# Quasi-Ordinary Power Series and Their Zeta Functions

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The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h, T)$ of a quasi-ordinary power series $h$ of arbitrary dimension over an algebraically closed field of characteristic zero from its characteristic exponents without using embedded resolution of singularities. This allows us to effectively represent $Z_{\text{DL}}(h, T)=P(T)/Q(T)$ such that almost all the candidate poles given by $Q(T)$ are poles. Anyway, these candidate poles give eigenvalues of the monodromy action on the complex $R\psi_h$ of nearby cycles on $h^{-1}(0).$ In particular we prove in this case the monodromy conjecture made by Denef-Loeser for the local motivic zeta function and the local topological zeta function. As a consequence, if $h$ is a quasi-ordinary polynomial defined over a number field we prove the Igusa monodromy conjecture for its local Igusa zeta function.Let 71, ..., 7r be the compact edges of the Newton diagram of h; for each q a 1, . . . , r, we have denoted by v(q) the number of non-zero distinct roots of h^q . Each one of these roots defines a Newton map and let hqj j = 1, . . . , v(q), be theanbsp;...

 Title : Quasi-Ordinary Power Series and Their Zeta Functions Author : Enrique Artal-Bartolo Publisher : American Mathematical Soc. - 2005-10-05

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