Local zeta functions and Newton polyhedra

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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, χtriv) in terms of Γ(f) when the distance between Γ(f) and the origin is at most one.