Local Zeta Functions for Rational Functions and Newton Polyhedra

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