Related papers: Bernstein-Sato polynomials for general ideals vs. …

Let $G$ be a linearly reductive group acting on a vector space $V$, and $f$ a (semi-)invariant polynomial on $V$. In this paper we study systematically decompositions of the Bernstein-Sato polynomial of $f$ in parallel with some…

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 study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated…

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…

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)…

The Bernstein-Sato polynomial, or the $b$-function, is an important invariant of singularities of hypersurfaces that is difficult to compute in general. We describe a few different results towards computing the $b$-function of the…

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the…

A log generic hypersurface in $\mathbb{P}^n$ with respect to a birational modification of $\mathbb{P}^n$ is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic…

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…

In analogy with the complex analytic case, Musta\c{t}\u{a} constructed (a family of) Bernstein-Sato polynomials for the structure sheaf $\mathcal{O}_X$ and a hypersurface $(f=0)$ in $X$, where $X$ is a regular variety over an $F$-finite…

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…

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…

This paper investigates the existence and properties of a Bernstein-Sato functional equation in nonregular settings. In particular, we construct $D$-modules in which such formal equations can be studied. The existence of the Bernstein-Sato…

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,…

We give estimates for the zero loci of Bernstein-Sato ideals. An upper bound is proved as a multivariate generalisation of the upper bound by Lichtin for the roots of Bernstein-Sato polynomials. The lower bounds generalise the fact that…

In this survey, we use (more or less) elementary means to establish the well-known result that for any given smooth multivariate function, the respective multivariate Bernstein polynomials converge to that function in all derivatives on…

Given a family of analytic functions near 0 \in C^n parametrized by a smooth space, we study the Bernstein polynomial of the fiber on an irreducible variety V of the space of parameters and we show that it is generically constant. We prove…

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…

It is be shown that the sequence of Bernstein polynomials for a function of several variables converges to this function uniformly along with every partial derivative of any order, provided that the latter derivative is well defined and…

We present a theory of the $b$-function (or Bernstein-Sato polynomial) in positive characteristic. Let $f$ be a non-constant polynomial with coefficients in a perfect field $k$ of characteristic $p>0.$ Its $b$-function $b_f$ is defined to…