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Generalized Brieskorn Modules II: Higher Bernstein Polynomials and Multiple Poles

Daniel Barlet

TL;DR

This work develops a deep link between higher Bernstein polynomials of geometric (a,b)-modules and the pole structure of meromorphic extensions tied to polar parts of holomorphic functions with isolated singularities. By introducing and exploiting the notion of frescos and their $[\\eta]$-primitive quotients, the authors show that roots of higher Bernstein polynomials bound the order of poles for associated meromorphic distributions, and conversely, the presence of poles forces lower bounds on nilpotent orders in the corresponding fresco. The results include precise statements for the case of Frescos, Jordan blocks, and the general geometric setting, with several corollaries translating root data into pole behavior for reduced Bernstein polynomials. Overall, the paper advances the understanding of how singularity theory, Brieskorn modules, and (a,b)-module theory interact to govern the analytic continuation properties of period-like integrals and their monodromy-determined poles.

Abstract

Our main result is to show that the existence of a root in. --$α$--Nfor the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphicform in the (convergent) Brieskorn (a,b)-module associated to f, under the hypothesis that f has an isolated singularity at the origin relative to the eigenvalue exp(2i$π$$α$) of the monodromy, produces poles of order at least p for themeromorphic extension of the (conjugate) analytic functional given by polar partsat points--$α$--N for N well chosen integer. This result is new, even forp= 1. As a corollary, this implies that, in the case of an isolated singularity for f,the existence of a root in. --$α$--N for the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphic form implies the existence of at leastp roots (counting multiplicities) for the usual reduced Bernstein polynomial of thegerm of f at the origin.In the case of an isolated singularity for f, we obtain that for each $α$ thebiggest root --$α$--m. of the reduced Bernstein polynomial of f in --$α$--N producesa pole at--$α$--m for the meromorphic extension of the associated distribution

Generalized Brieskorn Modules II: Higher Bernstein Polynomials and Multiple Poles

TL;DR

This work develops a deep link between higher Bernstein polynomials of geometric (a,b)-modules and the pole structure of meromorphic extensions tied to polar parts of holomorphic functions with isolated singularities. By introducing and exploiting the notion of frescos and their -primitive quotients, the authors show that roots of higher Bernstein polynomials bound the order of poles for associated meromorphic distributions, and conversely, the presence of poles forces lower bounds on nilpotent orders in the corresponding fresco. The results include precise statements for the case of Frescos, Jordan blocks, and the general geometric setting, with several corollaries translating root data into pole behavior for reduced Bernstein polynomials. Overall, the paper advances the understanding of how singularity theory, Brieskorn modules, and (a,b)-module theory interact to govern the analytic continuation properties of period-like integrals and their monodromy-determined poles.

Abstract

Our main result is to show that the existence of a root in. ----Nfor the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphicform in the (convergent) Brieskorn (a,b)-module associated to f, under the hypothesis that f has an isolated singularity at the origin relative to the eigenvalue exp(2i) of the monodromy, produces poles of order at least p for themeromorphic extension of the (conjugate) analytic functional given by polar partsat points----N for N well chosen integer. This result is new, even forp= 1. As a corollary, this implies that, in the case of an isolated singularity for f,the existence of a root in. ----N for the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphic form implies the existence of at leastp roots (counting multiplicities) for the usual reduced Bernstein polynomial of thegerm of f at the origin.In the case of an isolated singularity for f, we obtain that for each thebiggest root ----m. of the reduced Bernstein polynomial of f in ----N producesa pole at----m for the meromorphic extension of the associated distribution

Paper Structure

This paper contains 82 sections, 53 theorems, 101 equations.

Table of Contents

  1. Abstract.
  2. AMS classification.
  3. Key words.
  4. Introduction
  5. The aim of this article
  6. The main results
  7. Asymptotics and geometric (a,b)-modules
  8. Remarks.
  9. Some facts from [part I]
  10. Proof.
  11. Higher Bernstein Polynomials
  12. Remarks.
  13. Proof.
  14. Proof.
  15. Proof.
  16. ...and 67 more sections

Key Result

Theorem 1.2.1

Under the hypothesis $H(\alpha, 1)$ for $f$, let $\omega \in \Omega^{n+1}_0$ such that the nilpotent order of the fresco $\mathcal{F}^{[\alpha]}_\omega$ is equal to $p \in \mathbb{N}^*$, where $\mathcal{F}_\omega := B[a]\omega \subset H^{n+1}_0$. Then there exists $\omega' \in \Omega^{n+1}_0$ and $h holomorphic for $\Re(\lambda) \gg 1$, has a pole of order at least $p$ at a point in $-\alpha - \ma

Theorems & Definitions (58)

  • Theorem 1.2.1
  • Lemma 1.2.2
  • Corollary 1.2.3
  • Theorem 1.2.4
  • Theorem 1.2.5
  • Lemma 2.0.1
  • Definition 2.0.2
  • Lemma 2.1.1
  • Definition 2.2.1
  • Proposition 2.2.2
  • ...and 48 more