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C.5 Gauss-Manin connection
Let us consider as an example
.
First, we compute a matrix
such that
is a monodromy matrix of
and the Jordan normal form of
:
Now, we compute the V-fitration on
and the spectrum:
Here l[1] contains the spectral numbers, l[2] the corresponding multiplicities, l[3] a
-basis of the V-filtration on
in terms of the monomial basis of
in l[4] (seperated by degree).
Let us calculate one specific example, the maximal number
of triple points of type
of degree seven.
This calculation can be done over the rationals.
We choose a local ordering on
. Here we take the
negative degree lexicographical ordering, in SINGULAR denoted by ds :
The command spectrumnd(f) computes the spectrum of
and
returns a list with six entries:
The Milnor number
and the number of different spectrum numbers.
The other three entries are of type intvec .
They contain the numerators, denominators and
multiplicities of the spectrum numbers. So
has Milnor number 216 and geometrical
genus 35. Its spectrum consists of the 16 different rationals
appearing with multiplicities
1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1.
Therefore they have all the same spectrum, which we compute
for
| poly g=x^3+y^3+z^3;
list s2=spectrumnd(g);
s2;
==> [1]:
==> 8
==> [2]:
==> 1
==> [3]:
==> 4
==> [4]:
==> 1,4,5,2
==> [5]:
==> 1,3,3,1
==> [6]:
==> 1,3,3,1
| Evaluating semicontinuity is very easy:
This tells us that there are at most 18 singularities of type
is semiquasihomogeneous (sqh), so we can also apply the stronger
form of semicontinuity:
| semicontsqh(s1,s2);
==> 17
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So in fact a septic has at most 17 triple points of type
Note that spectrumnd(f) works only if
has a nondegenerate
principal part. In fact spectrumnd will detect a degenerate
principal part in many cases and print out an error message.
However if it is known in advance that
has nondegenerate
principal part, then the spectrum may be computed much faster
using spectrumnd(f,1) .
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