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