A computational approach to Milnor fiber cohomology

TL;DR

This paper introduces a computational method for determining the monodromy eigenvalues and eigenspaces of the Milnor fiber associated with a reduced projective plane curve, facilitating explicit calculations of the Alexander polynomial.

Contribution

It presents an effective algorithm for computing the monodromy characteristic polynomial and eigenspaces of Milnor fibers of plane curves, advancing computational algebraic geometry methods.

Findings

Algorithm successfully computes monodromy eigenvalues.

Explicit bases for eigenspaces are obtained in many cases.

The method enhances understanding of Milnor fiber cohomology.

Abstract

In this note we consider the Milnor fiber $F$ associated to a reduced projective plane curve $C$. A computational approach for the determination of the characteristic polynomial of the monodromy action on the first cohomology group of $F$, also known as the Alexander polynomial of the curve $C$, is presented. This leads to an effective algorithm to detect all the monodromy eigenvalues and, in many cases, explicit bases for the monodromy eigenspaces in terms of polynomial differential forms.