A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors

TL;DR

This paper proves a symmetry property of Bernstein-Sato polynomials for certain free divisors, extending understanding of their duality and behavior under specific resolutions.

Contribution

It establishes the symmetry of Bernstein-Sato polynomials for free divisors with Spencer logarithmic resolutions and relates polynomials of dual logarithmic connections.

Findings

Bernstein-Sato polynomial symmetry for free divisors with Spencer resolutions

Relation between polynomials of dual logarithmic connections

Applicability to locally quasi-homogeneous free divisors

Abstract

In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the $D[s]$-module $D[s] h^s$ admits a Spencer logarithmic resolution satisfies the symmetry property $b(-s-2) = \pm b(s)$. This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection $E$ and of its dual $E^*$ with respect to a free divisor of linear Jacobian type are related by the equality $b_{E}(s)=\pm b_{E^*}(-s-2)$. Our results are based on the behaviour of the modules $D[s] h^s$ and $D[s] E[s]h^s $ under duality.