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Transversals and Tangents:
SINGULAR
example
LIB "primdec.lib"; option(redSB);
Treating s and t as variables, we now define the ideal I and the excess
components:
ring R = 0, (s,t,a,b,c,d,e,f,g,h,k,l), (dp(2), dp(10));
ideal I = el-g^2, ek-gf, ak-dc, ah-c^2;
matrix M[2][5] = s , 1-s , -2 , 1-t , t ,
al-d^2, 2*(bl-dg), 2*(2bk-cg-df), 2*(bh-cf), eh-f^2;
I = I + minor(M,2);
I=std(I); // standard basis of I
matrix Q[4][4] = a , b , c , d ,
b , e , f , g ,
c , f , h , k ,
d , g , k , l ;
ideal E1 = std(minor(Q,2));
ideal E2 = g, f, e, d, c, b, a; // intersection at L1
ideal E3 = l, k, h, g, f, d, c; // intersection at L2
Now remove the excess components from I:
I=sat(I,E1)[1]; // approximately 80 seconds
I=sat(I,E2)[1]; // approximately 40 seconds
I=sat(I,E3)[1]; // approximately 30 seconds
Decompose the resulting ideal, making sure that computations will be done in the
complement of <st(t-1)(s-1)(s-t)>:
ideal L = s, t, t-1, s-1, s-t;
list F = facstd(I, L); // approximately 2-5 minutes
// depending on the random seed
size(F);
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==>
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8 // for some random seeds also 7 or 9
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Note that to a certain extent the output of facstd depends on the
random seed used at the start of
SINGULAR
.
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