Quasi-ordinary power series and their zeta functions


Quasi-ordinary power series and their zeta functions

Enrique Artal Bartolo ... [et al.]

(Memoirs of the American Mathematical Society, no. 841)

American Mathematical Society, 2005

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"November 2005, Volume 178, number 841 (end of volume)."

Includes bibliographical references (p. 83-85)



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.


Introduction Motivic integration Generating functions and Newton polyhedra Quasi-ordinary power series Denef-Loeser motivic zeta function under the Newton maps Consequences of the main theorems Monodromy conjecture for quasi-ordinary power series Bibliography.

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