D.5.5.14 delta

Procedure from library hnoether.lib (see hnoether_lib).

Usage:

delta(INPUT); INPUT a polynomial defining an isolated plane curve singularity at 0, or the Hamburger-Noether expansion thereof, i.e. the output of develop(f), or the output of hnexpansion(f[,"ess"]), or (one of the entries of) the list hne in the ring created by hnexpansion(f[,"ess"]).

Return:

the delta invariant of the singularity at 0, the vector space dimension of R~/R, where R~ is the normalization of the singularity R=basering/f

Note:

In case the Hamburger-Noether expansion of the curve f is needed for other purposes as well it is better to calculate this first with the aid of hnexpansion and use it as input instead of the polynomial itself.

Example:

LIB "hnoether.lib";
ring r = 32003,(x,y),ds;
poly f = x25+x24-4x23-1x22y+4x22+8x21y-2x21-12x20y-4x19y2+4x20+10x19y
+12x18y2-24x18y-20x17y2-4x16y3+x18+60x16y2+20x15y3-9x16y
-80x14y3-10x13y4+36x14y2+60x12y4+2x11y5-84x12y3-24x10y5
+126x10y4+4x8y6-126x8y5+84x6y6-36x4y7+9x2y8-1y9;
delta(f);
→ 96

See also: deltaLoc; invariants.