Bernstein-Sato polynomial and related invariants for meromorphic functions

TL;DR

This paper extends the theory of Bernstein-Sato polynomials to meromorphic functions, linking them to multiplier ideals and local zeta functions, thus advancing understanding of their analytic and algebraic properties.

Contribution

It introduces a new framework for Bernstein-Sato polynomials for meromorphic functions and connects these to multiplier ideals and local zeta functions.

Findings

Established Bernstein-Sato polynomials for meromorphic functions

Connected roots of Bernstein-Sato polynomials to jumping numbers of multiplier ideals

Analyzed the analytic continuation of local zeta functions in this context

Abstract

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 multiplier ideals for meromorphic functions and relate the jumping numbers of these multiplier ideals to the roots of the Bernstein-Sato polynomials.