on a conjecture on permutation rational functions over finite fields

Abstract

Let $p$ be a prime and $n$ be a positive integer, and consider $f_b(X)=X+(X^p-X+b{)}^{-1}\in {\mathbb{F}}_p(X)$, where $b\in {\mathbb{F}}_{p^n}$ is such that ${\text{Tr}}_{p^n/p}(b)\ne 0$. It is known that (i) $f_b$ permutes ${\mathbb{F}}_{p^n}$ for $p=2,3$ and all $n\ge 1$; (ii) for $p>3$ and $n=2$, $f_b$ permutes ${\mathbb{F}}_{p^2}$ if and only if ${\text{Tr}}_{p^2/p}(b)=\pm 1$; and (iii) for $p>3$ and $n\ge 5$, $f_b$ does not permute ${\mathbb{F}}_{p^n}$. It has been conjectured that for $p>3$ and $n=3,4$, $f_b$ does not permute ${\mathbb{F}}_{p^n}$. We prove this conjecture for sufficiently large $p$.

13 pages

References in corpus (2)

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