On the torsion of Brieskorn modules of homogeneous polynomials

TL;DR

This paper investigates the torsion properties of Brieskorn modules associated with homogeneous polynomials, revealing that for two variables the torsion is always of order one, while for higher variables, larger torsion orders can occur.

Contribution

It provides a detailed description of torsion in Brieskorn modules for n=2 and presents examples showing higher torsion orders for n>2.

Findings

Torsion in B(f) for n=2 always has order 1.

Examples exist where torsion order exceeds 1 for n>2.

The torsion structure varies with the number of variables.

Abstract

Let $f\in \mathbb{C}[X_1,..., X_n]$ be a homogeneous polynomial and B(f) be the corresponding Brieskorn module. We describe the torsion of the Brieskorn module B(f) for n=2 and show that any torsion element has order 1. For n>2, we find some examples in which the torsion order is strictly greater than 1.