An algebraic proof of the Milnor-Orlik theorem

TL;DR

This paper offers an algebraic proof of the Milnor-Orlik theorem, deriving the Milnor number formula for weighted-homogeneous polynomials with isolated singularities using commutative algebra techniques.

Contribution

It provides a new proof of the Milnor-Orlik theorem employing Koszul complexes and Hilbert series calculations, advancing algebraic methods in singularity theory.

Findings

Derived the Milnor number formula algebraically

Used Koszul complex to resolve the Milnor algebra

Calculated Hilbert series to obtain the formula

Abstract

A well-known theorem by Milnor-Orlik provides a formula for the Milnor number of a weighted-homogeneous polynomial having an isolated singularity that depends only on the weights. In this paper we present a proof of that result using techniques from commutative algebra. Our approach is to obtain a free resolution of the Milnor algebra through the Koszul complex. The desired formula is then obtained from a Hilbert series calculation.