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A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors. (English) Zbl 1327.14090
Summary: In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the \(\mathcal{D} [s]\)-module \(\mathcal{D} [s] h^s\) 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 \(\mathcal{E}\) and of its dual \(\mathcal{E}^\ast\) with respect to a free divisor of linear Jacobian type are related by the equality \(b_{\mathcal{E}}(s) = \pm b_{\mathcal{E}^\ast}(- s - 2)\). Our results are based on the behaviour of the modules \(\mathcal{D} [s] h^s\) and \(\mathcal{D} [s] \mathcal{E} [s] h^s\) under duality.

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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