Related papers: Bernstein-Sato polynomials in positive characteris…
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…
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 fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato polynomial for polynomials with $\mathbb{Z} / p^m$-coefficients, compatible with existing theory in the case $m = 1$. We show that the ``roots" of such polynomials are…
Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose…
In 1987, C. Sabbah proved the existence of Bernstein-Sato polynomials associated with several analytic functions. The purpose of this article is to give a more elementary and constructive proof of the result of C. Sabbah based on the notion…
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 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…
We introduce and study invariants of singularities in positive characteristic called F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals in characteristic zero. We discuss the connection between the…
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…
We describe the roots of the Bernstein-Sato polynomial of a monomial ideal using reduction mod p and invariants of singularities in positive chracteristic. We give in this setting a positive answer to a problem of Takagi, Watanabe and the…
Following work of Musta\c{t}\u{a} and Bitoun we recently developed a notion of Bernstein-Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein-Sato polynomial. Here we prove that for…
We overview numerous algorithms in computational $D$-module theory together with the theoretical background as well as the implementation in the computer algebra system \textsc{Singular}. We discuss new approaches to the computation 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…
We present a quite efficient method to calculate the roots of Bernstein-Sato polynomial for a defining polynomial $f$ of a projective hypersurface $Z\subset{\mathbb P}^{n-1}$ of degree $d$ having only weighted homogeneous isolated…
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…
Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset…
This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.
We prove some sufficient conditions in order that a root of the Bernstein-Sato polynomial contributes to a difference between certain D-modules generated by rational powers of a holomorphic function; for instance, this holds in the case of…
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 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…