Local Zeta Functions for Rational Functions and Newton Polyhedra

2017/02/22 by Bocardo-Gaspar, Miriam, Zúñiga-Galindo, W. A.
#11S40 (Primary) #14G10 #14M25 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1702.06938

Abstract

In this article, we introduce a notion of non-degeneracy, with respect to certain Newton polyhedra, for rational functions over non-Archimedean locals fields of arbitrary characteristic. We study the local zeta functions attached to non-degenerate rational functions, we show the existence of a meromorphic continuation for these zeta functions, as rational functions of q^-s, and give explicit formulas. In contrast with the classical local zeta functions, the meromorphic continuation of zeta functions for rational functions have poles with positive and negative real parts.

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