# Local Zeta Functions for Rational Functions and Newton Polyhedra

## TL;DR

This paper develops a framework for analyzing local zeta functions associated with non-degenerate rational functions over non-Archimedean fields, providing explicit formulas and meromorphic continuation with poles on both sides of the complex plane.

## Contribution

It introduces a new notion of non-degeneracy for rational functions relative to Newton polyhedra and establishes the meromorphic continuation of their local zeta functions.

## Key findings

- Existence of meromorphic continuation as rational functions of q^{-s}
- Explicit formulas for local zeta functions
- Poles with both positive and negative real parts

## Abstract

In this article, we introduce a notion of non-degeneracy, with respect to certain Newton polyhedra, for rational functions over non-Archimedean locals fields of arbitrary characteristic. We study the local zeta functions attached to non-degenerate rational functions, we show the existence of a meromorphic continuation for these zeta functions, as rational functions of $q^{-s}$, and give explicit formulas. In contrast with the classical local zeta functions, the meromorphic continuation of zeta functions for rational functions have poles with positive and negative real parts.