The Milnor Number of Plane Branches With Tame Semigroup of Values

TL;DR

This paper investigates the Milnor number of plane branches over fields of arbitrary characteristic, showing it equals the conductor of the semigroup of values when the semigroup is tame, extending Milnor's classical result.

Contribution

It proves that the Milnor number equals the conductor of the semigroup of values for plane branches with tame semigroups over any algebraically closed field.

Findings

Milnor number equals the conductor for tame semigroups

The result extends Milnor's classical theorem to positive characteristic fields

Tameness condition ensures invariance of the Milnor number

Abstract

The Milnor number of an isolated hypersurface singularity, defined as the codimension $\mu(f)$ of the ideal generated by the partial derivatives of a power series $f$ that represents locally the hypersurface, is an important topological invariant of the singularity over the complex numbers. However it may loose its significance when the base field is arbitrary. It turns out that if the ground field is of positive characteristic, this number depends upon the equation $f$ representing the hypersurface, hence it is not an invariant of the hypersurface. For a plane branch represented by an irreducible convergent power series $f$ in two indeterminates over the complex numbers, it was shown by Milnor that $\mu(f)$ always coincides with the conductor $c(f)$ of the semigroup of values $S(f)$ of the branch. This is not true anymore if the characteristic of the ground field is positive. In this paper we show that, over algebraically closed fields of arbitrary characteristic, this is true, provided that the semigroup $S(f)$ is tame, that is, the characteristic of the field does not divide any of its minimal generators.