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### Example

The SINGULAR library gaussman.lib [36] contains an implementation of the algorithm. We use it to compute an example. First, we have to load the library:


LIB "gaussman.lib";

We define the ring and :

ring R=0,(x,y),ds;
poly f=x5+x2y2+y5;

Finally, we compute the -filtration:

vfiltration(f);
==> [1]:  _[1]=-1/2  _[2]=-3/10  _[3]=-1/10  _[4]=0
==>       _[5]=1/10  _[6]=3/10   _[7]=1/2
==> [2]:  1,2,2,1,2,2,1
==> [3]:  [1]:  _[1]=gen(11)
==>       [2]:  _[1]=gen(10)  _[2]=gen(6)
==>       [3]:  _[1]=gen(9)   _[2]=gen(4)
==>       [4]:  _[1]=gen(5)
==>       [5]:  _[1]=gen(8)   _[2]=gen(3)
==>       [6]:  _[1]=gen(7)   _[2]=gen(2)
==>       [7]:  _[1]=gen(1)
==> [4]:  _[1]=y5  _[2]=y4  _[3]=y3   _[4]=y2   _[5]=xy
==>       _[6]=y   _[7]=x4  _[8]=x3   _[9]=x2  _[10]=x
==>      _[11]=1
==> [5]:  _[1]=2x2y+5y4  _[2]=2xy2+5x4  _[3]=x5-y5
==>       _[4]=y6

The result is a list with 5 entries: The first contains the spectral numbers, the second the corresponding multiplicities, the third -bases of the graded parts of the -filtration on in terms of the monomial -basis in the fourth entry, and the fifth a standard basis of the Jacobian ideal. A monomial in the fourth entry is considered as .

As an application of the implementation, the third author could verify Hertling's conjecture [21] about the variance of the spectral numbers for all isolated hypersurface singularities of Milnor number .

Christoph Lossen
2001-03-21