Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function ZΦ(s, f) associated to an analytic mapping f = (f1, . . . , fl) : U(⊆ K ) → K, and a locally constant function Φ, with support in U , in terms of a log-principalizaton of the K [x]−ideal If = (f1, . . . , fl). Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonical threshold of the subscheme given by If . We associate to an analytic mapping f = (f1, . . . , fl) a Newton polyhedron Γ (f) and a new notion of non-degeneracy with respect to Γ (f). The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii’s non-degeneracy notion depends on the Newton polyhedra of f1, . . . , fl . By constructing a log-principalization, we give an explicit list for the possible poles of ZΦ(s, f), l ≥ 1, in the case in which f is non-degenerate with respect to Γ (f).