On a special type of permutation rational functions

Abstract

Let p be a prime and n be a positive integer. We consider rational functions \(f_b(X)=X+1/(X^p-X+b)\) over \({\mathbb {F}}_{p^n}\) with \(\textrm{Tr}(b)\ne 0\). In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that \(f_b(X)\) is not a permutation for \(p>3\) and \(n\ge 5\), while it is for \(p=2,3\) and \(n\ge 1\). It is conjectured that \(f_b(X)\) is also not a permutation for \(p>3\) and \(n=3,4\), which was recently proved sufficiently large primes in Bartoli and Hou (Finite Fields Appl 76, Paper No. 101904, 2021). In this note, we give a new proof for the fact that \(f_b(X)\) is not a permutation for \(p>3\) and \(n\ge 5\). With this proof, we also show the existence of many elements \(b\in {\mathbb {F}}_{p^n}\) for which \(f_b(X)\) is not a permutation for \(n=3,4\).