Newton polyhedra, unstable faces and the poles of Igusa’s local zeta function

Abstract

In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial f is smaller than expected. We carry out this study in the case that f is sufficiently non-degenerate with respect to its Newton polyhedron Γ(f), and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial f to the same question about polynomials f μ , where μ are faces of Γ(f) depending on the examined pole and f μ is obtained from f by throwing away all monomials of f whose exponents do not belong to μ. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial f μ , with μ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than expected.