Related papers: Bernstein-Sato functional equations for ideals in …

Given two holomorphic functions $f$ and $g$ defined in two respective germs of complex analytic manifolds $(X,x)$ and $(Y,y)$, we know thanks to M. Saito that, as long as one of them is Euler homogeneous, the reduced (or microlocal)…

By using logarithmic $\mathcal D$-modules and Gr\"obner bases, we prove that Bernstein-Sato ideals satisfy some symmetric intersection property, answering a question posed by Budur. As an application, we obtain a formula for the…

We develop a theory of Bernstein-Sato polynomials for meromorphic functions and we use it to study the analytic continuation of Archimedian local zeta functions in this setting. We also introduce both an analytic and an algebraic theory of…

We give an introduction to a theory of b-functions, i.e. Bernstein-Sato polynomials. After reviewing some facts from D-modules, we introduce b-functions including the one for arbitrary ideals of the structure sheaf. We explain the relation…

In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider the analogous D-module in positive characteristic, and…

Bernstein's inequality is a central result in the theory of $D$-modules on smooth varieties. While Bernstein's inequality fails for rings of differential operators on general singularities, recent work of \`{A}lvarez Montaner, Hern\'andez,…

We consider the Bernstein--Sato polynomial of a locally quasi-homogeneous polynomial $f \in R = \mathbb{C}[x_{1}, x_{2}, x_{3}]$. We construct, in the analytic category, a complex of $\mathscr{D}_{X}[s]$-modules that can be used to compute…

We prove a conjecture of the first author relating the Bernstein-Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange…

For a central, not necessarily reduced, hyperplane arrangement $f$ equipped with any factorization $f = f_{1} \cdots f_{r}$ and for $f^{\prime}$ dividing $f$, we consider a more general type of Bernstein--Sato ideal consisting of the…

For a nonzero ideal I of C[x_1,...,x_n], with 0 in supp I, a generalization of a conjecture of Igusa - Denef - Loeser predicts that every pole of its topological zeta function is a root of its Bernstein-Sato polynomial. However, typically…

This is the second part of a work dedicated to the study of Bernstein-Sato polynomials for several analytic functions depending on parameters. In this part, we give constructive results generalizing previous ones obtained by the author in…

In 2006, Budur, Musta\c{t}\v{a} and Saito introduced the notion of Bernstein-Sato polynomial of an arbitrary scheme of finite type over fields of characteristic zero. Because of the strong monodromy conjecture, it should have a…

A function that is analytic on a domain of $\mathbb{C}^n$ is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein-Sato polynomial…

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely…

We show that the Bernstein-Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange's formula with the theory of Aomoto complexes due to Esnault,…

Given $p$ polynomials of $n$ variables over a field $k$ of characteristic 0 and a point $a \in k^n$, we propose an algorithm computing the local Bernstein-Sato ideal at $a$. Moreover with the same algorithm we compute a constructible…

Let $f$ be a semi-weighted-homogeneous polynomial having an isolated singularity at 0. Let $\alpha_{f,k}$ be the spectral numbers of $f$ at 0. By Malgrange and Varchenko there are non-negative integers $r_k$ such that the $\alpha_{f,k}-r_k$…

For a homogeneous polynomial of $n$ variables, we present a new method to compute the roots of Bernstein-Sato polynomial supported at the origin, assuming that general hyperplane sections of the associated projective hypersurface have at…

We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m x n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong…

The Bernstein-Sato polynomial (or global b-function) is an important invariant in singularity theory, which can be computed using symbolic methods in the theory of D-modules. After surveying algorithms for computing the global b-function,…