Bernstein-Sato polynomials of semi-weighted-homogeneous polynomials of nearly Brieskorn-Pham type

TL;DR

This paper presents a simple algorithm to determine shifts in Bernstein-Sato polynomials for semi-weighted-homogeneous polynomials of nearly Brieskorn-Pham type, refining classical results and enabling explicit computations.

Contribution

It introduces an efficient method to compute shifts in Bernstein-Sato polynomials for a specific class of polynomials, avoiding Gr"obner bases and refining existing stratification results.

Findings

Algorithm for shifts using Singular or C

Refinement of stratification control by weights

Examples with distant and nonconsecutive roots

Abstract

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 $\alpha_{f,k}-r_k$ are the roots up to sign of the local Bernstein-Sato polynomial $b_f(s)$ divided by $s+1$. However, it is quite difficult to determine these shifts $r_k$ explicitly on the parameter space of $\mu$-constant deformation of a weighted homogeneous polynomial. Assuming the latter is nearly Brieskorn-Pham type, we can obtain a very simple algorithm to determine these shifts, which can be realized by using Singular (or even C) without employing Gr\"obner bases. This implies a refinement of classical work of M. Kato and P. Cassou-Nogu\`es in two variable cases, showing that the stratification of the parameter space can be controlled by using the (partial) additive semigroup structure of the weights of parameters. As a corollary we get for instance a sufficient condition for all the shiftable roots of $b_f(s)$ to be shifted. We can also produce examples where the minimal root of $b_f(s)$ is quite distant from the others as well as examples of semi-homogeneous polynomials with roots of $b_f(s)$ nonconsecutive.