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 $R:=\mathbf{Q}[x,y]_{(x,y)}$ and $f=x^5\!+x^2y^2\!+y^5\in R$:

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

Finally, we compute the $V$-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 $\mathbf{C}$-bases of the graded parts of the $V$-filtration on $\Omega_f$ in terms of the monomial $\mathbf{C}$-basis in the fourth entry, and the fifth a standard basis of the Jacobian ideal. A monomial $x^\alpha y^\beta$ in the fourth entry is considered as $x^\alpha y^\beta{d\!\,} x\wedge{d\!\,} y\in\Omega_f$.

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 $\leq 16$.