No 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).

In this paper we construct and study a fundamental solution of Cauchy's problem for p−adic parabolic equations of the typewhere f (D, β), β > 0, is an elliptic pseudo-differential operator. We also show that the fundamental solution is the transition density of a Markov process with state space Q n p .

We prove that the p-adic Koba-Nielsen type string amplitudes are bona fide integrals. We attach to these amplitudes Igusa-type integrals depending on several complex parameters and show that these integrals admit meromorphic continuations as rational functions. Then we use these functions to regularize the Koba-Nielsen amplitudes. As far as we know, there is no a similar result for the Archimedean Koba-Nielsen amplitudes. We also discuss the existence of divergencies and the connections with multivariate Igusa's local zeta functions.