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The Milnor number of an isolated hypersurface singularity, defined as thecodimension $\mu(f)$ of the ideal generated by the partial derivatives of apower series $f$ that represents locally the hypersurface, is an importanttopological invariant of the singularity over the complex numbers. However itmay loose its significance when the base field is arbitrary. It turns out thatif the ground field is of positive characteristic, this number depends upon theequation $f$ representing the hypersurface, hence it is not an invariant of thehypersurface. For a plane branch represented by an irreducible convergent powerseries $f$ in two indeterminates over the complex numbers, it was shown byMilnor that $\mu(f)$ always coincides with the conductor $c(f)$ of thesemigroup of values $S(f)$ of the branch. This is not true anymore if thecharacteristic 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 thefield does not divide any of its minimal generators.

The Milnor Number of Plane Branches With Tame Semigroup of Values