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Spectrum
V-Filtration - An Example

LIB "gaussman.lib";
ring r=0,(x,y),ds;
poly f=x5+x2y2+y5;

The command vfilt(f) returns a list V :
  • V[1] contains the spectral numbers a[i] in increasing order.
  • V[2] contains the corresponding multiplicities.
  • V[3] contains a list of vector space bases for GrVa[i](H''/H') in terms of V[4].
  • V[4] contains a monomial vector space basis for (H''/H').
list V=vfilt(f);
print(V[1]);
==> -1/2, -3/10, -1/10, 0, 1/10, 3/10, 1/2
print(V[2])
==> 1, 2, 2, 1, 2, 2, 1

Vector space bases of the corresponding graded parts of the V-filtration:
for (int i=1; i<=7; i++) {print(matrix(V[4])*matrix(V[3][i],11,size(V[3][i]))); }
==> 1
==> x, y
==> x2, y2
==> xy
==> x2, y3
==> x4, y4
==> y5

0 0 0 0 0 0 0
7 0 0 0 0 0 0
6 0 0 0 0 0 0
5 0 0 0 0 0 0
3 0 0 0 0 0 0
2 4 0 0 0 0 0
1 2 3 5 6 0 0
Interprete: xayb = [xayb dx ^ dy] in H''/H'.

Spectrum and V-filtration

Sao Carlos, 08/02 http://www.singular.uni-kl.de