[720ff4] | 1 | // $Id: sing.lib,v 1.26 2005-04-25 16:58:46 Singular Exp $ |
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[6f2edc] | 2 | //(GMG/BM, last modified 26.06.96) |
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[3d124a7] | 3 | /////////////////////////////////////////////////////////////////////////////// |
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[720ff4] | 4 | version="$Id: sing.lib,v 1.26 2005-04-25 16:58:46 Singular Exp $"; |
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[fd3fb7] | 5 | category="Singularities"; |
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[5480da] | 6 | info=" |
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[8bb77b] | 7 | LIBRARY: sing.lib Invariants of Singularities |
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[80a0f0] | 8 | AUTHORS: Gert-Martin Greuel, email: greuel@mathematik.uni-kl.de @* |
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[091424] | 9 | Bernd Martin, email: martin@math.tu-cottbus.de |
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[3d124a7] | 10 | |
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[f34c37c] | 11 | PROCEDURES: |
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[0fbdd1] | 12 | codim (id1, id2); vector space dimension of of id2/id1 if finite |
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[3d124a7] | 13 | deform(i); infinitesimal deformations of ideal i |
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| 14 | dim_slocus(i); dimension of singular locus of ideal i |
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| 15 | is_active(f,id); is poly f an active element mod id? (id ideal/module) |
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| 16 | is_ci(i); is ideal i a complete intersection? |
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| 17 | is_is(i); is ideal i an isolated singularity? |
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| 18 | is_reg(f,id); is poly f a regular element mod id? (id ideal/module) |
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| 19 | is_regs(i[,id]); are gen's of ideal i regular sequence modulo id? |
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[c69ea5] | 20 | locstd(i); SB for local degree ordering without cancelling units |
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[3d124a7] | 21 | milnor(i); milnor number of ideal i; (assume i is ICIS in nf) |
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| 22 | nf_icis(i); generic combinations of generators; get ICIS in nf |
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| 23 | slocus(i); ideal of singular locus of ideal i |
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[cd6dbb2] | 24 | qhspectrum(f,w); spectrum numbers of w-homogeneous polynomial f |
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[3d124a7] | 25 | Tjurina(i); SB of Tjurina module of ideal i (assume i is ICIS) |
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| 26 | tjurina(i); Tjurina number of ideal i (assume i is ICIS) |
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[0b59f5] | 27 | T_1(i); T^1-module of ideal i |
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| 28 | T_2((i); T^2-module of ideal i |
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| 29 | T_12(i); T^1- and T^2-module of ideal i |
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[c69ea5] | 30 | tangentcone(id); compute tangent cone of id |
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| 31 | |
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[5480da] | 32 | "; |
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[3d124a7] | 33 | |
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[6f2edc] | 34 | LIB "inout.lib"; |
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[3d124a7] | 35 | LIB "random.lib"; |
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[c69ea5] | 36 | LIB "primdec.lib"; |
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[3d124a7] | 37 | /////////////////////////////////////////////////////////////////////////////// |
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| 38 | |
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| 39 | proc deform (ideal id) |
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[d2b2a7] | 40 | "USAGE: deform(id); id=ideal or poly |
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[3d124a7] | 41 | RETURN: matrix, columns are kbase of infinitesimal deformations |
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[6f2edc] | 42 | EXAMPLE: example deform; shows an example |
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[d2b2a7] | 43 | " |
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[6f2edc] | 44 | { |
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[0b59f5] | 45 | list L=T_1(id,""); |
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[6f2edc] | 46 | def K=L[1]; attrib(K,"isSB",1); |
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| 47 | return(L[2]*kbase(K)); |
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[3d124a7] | 48 | } |
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| 49 | example |
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| 50 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 51 | ring r = 32003,(x,y,z),ds; |
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| 52 | ideal i = xy,xz,yz; |
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| 53 | matrix T = deform(i); |
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| 54 | print(T); |
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| 55 | print(deform(x3+y5+z2)); |
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[3d124a7] | 56 | } |
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| 57 | /////////////////////////////////////////////////////////////////////////////// |
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| 58 | |
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| 59 | proc dim_slocus (ideal i) |
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[d2b2a7] | 60 | "USAGE: dim_slocus(i); i ideal or poly |
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[3d124a7] | 61 | RETURN: dimension of singular locus of i |
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| 62 | EXAMPLE: example dim_slocus; shows an example |
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[d2b2a7] | 63 | " |
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[3d124a7] | 64 | { |
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| 65 | return(dim(std(slocus(i)))); |
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| 66 | } |
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| 67 | example |
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| 68 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 69 | ring r = 32003,(x,y,z),ds; |
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| 70 | ideal i = x5+y6+z6,x2+2y2+3z2; |
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[3d124a7] | 71 | dim_slocus(i); |
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| 72 | } |
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| 73 | /////////////////////////////////////////////////////////////////////////////// |
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| 74 | |
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| 75 | proc is_active (poly f, id) |
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[d2b2a7] | 76 | "USAGE: is_active(f,id); f poly, id ideal or module |
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[3d124a7] | 77 | RETURN: 1 if f is an active element modulo id (i.e. dim(id)=dim(id+f*R^n)+1, |
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| 78 | if id is a submodule of R^n) resp. 0 if f is not active. |
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[6f2edc] | 79 | The basering may be a quotient ring |
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[3d124a7] | 80 | NOTE: regular parameters are active but not vice versa (id may have embedded |
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| 81 | components). proc is_reg tests whether f is a regular parameter |
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| 82 | EXAMPLE: example is_active; shows an example |
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[d2b2a7] | 83 | " |
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[3d124a7] | 84 | { |
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[6f2edc] | 85 | if( size(id)==0 ) { return(1); } |
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[3d124a7] | 86 | if( typeof(id)=="ideal" ) { ideal m=f; } |
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[6f2edc] | 87 | if( typeof(id)=="module" ) { module m=f*freemodule(nrows(id)); } |
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[3d124a7] | 88 | return(dim(std(id))-dim(std(id+m))); |
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| 89 | } |
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| 90 | example |
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| 91 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 92 | ring r =32003,(x,y,z),ds; |
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| 93 | ideal i = yx3+y,yz3+y3z; |
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| 94 | poly f = x; |
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[3d124a7] | 95 | is_active(f,i); |
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[6f2edc] | 96 | qring q = std(x4y5); |
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| 97 | poly f = x; |
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| 98 | module m = [yx3+x,yx3+y3x]; |
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[3d124a7] | 99 | is_active(f,m); |
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| 100 | } |
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| 101 | /////////////////////////////////////////////////////////////////////////////// |
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| 102 | |
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| 103 | proc is_ci (ideal i) |
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[d2b2a7] | 104 | "USAGE: is_ci(i); i ideal |
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[3d124a7] | 105 | RETURN: intvec = sequence of dimensions of ideals (j[1],...,j[k]), for |
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[6f2edc] | 106 | k=1,...,size(j), where j is minimal base of i. i is a complete |
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| 107 | intersection if last number equals nvars-size(i) |
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| 108 | NOTE: dim(0-ideal) = -1. You may first apply simplify(i,10); in order to |
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| 109 | delete zeroes and multiples from set of generators |
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| 110 | printlevel >=0: display comments (default) |
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[3d124a7] | 111 | EXAMPLE: example is_ci; shows an example |
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[d2b2a7] | 112 | " |
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[3d124a7] | 113 | { |
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| 114 | int n; intvec dimvec; ideal id; |
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| 115 | i=minbase(i); |
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| 116 | int s = ncols(i); |
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[6f2edc] | 117 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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[3d124a7] | 118 | //--------------------------- compute dimensions ------------------------------ |
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[6f2edc] | 119 | for( n=1; n<=s; n=n+1 ) |
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| 120 | { |
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[3d124a7] | 121 | id = i[1..n]; |
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| 122 | dimvec[n] = dim(std(id)); |
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| 123 | } |
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| 124 | n = dimvec[s]; |
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[6f2edc] | 125 | //--------------------------- output ------------------------------------------ |
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| 126 | if( n+s != nvars(basering) ) |
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| 127 | { dbprint(p,"// no complete intersection"); } |
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| 128 | if( n+s == nvars(basering) ) |
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| 129 | { dbprint(p,"// complete intersection of dim "+string(n)); } |
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| 130 | dbprint(p,"// dim-sequence:"); |
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[3d124a7] | 131 | return(dimvec); |
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| 132 | } |
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| 133 | example |
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[6f2edc] | 134 | { "EXAMPLE:"; echo = 2; |
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| 135 | int p = printlevel; |
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| 136 | printlevel = 1; // display comments |
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| 137 | ring r = 32003,(x,y,z),ds; |
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| 138 | ideal i = x4+y5+z6,xyz,yx2+xz2+zy7; |
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| 139 | is_ci(i); |
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| 140 | i = xy,yz; |
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[3d124a7] | 141 | is_ci(i); |
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[6f2edc] | 142 | printlevel = p; |
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[3d124a7] | 143 | } |
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| 144 | /////////////////////////////////////////////////////////////////////////////// |
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| 145 | |
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| 146 | proc is_is (ideal i) |
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[d2b2a7] | 147 | "USAGE: is_is(id); id ideal or poly |
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[3d124a7] | 148 | RETURN: intvec = sequence of dimensions of singular loci of ideals |
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[80a0f0] | 149 | generated by id[1]..id[i], k = 1..size(id); @* |
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| 150 | dim(0-ideal) = -1; |
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[3d124a7] | 151 | id defines an isolated singularity if last number is 0 |
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[6f2edc] | 152 | NOTE: printlevel >=0: display comments (default) |
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[3d124a7] | 153 | EXAMPLE: example is_is; shows an example |
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[d2b2a7] | 154 | " |
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[3d124a7] | 155 | { |
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| 156 | int l; intvec dims; ideal j; |
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[6f2edc] | 157 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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[3d124a7] | 158 | //--------------------------- compute dimensions ------------------------------ |
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[6f2edc] | 159 | for( l=1; l<=ncols(i); l=l+1 ) |
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[3d124a7] | 160 | { |
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[6f2edc] | 161 | j = i[1..l]; |
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[3d124a7] | 162 | dims[l] = dim(std(slocus(j))); |
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| 163 | } |
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[6f2edc] | 164 | dbprint(p,"// dim of singular locus = "+string(dims[size(dims)]), |
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| 165 | "// isolated singularity if last number is 0 in dim-sequence:"); |
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[3d124a7] | 166 | return(dims); |
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| 167 | } |
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| 168 | example |
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| 169 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 170 | int p = printlevel; |
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| 171 | printlevel = 1; |
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| 172 | ring r = 32003,(x,y,z),ds; |
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| 173 | ideal i = x2y,x4+y5+z6,yx2+xz2+zy7; |
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[3d124a7] | 174 | is_is(i); |
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[6f2edc] | 175 | poly f = xy+yz; |
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[3d124a7] | 176 | is_is(f); |
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[6f2edc] | 177 | printlevel = p; |
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[3d124a7] | 178 | } |
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| 179 | /////////////////////////////////////////////////////////////////////////////// |
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| 180 | |
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| 181 | proc is_reg (poly f, id) |
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[d2b2a7] | 182 | "USAGE: is_reg(f,id); f poly, id ideal or module |
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[3d124a7] | 183 | RETURN: 1 if multiplication with f is injective modulo id, 0 otherwise |
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| 184 | NOTE: let R be the basering and id a submodule of R^n. The procedure checks |
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| 185 | injectivity of multiplication with f on R^n/id. The basering may be a |
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[80a0f0] | 186 | quotient ring |
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[3d124a7] | 187 | EXAMPLE: example is_reg; shows an example |
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[d2b2a7] | 188 | " |
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[3d124a7] | 189 | { |
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| 190 | if( f==0 ) { return(0); } |
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| 191 | int d,ii; |
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| 192 | def q = quotient(id,ideal(f)); |
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| 193 | id=std(id); |
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| 194 | d=size(q); |
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[6f2edc] | 195 | for( ii=1; ii<=d; ii=ii+1 ) |
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[3d124a7] | 196 | { |
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| 197 | if( reduce(q[ii],id)!=0 ) |
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| 198 | { return(0); } |
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| 199 | } |
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| 200 | return(1); |
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| 201 | } |
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[6f2edc] | 202 | example |
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[3d124a7] | 203 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 204 | ring r = 32003,(x,y),ds; |
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| 205 | ideal i = x8,y8; |
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| 206 | ideal j = (x+y)^4; |
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| 207 | i = intersect(i,j); |
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| 208 | poly f = xy; |
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[3d124a7] | 209 | is_reg(f,i); |
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| 210 | } |
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| 211 | /////////////////////////////////////////////////////////////////////////////// |
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| 212 | |
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| 213 | proc is_regs (ideal i, list #) |
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[d2b2a7] | 214 | "USAGE: is_regs(i[,id]); i poly, id ideal or module (default: id=0) |
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[3d124a7] | 215 | RETURN: 1 if generators of i are a regular sequence modulo id, 0 otherwise |
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| 216 | NOTE: let R be the basering and id a submodule of R^n. The procedure checks |
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| 217 | injectivity of multiplication with i[k] on R^n/id+i[1..k-1]. |
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[6f2edc] | 218 | The basering may be a quotient ring |
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| 219 | printlevel >=0: display comments (default) |
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| 220 | printlevel >=1: display comments during computation |
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[3d124a7] | 221 | EXAMPLE: example is_regs; shows an example |
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[d2b2a7] | 222 | " |
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[3d124a7] | 223 | { |
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[6f2edc] | 224 | int d,ii,r; |
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| 225 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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[3d124a7] | 226 | if( size(#)==0 ) { ideal id; } |
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| 227 | else { def id=#[1]; } |
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| 228 | if( size(i)==0 ) { return(0); } |
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[6f2edc] | 229 | d=size(i); |
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[3d124a7] | 230 | if( typeof(id)=="ideal" ) { ideal m=1; } |
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[6f2edc] | 231 | if( typeof(id)=="module" ) { module m=freemodule(nrows(id)); } |
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| 232 | for( ii=1; ii<=d; ii=ii+1 ) |
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| 233 | { |
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| 234 | if( p>=2 ) |
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[3d124a7] | 235 | { "// checking whether element",ii,"is regular mod 1 ..",ii-1; } |
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[6f2edc] | 236 | if( is_reg(i[ii],id)==0 ) |
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| 237 | { |
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| 238 | dbprint(p,"// elements 1.."+string(ii-1)+" are regular, " + |
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| 239 | string(ii)+" is not regular mod 1.."+string(ii-1)); |
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| 240 | return(0); |
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[3d124a7] | 241 | } |
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[6f2edc] | 242 | id=id+i[ii]*m; |
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[3d124a7] | 243 | } |
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[6f2edc] | 244 | if( p>=1 ) { "// elements are a regular sequence of length",d; } |
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[3d124a7] | 245 | return(1); |
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| 246 | } |
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[6f2edc] | 247 | example |
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[3d124a7] | 248 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 249 | int p = printlevel; |
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| 250 | printlevel = 1; |
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| 251 | ring r1 = 32003,(x,y,z),ds; |
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| 252 | ideal i = x8,y8,(x+y)^4; |
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[3d124a7] | 253 | is_regs(i); |
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[6f2edc] | 254 | module m = [x,0,y]; |
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| 255 | i = x8,(x+z)^4;; |
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[3d124a7] | 256 | is_regs(i,m); |
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[6f2edc] | 257 | printlevel = p; |
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[3d124a7] | 258 | } |
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| 259 | /////////////////////////////////////////////////////////////////////////////// |
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| 260 | |
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| 261 | proc milnor (ideal i) |
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[d2b2a7] | 262 | "USAGE: milnor(i); i ideal or poly |
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[3d124a7] | 263 | RETURN: Milnor number of i, if i is ICIS (isolated complete intersection |
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[6f2edc] | 264 | singularity) in generic form, resp. -1 if not |
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[3d124a7] | 265 | NOTE: use proc nf_icis to put generators in generic form |
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[6f2edc] | 266 | printlevel >=0: display comments (default) |
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[3d124a7] | 267 | EXAMPLE: example milnor; shows an example |
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[d2b2a7] | 268 | " |
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[6f2edc] | 269 | { |
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| 270 | i = simplify(i,10); //delete zeroes and multiples from set of generators |
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[3d124a7] | 271 | int n = size(i); |
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| 272 | int l,q,m_nr; ideal t; intvec disc; |
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[6f2edc] | 273 | int p = printlevel-voice+3; // p=printlevel+1 (default: p=1) |
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| 274 | //---------------------------- hypersurface case ------------------------------ |
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[457d505] | 275 | if( n==1 or i==0 ) |
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[3d124a7] | 276 | { |
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| 277 | i = std(jacob(i[1])); |
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[6f2edc] | 278 | m_nr = vdim(i); |
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| 279 | if( m_nr<0 and p>=1 ) { "// no isolated singularity"; } |
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| 280 | return(m_nr); |
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[3d124a7] | 281 | } |
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| 282 | //------------ isolated complete intersection singularity (ICIS) -------------- |
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| 283 | for( l=n; l>0; l=l-1) |
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[6f2edc] | 284 | { t = minor(jacob(i),l); |
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| 285 | i[l] = 0; |
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[3d124a7] | 286 | q = vdim(std(i+t)); |
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| 287 | disc[l]= q; |
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| 288 | if( q ==-1 ) |
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[6f2edc] | 289 | { if( p>=1 ) |
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[3d124a7] | 290 | { "// not in generic form or no ICIS; use proc nf_icis to put"; |
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[6f2edc] | 291 | "// generators in generic form and then try milnor again!"; } |
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[3d124a7] | 292 | return(q); |
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| 293 | } |
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[6f2edc] | 294 | m_nr = q-m_nr; |
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[3d124a7] | 295 | } |
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[6f2edc] | 296 | //---------------------------- change sign ------------------------------------ |
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| 297 | if (m_nr < 0) { m_nr=-m_nr; } |
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| 298 | if( p>=1 ) { "//sequence of discriminant numbers:",disc; } |
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[3d124a7] | 299 | return(m_nr); |
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| 300 | } |
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| 301 | example |
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| 302 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 303 | int p = printlevel; |
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| 304 | printlevel = 1; |
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| 305 | ring r = 32003,(x,y,z),ds; |
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| 306 | ideal j = x5+y6+z6,x2+2y2+3z2,xyz+yx; |
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[3d124a7] | 307 | milnor(j); |
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[6f2edc] | 308 | poly f = x7+y7+(x-y)^2*x2y2+z2; |
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| 309 | milnor(f); |
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| 310 | printlevel = p; |
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[3d124a7] | 311 | } |
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| 312 | /////////////////////////////////////////////////////////////////////////////// |
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| 313 | |
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| 314 | proc nf_icis (ideal i) |
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[d2b2a7] | 315 | "USAGE: nf_icis(i); i ideal |
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[3d124a7] | 316 | RETURN: ideal = generic linear combination of generators of i if i is an ICIS |
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| 317 | (isolated complete intersection singularity), return i if not |
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| 318 | NOTE: this proc is useful in connection with proc milnor |
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[6f2edc] | 319 | printlevel >=0: display comments (default) |
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[3d124a7] | 320 | EXAMPLE: example nf_icis; shows an example |
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[d2b2a7] | 321 | " |
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[3d124a7] | 322 | { |
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| 323 | i = simplify(i,10); //delete zeroes and multiples from set of generators |
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[6f2edc] | 324 | int p,b = 100,0; |
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[3d124a7] | 325 | int n = size(i); |
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| 326 | matrix mat=freemodule(n); |
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[6f2edc] | 327 | int P = printlevel-voice+3; // P=printlevel+1 (default: P=1) |
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| 328 | //---------------------------- test: complete intersection? ------------------- |
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[3d124a7] | 329 | intvec sl = is_ci(i); |
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[6f2edc] | 330 | if( n+sl[n] != nvars(basering) ) |
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| 331 | { |
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| 332 | dbprint(P,"// no complete intersection"); |
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| 333 | return(i); |
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[3d124a7] | 334 | } |
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[6f2edc] | 335 | //--------------- test: isolated singularity in generic form? ----------------- |
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[3d124a7] | 336 | sl = is_is(i); |
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| 337 | if ( sl[n] != 0 ) |
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| 338 | { |
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[6f2edc] | 339 | dbprint(P,"// no isolated singularity"); |
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[3d124a7] | 340 | return(i); |
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| 341 | } |
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[6f2edc] | 342 | //------------ produce generic linear combinations of generators -------------- |
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[3d124a7] | 343 | int prob; |
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[6f2edc] | 344 | while ( sum(sl) != 0 ) |
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[3d124a7] | 345 | { prob=prob+1; |
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[6f2edc] | 346 | p=p-25; b=b+10; |
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[3d124a7] | 347 | i = genericid(i,p,b); // proc genericid from random.lib |
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| 348 | sl = is_is(i); |
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| 349 | } |
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[6f2edc] | 350 | dbprint(P,"// ICIS in generic form after "+string(prob)+" genericity loop(s)"); |
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| 351 | return(i); |
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[3d124a7] | 352 | } |
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| 353 | example |
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| 354 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 355 | int p = printlevel; |
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| 356 | printlevel = 1; |
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| 357 | ring r = 32003,(x,y,z),ds; |
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| 358 | ideal i = x3+y4,z4+yx; |
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| 359 | nf_icis(i); |
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| 360 | ideal j = x3+y4,xy,yz; |
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[3d124a7] | 361 | nf_icis(j); |
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[6f2edc] | 362 | printlevel = p; |
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[3d124a7] | 363 | } |
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| 364 | /////////////////////////////////////////////////////////////////////////////// |
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| 365 | |
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| 366 | proc slocus (ideal i) |
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[9f9f2c] | 367 | "USAGE: slocus(i); i ideal |
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[c69ea5] | 368 | RETURN: ideal of singular locus of i |
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| 369 | EXAMPLE: example slocus; shows an example |
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| 370 | " |
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| 371 | { |
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| 372 | def R=basering; |
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| 373 | int j,k; |
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| 374 | ideal res; |
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| 375 | |
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| 376 | if(ord_test(basering)!=1) |
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| 377 | { |
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| 378 | string va=varstr(basering); |
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| 379 | if( size( parstr(basering))>0){va=va+","+parstr(basering);} |
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| 380 | execute ("ring S = ("+charstr(basering)+"),("+va+"),dp;"); |
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| 381 | ideal i=imap(R,i); |
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| 382 | list l=equidim(i); |
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| 383 | setring R; |
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| 384 | list l=imap(S,l); |
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| 385 | } |
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| 386 | else |
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| 387 | { |
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| 388 | list l=equidim(i); |
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| 389 | } |
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| 390 | int n=size(l); |
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| 391 | if (n==1){return(slocusEqi(i));} |
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| 392 | res=slocusEqi(l[1]); |
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| 393 | for(j=2;j<=n;j++){res=intersect(res,slocusEqi(l[j]));} |
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| 394 | for(j=1;j<n;j++) |
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| 395 | { |
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| 396 | for(k=j+1;k<=n;k++){res=intersect(res,l[j]+l[k]);} |
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| 397 | } |
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| 398 | return(res); |
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| 399 | } |
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| 400 | example |
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| 401 | { "EXAMPLE:"; echo = 2; |
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| 402 | ring r = 0,(u,v,w,x,y,z),dp; |
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| 403 | ideal i = wx,wy,wz,vx,vy,vz,ux,uy,uz,y3-x2;; |
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| 404 | slocus(i); |
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| 405 | } |
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| 406 | /////////////////////////////////////////////////////////////////////////////// |
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| 407 | |
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| 408 | static proc slocusEqi (ideal i) |
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| 409 | "USAGE: slocus(i); i ideal |
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[9f9f2c] | 410 | RETURN: ideal of singular locus of i if i is pure dimensional |
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| 411 | NOTE: this proc returns i and c-minors of jacobian ideal of i where c is the |
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[b9b906] | 412 | codimension of i. Hence, if i is not pure dimensional, slocus may |
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| 413 | return an ideal such that its 0-locus is strictly contained in the |
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[9f9f2c] | 414 | singular locus of i |
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[3d124a7] | 415 | EXAMPLE: example slocus; shows an example |
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[d2b2a7] | 416 | " |
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[3d124a7] | 417 | { |
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[c69ea5] | 418 | ideal ist=std(i); |
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| 419 | if(deg(ist[1])==0){return(ist);} |
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| 420 | int cod = nvars(basering)-dim(ist); |
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[3d124a7] | 421 | i = i+minor(jacob(i),cod); |
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[6f2edc] | 422 | return(i); |
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[3d124a7] | 423 | } |
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| 424 | example |
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| 425 | { "EXAMPLE:"; echo = 2; |
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[9f9f2c] | 426 | ring r = 0,(x,y,z),ds; |
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[6f2edc] | 427 | ideal i = x5+y6+z6,x2+2y2+3z2; |
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[9f9f2c] | 428 | slocus(i); |
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[3d124a7] | 429 | } |
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| 430 | /////////////////////////////////////////////////////////////////////////////// |
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| 431 | |
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[cd6dbb2] | 432 | proc qhspectrum (poly f, intvec w) |
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| 433 | "USAGE: qhspectrum(f,w); f=poly, w=intvec; |
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[0fbdd1] | 434 | ASSUME: f is a weighted homogeneous isolated singularity w.r.t. the weights |
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| 435 | given by w; w must consist of as many positive integers as there |
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| 436 | are variables of the basering |
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| 437 | COMPUTE: the spectral numbers of the w-homogeneous polynomial f, computed in a |
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[80a0f0] | 438 | ring of characteristic 0 |
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[82716e] | 439 | RETURN: intvec d,s1,...,su where: |
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[c69ea5] | 440 | d = w-degree(f) and si/d = i-th spectral-number(f) |
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[82716e] | 441 | No return value if basering has parameters or if f is no isolated |
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[0fbdd1] | 442 | singularity, displays a warning in this case |
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[cd6dbb2] | 443 | EXAMPLE: example qhspectrum; shows an example |
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[d2b2a7] | 444 | " |
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[0fbdd1] | 445 | { |
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| 446 | int i,d,W; |
---|
| 447 | intvec sp; |
---|
| 448 | def r = basering; |
---|
| 449 | if( find(charstr(r),",")!=0 ) |
---|
| 450 | { |
---|
| 451 | "// coefficient field must not have parameters!"; |
---|
| 452 | return(); |
---|
| 453 | } |
---|
| 454 | ring s = 0,x(1..nvars(r)),ws(w); |
---|
| 455 | map phi = r,maxideal(1); |
---|
| 456 | poly f = phi(f); |
---|
| 457 | d = ord(f); |
---|
| 458 | W = sum(w)-d; |
---|
| 459 | ideal k = std(jacob(f)); |
---|
| 460 | if( vdim(k) == -1 ) |
---|
| 461 | { |
---|
| 462 | "// f is no isolated singuarity!"; |
---|
| 463 | return(); |
---|
| 464 | } |
---|
| 465 | k = kbase(k); |
---|
| 466 | for (i=1; i<=size(k); i++) |
---|
[82716e] | 467 | { |
---|
[0fbdd1] | 468 | sp[i]=W+ord(k[i]); |
---|
| 469 | } |
---|
| 470 | list L = sort(sp); |
---|
| 471 | sp = d,L[1]; |
---|
| 472 | return(sp); |
---|
| 473 | } |
---|
[82716e] | 474 | example |
---|
[0fbdd1] | 475 | { "EXAMPLE:"; echo = 2; |
---|
| 476 | ring r; |
---|
| 477 | poly f=x3+y5+z2; |
---|
| 478 | intvec w=10,6,15; |
---|
[cd6dbb2] | 479 | qhspectrum(f,w); |
---|
[0fbdd1] | 480 | // the spectrum numbers are: |
---|
| 481 | // 1/30,7/30,11/30,13/30,17/30,19/30,23/30,29/30 |
---|
| 482 | } |
---|
| 483 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 484 | |
---|
[3d124a7] | 485 | proc Tjurina (id, list #) |
---|
[d2b2a7] | 486 | "USAGE: Tjurina(id[,<any>]); id=ideal or poly |
---|
[6f2edc] | 487 | ASSUME: id=ICIS (isolated complete intersection singularity) |
---|
| 488 | RETURN: standard basis of Tjurina-module of id, |
---|
| 489 | of type module if id=ideal, resp. of type ideal if id=poly. |
---|
[80a0f0] | 490 | If a second argument is present (of any type) return a list: @* |
---|
[6f2edc] | 491 | [1] = Tjurina number, |
---|
| 492 | [2] = k-basis of miniversal deformation, |
---|
| 493 | [3] = SB of Tjurina module, |
---|
| 494 | [4] = Tjurina module |
---|
| 495 | DISPLAY: Tjurina number if printlevel >= 0 (default) |
---|
| 496 | NOTE: Tjurina number = -1 implies that id is not an ICIS |
---|
| 497 | EXAMPLE: example Tjurina; shows examples |
---|
[d2b2a7] | 498 | " |
---|
[3d124a7] | 499 | { |
---|
| 500 | //---------------------------- initialisation --------------------------------- |
---|
[6f2edc] | 501 | def i = simplify(id,10); |
---|
[3d124a7] | 502 | int tau,n = 0,size(i); |
---|
| 503 | if( size(ideal(i))==1 ) { def m=i; } // hypersurface case |
---|
| 504 | else { def m=i*freemodule(n); } // complete intersection case |
---|
| 505 | //--------------- compute Tjurina module, Tjurina number etc ------------------ |
---|
| 506 | def t1 = jacob(i)+m; // Tjurina module/ideal |
---|
| 507 | def st1 = std(t1); // SB of Tjurina module/ideal |
---|
| 508 | tau = vdim(st1); // Tjurina number |
---|
[6f2edc] | 509 | dbprint(printlevel-voice+3,"// Tjurina number = "+string(tau)); |
---|
| 510 | if( size(#)>0 ) |
---|
| 511 | { |
---|
| 512 | def kB = kbase(st1); // basis of miniversal deformation |
---|
| 513 | return(tau,kB,st1,t1); |
---|
| 514 | } |
---|
[3d124a7] | 515 | return(st1); |
---|
| 516 | } |
---|
| 517 | example |
---|
| 518 | { "EXAMPLE:"; echo = 2; |
---|
[6f2edc] | 519 | int p = printlevel; |
---|
| 520 | printlevel = 1; |
---|
| 521 | ring r = 0,(x,y,z),ds; |
---|
| 522 | poly f = x5+y6+z7+xyz; // singularity T[5,6,7] |
---|
| 523 | list T = Tjurina(f,""); |
---|
| 524 | show(T[1]); // Tjurina number, should be 16 |
---|
| 525 | show(T[2]); // basis of miniversal deformation |
---|
| 526 | show(T[3]); // SB of Tjurina ideal |
---|
| 527 | show(T[4]); ""; // Tjurina ideal |
---|
| 528 | ideal j = x2+y2+z2,x2+2y2+3z2; |
---|
| 529 | show(kbase(Tjurina(j))); // basis of miniversal deformation |
---|
| 530 | hilb(Tjurina(j)); // Hilbert series of Tjurina module |
---|
| 531 | printlevel = p; |
---|
[3d124a7] | 532 | } |
---|
| 533 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 534 | |
---|
| 535 | proc tjurina (ideal i) |
---|
[d2b2a7] | 536 | "USAGE: tjurina(id); id=ideal or poly |
---|
[6f2edc] | 537 | ASSUME: id=ICIS (isolated complete intersection singularity) |
---|
[3d124a7] | 538 | RETURN: int = Tjurina number of id |
---|
[6f2edc] | 539 | NOTE: Tjurina number = -1 implies that id is not an ICIS |
---|
[3d124a7] | 540 | EXAMPLE: example tjurina; shows an example |
---|
[d2b2a7] | 541 | " |
---|
[3d124a7] | 542 | { |
---|
[6f2edc] | 543 | return(vdim(Tjurina(i))); |
---|
[3d124a7] | 544 | } |
---|
| 545 | example |
---|
| 546 | { "EXAMPLE:"; echo = 2; |
---|
| 547 | ring r=32003,(x,y,z),(c,ds); |
---|
| 548 | ideal j=x2+y2+z2,x2+2y2+3z2; |
---|
[6f2edc] | 549 | tjurina(j); |
---|
[3d124a7] | 550 | } |
---|
| 551 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 552 | |
---|
[0b59f5] | 553 | proc T_1 (ideal id, list #) |
---|
| 554 | "USAGE: T_1(id[,<any>]); id = ideal or poly |
---|
| 555 | RETURN: T_1(id): of type module/ideal if id is of type ideal/poly. |
---|
| 556 | We call T_1(id) the T_1-module of id. It is a std basis of the |
---|
[3d124a7] | 557 | presentation of 1st order deformations of P/id, if P is the basering. |
---|
[6f2edc] | 558 | If a second argument is present (of any type) return a list of |
---|
| 559 | 3 modules: |
---|
[0b59f5] | 560 | [1]= T_1(id) |
---|
[3d124a7] | 561 | [2]= generators of normal bundle of id, lifted to P |
---|
[6f2edc] | 562 | [3]= module of relations of [2], lifted to P |
---|
| 563 | (note: transpose[3]*[2]=0 mod id) |
---|
| 564 | The list contains all non-easy objects which must be computed |
---|
[0b59f5] | 565 | to get T_1(id). |
---|
| 566 | DISPLAY: k-dimension of T_1(id) if printlevel >= 0 (default) |
---|
| 567 | NOTE: T_1(id) itself is usually of minor importance. Nevertheless, from it |
---|
[6f2edc] | 568 | all relevant information can be obtained. The most important are |
---|
[0b59f5] | 569 | probably vdim(T_1(id)); (which computes the Tjurina number), |
---|
| 570 | hilb(T_1(id)); and kbase(T_1(id)); |
---|
| 571 | If T_1 is called with two argument, then matrix([2])*(kbase([1])) |
---|
[6f2edc] | 572 | represents a basis of 1st order semiuniversal deformation of id |
---|
| 573 | (use proc 'deform', to get this in a direct way). |
---|
[3d124a7] | 574 | For a complete intersection the proc Tjurina is faster |
---|
[0b59f5] | 575 | EXAMPLE: example T_1; shows an example |
---|
[d2b2a7] | 576 | " |
---|
[3d124a7] | 577 | { |
---|
| 578 | ideal J=simplify(id,10); |
---|
| 579 | //--------------------------- hypersurface case ------------------------------- |
---|
[6f2edc] | 580 | if( size(J)<2 ) |
---|
| 581 | { |
---|
| 582 | ideal t1 = std(J+jacob(J[1])); |
---|
| 583 | module nb = [1]; module pnb; |
---|
[0b59f5] | 584 | dbprint(printlevel-voice+3,"// dim T_1 = "+string(vdim(t1))); |
---|
[82716e] | 585 | if( size(#)>0 ) |
---|
| 586 | { |
---|
| 587 | module st1 = t1*gen(1); |
---|
[0fbdd1] | 588 | attrib(st1,"isSB",1); |
---|
[82716e] | 589 | return(st1,nb,pnb); |
---|
[0fbdd1] | 590 | } |
---|
[3d124a7] | 591 | return(t1); |
---|
| 592 | } |
---|
| 593 | //--------------------------- presentation of J ------------------------------- |
---|
| 594 | int rk; |
---|
[3939bc] | 595 | def P = basering; |
---|
[3d124a7] | 596 | module jac, t1; |
---|
[6f2edc] | 597 | jac = jacob(J); // jacobian matrix of J converted to module |
---|
[3939bc] | 598 | list A=nres(J,2); // compute presentation of J |
---|
| 599 | def A(1..2)=A[1..2]; kill A; // A(2) = 1st syzygy module of J |
---|
[3d124a7] | 600 | //---------- go to quotient ring mod J and compute normal bundle -------------- |
---|
[3939bc] | 601 | qring R = std(J); |
---|
[6f2edc] | 602 | module jac = fetch(P,jac); |
---|
| 603 | module t1 = transpose(fetch(P,A(2))); |
---|
[fb9532f] | 604 | list B=nres(t1,2); // resolve t1, B(2)=(J/J^2)*=normal_bdl |
---|
[3939bc] | 605 | def B(1..2)=B[1..2]; kill B; |
---|
[6f2edc] | 606 | t1 = modulo(B(2),jac); // pres. of normal_bdl/trivial_deformations |
---|
| 607 | rk=nrows(t1); |
---|
[3d124a7] | 608 | //-------------------------- pull back to basering ---------------------------- |
---|
[3939bc] | 609 | setring P; |
---|
[0b59f5] | 610 | t1 = fetch(R,t1)+J*freemodule(rk); // T_1-module, presentation of T_1 |
---|
[6f2edc] | 611 | t1 = std(t1); |
---|
[0b59f5] | 612 | dbprint(printlevel-voice+3,"// dim T_1 = "+string(vdim(t1))); |
---|
[6f2edc] | 613 | if( size(#)>0 ) |
---|
| 614 | { |
---|
| 615 | module B2 = fetch(R,B(2)); // presentation of normal bundle |
---|
| 616 | list L = t1,B2,A(2); |
---|
| 617 | attrib(L[1],"isSB",1); |
---|
| 618 | return(L); |
---|
[3d124a7] | 619 | } |
---|
[6f2edc] | 620 | return(t1); |
---|
[3d124a7] | 621 | } |
---|
[6f2edc] | 622 | example |
---|
[3d124a7] | 623 | { "EXAMPLE:"; echo = 2; |
---|
[6f2edc] | 624 | int p = printlevel; |
---|
| 625 | printlevel = 1; |
---|
| 626 | ring r = 32003,(x,y,z),(c,ds); |
---|
| 627 | ideal i = xy,xz,yz; |
---|
[0b59f5] | 628 | module T = T_1(i); |
---|
| 629 | vdim(T); // Tjurina number = dim_K(T_1), should be 3 |
---|
| 630 | list L=T_1(i,""); |
---|
[6f2edc] | 631 | module kB = kbase(L[1]); |
---|
[3d124a7] | 632 | print(L[2]*kB); // basis of 1st order miniversal deformation |
---|
[6f2edc] | 633 | show(L[2]); // presentation of normal bundle |
---|
| 634 | print(L[3]); // relations of i |
---|
| 635 | print(transpose(L[3])*L[2]); // should be 0 (mod i) |
---|
| 636 | printlevel = p; |
---|
[3d124a7] | 637 | } |
---|
| 638 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 639 | |
---|
[0b59f5] | 640 | proc T_2 (ideal id, list #) |
---|
| 641 | "USAGE: T_2(id[,<any>]); id = ideal |
---|
| 642 | RETURN: T_2(id): T_2-module of id . This is a std basis of a presentation of |
---|
[6f2edc] | 643 | the module of obstructions of R=P/id, if P is the basering. |
---|
| 644 | If a second argument is present (of any type) return a list of |
---|
| 645 | 4 modules and 1 ideal: |
---|
[0b59f5] | 646 | [1]= T_2(id) |
---|
[3d124a7] | 647 | [2]= standard basis of id (ideal) |
---|
[80a0f0] | 648 | [3]= module of relations of id (=1st syzygy module of id) @* |
---|
[6f2edc] | 649 | [4]= presentation of syz/kos |
---|
| 650 | [5]= relations of Hom_P([3]/kos,R), lifted to P |
---|
| 651 | The list contains all non-easy objects which must be computed |
---|
[0b59f5] | 652 | to get T_2(id). |
---|
| 653 | DISPLAY: k-dimension of T_2(id) if printlevel >= 0 (default) |
---|
| 654 | NOTE: The most important information is probably vdim(T_2(id)). |
---|
[6f2edc] | 655 | Use proc miniversal to get equations of miniversal deformation. |
---|
[0b59f5] | 656 | EXAMPLE: example T_2; shows an example |
---|
[d2b2a7] | 657 | " |
---|
[3d124a7] | 658 | { |
---|
| 659 | //--------------------------- initialisation ---------------------------------- |
---|
| 660 | def P = basering; |
---|
[6f2edc] | 661 | ideal J = id; |
---|
| 662 | module kos,SK,B2,t2; |
---|
| 663 | list L; |
---|
[3d124a7] | 664 | int n,rk; |
---|
[6f2edc] | 665 | //------------------- presentation of non-trivial syzygies -------------------- |
---|
[3939bc] | 666 | list A=nres(J,2); // resolve J, A(2)=syz |
---|
| 667 | def A(1..2)=A[1..2]; kill A; |
---|
[3d124a7] | 668 | kos = koszul(2,J); // module of Koszul relations |
---|
[6f2edc] | 669 | SK = modulo(A(2),kos); // presentation of syz/kos |
---|
[3d124a7] | 670 | ideal J0 = std(J); // standard basis of J |
---|
[6f2edc] | 671 | //?*** sollte bei der Berechnung von res mit anfallen, zu aendern!! |
---|
[3d124a7] | 672 | //---------------------- fetch to quotient ring mod J ------------------------- |
---|
[3939bc] | 673 | qring R = J0; // make P/J the basering |
---|
[6f2edc] | 674 | module A2' = transpose(fetch(P,A(2))); // dual of syz |
---|
| 675 | module t2 = transpose(fetch(P,SK)); // dual of syz/kos |
---|
[3939bc] | 676 | list B=nres(t2,2); // resolve (syz/kos)* |
---|
| 677 | def B(1..2)=B[1..2]; kill B; |
---|
[0b59f5] | 678 | t2 = modulo(B(2),A2'); // presentation of T_2 |
---|
[6f2edc] | 679 | rk = nrows(t2); |
---|
[3d124a7] | 680 | //--------------------- fetch back to basering ------------------------------- |
---|
[3939bc] | 681 | setring P; |
---|
[3d124a7] | 682 | t2 = fetch(R,t2)+J*freemodule(rk); |
---|
[6f2edc] | 683 | t2 = std(t2); |
---|
[0b59f5] | 684 | dbprint(printlevel-voice+3,"// dim T_2 = "+string(vdim(t2))); |
---|
[6f2edc] | 685 | if( size(#)>0 ) |
---|
| 686 | { |
---|
| 687 | B2 = fetch(R,B(2)); // generators of Hom_P(syz/kos,R) |
---|
| 688 | L = t2,J0,A(2),SK,B2; |
---|
| 689 | return(L); |
---|
[3d124a7] | 690 | } |
---|
[6f2edc] | 691 | return(t2); |
---|
[3d124a7] | 692 | } |
---|
| 693 | example |
---|
| 694 | { "EXAMPLE:"; echo = 2; |
---|
[6f2edc] | 695 | int p = printlevel; |
---|
| 696 | printlevel = 1; |
---|
| 697 | ring r = 32003,(x,y),(c,dp); |
---|
| 698 | ideal j = x6-y4,x6y6,x2y4-x5y2; |
---|
[0b59f5] | 699 | module T = T_2(j); |
---|
[6f2edc] | 700 | vdim(T); |
---|
| 701 | hilb(T);""; |
---|
| 702 | ring r1 = 0,(x,y,z),dp; |
---|
| 703 | ideal id = xy,xz,yz; |
---|
[0b59f5] | 704 | list L = T_2(id,""); |
---|
| 705 | vdim(L[1]); // vdim of T_2 |
---|
[6f2edc] | 706 | print(L[3]); // syzygy module of id |
---|
| 707 | printlevel = p; |
---|
[3d124a7] | 708 | } |
---|
| 709 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 710 | |
---|
[0b59f5] | 711 | proc T_12 (ideal i, list #) |
---|
| 712 | "USAGE: T_12(i[,any]); i = ideal |
---|
[80a0f0] | 713 | RETURN: T_12(i): list of 2 modules: @* |
---|
| 714 | * standard basis of T_1-module =T_1(i), 1st order deformations @* |
---|
| 715 | * standard basis of T_2-module =T_2(i), obstructions of R=P/i @* |
---|
[6f2edc] | 716 | If a second argument is present (of any type) return a list of |
---|
[80a0f0] | 717 | 9 modules, matrices, integers: @* |
---|
[0b59f5] | 718 | [1]= standard basis of T_1-module |
---|
| 719 | [2]= standard basis of T_2-module |
---|
| 720 | [3]= vdim of T_1 |
---|
| 721 | [4]= vdim of T_2 |
---|
[80a0f0] | 722 | [5]= matrix, whose cols present infinitesimal deformations @* |
---|
| 723 | [6]= matrix, whose cols are generators of relations of i(=syz(i)) @* |
---|
| 724 | [7]= matrix, presenting Hom_P(syz/kos,R), lifted to P @* |
---|
[0b59f5] | 725 | [8]= presentation of T_1-module, no std basis |
---|
| 726 | [9]= presentation of T_2-module, no std basis |
---|
| 727 | DISPLAY: k-dimension of T_1 and T_2 if printlevel >= 0 (default) |
---|
[3d124a7] | 728 | NOTE: Use proc miniversal from deform.lib to get miniversal deformation of i, |
---|
| 729 | the list contains all objects used by proc miniversal |
---|
[0b59f5] | 730 | EXAMPLE: example T_12; shows an example |
---|
[d2b2a7] | 731 | " |
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[3d124a7] | 732 | { |
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| 733 | //--------------------------- initialisation ---------------------------------- |
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| 734 | int n,r1,r2,d1,d2; |
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[3bc8cd] | 735 | def P = basering; |
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[3d124a7] | 736 | i = simplify(i,10); |
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[6f2edc] | 737 | module jac,t1,t2,sbt1,sbt2; |
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[0b59f5] | 738 | matrix Kos,Syz,SK,kbT_1,Sx; |
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[6f2edc] | 739 | list L; |
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[3d124a7] | 740 | ideal i0 = std(i); |
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| 741 | //-------------------- presentation of non-trivial syzygies ------------------- |
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[3939bc] | 742 | list I= nres(i,2); // resolve i |
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[6f2edc] | 743 | Syz = matrix(I[2]); // syz(i) |
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[3d124a7] | 744 | jac = jacob(i); // jacobi ideal |
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[6f2edc] | 745 | Kos = koszul(2,i); // koszul-relations |
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| 746 | SK = modulo(Syz,Kos); // presentation of syz/kos |
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[3d124a7] | 747 | //--------------------- fetch to quotient ring mod i ------------------------- |
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[3bc8cd] | 748 | qring Ox = i0; // make P/i the basering |
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[6f2edc] | 749 | module Jac = fetch(P,jac); |
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| 750 | matrix No = transpose(fetch(P,Syz)); // ker(No) = Hom(syz,Ox) |
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| 751 | module So = transpose(fetch(P,SK)); // Hom(syz/kos,R) |
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[3939bc] | 752 | list resS = nres(So,2); |
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[6f2edc] | 753 | matrix Sx = resS[2]; |
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[3939bc] | 754 | list resN = nres(No,2); |
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[6f2edc] | 755 | matrix Nx = resN[2]; |
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[0b59f5] | 756 | module T_2 = modulo(Sx,No); // presentation of T_2 |
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| 757 | r2 = nrows(T_2); |
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| 758 | module T_1 = modulo(Nx,Jac); // presentation of T_1 |
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| 759 | r1 = nrows(T_1); |
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[3d124a7] | 760 | //------------------------ pull back to basering ------------------------------ |
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[3bc8cd] | 761 | setring P; |
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[0b59f5] | 762 | t1 = fetch(Ox,T_1)+i*freemodule(r1); |
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| 763 | t2 = fetch(Ox,T_2)+i*freemodule(r2); |
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[3d124a7] | 764 | sbt1 = std(t1); |
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| 765 | d1 = vdim(sbt1); |
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[6f2edc] | 766 | sbt2 = std(t2); |
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[3d124a7] | 767 | d2 = vdim(sbt2); |
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[0b59f5] | 768 | dbprint(printlevel-voice+3,"// dim T_1 = "+string(d1),"// dim T_2 = "+string(d2)); |
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[3d124a7] | 769 | if ( size(#)>0) |
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| 770 | { |
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[3bc8cd] | 771 | if (d1>0) |
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| 772 | { |
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[0b59f5] | 773 | kbT_1 = fetch(Ox,Nx)*kbase(sbt1); |
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[3bc8cd] | 774 | } |
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| 775 | else |
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| 776 | { |
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[0b59f5] | 777 | kbT_1 = 0; |
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[82716e] | 778 | } |
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[3bc8cd] | 779 | Sx = fetch(Ox,Sx); |
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[0b59f5] | 780 | L = sbt1,sbt2,d1,d2,kbT_1,Syz,Sx,t1,t2; |
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[3bc8cd] | 781 | return(L); |
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[3d124a7] | 782 | } |
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[6f2edc] | 783 | L = sbt1,sbt2; |
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| 784 | return(L); |
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[3d124a7] | 785 | } |
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| 786 | example |
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| 787 | { "EXAMPLE:"; echo = 2; |
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[6f2edc] | 788 | int p = printlevel; |
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| 789 | printlevel = 1; |
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[720ff4] | 790 | ring r = 199,(x,y,z,u,v),(c,ws(4,3,2,3,4)); |
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[6f2edc] | 791 | ideal i = xz-y2,yz2-xu,xv-yzu,yu-z3,z2u-yv,zv-u2; |
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| 792 | //a cyclic quotient singularity |
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[0b59f5] | 793 | list L = T_12(i,1); |
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[6f2edc] | 794 | print(L[5]); //matrix of infin. deformations |
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| 795 | printlevel = p; |
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[3d124a7] | 796 | } |
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| 797 | /////////////////////////////////////////////////////////////////////////////// |
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[1e745b] | 798 | proc codim (id1, id2) |
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[d2b2a7] | 799 | "USAGE: codim(id1,id2); id1,id2 ideal or module, both must be standard bases |
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[0fbdd1] | 800 | RETURN: int, which is: |
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| 801 | 1. the codimension of id2 in id1, i.e. the vectorspace dimension of |
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| 802 | id1/id2 if id2 is contained in id1 and if this number is finite |
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| 803 | 2. -1 if the dimension of id1/id2 is infinite |
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| 804 | 3. -2 if id2 is not contained in id1, |
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| 805 | COMPUTE: consider the two hilberseries iv1(t) and iv2(t), then, in case 1., |
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| 806 | q(t)=(iv2(t)-iv1(t))/(1-t)^n must be rational, and the result is the |
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| 807 | sum of the coefficients of q(t) (n dimension of basering) |
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| 808 | EXAMPLE: example codim; shows an example |
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[d2b2a7] | 809 | " |
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[1e745b] | 810 | { |
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| 811 | intvec iv1, iv2, iv; |
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| 812 | int i, d1, d2, dd, i1, i2, ia, ie; |
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[0fbdd1] | 813 | //--------------------------- check id2 < id1 ------------------------------- |
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| 814 | ideal led = lead(id1); |
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| 815 | attrib(led, "isSB",1); |
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| 816 | i = size(NF(lead(id2),led)); |
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[1e745b] | 817 | if ( i > 0 ) |
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| 818 | { |
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| 819 | return(-2); |
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| 820 | } |
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| 821 | //--------------------------- 1. check finiteness --------------------------- |
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| 822 | i1 = dim(id1); |
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| 823 | i2 = dim(id2); |
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| 824 | if (i1 < 0) |
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| 825 | { |
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| 826 | if (i2 == 0) |
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| 827 | { |
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| 828 | return vdim(id2); |
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| 829 | } |
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| 830 | else |
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| 831 | { |
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| 832 | return(-1); |
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| 833 | } |
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| 834 | } |
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| 835 | if (i2 != i1) |
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| 836 | { |
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| 837 | return(-1); |
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| 838 | } |
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| 839 | if (i2 <= 0) |
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| 840 | { |
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| 841 | return(vdim(id2)-vdim(id1)); |
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| 842 | } |
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[0fbdd1] | 843 | // if (mult(id2) != mult(id1)) |
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| 844 | //{ |
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| 845 | // return(-1); |
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| 846 | // } |
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[1e745b] | 847 | //--------------------------- module --------------------------------------- |
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| 848 | d1 = nrows(id1); |
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| 849 | d2 = nrows(id2); |
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| 850 | dd = 0; |
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| 851 | if (d1 > d2) |
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| 852 | { |
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| 853 | id2=id2,maxideal(1)*gen(d1); |
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| 854 | dd = -1; |
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| 855 | } |
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| 856 | if (d2 > d1) |
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| 857 | { |
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| 858 | id1=id1,maxideal(1)*gen(d2); |
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| 859 | dd = 1; |
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| 860 | } |
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| 861 | //--------------------------- compute first hilbertseries ------------------ |
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| 862 | iv1 = hilb(id1,1); |
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| 863 | i1 = size(iv1); |
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| 864 | iv2 = hilb(id2,1); |
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| 865 | i2 = size(iv2); |
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| 866 | //--------------------------- difference of hilbertseries ------------------ |
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| 867 | if (i2 > i1) |
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| 868 | { |
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| 869 | for ( i=1; i<=i1; i=i+1) |
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| 870 | { |
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| 871 | iv2[i] = iv2[i]-iv1[i]; |
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| 872 | } |
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| 873 | ie = i2; |
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| 874 | iv = iv2; |
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| 875 | } |
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| 876 | else |
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| 877 | { |
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| 878 | for ( i=1; i<=i2; i=i+1) |
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| 879 | { |
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| 880 | iv1[i] = iv2[i]-iv1[i]; |
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| 881 | } |
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| 882 | iv = iv1; |
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| 883 | for (ie=i1;ie>=0;ie=ie-1) |
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| 884 | { |
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| 885 | if (ie == 0) |
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| 886 | { |
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[82716e] | 887 | return(0); |
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[1e745b] | 888 | } |
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| 889 | if (iv[ie] != 0) |
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| 890 | { |
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| 891 | break; |
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| 892 | } |
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| 893 | } |
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| 894 | } |
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| 895 | ia = 1; |
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| 896 | while (iv[ia] == 0) { ia=ia+1; } |
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| 897 | //--------------------------- ia <= nonzeros <= ie ------------------------- |
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| 898 | iv1 = iv[ia]; |
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| 899 | for(i=ia+1;i<=ie;i=i+1) |
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| 900 | { |
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| 901 | iv1=iv1,iv[i]; |
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| 902 | } |
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| 903 | //--------------------------- compute second hilbertseries ----------------- |
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| 904 | iv2 = hilb(iv1); |
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| 905 | //--------------------------- check finitenes ------------------------------ |
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| 906 | i2 = size(iv2); |
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| 907 | i1 = ie - ia + 1 - i2; |
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| 908 | if (i1 != nvars(basering)) |
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| 909 | { |
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| 910 | return(-1); |
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| 911 | } |
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| 912 | //--------------------------- compute result ------------------------------- |
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| 913 | i1 = 0; |
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| 914 | for ( i=1; i<=i2; i=i+1) |
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| 915 | { |
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| 916 | i1 = i1 + iv2[i]; |
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| 917 | } |
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| 918 | return(i1+dd); |
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| 919 | } |
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[0fbdd1] | 920 | example |
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| 921 | { "EXAMPLE:"; echo = 2; |
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| 922 | ring r = 0,(x,y,z),dp; |
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| 923 | ideal j = y6,x4; |
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| 924 | ideal m = x,y; |
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| 925 | attrib(m,"isSB",1); //let Singular know that ideals are a standard basis |
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[82716e] | 926 | attrib(j,"isSB",1); |
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[0fbdd1] | 927 | codim(m,j); // should be 23 (Milnor number -1 of y7-x5) |
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| 928 | } |
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[c69ea5] | 929 | |
---|
| 930 | /////////////////////////////////////////////////////////////////////////////// |
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| 931 | |
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| 932 | proc tangentcone (id,list #) |
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| 933 | "USAGE: tangentcone(id [,n]); id = ideal, n = int |
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| 934 | RETURN: the tangent cone of id |
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| 935 | NOTE: the procedure works for any monomial ordering. |
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| 936 | If n=0 use std w.r.t. local ordering ds, if n=1 use locstd |
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| 937 | EXAMPLE: example tangentcone; shows an example |
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| 938 | " |
---|
| 939 | { |
---|
| 940 | int ii,n; |
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| 941 | def bas = basering; |
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| 942 | ideal tang; |
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| 943 | if (size(#) !=0) { n= #[1]; } |
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| 944 | if( n==0 ) |
---|
| 945 | { |
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| 946 | changeord("@newr@","ds"); |
---|
| 947 | ideal @id = imap(bas,id); |
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| 948 | @id = std(@id); |
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| 949 | setring bas; |
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| 950 | id = imap(@newr@,@id); |
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| 951 | kill @newr@; |
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| 952 | } |
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| 953 | else |
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| 954 | { |
---|
| 955 | id = locstd(id); |
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| 956 | } |
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| 957 | |
---|
| 958 | for(ii=1; ii<=size(id); ii++) |
---|
| 959 | { |
---|
| 960 | tang[ii]=jet(id[ii],mindeg(id[ii])); |
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| 961 | } |
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| 962 | return(tang); |
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| 963 | } |
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| 964 | example |
---|
| 965 | { "EXAMPLE:"; echo = 2; |
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| 966 | ring R = 0,(x,y,z),ds; |
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| 967 | ideal i = 7xyz+z5,x2+y3+z7,5z5+y5; |
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| 968 | tangentcone(i); |
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| 969 | } |
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| 970 | /////////////////////////////////////////////////////////////////////////////// |
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| 971 | |
---|
| 972 | proc locstd (id) |
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| 973 | "USAGE: locstd (id); id = ideal |
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| 974 | RETURN: a standard basis for a local degree ordering |
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| 975 | NOTE: the procedure homogenizes id w.r.t. a new 1st variable @t@, computes |
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| 976 | a SB wrt (dp(1),dp) and substitutes @t@ by 1. |
---|
| 977 | Hence the result is a SB with respect to an ordering which sorts |
---|
| 978 | first w.r.t. the order and then refines it with dp. This is a |
---|
| 979 | local degree ordering. |
---|
| 980 | This is done in order to avoid cancellation of units and thus |
---|
| 981 | be able to use option(contentSB); |
---|
| 982 | EXAMPLE: example locstd; shows an example |
---|
| 983 | " |
---|
| 984 | { |
---|
| 985 | int ii; |
---|
| 986 | def bas = basering; |
---|
| 987 | execute("ring @r_locstd |
---|
| 988 | =("+charstr(bas)+"),(@t@,"+varstr(bas)+"),(dp(1),dp);"); |
---|
| 989 | ideal @id = imap(bas,id); |
---|
| 990 | ideal @hid = homog(@id,@t@); |
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| 991 | @hid = std(@hid); |
---|
| 992 | @hid = subst(@hid,@t@,1); |
---|
| 993 | setring bas; |
---|
| 994 | def @hid = imap(@r_locstd,@hid); |
---|
| 995 | attrib(@hid,"isSB",1); |
---|
| 996 | kill @r_locstd; |
---|
| 997 | return(@hid); |
---|
| 998 | } |
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| 999 | example |
---|
| 1000 | { "EXAMPLE:"; echo = 2; |
---|
| 1001 | ring R = 0,(x,y,z),ds; |
---|
| 1002 | ideal i = xyz+z5,2x2+y3+z7,3z5+y5; |
---|
| 1003 | locstd(i); |
---|
| 1004 | } |
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