New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education
Quasi-Ordinary Power Series and Their Zeta Functions
Enrique Artal Bartolo, University of Zaragoza, Spain, Pierrette Cassou-Noguès, Bordeaux, France, and Ignacio Luengo and Alejandro Melle Hernández, Universidad Complutense de Madrid, Spain
 SEARCH THIS BOOK:
Memoirs of the American Mathematical Society
2005; 85 pp; softcover
Volume: 178
ISBN-10: 0-8218-3876-8
ISBN-13: 978-0-8218-3876-1
List Price: US$58 Individual Members: US$34.80
Institutional Members: US\$46.40
Order Code: MEMO/178/841

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.