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On the Carlitz Rank of Permutation Polynomials Over Finite Fields: Recent Developments

  • Nurdagül Anbar
  • Almasa Odžak
  • Vandita Patel
  • Luciane Quoos
  • Anna Somoza
  • Alev TopuzoğluEmail author
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 11)

Abstract

The Carlitz rank of a permutation polynomial over a finite field \({\mathbb {F}_q}\) is a simple concept that was introduced in the last decade. In this survey article, we present various interesting results obtained by the use of this notion in the last few years. We emphasize the recent work of the authors on the permutation behavior of polynomials f + g, where f is a permutation over \({\mathbb {F}_q}\) of a given Carlitz rank, and \(g\in {\mathbb {F}_q}[x]\) is of prescribed degree. The relation of this problem to the well-known Chowla–Zassenhaus conjecture is described. We also present some initial observations on the iterations of a permutation polynomial \(f \in {\mathbb {F}_q}[x]\) and hence on the order of f as an element of the symmetric group S q .

Keywords

Carlitz rank Permutation polynomials Finite fields 

Notes

Acknowledgements

The initial work on this project began during “Women in Numbers Europe 2 (WIN-E2)” workshop, held in the Lorentz Center, Leiden, in September 2016. The authors are grateful to the Lorentz Center and all supporting institutions for making this conference and collaboration possible. They would especially like to thank the organizers of WIN-E2, Irene Bouw, Rachel Newton, and Ekin Ozman for all of their hard work, as this resulted in an extremely fruitful and enjoyable meeting.

The authors N.A., A.O., V.P., L.Q., and A.T. are partially supported by H.C. Ørsted COFUND Post-doc Fellowship from the project “Algebraic curves with many rational points” and the Austrian Science Fund FWF Project P 30405-N32; Federal Ministry of Education and Science, grant No.05-39-3663-1/14; an EPSRC studentship; CNPq, PDE grant number 200434/2015-2; and TUBITAK project number 114F432, respectively.

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Copyright information

© The Author(s) and the Association for Women in Mathematics 2018

Authors and Affiliations

  • Nurdagül Anbar
    • 1
  • Almasa Odžak
    • 2
  • Vandita Patel
    • 3
  • Luciane Quoos
    • 4
  • Anna Somoza
    • 5
    • 6
  • Alev Topuzoğlu
    • 7
    Email author
  1. 1.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria
  2. 2.University of SarajevoSarajevoBosnia and Herzegovina
  3. 3.University of WarwickCoventryUK
  4. 4.Universidade Federal do Rio de Janeiro, Cidade UniversitáriaRio de JaneiroBrazil
  5. 5.Universitat Politècnica de CatalunyaBarcelonaSpain
  6. 6.Leiden UniversityLeidenNetherlands
  7. 7.Sabancı UniversityMDBFOrhanlı, Tuzla, IstanbulTurkey

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