On a conjecture on permutation rational functions over finite fields

Let p be a prime and n be a positive integer, and consider f(b)(X) = X +(X-p-X+b)(-1) is an element of F-pn(X), where b is an element of F-pn is such that Tr-pn/p(b) not equal 0. It is known that (i) f(b) permutes F-pn for p = 2, 3 and all n >= 1; (ii) for p > 3 and n = 2, f(b) permutes F-p2 if and only if Tr-p2/p(b) = +/- 1; and (iii) for p > 3 and n > 5, fb does not permute F-pn. It has been conjectured that for p > 3 and n = 3, 4, f(b) does not permute F-pn. We prove this conjecture for sufficiently large p. (C) 2021 Published by Elsevier Inc.