Related papers: Decompositions of Bernstein-Sato polynomials and s…
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…
In this paper we compute b-functions (or Bernstein-Sato polynomials) of various semi-invariants of quivers. The main tool is an explicit relation for the b-functions between semi-invariants that correspond to each other under reflection…
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 introduce the notion of 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…
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…
We compute the Bernstein-Sato polynomial of $f$, a function which given a pair $(M,v)$ in $X = M_n(\mathbf{C}) \times \mathbf{C}^n$ tests whether $v$ is a cyclic vector for $M$. The proof includes a description of shift operators…
This paper is the widely extended version of the publication, appeared in Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on proofs, present new algorithms and examples. We present a general algorithm for…
We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the…
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 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 $\alpha_{f,k}-r_k$…
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 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,…
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…
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,…
The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f(x) = g(h(x)) in F[x] over a field F is well understood in…
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…
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…
Let $f_1$ and $f_2$ be two semi-universal deformations of quasi homogeneous polynomials in two variables respectively for the weight vectors $\rho_1$ and $\rho_2$ such that they satisfy similar conditions to that of semi quasi homogeneous…
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…