Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra

Abstract

In this paper we provide a geometric description of the possible poles of the Igusa local zeta function Z Φ (s, f) associated to an analytic mapping f = (f 1 ,. .. , f l) : U (⊆ K n) → K l , and a locally constant function Φ, with support in U , in terms of a log-principalizaton of the K [x] −ideal I f = (f 1 ,. .. , f l). 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 I f. We associate to an analytic mapping f = (f 1 ,. .. , f l) 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 f 1 ,. .. , f l. By constructing a log-principalization, we give an explicit list for the possible poles of Z Φ (s, f), l ≥ 1, in the case in which f is non-degenerate with respect to Γ (f).

Igusa’s p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

“…In this section, we give an explicit formula for Igusa's local zeta function Z f ,gdx , associated to f and the integration measure |g(x)||dx| on Z n p , in the same style as the previous ones. This time we adapt a formula for Igusa's local zeta function of a polynomial mapping, given in [16], which is a generalization of the formulas proven in [6] and [9]. In the same way the formula we will state here, generalizes the main results of Sections 2 and 4.…”

“…For instance, we now associate these zeta functions to several polynomials or to an ideal in a polynomial ring. See, e.g., [9,16,14,15].…”

“…In this respect, it is useful to examine which known results and formulas have their analogue in this generalized context. In this paper, we adapt results from [6,9,16] and prove an explicit formula for Igusa's p-adic zeta function of a strongly non-degenerated polynomial mapping and a 'polynomial measure'. Although we state the formula for the p-adic one, the result can also be formulated for the topological and the motivic zeta function.…”

“…, x n ] and the usual integration measure. In 2008, Veys and Zúñiga-Galindo [16] generalized those two formulas to a formula for Igusa's local zeta function Z f of a polynomial mapping f , that is strongly non-degenerated over F p with respect to its Newton polyhedron (Definition 1.5).…”

“…In the first section, we list all the needed definitions, notations and results that are already known, without much explanation. For more details on Igusa's zeta function and Newton polyhedra, we refer to [6], [9], and [16]; for more background on convex geometry, we refer to [13]. In Section 2, we derive a formula for Igusa's zeta function associated to a monomial ideal and a measure |g(x)||dx|.…”