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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra


Authors: Willem Veys and W. A. Zúñiga-Galindo
Journal: Trans. Amer. Math. Soc. 360 (2008), 2205-2227
MSC (2000): Primary 11S40, 11D79, 14M25; Secondary 32S45
Published electronically: November 28, 2007
MathSciNet review: 2366980
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Abstract: In this paper we provide a geometric description of the possible poles of the Igusa local zeta function $ Z_{\Phi }(s,\mathbf{f}) $ associated to an analytic mapping $ \mathbf{f}=$ $ \left(f_{1},\ldots ,f_{l}\right) :U(\subseteq K^{n})\rightarrow K^{l} $, and a locally constant function $ \Phi$, with support in $ U $, in terms of a log-principalizaton of the $ K\left[x \right] - $ideal $ \mathcal{I}_{\mathbf{f}}=\left(f_{1},\ldots ,f_{l}\right)$. Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in terms of the log-canonical threshold of the subscheme given by $ \mathcal{I}_{\mathbf{f}} $. We associate to an analytic mapping $ \boldsymbol{f} $ $ = $ $ \left(f_{1},\ldots ,f_{l}\right) $ a Newton polyhedron $ \Gamma \left(\boldsymbol{f}\right) $ and a new notion of non-degeneracy with respect to $ \Gamma \left(\boldsymbol{f}\right) $. The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii's non-degeneracy notion depends on the Newton polyhedra of $ f_{1},\ldots ,f_{l} $ . By constructing a log-principalization, we give an explicit list for the possible poles of $ Z_{\Phi }(s,\mathbf{f}) $, $ l\geq 1 $, in the case in which $ \mathbf{f} $ is non-degenerate with respect to $ \Gamma \left(\boldsymbol{f}\right) $.


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Additional Information

Willem Veys
Affiliation: Department of Mathematics, University of Leuven, Celestijnenlaan 200 B, B-3001 Leuven (Heverlee), Belgium
Email: wim.veys@wis.kuleuven.be

W. A. Zúñiga-Galindo
Affiliation: Department of Mathematics and Computer Science, Barry University, 11300 N.E. Second Avenue, Miami Shores, Florida 33161
Address at time of publication: Departamento de Matemáticas, Centro de Investigacion y Estudios Avanzados del I.P.N., Av. Inst. Politécnico Nacional 2508, C.P. 07360, México D.F., México
Email: wzuniga@mail.barry.edu, wzuniga@math.cinvestav.mx

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04422-4
PII: S 0002-9947(07)04422-4
Keywords: Igusa zeta functions, congruences in many variables, topological zeta functions, motivic zeta functions, Newton polyhedra, toric varieties, log-principalization of ideals
Received by editor(s): January 9, 2006
Received by editor(s) in revised form: September 1, 2006
Published electronically: November 28, 2007
Additional Notes: The first author was partially supported by the Fund of Scientific Research – Flanders (G.0318.06).
The second author thanks the financial support of the NSA. Project sponsored by the National Security Agency under Grant Number H98230-06-1-0040. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.