(The published version is available online.)
A polynomial g over Fq is called exceptional if the map a→g(a) induces a bijection on Fqn for infinitely many n. We construct a new family of exceptional polynomials over finite fields of characteristic 3.
It is known that the composition of two polynomials is exceptional if and only if both polynomials are exceptional, so it suffices to study the indecomposable exceptional polynomials. Letting p denote the characteristic of Fq, Fried, Guralnick and Saxl showed that any indecomposable exceptional polynomial has degree either prime or a power of p, except perhaps when p ≤ 3 in which case they could not rule out the possibility that the degree is pr(pr-1)/2 with r > 1 odd. In case p=2, indecomposable exceptional polynomials of the latter degrees were produced by Müller and Cohen–Matthews. We give a new approach which recovers these polynomials and also yields polynomials in characteristic 3 with analogous properties.
Additional comment from May 1998: Exceptional polynomials are distinguished by a property of the factorization of Rg := g(x)-g(y). Letting K denote the algebraic closure of Fq, a polynomial g over Fq is exceptional if and only if the scalar multiples of x-y are the only irreducible factors of Rg in Fq[x, y] which remain irreducible in K[x, y]. In a subsequent paper, I give a general method for factoring Rg when g varies over an infinite family of polynomials. In that paper I factor Rg for each of the exceptional polynomials g discussed in this paper.
Additional comment from September 2008: This paper initiated a systematic approach to exceptional polynomials of non-prime power degree. A complete classification of indecomposable exceptional polynomials of non-prime power degree was obtained by Guralnick–Zieve and Guralnick–Rosenberg–Zieve. The occurring polynomials include twists of the two families of exceptional polynomials discussed in this paper, as well as a new family of exceptional polynomials in characteristic 2.
Michael Zieve: home page publication list