On the difference between permutation polynomials

Anbar, Nurdagül and Odz̆ak, Almasa and Patel, Vandita and Quoos, Luciane and Somoza, Anna and Topuzoğlu, Alev (2018) On the difference between permutation polynomials. Finite Fields and Their Applications, 49 . pp. 132-142. ISSN 1071-5797 (Print) 1090-2465 (Online)

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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).
Item Type: Article
Uncontrolled Keywords: Carlitz rank; Chowla–Zassenhaus conjecture; Curves over finite fields; Permutation polynomials
Divisions: Faculty of Engineering and Natural Sciences
Depositing User: Alev Topuzoğlu
Date Deposited: 17 May 2023 11:58
Last Modified: 17 May 2023 11:58
URI: https://research.sabanciuniv.edu/id/eprint/45592

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