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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local zeta function for curves, non-degeneracy conditions and Newton polygons

Authors: M. J. Saia and W. A. Zuniga-Galindo
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 59-88
MSC (2000): Primary 11D79, 14G20, 14M25
Published electronically: December 15, 2003
MathSciNet review: 2098087
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Abstract: This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left\{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right\} $called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non-degeneracy with respect to $\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right] $ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$.

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Additional Information

M. J. Saia
Affiliation: Instituto de Matemática E Computaçao, Universidade de São Paulo at São Carlos, Av. do Trabalhador São-Carlense 400, CEP 13560-970, São Carlos - SP, Brasil

W. A. Zuniga-Galindo
Affiliation: Department of Mathematics and Computer Science, Barry University, 11300 N.E. Second Avenue, Miami Shores, Florida 33161

Keywords: Igusa local zeta functions, Newton polygons, degenerate curves, non-degeneracy conditions, polynomial congruences
Received by editor(s): July 10, 2001
Received by editor(s) in revised form: May 6, 2003
Published electronically: December 15, 2003
Additional Notes: The first named author was partially supported by CNPq-Grant 300556/92-6
The second named author was supported by COLCIENCIAS-Grant # 089-2000. The second named author also thanks the partial support given by FAPESP for visiting the Instituto de Matemática e Computaçao, Universidade de São Paulo, Campus São Carlos, in January 2000
Article copyright: © Copyright 2003 American Mathematical Society