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

Willem Veys·W. A. Zuniga-Galindo
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Abstract

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function associated to an analytic mapping and a locally constant function, in terms of a log-principalizaton of an ideal naturally attached to the mapping. 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 the ideal attached to the mapping. We associate to an analytic mapping a Newton polyhedron and a new notion of non-degeneracy with respect to it. By constructing a log-principalization, we give an explicit list for the possible poles of the Igusa zeta function associated to a non-degenerate mapping.

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Willem Veys, W. A. Zuniga-Galindo. 2006-09-08. Zeta Functions for Analytic Mappings, Log-principalization of Ideals, and Newton Polyhedra. https://arxiv.org/abs/math/0601336

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