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Bernstein-Sato polynomial and related invariants for meromorphic functions

Josep Àlvarez Montaner, Manuel González Villa, Edwin León-Cardenal, Luis Núñez-Betancourt

TL;DR

The paper develops a Bernstein-Sato theory for meromorphic functions by defining order-$\alpha$ polynomials $b_{f/g}^\alpha(s)$ and the standard $b_{f/g}(s)$, and builds both analytic and algebraic multiplier-ideals frameworks for meromorphic germs. Using $D$-module methods in the Sabbah framework, it establishes functional equations, proves rationality of roots, and derives the meromorphic continuation of Archimedean local zeta functions with poles controlled by the $b_{f/g}^\alpha(s)$, while relating jumping numbers to these roots. It further extends multiplier-ideal theory to meromorphic contexts, showing rational jumping numbers and Skoda-type relations, and presents Takeuchi’s polynomial within a holonomic argument, clarifying the connections among zeta poles, monodromy invariants, and divisor data. Collectively, these results link singularity invariants of meromorphic functions to zeta-function analytic properties and to multiplier-ideal thresholds, providing a cohesive toolkit for studying meromorphic hypersurface invariants.

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.

Bernstein-Sato polynomial and related invariants for meromorphic functions

TL;DR

The paper develops a Bernstein-Sato theory for meromorphic functions by defining order- polynomials and the standard , and builds both analytic and algebraic multiplier-ideals frameworks for meromorphic germs. Using -module methods in the Sabbah framework, it establishes functional equations, proves rationality of roots, and derives the meromorphic continuation of Archimedean local zeta functions with poles controlled by the , while relating jumping numbers to these roots. It further extends multiplier-ideal theory to meromorphic contexts, showing rational jumping numbers and Skoda-type relations, and presents Takeuchi’s polynomial within a holonomic argument, clarifying the connections among zeta poles, monodromy invariants, and divisor data. Collectively, these results link singularity invariants of meromorphic functions to zeta-function analytic properties and to multiplier-ideal thresholds, providing a cohesive toolkit for studying meromorphic hypersurface invariants.

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.
Paper Structure (11 sections, 22 theorems, 92 equations, 1 figure)

This paper contains 11 sections, 22 theorems, 92 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Log-resolution of meromorphic functions
  3. Meromorphic Bernstein-Sato polynomial
  4. Rings of differential operators and its modules
  5. A $D_{R|\mathbb{K}}[s]$-module for meromorphic functions
  6. Bernstein-Sato polynomials for meromorphic functions
  7. Rationality of the roots of the meromorphic Bernstein-Sato polynomial.
  8. Takeuchi's meromorphic Bernstein-Sato polynomial
  9. Archimedean Local Zeta Functions
  10. Multiplier ideals: analytic construction
  11. Multiplier ideals: algebraic construction

Key Result

Theorem 1

Let $R$ be either a polynomial ring $R=\mathbb{K}[x_1,\dots,x_n]$ or a ring of holomorphic functions $R=\mathbb{C}\{x_1,\dots,x_n\}$. Let $f,g\in R$ be nonzero elements. Then:

Figures (1)

  • Figure :

Theorems & Definitions (61)

  • Theorem 1: Theorem \ref{['Thm:GeneralizedFuncEq']}
  • Theorem 2: Theorem \ref{['Thm:Kashiwara']}, Corollary \ref{['Cor:Kashiwara']}
  • Theorem 3: Theorem \ref{['Thm:MerContZetaF']}
  • Theorem 4
  • Proposition 5
  • Definition 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 51 more