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Quasi-Ordinary Power Series and Their Zeta Functions

Enrique Artal Bartolo University of Zaragoza, Zaragoza, Spain
Pierrette Cassou-Noguès Bordeaux, France
Available Formats:
Electronic ISBN: 978-1-4704-0442-0
Product Code: MEMO/178/841.E
85 pp
List Price: $61.00 MAA Member Price:$54.90
AMS Member Price: $36.60 Click above image for expanded view Quasi-Ordinary Power Series and Their Zeta Functions Enrique Artal Bartolo University of Zaragoza, Zaragoza, Spain Pierrette Cassou-Noguès Bordeaux, France Ignacio Luengo Universidad Complutense de Madrid, Madrid, Spain Alejandro Melle Hernández Universidad Complutense de Madrid, Madrid, Spain Available Formats:  Electronic ISBN: 978-1-4704-0442-0 Product Code: MEMO/178/841.E 85 pp  List Price:$61.00 MAA Member Price: $54.90 AMS Member Price:$36.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1782005
MSC: Primary 14; 32;

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.

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

• 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
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Volume: 1782005
MSC: Primary 14; 32;

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.