpose(preim)); |
---|
| 4441 | } |
---|
| 4442 | |
---|
| 4443 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4444 | |
---|
| 4445 | static proc divdr(matrix m,matrix n, list #) |
---|
| 4446 | { |
---|
| 4447 | if (n!=matrix(0,nrows(n),ncols(n))) |
---|
| 4448 | { |
---|
| 4449 | m=transpose(m); |
---|
| 4450 | n=transpose(n); |
---|
| 4451 | matrix con=concat(m,n); |
---|
| 4452 | matrix s=syz(con); |
---|
| 4453 | s=submat(s,1..ncols(m),1..ncols(s)); |
---|
| 4454 | s=transpose(compress(s)); |
---|
| 4455 | } |
---|
| 4456 | else |
---|
| 4457 | { |
---|
| 4458 | matrix s=transpose(syz(transpose(m))); |
---|
| 4459 | } |
---|
| 4460 | int i; |
---|
| 4461 | matrix g; |
---|
| 4462 | matrix sm; |
---|
| 4463 | if (size(#)!=0) |
---|
| 4464 | { |
---|
| 4465 | for (i=1; i<=nrows(s); i++) |
---|
| 4466 | { |
---|
| 4467 | g=deletecol(transpose(s),i); |
---|
| 4468 | sm=transpose(submat(s,i,intvec(1..ncols(s)))); |
---|
| 4469 | sm=reduce(sm,slimgb(g)); |
---|
| 4470 | if (sm==matrix(0,nrows(sm),ncols(sm))) |
---|
| 4471 | { |
---|
| 4472 | s=g; |
---|
| 4473 | s=transpose(s); |
---|
| 4474 | i=i-1; |
---|
| 4475 | } |
---|
| 4476 | } |
---|
| 4477 | } |
---|
[7fe9f8b] | 4478 | return(s); |
---|
| 4479 | } |
---|
| 4480 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4481 | |
---|
| 4482 | static proc matrixLift(matrix M,matrix N) |
---|
| 4483 | { |
---|
| 4484 | intvec v=option(get); |
---|
| 4485 | option(none); |
---|
| 4486 | matrix l=transpose(lift(transpose(M),transpose(N))); |
---|
| 4487 | option(set,v); |
---|
| 4488 | return(l); |
---|
| 4489 | } |
---|
| 4490 | |
---|
| 4491 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4492 | |
---|
| 4493 | static proc VdStrictGB (matrix M,int d,list #) |
---|
| 4494 | "USAGE:VdStrictGB(M,d[,v]); M a matrix, d an integer, v an optional intvec |
---|
| 4495 | ASSUME:-basering is the nth Weyl algebra D_n @* |
---|
| 4496 | -1<=d<=n @* |
---|
| 4497 | -v (if given) is the shift vector on the range of M (in particular, |
---|
| 4498 | size(v)=ncols(M)); otherwise v is assumed to be the zero shift vector |
---|
[0e8a5a] | 4499 | RETURN:matrix N; the rows of N form a V_d-strict Groebner basis with respect to v |
---|
[7fe9f8b] | 4500 | for the module generated by the rows of M |
---|
| 4501 | " |
---|
[0e8a5a] | 4502 | { |
---|
[7fe9f8b] | 4503 | if (M==matrix(0,nrows(M),ncols(M))) |
---|
| 4504 | { |
---|
| 4505 | return (matrix(0,1,ncols(M))); |
---|
| 4506 | } |
---|
| 4507 | intvec op=option(get); |
---|
| 4508 | def W =basering; |
---|
| 4509 | int ncM=ncols(M); |
---|
| 4510 | list Data=ringlist(W); |
---|
| 4511 | Data[2]=list("nhv")+Data[2]; |
---|
| 4512 | Data[3][3]=Data[3][1]; |
---|
| 4513 | Data[3][1]=list("dp",intvec(1)); |
---|
| 4514 | matrix re[size(Data[2])][size(Data[2])]=UpOneMatrix(size(Data[2])); |
---|
| 4515 | Data[5]=re; |
---|
| 4516 | int k,l; |
---|
| 4517 | Data[6]=transpose(concat(matrix(0,1,1),transpose(concat(matrix(0,1,1),Data[6])))); |
---|
| 4518 | def Whom=ring(Data);// D_n[nhv] with the new commuative variable nhv |
---|
| 4519 | setring Whom; |
---|
| 4520 | matrix Mnew=imap(W,M); |
---|
| 4521 | intvec v; |
---|
| 4522 | if (size(#)!=0) |
---|
| 4523 | { |
---|
| 4524 | v=#[1]; |
---|
| 4525 | } |
---|
| 4526 | if (size(v) < ncM) |
---|
| 4527 | { |
---|
| 4528 | v=v,0:(ncM-size(v)); |
---|
| 4529 | } |
---|
| 4530 | Mnew=homogenize(Mnew, d, v);//homogenization of M with respect to the new variable |
---|
| 4531 | Mnew=transpose(Mnew); |
---|
| 4532 | Mnew=slimgb(Mnew);// computes a Groebner basis of the homogenzition of M |
---|
| 4533 | Mnew=subst(Mnew,nhv,1);// substitution of 1 gives V_d-strict Groebner basis of M |
---|
| 4534 | Mnew=compress(Mnew); |
---|
| 4535 | Mnew=transpose(Mnew); |
---|
| 4536 | setring W; |
---|
| 4537 | M=imap(Whom,Mnew); |
---|
| 4538 | option(set,op); |
---|
| 4539 | return(M); |
---|
| 4540 | } |
---|
| 4541 | |
---|
| 4542 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4543 | |
---|
| 4544 | static proc VdNormalForm(matrix F,matrix M,int d,intvec v,list #) |
---|
[0e8a5a] | 4545 | "USAGE:VdNormalForm(F,M,d,v[,w]); F and M matrices, d int, v intvec, w an optional |
---|
[7fe9f8b] | 4546 | intvec |
---|
| 4547 | ASSUME:-basering is the nth Weyl algebra D_n @* |
---|
| 4548 | -F a n_1 x n_2-matrix and M a m_1 x m_2-matrix with m_2<=n_2 @* |
---|
| 4549 | -d is an integer between 1 and n @* |
---|
| 4550 | -v is a shift vector for D_n^(m_2) and hence size(v)=m_2 @* |
---|
| 4551 | -w is a shift vector for D_n^(m_1-m_2) and hence size(v)=m_1-m_2 @* |
---|
| 4552 | RETURN:a n_1 x n_2-matrix N such that:@* |
---|
[0e8a5a] | 4553 | -If no optional intvec w is given:(N[i,1],..,N[i,m_2]) is a V_d-strict normal |
---|
| 4554 | form of (F[i,1],...,F[i,m_2]) with respect to a V_d-strict Groebner basis of |
---|
[7fe9f8b] | 4555 | the rows of M and the shift vector v |
---|
[0e8a5a] | 4556 | -If w is given:(N[i,1],..,N[i,m_2]) is chosen such that |
---|
[7fe9f8b] | 4557 | Vddeg((N[i,1],...,N[i,m_2])[v])<=Vddeg((F[i,m_2+1],...,F[i,m_1])[v]); |
---|
| 4558 | -N[i,j]=F[i,j] for j>m_2 |
---|
| 4559 | " |
---|
[0e8a5a] | 4560 | { |
---|
[7fe9f8b] | 4561 | int SBcom; |
---|
| 4562 | def W =basering; |
---|
| 4563 | int c=ncols(M); |
---|
| 4564 | matrix keepF=F; |
---|
| 4565 | if (size(#)!=0) |
---|
[0e8a5a] | 4566 | { |
---|
| 4567 | intvec w=#[1]; |
---|
| 4568 | } |
---|
[7fe9f8b] | 4569 | F=submat(F,intvec(1..nrows(F)),intvec(1..c)); |
---|
| 4570 | list Data=ringlist(W); |
---|
| 4571 | Data[2]=list("nhv")+Data[2]; |
---|
| 4572 | Data[3][3]=Data[3][1]; |
---|
| 4573 | Data[3][1]=list("dp",intvec(1)); |
---|
| 4574 | matrix re[size(Data[2])][size(Data[2])]=UpOneMatrix(size(Data[2])); |
---|
| 4575 | Data[5]=re; |
---|
| 4576 | int k,l,nr,nc; |
---|
| 4577 | matrix rep[size(Data[2])][size(Data[2])]; |
---|
| 4578 | for (l=size(Data[2])-1;l>=1; l--) |
---|
| 4579 | { |
---|
[0e8a5a] | 4580 | for (k=l-1; k>=1;k--) |
---|
| 4581 | { |
---|
| 4582 | rep[k+1,l+1]=Data[6][k,l]; |
---|
| 4583 | } |
---|
[7fe9f8b] | 4584 | } |
---|
| 4585 | Data[6]=rep; |
---|
| 4586 | def Whom=ring(Data);//new ring D_n[nvh] this new commuative variable nhv |
---|
| 4587 | setring Whom; |
---|
| 4588 | matrix Mnew=imap(W,M); |
---|
| 4589 | list forMnew=homogenize(Mnew,d,v,1);//commputes homogenization of M; |
---|
| 4590 | Mnew=forMnew[1]; |
---|
| 4591 | int rightexp=forMnew[2]; |
---|
| 4592 | matrix Fnew=imap(W,F); |
---|
| 4593 | matrix keepF=imap(W,keepF); |
---|
| 4594 | matrix Fb; |
---|
| 4595 | int cc; |
---|
| 4596 | intvec i1,i2; |
---|
| 4597 | matrix zeromat,subm1,subm2,zeromat2; |
---|
| 4598 | for (l=1; l<=nrows(Fnew); l++) |
---|
| 4599 | { |
---|
[0e8a5a] | 4600 | if (size(#)!=0) |
---|
| 4601 | { |
---|
| 4602 | subm2=submat(keepF,l,((ncols(Fnew)+1)..ncols(keepF))); |
---|
| 4603 | zeromat2=matrix(0,1,ncols(subm2)); |
---|
| 4604 | if (submat(keepF,l,((ncols(Fnew)+1)..ncols(keepF)))==zeromat2) |
---|
| 4605 | { |
---|
| 4606 | for (cc=1; cc<=ncols(Fnew); c++) |
---|
| 4607 | { |
---|
| 4608 | Fnew[l,cc]=0; |
---|
| 4609 | } |
---|
| 4610 | } |
---|
| 4611 | i1=intvec(1..ncols(Fnew)); |
---|
| 4612 | subm1=submat(Fnew,l,i1); |
---|
| 4613 | subm2=submat(keepF,l,(ncols(Fnew)+1)..ncols(keepF)); |
---|
| 4614 | zeromat=matrix(0,1,ncols(Fnew)); |
---|
| 4615 | if (VdDegnhv(subm1,d,v)>VdDegnhv(subm2,d,w) |
---|
| 4616 | and submat(Fnew,l,intvec(1..ncols(Fnew)))!=zeromat) |
---|
| 4617 | { |
---|
[08fa62] | 4618 | //print("Reduzierung des V_d-Grades noetig"); |
---|
[0e8a5a] | 4619 | /*We need to reduce the V_d-degree. First we homogenize the |
---|
| 4620 | lth row of Fnew*/ |
---|
| 4621 | Fb=homogenize(subm1,d,v)*(nhv^rightexp); |
---|
| 4622 | if (SBcom==0) |
---|
| 4623 | { |
---|
| 4624 | /*computes a V_d-strict standard basis*/ |
---|
| 4625 | Mnew=slimgb(transpose(Mnew));// |
---|
| 4626 | SBcom=1; |
---|
| 4627 | } |
---|
| 4628 | /*computes a V_d-strict normal form for FB*/ |
---|
| 4629 | Fb=transpose(reduce(transpose(Fb),Mnew)); |
---|
| 4630 | if (VdDegnhv(Fb,d,v)> VdDegnhv(subm2,d,w) |
---|
| 4631 | and Fb!=matrix(0,nrows(Fb),ncols(Fb)))//should not happen |
---|
| 4632 | { |
---|
| 4633 | //print("Reduzierung fehlgeschlagen!!!!!!!!!!!!!!!!"); |
---|
| 4634 | } |
---|
| 4635 | } |
---|
| 4636 | else |
---|
| 4637 | { |
---|
| 4638 | /*condition on V_ddeg already satisfied -> no normal form |
---|
| 4639 | computation is needed*/ |
---|
| 4640 | Fb=submat(Fnew,l,intvec(1..ncols(Fnew))); |
---|
| 4641 | } |
---|
| 4642 | } |
---|
| 4643 | else |
---|
| 4644 | { |
---|
| 4645 | Fb=homogenize(submat(Fnew,l,intvec(1..ncols(Fnew))),d,v); |
---|
| 4646 | if (SBcom==0) |
---|
| 4647 | { |
---|
| 4648 | Mnew=slimgb(transpose(Mnew));// computes a V_d-strict Groebner basis |
---|
| 4649 | SBcom=1; |
---|
| 4650 | } |
---|
| 4651 | Fb=transpose(reduce(transpose(Fb),Mnew));//normal form |
---|
| 4652 | } |
---|
| 4653 | for (k=1; k<=ncols(Fnew);k++) |
---|
| 4654 | { |
---|
| 4655 | Fnew[l,k]=Fb[1,k]; |
---|
| 4656 | } |
---|
[7fe9f8b] | 4657 | } |
---|
| 4658 | Fnew=subst(Fnew,nhv,1);//obtain normal form in D_n |
---|
| 4659 | setring W; |
---|
| 4660 | F=imap(Whom,Fnew); |
---|
| 4661 | return(F); |
---|
| 4662 | } |
---|
| 4663 | |
---|
| 4664 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4665 | |
---|
| 4666 | static proc homogenize (matrix M,int d,intvec v,list #) |
---|
| 4667 | { |
---|
| 4668 | /* we compute the F[v]-homogenization of each row of M (cf. Def. 3.4 in [OT])*/ |
---|
| 4669 | if (M==matrix(0,nrows(M),ncols(M))) |
---|
[0e8a5a] | 4670 | { |
---|
| 4671 | return(M); |
---|
| 4672 | } |
---|
[7fe9f8b] | 4673 | int i,l,s, kmin, nhvexp; |
---|
[0e8a5a] | 4674 | poly f; |
---|
[7fe9f8b] | 4675 | intvec vnm; |
---|
| 4676 | list findmin,maxnhv,rempoly,remk,rem1,rem2; |
---|
| 4677 | int n=(nvars(basering)-1) div 2; |
---|
[0e8a5a] | 4678 | for (int k=1; k<=nrows(M); k++) |
---|
| 4679 | { |
---|
| 4680 | for (l=1; l<=ncols (M); l++) |
---|
| 4681 | { |
---|
| 4682 | f=M[k,l]; |
---|
| 4683 | s=size(f); |
---|
| 4684 | for (i=1; i<=s; i++) |
---|
| 4685 | { |
---|
| 4686 | vnm=leadexp(f); |
---|
| 4687 | vnm=vnm[n+2..n+d+1]-vnm[2..d+1]; |
---|
| 4688 | kmin=sum(vnm)+v[l]; |
---|
| 4689 | rem1[size(rem1)+1]=lead(f); |
---|
| 4690 | rem2[size(rem2)+1]=kmin; |
---|
| 4691 | findmin=insert(findmin,kmin); |
---|
| 4692 | f=f-lead(f); |
---|
| 4693 | } |
---|
| 4694 | rempoly[l]=rem1; |
---|
| 4695 | remk[l]=rem2; |
---|
| 4696 | rem1=list(); |
---|
| 4697 | rem2=list(); |
---|
| 4698 | } |
---|
| 4699 | if (size(findmin)!=0) |
---|
| 4700 | { |
---|
| 4701 | kmin=Min(findmin); |
---|
| 4702 | } |
---|
| 4703 | for (l=1; l<=ncols(M); l++) |
---|
| 4704 | { |
---|
| 4705 | if (M[k,l]!=0) |
---|
| 4706 | { |
---|
| 4707 | M[k,l]=0; |
---|
| 4708 | for (i=1; i<=size(rempoly[l]);i++) |
---|
| 4709 | { |
---|
| 4710 | nhvexp=remk[l][i]-kmin; |
---|
| 4711 | M[k,l]=M[k,l]+nhv^(nhvexp)*rempoly[l][i]; |
---|
| 4712 | maxnhv[size(maxnhv)+1]=nhvexp; |
---|
| 4713 | } |
---|
| 4714 | } |
---|
| 4715 | } |
---|
| 4716 | rempoly=list(); |
---|
| 4717 | remk=list(); |
---|
| 4718 | findmin=list(); |
---|
| 4719 | } |
---|
[7fe9f8b] | 4720 | maxnhv=Max(maxnhv); |
---|
| 4721 | nhvexp=maxnhv[1]; |
---|
| 4722 | if (size(#)!=0) |
---|
[0e8a5a] | 4723 | { |
---|
| 4724 | return(list(M,nhvexp));//only needed for normal form computations |
---|
| 4725 | } |
---|
[7fe9f8b] | 4726 | return(M); |
---|
| 4727 | } |
---|
| 4728 | |
---|
| 4729 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4730 | |
---|
| 4731 | static proc soldr (matrix M,matrix N) |
---|
[0e8a5a] | 4732 | { |
---|
| 4733 | /* We compute a ncols(M) x nrows(M)-matrix C such that |
---|
| 4734 | C[i,1]M_1+...+C[i,nrows(M)]M_(nrows(M))= e_i mod im(N), |
---|
| 4735 | where e_i is the ith basis element on the range of M, M_j denotes the jth row |
---|
| 4736 | of M and im(N) is generated by the rows of N */ |
---|
[7fe9f8b] | 4737 | int n=nrows(M); |
---|
| 4738 | int q=ncols(M); |
---|
| 4739 | matrix S=concat(transpose(M),transpose(N)); |
---|
| 4740 | def W=basering; |
---|
| 4741 | list Data=ringlist(W); |
---|
| 4742 | list Save=Data[3]; |
---|
| 4743 | Data[3]=list(list("c",0),list("dp",intvec(1..nvars(W)))); |
---|
| 4744 | def Wmod=ring(Data); |
---|
| 4745 | setring Wmod; |
---|
| 4746 | matrix Smod=imap(W,S); |
---|
| 4747 | matrix E[q][1]; |
---|
| 4748 | matrix Smod2,Smodnew; |
---|
| 4749 | option(returnSB); |
---|
| 4750 | int i,j; |
---|
| 4751 | for (i=1;i<=q;i++) |
---|
[0e8a5a] | 4752 | { |
---|
| 4753 | E[i,1]=1; |
---|
| 4754 | Smod2=concat(E,Smod); |
---|
| 4755 | Smod2=syz(Smod2); |
---|
| 4756 | E[i,1]=0; |
---|
| 4757 | for (j=1;j<=ncols(Smod2);j++) |
---|
| 4758 | { |
---|
| 4759 | if (Smod2[1,j]==1) |
---|
| 4760 | { |
---|
| 4761 | Smodnew=concat(Smodnew,(-1)*(submat(Smod2,intvec(2..n+1),j))); |
---|
| 4762 | break; |
---|
| 4763 | } |
---|
| 4764 | } |
---|
[7fe9f8b] | 4765 | } |
---|
| 4766 | Smodnew=transpose(submat(Smodnew,intvec(1..n),intvec(2..q+1))); |
---|
| 4767 | setring W; |
---|
| 4768 | matrix Snew=imap(Wmod,Smodnew); |
---|
| 4769 | option(none); |
---|
| 4770 | return (Snew); |
---|
| 4771 | } |
---|
| 4772 | |
---|
| 4773 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4774 | |
---|
| 4775 | static proc prodr (int k,int l) |
---|
| 4776 | { |
---|
| 4777 | if (k==0) |
---|
[0e8a5a] | 4778 | { |
---|
| 4779 | matrix P=unitmat(l); |
---|
| 4780 | return (P); |
---|
| 4781 | } |
---|
[7fe9f8b] | 4782 | matrix O[l][k]; |
---|
| 4783 | matrix P=transpose(concat(O,unitmat(l))); |
---|
| 4784 | return (P); |
---|
| 4785 | } |
---|
| 4786 | |
---|
| 4787 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4788 | |
---|
| 4789 | static proc VdDeg(matrix M,int d,intvec v,list #) |
---|
| 4790 | { |
---|
| 4791 | /* We assume that the basering it the nth Weyl algebra and that M is a 1 x r- |
---|
[0e8a5a] | 4792 | matrix. |
---|
| 4793 | We compute the V_d-deg of M with respect to the shift vector v, |
---|
| 4794 | i.e V_ddeg(M)=max (V_ddeg(M_i)+v[i]), where k=V_ddeg(M_i) if k is the minimal |
---|
| 4795 | integer, such that M_i can be expressed as a sum of operators |
---|
| 4796 | x(1)^(a_1)*...*x(n)^(a_n)*D(1)^(b_1)*...*D(n)^(b_n) with |
---|
| 4797 | a_1+..+a_d+k>=b_1+..+b_d*/ |
---|
[7fe9f8b] | 4798 | int i, j, etoint; |
---|
| 4799 | int n=nvars(basering) div 2; |
---|
| 4800 | intvec e; |
---|
| 4801 | list findmax; |
---|
| 4802 | int c=ncols(M); |
---|
| 4803 | poly l; |
---|
| 4804 | list positionpoly,positionVd; |
---|
| 4805 | for (i=1; i<=c; i++) |
---|
| 4806 | { |
---|
[0e8a5a] | 4807 | positionpoly[i]=list(); |
---|
| 4808 | positionVd[i]=list(); |
---|
| 4809 | while (M[1,i]!=0) |
---|
| 4810 | { |
---|
| 4811 | l=lead(M[1,i]); |
---|
| 4812 | positionpoly[i][size(positionpoly[i])+1]=l; |
---|
| 4813 | e=leadexp(l); |
---|
| 4814 | e=-e[1..d]+e[n+1..n+d]; |
---|
| 4815 | e=sum(e)+v[i]; |
---|
| 4816 | etoint=e[1]; |
---|
| 4817 | positionVd[i][size(positionVd[i])+1]=etoint; |
---|
| 4818 | findmax[size(findmax)+1]=etoint; |
---|
| 4819 | M[1,i]=M[1,i]-l; |
---|
| 4820 | } |
---|
| 4821 | } |
---|
| 4822 | if (size(findmax)!=0) |
---|
| 4823 | { |
---|
| 4824 | int maxVd=Max(findmax); |
---|
| 4825 | if (size(#)==0) |
---|
| 4826 | { |
---|
| 4827 | return (maxVd); |
---|
| 4828 | } |
---|
| 4829 | } |
---|
| 4830 | else // M is 0-modul |
---|
| 4831 | { |
---|
| 4832 | return(int(0)); |
---|
| 4833 | } |
---|
| 4834 | l=0; |
---|
| 4835 | for (i=c; i>=1; i--) |
---|
| 4836 | { |
---|
| 4837 | for (j=1; j<=size(positionVd[i]); j++) |
---|
| 4838 | { |
---|
| 4839 | if (positionVd[i][j]==maxVd) |
---|
| 4840 | { |
---|
| 4841 | l=l+positionpoly[i][j]; |
---|
| 4842 | } |
---|
| 4843 | } |
---|
| 4844 | if (l!=0) |
---|
| 4845 | { |
---|
| 4846 | /*returns the largest component that has maximal V_d-degree |
---|
| 4847 | and its terms of maximal V_d-deg (needed for globalBFun)*/ |
---|
| 4848 | return (list(l,i)); |
---|
| 4849 | } |
---|
| 4850 | } |
---|
| 4851 | } |
---|
| 4852 | |
---|
| 4853 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4854 | |
---|
| 4855 | static proc VdDegTilde(matrix M,int d,intvec v,list #) |
---|
| 4856 | { |
---|
| 4857 | /* We assume that the basering it the nth Weyl algebra and that M is a 1 x r- |
---|
| 4858 | matrix. |
---|
| 4859 | We compute the \tilde(V_d)-deg of M with respect to the shift vector v, |
---|
| 4860 | i.e \tilde(V_d)deg(M)=max (\tilde(V_d)deg(M_i)+v[i]), where k=\tilde(V_d)deg(M_i) if k is the minimal |
---|
| 4861 | integer, such that M_i can be expressed as a sum of operators |
---|
| 4862 | x(1)^(a_1)*...*x(n)^(a_n)*D(1)^(b_1)*...*D(n)^(b_n) with |
---|
| 4863 | a_1+..+a_d<=b_1+..+b_d+k*/ |
---|
| 4864 | int i, j, etoint; |
---|
| 4865 | int n=nvars(basering) div 2; |
---|
| 4866 | intvec e; |
---|
| 4867 | list findmax; |
---|
| 4868 | int c=ncols(M); |
---|
| 4869 | poly l; |
---|
| 4870 | list positionpoly,positionVd; |
---|
| 4871 | for (i=1; i<=c; i++) |
---|
| 4872 | { |
---|
| 4873 | positionpoly[i]=list(); |
---|
| 4874 | positionVd[i]=list(); |
---|
| 4875 | while (M[1,i]!=0) |
---|
| 4876 | { |
---|
| 4877 | l=lead(M[1,i]); |
---|
| 4878 | positionpoly[i][size(positionpoly[i])+1]=l; |
---|
| 4879 | e=leadexp(l); |
---|
| 4880 | e=e[1..d]-e[n+1..n+d]; |
---|
| 4881 | e=sum(e)+v[i]; |
---|
| 4882 | etoint=e[1]; |
---|
| 4883 | positionVd[i][size(positionVd[i])+1]=etoint; |
---|
| 4884 | findmax[size(findmax)+1]=etoint; |
---|
| 4885 | M[1,i]=M[1,i]-l; |
---|
| 4886 | } |
---|
[7fe9f8b] | 4887 | } |
---|
| 4888 | if (size(findmax)!=0) |
---|
| 4889 | { |
---|
[0e8a5a] | 4890 | int maxVd=Max(findmax); |
---|
| 4891 | if (size(#)==0) |
---|
| 4892 | { |
---|
| 4893 | return (maxVd); |
---|
| 4894 | } |
---|
[7fe9f8b] | 4895 | } |
---|
| 4896 | else // M is 0-modul |
---|
| 4897 | { |
---|
[0e8a5a] | 4898 | return(int(0)); |
---|
[7fe9f8b] | 4899 | } |
---|
[0e8a5a] | 4900 | l=0; |
---|
| 4901 | for (i=c; i>=1; i--) |
---|
[7fe9f8b] | 4902 | { |
---|
[0e8a5a] | 4903 | for (j=1; j<=size(positionVd[i]); j++) |
---|
| 4904 | { |
---|
| 4905 | if (positionVd[i][j]==maxVd) |
---|
| 4906 | { |
---|
| 4907 | l=l+positionpoly[i][j]; |
---|
| 4908 | } |
---|
| 4909 | } |
---|
| 4910 | if (l!=0) |
---|
| 4911 | { |
---|
| 4912 | /*returns the largest component that has maximal V_d-degree |
---|
| 4913 | and its terms of maximal V_d-deg (needed for globalBFun)*/ |
---|
| 4914 | return (list(l,i)); |
---|
| 4915 | } |
---|
[7fe9f8b] | 4916 | } |
---|
| 4917 | } |
---|
[0e8a5a] | 4918 | |
---|
[7fe9f8b] | 4919 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4920 | |
---|
| 4921 | static proc VdDegnhv(matrix M,int d,intvec v,list #) |
---|
| 4922 | { |
---|
[0e8a5a] | 4923 | /* As the procedure VdDeg, but the basering is the nth Weyl algebra |
---|
| 4924 | with a commutative variable nhv*/ |
---|
[7fe9f8b] | 4925 | int i,j,etoint; |
---|
| 4926 | int n=nvars(basering) div 2; |
---|
| 4927 | intvec e; |
---|
| 4928 | int etoint; |
---|
| 4929 | list findmax; |
---|
| 4930 | int c=ncols(M); |
---|
| 4931 | poly l; |
---|
| 4932 | list positionpoly; |
---|
| 4933 | list positionVd; |
---|
| 4934 | for (i=1; i<=c; i++) |
---|
| 4935 | { |
---|
[0e8a5a] | 4936 | positionpoly[i]=list(); |
---|
| 4937 | positionVd[i]=list(); |
---|
| 4938 | while (M[1,i]!=0) |
---|
| 4939 | { |
---|
| 4940 | l=lead(M[1,i]); |
---|
| 4941 | positionpoly[i][size(positionpoly[i])+1]=l; |
---|
| 4942 | e=leadexp(l); |
---|
| 4943 | e=-e[2..d+1]+e[n+2..n+d+1]; |
---|
| 4944 | e=sum(e)+v[i]; |
---|
| 4945 | etoint=e[1]; |
---|
| 4946 | positionVd[i][size(positionVd[i])+1]=etoint; |
---|
| 4947 | findmax[size(findmax)+1]=etoint; |
---|
| 4948 | M[1,i]=M[1,i]-l; |
---|
| 4949 | } |
---|
[7fe9f8b] | 4950 | } |
---|
| 4951 | if (size(findmax)!=0) |
---|
| 4952 | { |
---|
[0e8a5a] | 4953 | int maxVd=Max(findmax); |
---|
| 4954 | if (size(#)==0) |
---|
| 4955 | { |
---|
| 4956 | return (maxVd); |
---|
| 4957 | } |
---|
[7fe9f8b] | 4958 | } |
---|
| 4959 | else // M is 0-modul |
---|
[0e8a5a] | 4960 | { |
---|
| 4961 | return(int(0)); |
---|
| 4962 | } |
---|
[7fe9f8b] | 4963 | } |
---|
| 4964 | |
---|
| 4965 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4966 | |
---|
| 4967 | static proc deletecol(matrix M,int l) |
---|
| 4968 | { |
---|
[0e8a5a] | 4969 | if (ncols(M)==1) |
---|
| 4970 | { |
---|
| 4971 | return(M); |
---|
| 4972 | } |
---|
[7fe9f8b] | 4973 | int s=ncols(M); |
---|
| 4974 | if (l==1) |
---|
[0e8a5a] | 4975 | { |
---|
| 4976 | M=submat(M,(1..nrows(M)),(2..ncols(M))); |
---|
| 4977 | return(M); |
---|
| 4978 | } |
---|
[7fe9f8b] | 4979 | if (l==s) |
---|
[0e8a5a] | 4980 | { |
---|
| 4981 | M=submat(M,(1..nrows(M)),(1..(ncols(M)-1))); |
---|
| 4982 | return(M); |
---|
| 4983 | } |
---|
[7fe9f8b] | 4984 | intvec v=(1..(l-1)),((l+1)..s); |
---|
| 4985 | M=submat(M,(1..nrows(M)),v); |
---|
| 4986 | return(M); |
---|
| 4987 | } |
---|
| 4988 | |
---|
| 4989 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 4990 | |
---|
| 4991 | static proc mHom(poly f) |
---|
| 4992 | {/*for globalBFunOT*/ |
---|
| 4993 | poly g; |
---|
| 4994 | poly l; |
---|
| 4995 | poly add; |
---|
| 4996 | intvec e; |
---|
| 4997 | list minint; |
---|
| 4998 | list remf; |
---|
| 4999 | int i; |
---|
| 5000 | int j; |
---|
| 5001 | int n=nvars(basering) div 4; |
---|
| 5002 | if (f==0) |
---|
[0e8a5a] | 5003 | { |
---|
| 5004 | return(f); |
---|
| 5005 | } |
---|
[7fe9f8b] | 5006 | while (f!=0) |
---|
[0e8a5a] | 5007 | { |
---|
| 5008 | l=lead(f); |
---|
| 5009 | e=leadexp(l); |
---|
| 5010 | remf[size(remf)+1]=list(); |
---|
| 5011 | remf[size(remf)][1]=l; |
---|
| 5012 | for (i=1; i<=n; i++) |
---|
| 5013 | { |
---|
| 5014 | remf[size(remf)][i+1]=-e[2*n+i]+e[3*n+i]; |
---|
| 5015 | if (size(minint)<i) |
---|
| 5016 | { |
---|
| 5017 | minint[i]=list(); |
---|
| 5018 | } |
---|
| 5019 | minint[i][size(minint[i])+1]=-e[2*n+i]+e[3*n+i]; |
---|
| 5020 | } |
---|
| 5021 | f=f-l; |
---|
| 5022 | } |
---|
[7fe9f8b] | 5023 | for (i=1; i<=n; i++) |
---|
[0e8a5a] | 5024 | { |
---|
| 5025 | minint[i]=Min(minint[i]); |
---|
| 5026 | } |
---|
[7fe9f8b] | 5027 | for (i=1; i<=size(remf); i++) |
---|
| 5028 | { |
---|
[0e8a5a] | 5029 | add=remf[i][1]; |
---|
| 5030 | for (j=1; j<=n; j++) |
---|
| 5031 | { |
---|
| 5032 | add=v(j)^(remf[i][j+1]-minint[j])*add; |
---|
| 5033 | } |
---|
| 5034 | g=g+add; |
---|
[7fe9f8b] | 5035 | } |
---|
| 5036 | return (g); |
---|
| 5037 | } |
---|
| 5038 | |
---|
| 5039 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5040 | |
---|
| 5041 | static proc permuteVar(list L,int n) |
---|
| 5042 | {/*for globalBFunOT*/ |
---|
| 5043 | if (typeof(L[1])=="intvec") |
---|
[0e8a5a] | 5044 | { |
---|
| 5045 | intvec v=L[1]; |
---|
| 5046 | } |
---|
[7fe9f8b] | 5047 | else |
---|
[0e8a5a] | 5048 | { |
---|
| 5049 | intvec v=(1:L[1]),(0:L[1]); |
---|
| 5050 | } |
---|
[7fe9f8b] | 5051 | int i;int k; int indi=0; |
---|
| 5052 | int j; |
---|
| 5053 | int s=size(v); |
---|
| 5054 | poly e; |
---|
| 5055 | intvec fore; |
---|
| 5056 | for (i=2; i<=size(v); i=i+2) |
---|
[0e8a5a] | 5057 | { |
---|
[7fe9f8b] | 5058 | |
---|
[0e8a5a] | 5059 | if (v[i]!=0) |
---|
| 5060 | { |
---|
| 5061 | j=i+1; |
---|
| 5062 | while (v[j]!=0) |
---|
| 5063 | { |
---|
| 5064 | j=j+1; |
---|
| 5065 | } |
---|
| 5066 | v[i]=0; |
---|
| 5067 | v[j]=1; |
---|
| 5068 | fore=0; |
---|
| 5069 | indi=0; |
---|
| 5070 | for (k=1; k<=size(v); k++) |
---|
| 5071 | { |
---|
| 5072 | if (k!=i and k!=j) |
---|
| 5073 | { |
---|
| 5074 | if (indi==0) |
---|
| 5075 | { |
---|
| 5076 | indi=1; |
---|
| 5077 | fore[1]=v[k]; |
---|
| 5078 | } |
---|
| 5079 | else |
---|
| 5080 | { |
---|
| 5081 | fore[size(fore)+1]=v[k]; |
---|
| 5082 | } |
---|
| 5083 | } |
---|
| 5084 | } |
---|
| 5085 | e=e-(j-i)*permutevar(list(fore),n); |
---|
[7fe9f8b] | 5086 | } |
---|
| 5087 | } |
---|
| 5088 | e=e+s(n)^(size(v) div 2); |
---|
| 5089 | return (e); |
---|
| 5090 | } |
---|
| 5091 | |
---|
| 5092 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5093 | |
---|
| 5094 | static proc makeHomogenizedWeyl(int n,list #) |
---|
| 5095 | { |
---|
| 5096 | /*modified version of the procedure makeWeyl() from the library nctools.lib*/ |
---|
| 5097 | /*Creates the nth homogenized Weyl algebra with variables x(1),..,x(n),D(1),.., |
---|
[0e8a5a] | 5098 | D(n) and homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
| 5099 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
[7fe9f8b] | 5100 | if (n<1) |
---|
[0e8a5a] | 5101 | { |
---|
| 5102 | print*("Incorrect input"); |
---|
| 5103 | return(); |
---|
| 5104 | } |
---|
[7fe9f8b] | 5105 | if (n ==1) |
---|
[0e8a5a] | 5106 | { |
---|
| 5107 | ring @rr = 0,(x(1),D(1),h),dp; |
---|
| 5108 | } |
---|
[7fe9f8b] | 5109 | else |
---|
[0e8a5a] | 5110 | { |
---|
| 5111 | ring @rr = 0,(x(1..n),D(1..n),h),dp; |
---|
| 5112 | } |
---|
[7fe9f8b] | 5113 | setring @rr; |
---|
[0e8a5a] | 5114 | int i=0; |
---|
[7fe9f8b] | 5115 | if (size(#)==0) |
---|
[0e8a5a] | 5116 | { |
---|
| 5117 | def @rrr = homogenizedWeyl(i); |
---|
| 5118 | } |
---|
[7fe9f8b] | 5119 | else |
---|
[0e8a5a] | 5120 | { |
---|
| 5121 | def @rrr=homogenizedWeyl(i,#); |
---|
| 5122 | } |
---|
| 5123 | return(@rrr); |
---|
| 5124 | } |
---|
| 5125 | |
---|
| 5126 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5127 | |
---|
| 5128 | static proc makeHomogenizedWeylTilde(int n,list #) |
---|
| 5129 | { |
---|
| 5130 | /*modified version of the procedure makeWeyl() from the library nctools.lib*/ |
---|
| 5131 | /*Creates the nth homogenized Weyl algebra with variables x(1),..,x(n),D(1),.., |
---|
| 5132 | D(n) and homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
| 5133 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
| 5134 | if (n<1) |
---|
| 5135 | { |
---|
| 5136 | print*("Incorrect input"); |
---|
| 5137 | return(); |
---|
| 5138 | } |
---|
| 5139 | if (n ==1) |
---|
| 5140 | { |
---|
| 5141 | ring @rr = 0,(x(1),D(1),h),dp; |
---|
| 5142 | } |
---|
| 5143 | else |
---|
| 5144 | { |
---|
| 5145 | ring @rr = 0,(x(1..n),D(1..n),h),dp; |
---|
| 5146 | } |
---|
| 5147 | setring @rr; |
---|
| 5148 | int i=1; |
---|
| 5149 | if (size(#)==0) |
---|
| 5150 | { |
---|
| 5151 | def @rrr = homogenizedWeyl(i); |
---|
| 5152 | } |
---|
| 5153 | else |
---|
| 5154 | { |
---|
| 5155 | def @rrr=homogenizedWeyl(i,#); |
---|
| 5156 | } |
---|
| 5157 | return(@rrr); |
---|
| 5158 | } |
---|
| 5159 | |
---|
| 5160 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5161 | |
---|
| 5162 | static proc makeConverseHomogenizedWeylTilde(int n,list #) |
---|
| 5163 | { |
---|
| 5164 | /*modified version of the procedure makeWeyl() from the library nctools.lib*/ |
---|
| 5165 | /*Creates the nth homogenized Weyl algebra with variables x(1),..,x(n),D(1),.., |
---|
| 5166 | D(n) and homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
| 5167 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
| 5168 | if (n<1) |
---|
| 5169 | { |
---|
| 5170 | print*("Incorrect input"); |
---|
| 5171 | return(); |
---|
| 5172 | } |
---|
| 5173 | if (n ==1) |
---|
| 5174 | { |
---|
| 5175 | ring @rr = 0,(D(1),x(1),h),dp; |
---|
| 5176 | } |
---|
| 5177 | else |
---|
| 5178 | { |
---|
| 5179 | ring @rr = 0,(D(1..n),x(1..n),h),dp; |
---|
| 5180 | } |
---|
| 5181 | setring @rr; |
---|
| 5182 | int i=1; |
---|
| 5183 | if (size(#)==0) |
---|
| 5184 | { |
---|
| 5185 | def @rrr = converseHomogenizedWeyl(i); |
---|
| 5186 | } |
---|
| 5187 | else |
---|
| 5188 | { |
---|
| 5189 | def @rrr=converseHomogenizedWeyl(i,#); |
---|
| 5190 | } |
---|
[7fe9f8b] | 5191 | return(@rrr); |
---|
| 5192 | } |
---|
| 5193 | |
---|
| 5194 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5195 | |
---|
[0e8a5a] | 5196 | static proc converseHomogenizedWeyl (int tilde,list #) |
---|
[7fe9f8b] | 5197 | { |
---|
| 5198 | /*modified version of the procedure Weyl() from the library nctools.lib*/ |
---|
[0e8a5a] | 5199 | /*Creates a homogenized Weyl algebra structure on the basering. We assume |
---|
| 5200 | n=nvars(basering) is odd. The first (n-1)/2 variables will be treated as the |
---|
| 5201 | x(i), the next (n-1)/2 as the corresponding differentials D(i) and the last as |
---|
| 5202 | the homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
| 5203 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
[7fe9f8b] | 5204 | string rname=nameof(basering); |
---|
| 5205 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
[0e8a5a] | 5206 | { |
---|
| 5207 | "You have to call the procedure from the ring"; |
---|
| 5208 | return(); |
---|
| 5209 | } |
---|
[7fe9f8b] | 5210 | int nv = nvars(basering); |
---|
| 5211 | int N = (nv-1) div 2; |
---|
| 5212 | if (((nv-1) % 2) != 0) |
---|
[0e8a5a] | 5213 | { |
---|
| 5214 | "Cannot create homogenized Weyl structure for an even number of generators"; |
---|
| 5215 | return(); |
---|
| 5216 | } |
---|
[7fe9f8b] | 5217 | matrix @D[nv][nv]; |
---|
| 5218 | int i; |
---|
| 5219 | for ( i=1; i<=N; i++ ) |
---|
[0e8a5a] | 5220 | { |
---|
| 5221 | @D[i,N+i]=-h^2; |
---|
| 5222 | } |
---|
[7fe9f8b] | 5223 | def @R = nc_algebra(1,@D); |
---|
| 5224 | setring @R; |
---|
| 5225 | list RL=ringlist(@R); |
---|
| 5226 | intvec v; |
---|
| 5227 | /*we need this ordering for Groebner basis computations*/ |
---|
[0e8a5a] | 5228 | if (tilde==0) |
---|
| 5229 | { |
---|
| 5230 | for (i=1; i<=N; i++) |
---|
| 5231 | { |
---|
| 5232 | v[i]=-1; |
---|
| 5233 | v[N+i]=1; |
---|
| 5234 | } |
---|
| 5235 | } |
---|
| 5236 | else |
---|
| 5237 | { |
---|
| 5238 | for (i=1; i<=N; i++) |
---|
| 5239 | { |
---|
| 5240 | v[i]=1; |
---|
| 5241 | v[N+i]=-1; |
---|
| 5242 | } |
---|
| 5243 | } |
---|
| 5244 | v[nv]=0; |
---|
| 5245 | /* we assign weights to module components*/ |
---|
| 5246 | if (size(#)!=0) |
---|
| 5247 | { |
---|
| 5248 | if (typeof(#[1])=="intvec") |
---|
| 5249 | { |
---|
| 5250 | intvec m=#[1]; |
---|
| 5251 | for (i=1; i<=size(m); i++) |
---|
| 5252 | { |
---|
| 5253 | v[size(v)+1]=m[i];//assigns weight m[i] to the ith module component |
---|
| 5254 | } |
---|
| 5255 | RL[3]=insert(RL[3],list("am",v)); |
---|
| 5256 | } |
---|
| 5257 | else |
---|
| 5258 | { |
---|
| 5259 | RL[3]=insert(RL[3],list("a",v)); |
---|
| 5260 | } |
---|
| 5261 | } |
---|
| 5262 | else |
---|
| 5263 | { |
---|
| 5264 | RL[3]=insert(RL[3],list("a",v)); |
---|
| 5265 | } |
---|
| 5266 | intvec w=(1:nv); |
---|
| 5267 | if (size(#)>=2) |
---|
| 5268 | { |
---|
| 5269 | if (typeof(#[2])=="intvec") |
---|
| 5270 | { |
---|
| 5271 | intvec n=#[2]; |
---|
| 5272 | for (i=1; i<=size(n); i++) |
---|
| 5273 | { |
---|
| 5274 | w[size(w)+1]=n[i]; |
---|
| 5275 | } |
---|
| 5276 | RL[3]=insert(RL[3],list("am",w)); |
---|
| 5277 | } |
---|
| 5278 | else |
---|
| 5279 | { |
---|
| 5280 | RL[3]=insert(RL[3],list("a",w)); |
---|
| 5281 | } |
---|
| 5282 | } |
---|
| 5283 | else |
---|
| 5284 | { |
---|
| 5285 | RL[3]=insert(RL[3],list("a",w)); |
---|
| 5286 | } |
---|
| 5287 | /*this ordering is needed for globalBFun and globalBFunOT*/ |
---|
| 5288 | list saveord=RL[3][3]; |
---|
| 5289 | RL[3][3]=RL[3][4]; |
---|
| 5290 | RL[3][4]=saveord; |
---|
| 5291 | intvec notforh=(1:(size(RL[3][4][2])-1)); |
---|
| 5292 | RL[3][4][2]=notforh; |
---|
| 5293 | RL[3][5]=list("dp",1); |
---|
| 5294 | def @@R=ring(RL); |
---|
| 5295 | return(@@R); |
---|
| 5296 | } |
---|
| 5297 | /////////////////////////////////////////////////////////////////////////////////// |
---|
| 5298 | |
---|
| 5299 | static proc homogenizedWeyl (int tilde,list #) |
---|
| 5300 | { |
---|
| 5301 | /*modified version of the procedure Weyl() from the library nctools.lib*/ |
---|
| 5302 | /*Creates a homogenized Weyl algebra structure on the basering. We assume |
---|
| 5303 | n=nvars(basering) is odd. The first (n-1)/2 variables will be treated as the |
---|
| 5304 | x(i), the next (n-1)/2 as the corresponding differentials D(i) and the last as |
---|
| 5305 | the homogenization variable h, i.e. it holds x(i)*D(i)=D(i)*x(1)+h^2. |
---|
| 5306 | If # contains on intvec v, we assign weight v[i] to the ith module component.*/ |
---|
| 5307 | string rname=nameof(basering); |
---|
| 5308 | if ( rname == "basering") // i.e. no ring has been set yet |
---|
| 5309 | { |
---|
| 5310 | "You have to call the procedure from the ring"; |
---|
| 5311 | return(); |
---|
| 5312 | } |
---|
| 5313 | int nv = nvars(basering); |
---|
| 5314 | int N = (nv-1) div 2; |
---|
| 5315 | if (((nv-1) % 2) != 0) |
---|
| 5316 | { |
---|
| 5317 | "Cannot create homogenized Weyl structure for an even number of generators"; |
---|
| 5318 | return(); |
---|
| 5319 | } |
---|
| 5320 | matrix @D[nv][nv]; |
---|
| 5321 | int i; |
---|
| 5322 | for ( i=1; i<=N; i++ ) |
---|
| 5323 | { |
---|
| 5324 | @D[i,N+i]=h^2; |
---|
| 5325 | } |
---|
| 5326 | def @R = nc_algebra(1,@D); |
---|
| 5327 | setring @R; |
---|
| 5328 | list RL=ringlist(@R); |
---|
| 5329 | intvec v; |
---|
| 5330 | /*we need this ordering for Groebner basis computations*/ |
---|
| 5331 | if (tilde==0) |
---|
| 5332 | { |
---|
| 5333 | for (i=1; i<=N; i++) |
---|
| 5334 | { |
---|
| 5335 | v[i]=-1; |
---|
| 5336 | v[N+i]=1; |
---|
| 5337 | } |
---|
| 5338 | } |
---|
| 5339 | else |
---|
| 5340 | { |
---|
| 5341 | for (i=1; i<=N; i++) |
---|
| 5342 | { |
---|
| 5343 | v[i]=1; |
---|
| 5344 | v[N+i]=-1; |
---|
| 5345 | } |
---|
| 5346 | } |
---|
[7fe9f8b] | 5347 | v[nv]=0; |
---|
| 5348 | /* we assign weights to module components*/ |
---|
| 5349 | if (size(#)!=0) |
---|
| 5350 | { |
---|
[0e8a5a] | 5351 | if (typeof(#[1])=="intvec") |
---|
| 5352 | { |
---|
| 5353 | intvec m=#[1]; |
---|
| 5354 | for (i=1; i<=size(m); i++) |
---|
| 5355 | { |
---|
| 5356 | v[size(v)+1]=m[i];//assigns weight m[i] to the ith module component |
---|
| 5357 | } |
---|
| 5358 | RL[3]=insert(RL[3],list("am",v)); |
---|
| 5359 | } |
---|
| 5360 | else |
---|
| 5361 | { |
---|
| 5362 | RL[3]=insert(RL[3],list("a",v)); |
---|
| 5363 | } |
---|
[7fe9f8b] | 5364 | } |
---|
| 5365 | else |
---|
[0e8a5a] | 5366 | { |
---|
| 5367 | RL[3]=insert(RL[3],list("a",v)); |
---|
| 5368 | } |
---|
[7fe9f8b] | 5369 | intvec w=(1:nv); |
---|
| 5370 | if (size(#)>=2) |
---|
| 5371 | { |
---|
[0e8a5a] | 5372 | if (typeof(#[2])=="intvec") |
---|
| 5373 | { |
---|
| 5374 | intvec n=#[2]; |
---|
| 5375 | for (i=1; i<=size(n); i++) |
---|
| 5376 | { |
---|
| 5377 | w[size(w)+1]=n[i]; |
---|
| 5378 | } |
---|
| 5379 | RL[3]=insert(RL[3],list("am",w)); |
---|
| 5380 | } |
---|
| 5381 | else |
---|
| 5382 | { |
---|
| 5383 | RL[3]=insert(RL[3],list("a",w)); |
---|
| 5384 | } |
---|
[7fe9f8b] | 5385 | } |
---|
| 5386 | else |
---|
[0e8a5a] | 5387 | { |
---|
| 5388 | RL[3]=insert(RL[3],list("a",w)); |
---|
| 5389 | } |
---|
[7fe9f8b] | 5390 | /*this ordering is needed for globalBFun and globalBFunOT*/ |
---|
| 5391 | list saveord=RL[3][3]; |
---|
| 5392 | RL[3][3]=RL[3][4]; |
---|
| 5393 | RL[3][4]=saveord; |
---|
| 5394 | intvec notforh=(1:(size(RL[3][4][2])-1)); |
---|
| 5395 | RL[3][4][2]=notforh; |
---|
| 5396 | RL[3][5]=list("dp",1); |
---|
| 5397 | def @@R=ring(RL); |
---|
| 5398 | return(@@R); |
---|
| 5399 | } |
---|
| 5400 | |
---|
| 5401 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5402 | |
---|
| 5403 | static proc nHomogenize (matrix M,list #) |
---|
| 5404 | { |
---|
| 5405 | /* # may contain an intvec v, if no intvec is given, we assume that v=(0:ncols(M)) |
---|
[0e8a5a] | 5406 | We compute the h[v]-homogenization of the rows of M as in Definition 9.2 [OT]*/ |
---|
[7fe9f8b] | 5407 | int l; poly f; int s; int i; intvec vnm;int kmin; list findmax; |
---|
| 5408 | int n=(nvars(basering)-1) div 2; |
---|
| 5409 | list rempoly; |
---|
| 5410 | list remk; |
---|
| 5411 | list rem1; |
---|
| 5412 | list rem2; |
---|
| 5413 | list maxhexp; |
---|
| 5414 | int hexp; |
---|
| 5415 | intvec v=(0:ncols(M)); |
---|
| 5416 | if (size(#)!=0) |
---|
| 5417 | { |
---|
[0e8a5a] | 5418 | if (typeof(#[1])=="intvec") |
---|
| 5419 | { |
---|
| 5420 | v=#[1]; |
---|
| 5421 | } |
---|
[7fe9f8b] | 5422 | } |
---|
| 5423 | if (size(v)<ncols(M)) |
---|
| 5424 | { |
---|
[0e8a5a] | 5425 | for (i=size(v)+1; i<=ncols(M); i++) |
---|
| 5426 | { |
---|
| 5427 | v[i]=0; |
---|
| 5428 | } |
---|
[7fe9f8b] | 5429 | } |
---|
[0e8a5a] | 5430 | for (int k=1; k<=nrows(M); k++) |
---|
[7fe9f8b] | 5431 | { |
---|
[0e8a5a] | 5432 | for (l=1; l<=ncols (M); l++) |
---|
| 5433 | { |
---|
| 5434 | f=M[k,l]; |
---|
| 5435 | s=size(f); |
---|
| 5436 | for (i=1; i<=s; i++) |
---|
| 5437 | { |
---|
| 5438 | vnm=leadexp(f); |
---|
| 5439 | kmin=sum(vnm)+v[l]; |
---|
| 5440 | rem1[size(rem1)+1]=lead(f); |
---|
| 5441 | rem2[size(rem2)+1]=kmin; |
---|
| 5442 | findmax=insert(findmax,kmin); |
---|
| 5443 | f=f-lead(f); |
---|
| 5444 | } |
---|
| 5445 | rempoly[l]=rem1; |
---|
| 5446 | remk[l]=rem2; |
---|
| 5447 | rem1=list(); |
---|
| 5448 | rem2=list(); |
---|
| 5449 | } |
---|
| 5450 | if (size(findmax)!=0) |
---|
| 5451 | { |
---|
| 5452 | kmin=Max(findmax); |
---|
| 5453 | } |
---|
| 5454 | else |
---|
[7fe9f8b] | 5455 | { |
---|
[0e8a5a] | 5456 | kmin=0; |
---|
| 5457 | } |
---|
| 5458 | for (l=1; l<=ncols(M); l++) |
---|
| 5459 | { |
---|
| 5460 | if (M[k,l]!=0) |
---|
| 5461 | { |
---|
| 5462 | M[k,l]=0; |
---|
| 5463 | for (i=1; i<=size(rempoly[l]);i++) |
---|
| 5464 | { |
---|
| 5465 | hexp=kmin-remk[l][i]; |
---|
| 5466 | maxhexp[size(maxhexp)+1]=hexp; |
---|
| 5467 | M[k,l]=M[k,l]+h^hexp*rempoly[l][i]; |
---|
| 5468 | } |
---|
| 5469 | } |
---|
[7fe9f8b] | 5470 | } |
---|
[0e8a5a] | 5471 | rempoly=list(); |
---|
| 5472 | remk=list(); |
---|
| 5473 | findmax=list(); |
---|
[7fe9f8b] | 5474 | } |
---|
| 5475 | if (size(maxhexp)!=0) |
---|
[0e8a5a] | 5476 | { |
---|
| 5477 | maxhexp=Max(maxhexp); |
---|
| 5478 | hexp=maxhexp[1]; |
---|
| 5479 | } |
---|
[7fe9f8b] | 5480 | else |
---|
[0e8a5a] | 5481 | { |
---|
| 5482 | hexp=0; |
---|
| 5483 | } |
---|
[7fe9f8b] | 5484 | if (size(#)>1) |
---|
[0e8a5a] | 5485 | { |
---|
| 5486 | list forreturn=M,hexp; |
---|
[7fe9f8b] | 5487 | |
---|
[0e8a5a] | 5488 | return(forreturn); |
---|
| 5489 | } |
---|
[7fe9f8b] | 5490 | return(M); |
---|
| 5491 | } |
---|
| 5492 | |
---|
| 5493 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5494 | |
---|
| 5495 | static proc max(int i,int j) |
---|
| 5496 | { |
---|
| 5497 | if(i>j){return(i);} |
---|
| 5498 | return(j); |
---|
| 5499 | } |
---|
| 5500 | |
---|
| 5501 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5502 | |
---|
| 5503 | static proc nDeg (matrix M,intvec m) |
---|
[0e8a5a] | 5504 | {/*we compute an intvec n such that n[i]=max(deg(M[i,j])+m[j]|M[i,j]!=0) (where deg |
---|
| 5505 | stands for the total degree) if (M[i,j]!=0 for some j) and n[i]=0 else*/ |
---|
[7fe9f8b] | 5506 | int i; int j; |
---|
| 5507 | intvec n; |
---|
| 5508 | list L; |
---|
| 5509 | for (i=1; i<=nrows(M); i++) |
---|
| 5510 | { |
---|
[0e8a5a] | 5511 | L=list(); |
---|
| 5512 | for (j=1; j<=ncols(M); j++) |
---|
| 5513 | { |
---|
| 5514 | if (M[i,j]!=0) |
---|
| 5515 | { |
---|
| 5516 | L=insert(L,deg(M[i,j])+m[j]); |
---|
| 5517 | } |
---|
| 5518 | } |
---|
| 5519 | if (size(L)==0) |
---|
| 5520 | { |
---|
| 5521 | n[i]=0; |
---|
| 5522 | } |
---|
| 5523 | else |
---|
| 5524 | { |
---|
| 5525 | n[i]=Max(L); |
---|
| 5526 | } |
---|
[7fe9f8b] | 5527 | } |
---|
| 5528 | return(n); |
---|
| 5529 | } |
---|
| 5530 | |
---|
| 5531 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5532 | |
---|
[76d26c] | 5533 | static proc minIntRootD(list L,list #) |
---|
| 5534 | "USAGE:minIntRootD(L [,M]); L list, M optinonal list |
---|
[7fe9f8b] | 5535 | ASSUME:L a list of univariate polynomials with rational coefficients @* |
---|
[0e8a5a] | 5536 | the variable of the polynomial is s if size(#)==0 (needed for proc |
---|
[7fe9f8b] | 5537 | MVComplex) and t else (needed for globalBFun) |
---|
[0e8a5a] | 5538 | RETURN:-if size(#)==0: int i, where i is an integer root of one of the polynomials |
---|
[7fe9f8b] | 5539 | and it is minimal with respect to that property@* |
---|
[0e8a5a] | 5540 | -if size(#)!=0: list L=(i,j), where i is as above and j is an integer root |
---|
| 5541 | of one of the polynomials and is maximal with respect to that property (if |
---|
[7fe9f8b] | 5542 | an integer root exists) or L=list() else |
---|
| 5543 | " |
---|
| 5544 | { |
---|
| 5545 | def B=basering; |
---|
| 5546 | if (size(#)==0) |
---|
[0e8a5a] | 5547 | { |
---|
| 5548 | ring rnew=0,s,dp; |
---|
| 5549 | } |
---|
[7fe9f8b] | 5550 | else |
---|
[0e8a5a] | 5551 | { |
---|
| 5552 | ring rnew=0,t,dp; |
---|
| 5553 | } |
---|
[7fe9f8b] | 5554 | list L=imap(B,L); |
---|
| 5555 | |
---|
| 5556 | int i; |
---|
| 5557 | int j; |
---|
| 5558 | number isint; |
---|
| 5559 | list possmin; |
---|
| 5560 | ideal allfac; |
---|
| 5561 | list allfacs; |
---|
| 5562 | for (i=1; i<=size(L); i++) |
---|
| 5563 | { |
---|
[0e8a5a] | 5564 | allfac=factorize(L[i],1); |
---|
| 5565 | for (j=1; j<=ncols(allfac); j++) |
---|
| 5566 | { |
---|
| 5567 | allfacs[j]=allfac[j]; |
---|
| 5568 | } |
---|
| 5569 | for (j=1; j<=size(allfacs); j++) |
---|
[7fe9f8b] | 5570 | { |
---|
[0e8a5a] | 5571 | if (deg(allfacs[j])==1) |
---|
| 5572 | { |
---|
| 5573 | isint=number(subst(allfacs[j],var(1),0)/leadcoef(allfacs[j])); |
---|
| 5574 | if (isint-int(isint)==0) |
---|
| 5575 | { |
---|
| 5576 | possmin[size(possmin)+1]=int(isint); |
---|
| 5577 | } |
---|
| 5578 | } |
---|
[7fe9f8b] | 5579 | } |
---|
[0e8a5a] | 5580 | allfacs=list(); |
---|
[7fe9f8b] | 5581 | } |
---|
| 5582 | int zerolist; |
---|
| 5583 | if (size(possmin)!=0) |
---|
[0e8a5a] | 5584 | { |
---|
| 5585 | int miniroot=(-1)*Max(possmin); |
---|
| 5586 | int maxiroot=(-1)*Min(possmin); |
---|
| 5587 | } |
---|
[7fe9f8b] | 5588 | else |
---|
[0e8a5a] | 5589 | { |
---|
| 5590 | zerolist=1; |
---|
| 5591 | } |
---|
[7fe9f8b] | 5592 | setring B; |
---|
| 5593 | if (size(#)==0) |
---|
[0e8a5a] | 5594 | { |
---|
| 5595 | return(miniroot); |
---|
| 5596 | } |
---|
| 5597 | else |
---|
| 5598 | { |
---|
| 5599 | if (zerolist==0) |
---|
| 5600 | { |
---|
| 5601 | return(list(miniroot,maxiroot)); |
---|
| 5602 | } |
---|
| 5603 | else |
---|
| 5604 | { |
---|
| 5605 | return(list()); |
---|
| 5606 | } |
---|
| 5607 | } |
---|
| 5608 | } |
---|
| 5609 | |
---|
| 5610 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5611 | |
---|
| 5612 | proc converseWeyl(list #) |
---|
| 5613 | { |
---|
| 5614 | string rname=nameof(basering); |
---|
| 5615 | int @chr = 0; |
---|
| 5616 | int nv = nvars(basering); |
---|
| 5617 | int N = nv div 2; |
---|
| 5618 | matrix @D[nv][nv]; |
---|
| 5619 | int i; |
---|
| 5620 | for ( i=1; i<=N; i++) |
---|
[7fe9f8b] | 5621 | { |
---|
[0e8a5a] | 5622 | @D[i,N+i]=-1; |
---|
[7fe9f8b] | 5623 | } |
---|
[0e8a5a] | 5624 | def @R = nc_algebra(1,@D); |
---|
| 5625 | return(@R); |
---|
| 5626 | } |
---|
| 5627 | |
---|
| 5628 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5629 | |
---|
| 5630 | proc makeConverseWeyl(int n, list #) |
---|
| 5631 | { |
---|
| 5632 | if (n==1) |
---|
| 5633 | { |
---|
| 5634 | ring @rr = 0,(D(1),x(1)),dp; |
---|
| 5635 | } |
---|
[7fe9f8b] | 5636 | else |
---|
| 5637 | { |
---|
[0e8a5a] | 5638 | ring @rr = 0,(D(1..n),x(1..n)),dp; |
---|
| 5639 | } |
---|
| 5640 | setring @rr; |
---|
| 5641 | def @rrr = converseWeyl(); |
---|
| 5642 | return(@rrr); |
---|
| 5643 | } |
---|
| 5644 | |
---|
| 5645 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5646 | |
---|
| 5647 | proc makeOmega(int n) |
---|
| 5648 | { |
---|
| 5649 | def R=basering; |
---|
| 5650 | int i; |
---|
| 5651 | int j,k,l; |
---|
| 5652 | list omega; |
---|
| 5653 | omega[1]=list(list(list())); |
---|
| 5654 | omega[2]=list(); |
---|
| 5655 | for (i=1; i<=n; i++) |
---|
[7fe9f8b] | 5656 | { |
---|
[0e8a5a] | 5657 | omega[2][i]=list(i); |
---|
[7fe9f8b] | 5658 | } |
---|
[0e8a5a] | 5659 | for (i=2; i<=n; i++) |
---|
[7fe9f8b] | 5660 | { |
---|
[0e8a5a] | 5661 | omega[i+1]=list(); |
---|
| 5662 | for (j=1; j<=size(omega[i]); j++) |
---|
| 5663 | { |
---|
| 5664 | if (omega[i][j][size(omega[i][j])]<n) |
---|
| 5665 | { |
---|
| 5666 | for (k=omega[i][j][size(omega[i][j])]+1; k<=n; k++) |
---|
| 5667 | { |
---|
| 5668 | omega[i+1][size(omega[i+1])+1]=omega[i][j]; |
---|
| 5669 | omega[i+1][size(omega[i+1])][size( omega[i+1][size(omega[i+1])])+1]=k; |
---|
| 5670 | } |
---|
| 5671 | } |
---|
| 5672 | } |
---|
[7fe9f8b] | 5673 | } |
---|
[0e8a5a] | 5674 | list omegamaps; |
---|
| 5675 | matrix om; |
---|
| 5676 | list lms; |
---|
| 5677 | omegamaps[1]=matrix(0,n,1); |
---|
| 5678 | for (i=1; i<=n; i++) |
---|
| 5679 | { |
---|
| 5680 | omegamaps[1][i,1]=var(n+i); |
---|
| 5681 | } |
---|
| 5682 | for (i=2; i<=n; i++) |
---|
| 5683 | { |
---|
| 5684 | om=matrix(0,size(omega[i+1]),size(omega[i])); |
---|
| 5685 | for (k=1; k<=size(omega[i]); k++) |
---|
| 5686 | { |
---|
| 5687 | for (l=1; l<=size(omega[i+1]); l++) |
---|
| 5688 | { |
---|
| 5689 | lms=LMSubset(omega[i][k],omega[i+1][l],1); |
---|
| 5690 | om[l,k]=lms[2]*var(n+lms[1]); |
---|
| 5691 | } |
---|
| 5692 | } |
---|
| 5693 | omegamaps[i]=om; |
---|
| 5694 | } |
---|
| 5695 | omegamaps[n+1]=matrix(0,1,1); |
---|
| 5696 | list allomega; |
---|
| 5697 | for (i=1; i<=n+1; i++) |
---|
| 5698 | { |
---|
| 5699 | allomega[2*i]=omega[n+2-i]; |
---|
| 5700 | allomega[2*i-1]=omegamaps[n+2-i]; |
---|
| 5701 | } |
---|
| 5702 | return(allomega); |
---|
| 5703 | } |
---|
| 5704 | |
---|
| 5705 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5706 | |
---|
| 5707 | static proc makeDoubleComplex(list L, list M, list Q, list G) |
---|
| 5708 | { |
---|
| 5709 | list doublecomplex; |
---|
| 5710 | int i,j,k,l; |
---|
| 5711 | int s1; |
---|
| 5712 | int s2; |
---|
| 5713 | int c; |
---|
| 5714 | int d; |
---|
| 5715 | list gens=list(); |
---|
| 5716 | for (i=1; i<=size(L) div 2; i++) |
---|
| 5717 | { |
---|
| 5718 | doublecomplex[i]=list(); |
---|
| 5719 | for (j=1; j<=size(M) div 2; j++) |
---|
| 5720 | { |
---|
| 5721 | doublecomplex[i][j]=list(); |
---|
| 5722 | doublecomplex[i][j]=list(M[2*j]+list(L[2*i-1])); |
---|
| 5723 | gens=list(); |
---|
| 5724 | doublecomplex[i][j][6]=G[i]; |
---|
| 5725 | if (size(Q[i])!=0) |
---|
| 5726 | { |
---|
| 5727 | doublecomplex[i][j][4]=tensor(unitmat(size(M[2*j])),Q[i]); |
---|
| 5728 | for (c=1; c<=size(M[2*j]); c++) |
---|
| 5729 | { |
---|
| 5730 | for (d=1; d<=ncols(Q[i]); d++) |
---|
| 5731 | { |
---|
| 5732 | gens[size(gens)+1]=list(M[2*j][c],d); |
---|
| 5733 | } |
---|
| 5734 | } |
---|
| 5735 | doublecomplex[i][j][5]=gens; |
---|
| 5736 | } |
---|
| 5737 | else |
---|
| 5738 | { |
---|
| 5739 | doublecomplex[i][j][4]=list(); |
---|
| 5740 | doublecomplex[i][j][5]=list(); |
---|
| 5741 | } |
---|
| 5742 | if (size(Q[i])!=0) |
---|
| 5743 | { |
---|
| 5744 | if (Q[i]==matrix(0,nrows(Q[i]),ncols(Q[i]))) |
---|
| 5745 | { |
---|
| 5746 | doublecomplex[i][j][4]=list(); |
---|
| 5747 | } |
---|
| 5748 | } |
---|
| 5749 | if (j!=1) |
---|
| 5750 | { |
---|
| 5751 | s1=(size(doublecomplex[i][j-1][1])-1)*doublecomplex[i][j-1][1][size(doublecomplex[i][j-1][1])]; |
---|
| 5752 | s2=(size(doublecomplex[i][j][1])-1)*doublecomplex[i][j][1][size(doublecomplex[i][j][1])]; |
---|
| 5753 | if (s1==0 or s2==0) |
---|
| 5754 | { |
---|
| 5755 | doublecomplex[i][j-1][3]=list(); |
---|
| 5756 | } |
---|
| 5757 | else |
---|
| 5758 | { |
---|
| 5759 | doublecomplex[i][j-1][3]=tensor(M[2*j-1],unitmat(L[2*i-1])); |
---|
| 5760 | } |
---|
| 5761 | |
---|
| 5762 | } |
---|
| 5763 | if (j==size(M) div 2) |
---|
| 5764 | { |
---|
| 5765 | doublecomplex[i][j][3]=list(); |
---|
| 5766 | } |
---|
| 5767 | if (i!=1) |
---|
| 5768 | { |
---|
| 5769 | s1=(size(doublecomplex[i-1][j][1])-1)*doublecomplex[i-1][j][1][size(doublecomplex[i-1][j][1])]; |
---|
| 5770 | s2=(size(doublecomplex[i][j][1])-1)*doublecomplex[i][j][1][size(doublecomplex[i][j][1])]; |
---|
| 5771 | if (s1==0 or s2==0) |
---|
| 5772 | { |
---|
| 5773 | doublecomplex[i-1][j][2]=list(); |
---|
| 5774 | } |
---|
| 5775 | else |
---|
| 5776 | { |
---|
| 5777 | doublecomplex[i-1][j][2]=tensor(unitmat(size(M[2*j])),L[2*(i-1)]); |
---|
| 5778 | } |
---|
| 5779 | } |
---|
| 5780 | if (i==size(L) div 2) |
---|
| 5781 | { |
---|
| 5782 | doublecomplex[i][j][2]=list(); |
---|
| 5783 | } |
---|
| 5784 | } |
---|
| 5785 | } |
---|
| 5786 | return(doublecomplex); |
---|
[7fe9f8b] | 5787 | } |
---|
| 5788 | |
---|
[0e8a5a] | 5789 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5790 | |
---|
| 5791 | static proc transferDiffforms(matrix m, list L) |
---|
| 5792 | { |
---|
| 5793 | int i; |
---|
| 5794 | list transfered; |
---|
| 5795 | if (size(L[4])==0) |
---|
| 5796 | { |
---|
| 5797 | return(list()); |
---|
| 5798 | } |
---|
| 5799 | if (size(L[5])==0) |
---|
| 5800 | { |
---|
| 5801 | return(list()); |
---|
| 5802 | } |
---|
| 5803 | m=m*L[4]; |
---|
| 5804 | list transferedm=list(); |
---|
| 5805 | int si=L[5][size(L[5])][2];//Anzahl der direkten Summanden in \oplus R_F_I |
---|
| 5806 | matrix fortrans=matrix(0,1,si); |
---|
| 5807 | list omegagen=list(); |
---|
| 5808 | list save=list(); |
---|
| 5809 | int t; |
---|
| 5810 | int c; |
---|
| 5811 | int j; |
---|
| 5812 | list converteddiff; |
---|
| 5813 | vector w; |
---|
| 5814 | poly p=1; |
---|
| 5815 | for (i=1; i<=ncols(m); i++) |
---|
| 5816 | { |
---|
| 5817 | if (m[1,i]!=0) |
---|
| 5818 | { |
---|
| 5819 | if (size(omegagen)==0) |
---|
| 5820 | { |
---|
| 5821 | omegagen=L[5][i][1]; |
---|
| 5822 | fortrans[1,L[5][i][2]]= fortrans[1,L[5][i][2]]+m[1,i]; |
---|
| 5823 | } |
---|
| 5824 | else |
---|
| 5825 | { |
---|
| 5826 | t=0; |
---|
| 5827 | for (j=1; j<=size(omegagen);j++) |
---|
| 5828 | { |
---|
| 5829 | if (size(omegagen[j])!=0) |
---|
| 5830 | { |
---|
| 5831 | if (omegagen[j]!=L[5][i][1][j]) |
---|
| 5832 | { |
---|
| 5833 | t=1; |
---|
| 5834 | } |
---|
| 5835 | } |
---|
| 5836 | } |
---|
| 5837 | if (t==0) |
---|
| 5838 | { |
---|
| 5839 | fortrans[1,L[5][i][2]]= fortrans[1,L[5][i][2]]+m[1,i]; |
---|
| 5840 | } |
---|
| 5841 | else |
---|
| 5842 | { |
---|
| 5843 | converteddiff=list(); |
---|
| 5844 | for (j=1; j<=ncols(fortrans); j++) |
---|
| 5845 | { |
---|
| 5846 | if (fortrans[1,j]!=0) |
---|
| 5847 | { |
---|
| 5848 | w=[p,L[6][j]]; |
---|
| 5849 | converteddiff[j]=dmodActionRat(fortrans[1,j],w); |
---|
| 5850 | } |
---|
| 5851 | else |
---|
| 5852 | { |
---|
| 5853 | converteddiff[j]=0; |
---|
| 5854 | } |
---|
| 5855 | |
---|
| 5856 | } |
---|
| 5857 | save[size(save)+1]=list(converteddiff,omegagen); |
---|
| 5858 | omegagen=L[5][i][1]; |
---|
| 5859 | fortrans=matrix(0,1,si); |
---|
| 5860 | fortrans[1,L[5][i][2]]= fortrans[1,L[5][i][2]]+m[1,i]; |
---|
| 5861 | } |
---|
| 5862 | } |
---|
| 5863 | } |
---|
| 5864 | } |
---|
| 5865 | if (fortrans==matrix(0,1,si)) |
---|
| 5866 | { |
---|
| 5867 | return(list()); |
---|
| 5868 | } |
---|
| 5869 | converteddiff=list(); |
---|
| 5870 | for (j=1; j<=ncols(fortrans); j++) |
---|
| 5871 | { |
---|
| 5872 | if (fortrans[1,j]!=0) |
---|
| 5873 | { |
---|
| 5874 | w=[p,L[6][j]]; |
---|
| 5875 | converteddiff[j]=dmodActionRat(fortrans[1,j],w); |
---|
| 5876 | } |
---|
| 5877 | else |
---|
| 5878 | { |
---|
| 5879 | converteddiff[j]=0; |
---|
| 5880 | } |
---|
| 5881 | } |
---|
| 5882 | save[size(save)+1]=list(converteddiff,omegagen); |
---|
| 5883 | return(save); |
---|
| 5884 | } |
---|
[7fe9f8b] | 5885 | |
---|
| 5886 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5887 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5888 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5889 | /* |
---|
| 5890 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5891 | FURTHER EXAMPLES FOR TESTING THE PROCEDURES |
---|
| 5892 | //////////////////////////////////////////////////////////////////////////////////// |
---|
| 5893 | LIB "derham.lib"; |
---|
| 5894 | |
---|
| 5895 | //---------------------------------------- |
---|
| 5896 | //EXAMPLE 1 |
---|
| 5897 | //---------------------------------------- |
---|
| 5898 | ring r=0,(x,y),dp; |
---|
| 5899 | poly f=y2-x3-2x+3; |
---|
| 5900 | list L=deRhamCohomology(f); |
---|
| 5901 | L; |
---|
| 5902 | kill r; |
---|
| 5903 | |
---|
| 5904 | //---------------------------------------- |
---|
| 5905 | //EXAMPLE 2 |
---|
| 5906 | //---------------------------------------- |
---|
| 5907 | ring r=0,(x,y),dp; |
---|
| 5908 | poly f=y2-x3-x; |
---|
| 5909 | list L=deRhamCohomology(f); |
---|
| 5910 | L; |
---|
| 5911 | kill r; |
---|
| 5912 | |
---|
| 5913 | //---------------------------------------- |
---|
| 5914 | //EXAMPLE 3 |
---|
| 5915 | //---------------------------------------- |
---|
| 5916 | ring r=0,(x,y),dp; |
---|
| 5917 | list C=list(x2-1,(x+1)*y,y*(y2+2x+1)); |
---|
| 5918 | list L=deRhamCohomology(C); |
---|
| 5919 | L; |
---|
| 5920 | kill r; |
---|
| 5921 | |
---|
| 5922 | //---------------------------------------- |
---|
| 5923 | //EXAMPLE 4 |
---|
| 5924 | //---------------------------------------- |
---|
| 5925 | ring r=0,(x,y,z),dp; |
---|
| 5926 | list C=list(x*(x-1),y,z*(z-1),z*(x-1)); |
---|
| 5927 | list L=deRhamCohomology(C); |
---|
| 5928 | L; |
---|
| 5929 | kill r; |
---|
| 5930 | |
---|
| 5931 | //---------------------------------------- |
---|
| 5932 | //EXAMPLE 5 |
---|
| 5933 | //---------------------------------------- |
---|
| 5934 | ring r=0,(x,y,z),dp; |
---|
| 5935 | list C=list(x*y,y*z); |
---|
| 5936 | list L=deRhamCohomology(C,"Vdres"); |
---|
| 5937 | L; |
---|
| 5938 | kill r; |
---|
| 5939 | |
---|
| 5940 | //---------------------------------------- |
---|
| 5941 | //EXAMPLE 6 |
---|
| 5942 | //---------------------------------------- |
---|
| 5943 | ring r=0,(x,y,z,u),dp; |
---|
| 5944 | list C=list(x,y,z,u); |
---|
| 5945 | list L=deRhamCohomology(C); |
---|
| 5946 | L; |
---|
| 5947 | kill r; |
---|
| 5948 | |
---|
| 5949 | //---------------------------------------- |
---|
| 5950 | //EXAMPLE 7 |
---|
| 5951 | //---------------------------------------- |
---|
| 5952 | ring r=0,(x,y,z),dp; |
---|
| 5953 | poly f=x3+y3+z3; |
---|
| 5954 | list L=deRhamCohomology(f); |
---|
| 5955 | L; |
---|
| 5956 | kill r; |
---|
| 5957 | |
---|
| 5958 | //---------------------------------------- |
---|
| 5959 | //EXAMPLE 8 |
---|
| 5960 | //---------------------------------------- |
---|
| 5961 | ring r=0,(x,y,z),dp; |
---|
| 5962 | poly f=x2+y2+z2; |
---|
| 5963 | list L=deRhamCohomology(f,"Vdres"); |
---|
| 5964 | L; |
---|
| 5965 | kill r; |
---|
| 5966 | |
---|
| 5967 | //---------------------------------------- |
---|
| 5968 | //EXAMPLE 9 |
---|
| 5969 | //---------------------------------------- |
---|
| 5970 | ring r=0,(x,y,z,u),dp; |
---|
| 5971 | list C=list(x2+y2+z2,u); |
---|
| 5972 | list L=deRhamCohomology(C); |
---|
| 5973 | L; |
---|
| 5974 | kill r; |
---|
| 5975 | |
---|
| 5976 | |
---|
| 5977 | //---------------------------------------- |
---|
| 5978 | //EXAMPLE 10 |
---|
| 5979 | //---------------------------------------- |
---|
| 5980 | ring r=0,(x,y,z),dp; |
---|
| 5981 | list C=list((x*(y-1),y2-1)); |
---|
| 5982 | list L=deRhamCohomology(C); |
---|
| 5983 | L; |
---|
| 5984 | kill r; |
---|
| 5985 | |
---|
| 5986 | |
---|
| 5987 | */ |
---|