A bound for the Milnor number of plane curve singularities

TL;DR

This paper establishes an upper bound for the Milnor number of plane algebraic curves with isolated singularities, characterizes cases of equality, and provides insights into the singularity structure of such curves.

Contribution

It introduces a new upper bound for the Milnor number of plane curve singularities and characterizes the curves that attain this bound.

Findings

The Milnor number is bounded above by (d-1)^2 - [d/2].

Equality holds precisely for certain configurations of d concurrent lines.

Provides a classification of curves with maximal Milnor number.

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 $\mu_0(f)=(d-1)^2-\left[\frac{d}{2}\right]$.