Recent zbMATH articles in MSC 12E12 https://www.zbmath.org/atom/cc/12E12 2021-04-16T16:22:00+00:00 Werkzeug On a type of permutation rational functions over finite fields. https://www.zbmath.org/1456.11219 2021-04-16T16:22:00+00:00 "Hou, Xiang-dong" https://www.zbmath.org/authors/?q=ai:hou.xiang-dong "Sze, Christopher" https://www.zbmath.org/authors/?q=ai:sze.christopher Summary: Let $$p$$ be a prime and $$n$$ be a positive integer. Let $$f_b(X)=X+(X^p-X+b)^{-1}$$, where $$b\in\mathbb{F}_{p^n}$$ is such that $$\text{Tr}_{p^n/p}(b)\neq 0$$. In 2008, \textit{J. Yuan} et al. [Finite Fields Appl. 14, No. 2, 482--493 (2008; Zbl 1211.11136)] showed that for $$p=2,3,f_b$$ permutes $$\mathbb{F}_{p^n}$$ for all $$n\geq 1$$. Using the Hasse-Weil bound, we show that when $$p>3$$ and $$n\geq 5, f_b$$ does not permute $$\mathbb{F}_{p^n}$$. For $$p>3$$ and $$n=2$$, we prove that $$f_b$$ permutes $$\mathbb{F}_{p^2}$$ if and only if $$\text{Tr}_{p^2/p}(b)=\pm 1$$. We conjecture that for $$p>3$$ and $$n=3,4,f_b$$ does not permute $$\mathbb{F}_{p^n}$$.