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 $b(-s-2)=\pm b(s)$. 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 $E^{\ast }$ with respect to a free divisor of linear Jacobian type are related by the equality $b_E(s)=\pm b_{E^{\ast }}(-s-2)$. Our results are based on the behaviour of the modules $D[s]h^s$ and $D[s]E[s]h^s$ under duality.

Final version

References in corpus (3)

• Linearity conditions on the Jacobian ideal and logarithmic--meromorphic comparison for free divisors
• The module $D{f}^s$ for locally quasi-homogeneous free divisors
• Dualité et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres

Cited by in corpus (9)

• On Hasse--Schmidt derivations: the action of substitution maps
• Bernstein-Sato ideals and hyperplane arrangements
• Survey on the $D$-module $f^s$
• Chern Classes of Logarithmic Derivations for Free Divisors with Jacobian Ideal of Linear Type
• Families of twisted $\mathcal{D}$-modules and arithmetic models of Harish-Chandra modules
• Hodge ideals of free divisors
• The Bernstein-Sato $b$-function of the Vandermonde determinant
• Tautological systems and free divisors
• On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors