Let $f$ be a quasi-homogeneous polynomial with an isolated singularity in $\mathbf{C}^{n}$. We compute the length of the ${\mathcal{D}}$-modules ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers of $f$ in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When $\unicode[STIX]{x1D706}=-1$ we obtain one more than the reduced genus of the singularity ($\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for $Z$ the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ is nonzero when $\unicode[STIX]{x1D706}$ is a root of the $b$-function of $f$ (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these ${\mathcal{D}}$-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.

We introduce the notion of the Bernstein–Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible) using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V-filtration. This implies a relation between the roots of the Bernstein–Sato polynomial and the jumping coefficients of the multiplier ideals, and also a criterion for rational singularities in terms of the maximal root of the polynomial in the case of a reduced complete intersection. These are generalizations of the hypersurface case. We can calculate the polynomials explicitly in the case of monomial ideals.

In 1987, Sabbah proved the existence of Bernstein–Sato polynomials associated with several analytic functions. The purpose of this paper is to give a more elementary and constructive proof of Sabbah's result based on the notion of the analytic Gröbner fan of a $\mathcal D$-module.

Let $Q\in{\mathbb C}[x_1,\dotsc,x_n]$ be a homogeneous polynomial of degree $k>0$. We establish a connection between the Bernstein–Sato polynomial $b_Q(s)$ and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integer $u_Q={\rm max}\{i\in{\mathbb Z}:b_Q(-(i+n)/k)=0\}$ bounds the top degree (as differential form) of the elements in $H^{n-1}_{\rm DR}(Q^{-1}(1),{\mathbb C})$. The link is provided by the relative de Rham complex and ${\mathcal D}$-module algorithms for computing integration functors.

As an application we determine the Bernstein–Sato polynomial $b_Q(s)$ of a generic central arrangement $Q=\prod_{i=1}^kH_i$ of hyperplanes. In turn, we obtain information about the cohomology of the Milnor fiber of such arrangements related to results of Orlik and Randell who investigated the monodromy.

We also introduce certain subschemes of the arrangement determined by the roots of $b_Q(s)$. They appear to correspond to iterated singular loci.