Quasi-Ordinary Power Series and Their Zeta Functions

Released on by Enrique Artal-Bartolo
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  • Quasi-Ordinary Power Series and Their Zeta Functions Book Detail

  • Author : Enrique Artal-Bartolo
  • Release Date : 2005-10-05
  • Publisher : American Mathematical Soc.
  • Genre : Functions, Zeta
  • Pages : 100
  • ISBN 13 : 9780821865637
  • File Size : 69,69 MB
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Quasi-Ordinary Power Series and Their Zeta Functions by Enrique Artal-Bartolo PDF Summary

Book Description: 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.

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Zeta Functions in Algebra and Geometry

Zeta Functions in Algebra and Geometry

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Contains the proceedings of the Second International Workshop on Zeta Functions in Algebra and Geometry held May 3-7, 2010 at the Universitat de les Illes Balea