Newton Polyhedra and Igusa's Local Zeta Function

Abstract

We give a very explicit formula for Igusa's local zeta function Zf(s) associated to a polynomial f in several variables over the p-adic numbers, when f is sufficiently non-degenerated with respect to its Newton polyhedron Γ(f). Using this formula and the estimates of Adolphson and Sperber on exponential sums over finite fields, we study the largest real pole different from −1 of Zf(s).