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 $Z_{\Phi }(s,\mathbf {f})$ 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 $\Gamma \left (\boldsymbol {f}\right )$.