On some permutation polynomials over finite fields

Let Fq be a finite field of q = pm elements with characteristic p. A polynomial P(x) ∈ Fq[x] is called a permutation polynomial of Fq if P(x) induces a bijective map from Fq to itself. In general, finding classes of permutation polynomials of Fq is a difficult problem (see [3, Chapter 7] for a survey of some known classes). An important class of permutation polynomials consists of permutation polynomials of the form P(x) = xr f (x(q−1)/l), where l is a positive divisor of q− 1 and f (x) ∈ Fq[x]. These polynomials were first studied by Rogers and Dickson for the case f (x) = g(x)l, where g(x) ∈ Fq[x] [3, Theorem 7.10]. A very general result regarding these polynomials is given in [8]. In recent years, several authors have considered the case that f (x) is a binomial (e.g., [2, 9] and [1]). Here we consider the binomial P(x) = xr + xu with r < u. Let s= (u− r,q− 1) and l = (q− 1)/s. Then we can rewrite P(x) as P(x) = xr(1 + xes), where s= (q− 1)/l and (e, l) = 1. If P(x) = xr(1 + xes) is a permutation binomial of Fq, then P(x) has exactly one root in Fq and thus l is odd. When l = 3,5, the permutation behavior of P(x) was studied by Wang [9]. In the case l = 5, the permutation binomial P(x) is determined in terms of the Lucas sequence {Ln}, where