Poles of Non-Archimedean Zeta Functions for Non-degenerate Rational Functions

Abstract

In this article, we study local zeta functions over non-Archimedean locals fields of arbitrary characteristic attached to rational functions and characters $\chi$ of the units of the ring of integers $\mathcal{O}_{K}$, by using an approach based on the multivariate $\pi$-adic stationary phase formula and Newton polyhedra. When the rational function is non-degenerate with respect to its Newton polyhedron, we give an explicit formula for the local zeta function and a list of the possible poles in terms of the normal vectors of the supporting hyperplanes of the Newton polyhedron attached to the rational function and their expected multiplicities. Furthermore, we obtain some conditions under which the local zeta function attached to the trivial character has at least one real pole by describing the largest negative real pole and the smallest positive one.