Open Access
2003 Local zeta functions and Newton polyhedra
W. A. Zuniga-Galindo
Nagoya Math. J. 172: 31-58 (2003).


To a polynomial $f$ over a non-archimedean local field $K$ and a character $\chi$ of the group of units of the valuation ring of $K$ one associates Igusa's local zeta function $Z(s, f, \chi)$. In this paper, we study the local zeta function $Z(s, f, \chi)$ associated to a non-degenerate polynomial $f$, by using an approach based on the $p$-adic stationary phase formula and Néron $p$-desingularization. We give a small set of candidates for the poles of $Z(s, f, \chi)$ in terms of the Newton polyhedron $\Gamma(f)$ of $f$. We also show that for almost all $\chi$, the local zeta function $Z(s, f, \chi)$ is a polynomial in $q^{-s}$ whose degree is bounded by a constant independent of $\chi$. Our second result is a description of the largest pole of $Z(s, f, \chi_{triv})$ in terms of $\Gamma(f)$ when the distance between $\Gamma(f)$ and the origin is at most one.


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W. A. Zuniga-Galindo. "Local zeta functions and Newton polyhedra." Nagoya Math. J. 172 31 - 58, 2003.


Published: 2003
First available in Project Euclid: 27 April 2005

zbMATH: 1063.11048
MathSciNet: MR2019519

Primary: 11S40
Secondary: 11L05

Rights: Copyright © 2003 Editorial Board, Nagoya Mathematical Journal

Vol.172 • 2003
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