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]=y6The 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 .