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Local Zeta Functions for Rational Functions and Newton Polyhedra

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Singularities and Foliations. Geometry, Topology and Applications (NBMS 2015, BMMS 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((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.

The second author was partially supported by Conacyt Grant No. 250845.

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Acknowledgements

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

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Authors and Affiliations

  1. Centro de Investigacion y de Estudios Avanzados del I.P.N., Departamento de Matematicas, Av. Instituto Politecnico Nacional 2508 Col, C. P. 07360, San Pedro Zacatenco, Mexico D.F., Mexico

    Miriam Bocardo–Gaspar & W. A. Zúñiga–Galindo

Authors
  1. Miriam Bocardo–Gaspar
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  2. W. A. Zúñiga–Galindo
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Corresponding author

Correspondence to W. A. Zúñiga–Galindo .

Editor information

Editors and Affiliations

  1. ICMC, University of São Paulo, São Carlos, São Paulo, Brazil

    Raimundo Nonato Araújo dos Santos

  2. Department of Mathematics, Federal University of Paraíba, João Pessoa, Paraíba, Brazil

    Aurélio Menegon Neto

  3. Mathematics Institute, University of Warwick, Coventry, United Kingdom

    David Mond

  4. ICMC, University of São Paulo, São Carlos, São Paulo, Brazil

    Marcelo J. Saia

  5. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, Mexico

    Jawad Snoussi

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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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