Local zeta functions and Newton polyhedra

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