On the difference between permutation polynomials

Nurdagül Anbar, Almasa Odz̆ak, Vandita Patel, Luciane Quoos, Anna Somoza, Alev Topuzoğlu

Research output: Contribution to journalArticlepeer-review

Abstract

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p>(d2−3d+4)2, then there is no complete mapping polynomial f in Fp[x] of degree d≥2. For arbitrary finite fields Fq, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f. Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥2 over Fp, with p>(d2−3d+4)2, then the degree k of g satisfies k≥3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in Fq[x], we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n≥1 that if f+xk is a permutation over Fq, with gcd⁡(k+1,q−1)=1, then k≥(q−n)/(n+3).

Original languageEnglish
Pages (from-to)132-142
Number of pages11
JournalFinite Fields and their Applications
Volume49
Early online date3 Oct 2017
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Carlitz rank
  • Chowla–Zassenhaus conjecture
  • Curves over finite fields
  • Permutation polynomials

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