Abstract
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.
Keywords
2000 Mathematics Subject Classification.
The second author was partially supported by Conacyt Grant No. 250845.
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The authors wish to thank the referee for his/her careful reading of the original manuscript.
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Bocardo–Gaspar, M., Zúñiga–Galindo, W.A. (2018). Local Zeta Functions for Rational Functions and Newton Polyhedra. In: Araújo dos Santos, R., Menegon Neto, A., Mond, D., Saia, M., Snoussi, J. (eds) Singularities and Foliations. Geometry, Topology and Applications. NBMS BMMS 2015 2015. Springer Proceedings in Mathematics & Statistics, vol 222. Springer, Cham. https://doi.org/10.1007/978-3-319-73639-6_12
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