A bound for the Milnor number of plane curve singularities

Abstract

Let $f=0$ be a plane algebraic curve of degree $d>1$ with an isolated singular point at the origin of the complex plane. We show that the Milnor number ${\mu }_0(f)$ is less than or equal to $(d-1{)}^2-\left[\frac{d}{2}\right]$, unless $f=0$ is a set of $d$ concurrent lines passing through 0. Then we characterize the curves $f=0$ for which ${\mu }_0(f)=(d-1{)}^2-\left[\frac{d}{2}\right]$.