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 $Z_{\Phi }(s,\mathbf{f})$ associated to an analytic mapping $\mathbf{f}=$ $\left(f_{1},\ldots ,f_{l}\right) :U(\subseteq K^{n})\rightarrow K^{l}$, and a locally constant function $\Phi$, with support in $U$, in terms of a log-principalizaton of the $K\left[x \right] -$ideal $\mathcal{I}_{\mathbf{f}}=\left(f_{1},\ldots ,f_{l}\right)$. 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 $\mathcal{I}_{\mathbf{f}}$. We associate to an analytic mapping $\boldsymbol{f}$ $=$ $\left(f_{1},\ldots ,f_{l}\right)$ a Newton polyhedron $\Gamma \left(\boldsymbol{f}\right)$ and a new notion of non-degeneracy with respect to $\Gamma \left(\boldsymbol{f}\right)$. 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 $f_{1},\ldots ,f_{l}$ . By constructing a log-principalization, we give an explicit list for the possible poles of $Z_{\Phi }(s,\mathbf{f})$, $l\geq 1$, in the case in which $\mathbf{f}$ is non-degenerate with respect to $\Gamma \left(\boldsymbol{f}\right)$.