Local Zeta Functions for Rational Functions and Newton Polyhedra

  • Miriam Bocardo–Gaspar
  • W. A. Zúñiga–Galindo
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 222)

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

Igusa local zeta functions Newton polyhedra Non-degeneracy conditions 

2000 Mathematics Subject Classification.

Primary 14G10 11S40 Secondary 14M25 

Notes

Acknowledgements

The authors wish to thank the referee for his/her careful reading of the original manuscript.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Miriam Bocardo–Gaspar
    • 1
  • W. A. Zúñiga–Galindo
    • 1
  1. 1.Centro de Investigacion y de Estudios Avanzados del I.P.N.Departamento de Matematicas, Av. Instituto Politecnico Nacional 2508 ColSan Pedro Zacatenco, Mexico D.F.Mexico

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