(The published version is available online.)

A polynomial *g* over **F**_{q} is called
*exceptional* if the map *a*→*g*(*a*)
induces a bijection on **F**_{qn}
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 **F**_{q},
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 *p ^{r}*(

*Additional comment from May 1998*:
Exceptional polynomials are distinguished by a property of the factorization of
*R _{g}* :=

*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*:
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