Singular

D.4.23.9 deltaLoc

Usage:

deltaLoc(f,J); f poly, J ideal

Assume:

f is reduced bivariate polynomial; basering has exactly two variables; J is irreducible prime component of the singular locus of f (e.g., one entry of the output of minAssGTZ(I);, I = <f,jacob(f)>).

Return:

list L:

L[1]; int:

the sum of (local) delta invariants of f at the (conjugated) singular points given by J.

L[2]; int:

the sum of (local) Tjurina numbers of f at the (conjugated) singular points given by J.

L[3]; int:

the sum of (local) number of branches of f at the (conjugated) singular points given by J.

Note:

procedure makes use of execute; increasing printlevel displays more comments (default: printlevel=0).

LIB "normal.lib";
ring r=0,(x,y),dp;
poly f=(x2+y^2-1)^3 +27x2y2;
ideal I=f,jacob(f);
I=std(I);
list qr=minAssGTZ(I);
size(qr);
==> 6
// each component of the singular locus either describes a cusp or a pair
// of conjugated nodes:
deltaLoc(f,qr[1]);
==> [1]:
==>    1
==> [2]:
==>    2
==> [3]:
==>    1
deltaLoc(f,qr[2]);
==> [1]:
==>    1
==> [2]:
==>    2
==> [3]:
==>    1
deltaLoc(f,qr[3]);
==> [1]:
==>    1
==> [2]:
==>    2
==> [3]:
==>    1
deltaLoc(f,qr[4]);
==> [1]:
==>    1
==> [2]:
==>    2
==> [3]:
==>    1
deltaLoc(f,qr[5]);
==> [1]:
==>    2
==> [2]:
==>    2
==> [3]:
==>    4
deltaLoc(f,qr[6]);
==> [1]:
==>    2
==> [2]:
==>    2
==> [3]:
==>    4