On Igusa local zeta functions of Hauser hybrid polynomials

TL;DR

This paper proves the rationality of Igusa local zeta functions for a class of hybrid polynomials in positive characteristic local fields, extending previous results and explicitly identifying potential poles.

Contribution

It establishes the rationality of local zeta functions for Hauser hybrid polynomials in positive characteristic, a previously unknown case, and explicitly lists candidate poles.

Findings

Proves rationality of local zeta functions for hybrid polynomials in positive characteristic.

Explicitly lists all candidate poles for these zeta functions.

Generalizes previous work by León-Cardenal, Ibadula, Segers, Yin, and Hong.

Abstract

Let $K$ be a local field and $f(x)\in K[x]$ be a non-constant polynomial. When ${\rm char}K=0$, Igusa showed the local zeta function is a rational function. However, when ${\rm char}K>0$, the rationality of the local zeta function is unknown in general. In this paper, we study the local zeta functions for the so-called hybrid polynomials in three variables with coefficients in a non-archimedean local field of positive characteristic. These hybrid polynomials were first introduced by Hauser in 2003 to study the resolution of singularities in positive characteristic. We establish the rationality theorem for these local zeta functions and list explicitly all the candidate poles. Our result generalizes the work of Le$\acute{o}$n-Cardenal, Ibadula and Segers and that of Yin and Hong.