A duality approach to the symmetry of Bernstein-Sato polynomials of 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) = ±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 bE(s) = ±bE∗ (−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.