Igusa local zeta functions of a class of hybrid polynomials

TL;DR

This paper investigates Igusa's local zeta functions for a specific class of hybrid polynomials over non-archimedean local fields of positive characteristic, proving their rationality and explicitly describing their poles.

Contribution

It establishes the rationality and pole structure of Igusa's local zeta functions for hybrid polynomials, extending previous work in positive characteristic.

Findings

Proves the rationality of the local zeta functions.

Explicitly describes the poles of these functions.

Uses Igusa's stationary phase formula in the proof.

Abstract

In this paper, we study the Igusa's local zeta functions of a class of hybrid polynomials with coefficients in a non-archimedean local field of positive characteristic. Such class of hybrid polynomial was first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We prove the rationality of these local zeta functions and describe explicitly their poles. The proof is based on Igusa's stationary phase formula.