Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
The following link can be shared to navigate to this page. You can select the link to copy or click the 'Copy To Clipboard' button below.
Copy To Clipboard
Successfully Copied!
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
Front Cover for Quasi-Ordinary Power Series and Their Zeta Functions
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
Front Cover for Quasi-Ordinary Power Series and Their Zeta Functions
Click above image for expanded view
  • Front Cover for Quasi-Ordinary Power Series and Their Zeta Functions
  • Back Cover for Quasi-Ordinary Power Series and Their Zeta Functions
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.

    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
  • Request Review Copy
  • Get Permissions
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.

Readership

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
Please select which format for which you are requesting permissions.