On the reduced Bernstein-Sato polynomial of Thom-Sebastiani singularities

TL;DR

This paper provides an algebraic proof of a relation between Bernstein-Sato polynomials of Thom-Sebastiani sums, extending previous results to more general algebraic settings beyond analytic functions.

Contribution

It introduces a purely algebraic proof of the relation between Bernstein-Sato polynomials for Thom-Sebastiani sums, applicable in broader algebraic contexts.

Findings

Established a general algebraic relation for Bernstein-Sato polynomials of Thom-Sebastiani sums.

Extended the relation beyond analytic functions to more general algebraic settings.

Provided a new proof technique that does not rely on analytic assumptions.

Abstract

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) Bernstein-Sato polynomial of the Thom-Sebastiani sum $f+g$ can be expressed in terms of those of $f$ and $g$. In this note we give a purely algebraic proof of a similar relation between the whole functional equations that can be applied to any setting (not necessarily analytic) in which Bernstein-Sato polynomials can be defined.