Igusa's Local Zeta Functions and Exponential Sums for Arithmetically Non Degenerate Polynomials

TL;DR

This paper analyzes the poles of twisted local zeta functions for arithmetically non degenerate polynomials in two variables over non-Archimedean fields, providing explicit pole candidates and asymptotic exponential sum expansions.

Contribution

It introduces explicit pole candidates for local zeta functions based on arithmetic Newton polygons, extending non degeneracy notions and deriving exponential sum asymptotics.

Findings

Explicit pole candidates derived from arithmetic Newton polygons

Asymptotic expansions for exponential sums attached to these polynomials

Extension of non degeneracy concepts to broader polynomial classes

Abstract

We study the twisted local zeta function associated to a polynomial in two variables with coefficients in a non-Archimedean local field of arbitrary characteristic. Under the hypothesis that the polynomial is arithmetically non degenerate, we obtain an explicit list of candidates for the poles in terms of geometric data obtained from a family of arithmetic Newton polygons attached to the polynomial. The notion of arithmetical non degeneracy due to Saia and Z\'u\~niga-Galindo is weaker than the usual notion of non degeneracy due to Kouchnirenko. As an application we obtain asymptotic expansions for certain exponential sums attached to these polynomials.