Hendrik W. Lenstra, Jr., and Michael E. Zieve:
A family of exceptional polynomials in characteristic three,
Finite Fields and Applications, London Math. Soc. Lecture Note Ser. 233, Cambridge Univ. Press (1996), 209–218. MR 98a:11174

(The published version is available online.)

A polynomial  g  over  Fq  is called exceptional if the map  ag(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  FqFried, 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[xy]  which remain irreducible in  K[xy].  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.


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