Local zeta functions and Newton polyhedra

Abstract

To a polynomial $f$ over a non-archimedean local field $K$ and a character $χ$ of the group of units of the valuation ring of $K$ one associates Igusa's local zeta function $Z(s,f,χ)$. In this paper, we study the local zeta function $Z(s,f,χ)$ associated to a non-degenerate polynomial $f$, by using an approach based on the p-adic stationary phase formula and Néron p-desingularization. We give a small set of candidates for the poles of $Z(s,f,χ)$ in terms of the Newton polyhedron $ Γ(f)$ of $f$. We also show that for almost all $χ$, the local zeta function $Z(s,f,χ)$ is a polynomial in $q^{-s}$ whose degree is bounded by a constant independent of $χ$. Our second result is a description of the largest pole of $Z(s,f, χ_{\text{triv}})$ in terms of $ Γ(f)$ when the distance between $Γ(f)$ and the origin is at most one.