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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 :
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V[1] contains the spectral numbers
a[i] in increasing order.
-
V[2] contains the corresponding
multiplicities.
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V[3] contains a list of vector space bases for
GrVa[i](H''/H') in
terms of V[4].
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V[4] contains a monomial vector space basis for
(H''/H').
list V=vfilt(f);
print(V[1]);
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==>
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-1/2, -3/10, -1/10, 0, 1/10, 3/10, 1/2
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print(V[2])
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]))); }
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| 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 |
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Interprete:
xayb =
[xayb dx ^ dy]
in
H''/H'.
Spectrum and V-filtration
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