---
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.'
author:
- Christian Sevenheck
title: |
    Bernstein polynomials and spectral numbers\
    for linear free divisors
---

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Introduction {#sec:Introduction}
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In this note, we show that for reductive linear free divisors $D\subset {{\mathds C}}^n$, which were studied in a number of recent papers (see [@BM] [@GMNS], [@dGMS] and [@GS]), the roots of the Bernstein polynomial of a defining equation $h$ of $D$ can be recovered as a certain set of eigenvalues of a residue endomorphism. More precisely, for a generic linear form $f$ on ${{\mathds C}}^n$, one defines a family of Gauß-Manin systems for $f$, seen as a function on the fibres of $h$. This family has a specific (logarithmic) extension over $D$, which gives the set of residue eigenvalues we are interested in.

This relation to the Bernstein polynomial has a number of consequences: First, while the definition of the Bernstein polynomial is rather simple, it is in general very hard to calculate its 