On a special type of permutation rational functions

Official URL: http://dx.doi.org/10.1007/s00200-022-00592-1

Let p be a prime and n be a positive integer. We consider rational functions fb(X) = X + 1∕(Xp − X + b) over 픽pn with Tr(b) ≠ 0. In Hou and Sze (Finite Fields Appl 68, Paper No. 10175, 2020), it is shown that fb(X) is not a permutation for p > 3 and n ≥ 5, while it is for p = 2, 3 and n ≥ 1. It is conjectured that fb(X) is also not a permutation for p > 3 and n = 3, 4, which was recently proved sufciently 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 fb(X) is not a permutation for p > 3 and n ≥ 5. With this proof, we also show the existence of many elements b ∈ 픽pn for which fb(X) is not a permutation for n = 3, 4.