# Quasi-ordinary power series and their zeta functions

### About this Title

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

Publication: Memoirs of the American Mathematical Society

Publication Year
2005: Volume 178, Number 841

ISBNs: 978-0-8218-3876-1 (print); 978-1-4704-0442-0 (online)

DOI: http://dx.doi.org/10.1090/memo/0841

MathSciNet review: 2172403

MSC: Primary 14B05; Secondary 11S40, 14E15, 32S50

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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.

Readership

Graduate students and research mathematicians
interested in analysis and number theory.

### Table of Contents

**Chapters**

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