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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:
 
semicont(s1,s2);
==> 18

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

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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