on a conjecture on permutation rational functions over finite fields

2020/08/08 by Daniele Bartoli, Bartoli, Daniele, Xiang‐dong Hou +1
Computer Science · Engineering · Mathematics · #11R58 #11T06 #11T55 #14H05 #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2008.03432

openalex publication_date 2020/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a prime and n be a positive integer, and consider f_b(X)=X+(X^p-X+b)^-1∈ \Bbb F_p(X), where b∈\Bbb F_pⁿ is such that Tr_pⁿ/p(b)≠ 0. It is known that (i) f_b permutes \Bbb F_pⁿ for p=2,3 and all n≥ 1; (ii) for p>3 and n=2, f_b permutes \Bbb F_p² if and only if Tr_p²/p(b)=± 1; and (iii) for p>3 and n≥ 5, f_b does not permute \Bbb F_pⁿ. It has been conjectured that for p>3 and n=3,4, f_b does not permute \Bbb F_pⁿ. We prove this conjecture for sufficiently large p.

Citations

• On the Tu-Zeng Permutation Trinomial of Type (1/4,3/4)
• On a Type of Permutation Rational Functions over Finite Fields

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