Singular Manual: esIdeal

D.6.8.2 esIdeal

Procedure from library equising.lib (see equising_lib).

Usage:

esIdeal(f[,any]]); f poly

Assume:

f is a reduced bivariate polynomial, the basering has precisely two variables, is local and no qring, and the characteristic of the ground field does not divide mult(f).

Return:

if called with only one parameter: list of two ideals,

          _[1]:  equisingularity ideal of f (in sense of Wahl),
          _[2]:  ideal of equisingularity with fixed position of the
                 singularity;

if called with more than one parameter: list of three ideals,

          _[1]:  equisingularity ideal of f (in sense of Wahl)
          _[2]:  ideal of equisingularity with fixed position of the
                 singularity;
          _[3]:  ideal of all g such that the deformation defined by f+eg
                 (e^2=0) is isomorphic to an equisingular deformation
                 of V(f) with all equimultiple sections being trivial.

Note:

if some of the above condition is not satisfied then return value is list(0,0).

Example:

LIB "equising.lib";
ring r=0,(x,y),ds;
poly f=x7+y7+(x-y)^2*x2y2;
list K=esIdeal(f);
==> polynomial is Newton degenerate !
==> 
==> // 
==> // versal deformation with triv. section
==> // =====================================
==> // 
==> // 
==> // Compute equisingularity Stratum over Spec(C[t]/t^2)
==> // ===================================================
==> // 
==> // finished
==> // 
option(redSB);
// Wahl's equisingularity ideal:
std(K[1]);
==> _[1]=4x4y-10x2y3+6xy4+21x6+14y6
==> _[2]=4x3y2-6x2y3+2xy4+7x6
==> _[3]=x2y4-xy5
==> _[4]=x7
==> _[5]=xy6
==> _[6]=y7
ring rr=0,(x,y),ds;
poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31;
list K=esIdeal(f);
==> polynomial is Newton degenerate !
==> 
==> // 
==> // versal deformation with triv. section
==> // =====================================
==> // 
==> // 
==> // Compute equisingularity Stratum over Spec(C[t]/t^2)
==> // ===================================================
==> // 
==> // finished
==> // 
vdim(std(K[1]));
==> 68
// the latter should be equal to:
tau_es(f);
==> 68