on a conjecture on permutation rational functions over finite fields

TL;DR

This paper investigates permutation properties of a specific rational function over finite fields, proving a conjecture that it does not permute certain fields for large primes, thereby advancing understanding of permutation rational functions.

Contribution

The paper proves a conjecture that the function $f_b$ does not permute $F_{p^n}$ for $p>3$ and $n=3,4$ when $p$ is sufficiently large, filling a gap in the existing knowledge.

Findings

For large primes $p$, $f_b$ does not permute $F_{p^3}$ and $F_{p^4}$.

Confirmed the conjecture for all sufficiently large $p$.

Extended understanding of permutation rational functions over finite fields.

Abstract

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