# Quasi-Ordinary Power Series and Their Zeta Functions

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*Enrique Artal Bartolo; Pierrette Cassou-Noguès; Ignacio Luengo; Alejandro Melle Hernández*

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.

#### Table of Contents

# Table of Contents

## Quasi-Ordinary Power Series and Their Zeta Functions

- Contents v6 free
- Introduction 18 free
- Chapter 1. Motivic integration 714 free
- Chapter 2. Generating functions and Newton polyhedra 1118
- Chapter 3. Quasi–ordinary power series 2128
- Chapter 4. Denef–Loeser motivic zeta function under the Newton maps 4148
- Chapter 5. Consequences of the main theorems 6370
- Chapter 6. Monodromy conjecture for quasi–ordinary power series 7784
- Bibliography 8390