On the local zeta functions and the b-functions of certain hyperplane arrangements

TL;DR

This paper proves conjectures relating the poles of p-adic and topological local zeta functions to roots of Bernstein-Sato polynomials for specific hyperplane arrangements, including in three-dimensional space.

Contribution

It establishes the validity of these conjectures for certain hyperplane arrangements, advancing understanding of the connection between zeta functions and b-functions.

Findings

Conjectures confirmed for specific hyperplane arrangements.

Includes the case of reduced hyperplane arrangements in three dimensions.

Provides new proofs linking local zeta functions and Bernstein-Sato polynomials.

Abstract

Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef and F. Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein-Sato polynomials (i.e. the b-functions). We prove these conjectures for certain hyperplane arrangements, including the case of reduced hyperplane arrangements in three-dimensional affine space.