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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
Ignacio Luengo Universidad Complutense de Madrid, Madrid, Spain
Alejandro Melle Hernández Universidad Complutense de Madrid, Madrid, Spain
Quasi-Ordinary Power Series and Their Zeta Functions
eBook ISBN:  978-1-4704-0442-0
Product Code:  MEMO/178/841.E
List Price: $61.00
MAA Member Price: $54.90
AMS Member Price: $36.60
Quasi-Ordinary Power Series and Their Zeta Functions
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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
Ignacio Luengo Universidad Complutense de Madrid, Madrid, Spain
Alejandro Melle Hernández Universidad Complutense de Madrid, Madrid, Spain
eBook ISBN:  978-1-4704-0442-0
Product Code:  MEMO/178/841.E
List Price: $61.00
MAA Member Price: $54.90
AMS Member Price: $36.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1782005; 85 pp
    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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1782005; 85 pp
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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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