Bernstein polynomials and spectral numbers for linear free divisors

TL;DR

This paper explores Bernstein polynomials and spectral numbers associated with reductive linear free divisors, establishing new links between roots of Bernstein polynomials and residue eigenvalues through the construction of specialized Brieskorn lattices.

Contribution

It introduces a novel framework for defining Brieskorn lattices for non-isolated singularities and extends Malgrange's results to this broader context.

Findings

Established a correspondence between Bernstein polynomial roots and residue eigenvalues.

Defined suitable Brieskorn lattices for non-isolated singularities.

Extended Malgrange's theorem to linear free divisors.

Abstract

We discuss Bernstein polynomials of reductive linear free divisors. We define suitable Brieskorn lattices for these non-isolated singularities, and show the analogue of Malgrange's result relating the roots of the Bernstein polynomial to the residue eigenvalues on the saturation of these Brieskorn lattices.