(Both the published version and the arXiv version are 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*. 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}*(

The strategy of our proof is to identify the curves *C* which can occur
as the Galois closure of the cover
*g* : **P**^{1}→**P**^{1}
for a polynomial *g* satisfying the required properties. It turns out
that any such *C* is geometrically isomorphic to the smooth plane curve
*y*^{q+1}+*z*^{q+1}=**T**(*yz*)+*c*,
where *Q*=2^{r} with *r* > 1
odd, and where
**T**(*X*)=*X*^{q/2}+*X*^{q/4}+...+*X*.
This family of curves is of independent interest, since each curve in the family
has ordinary Jacobian and has automorphism group significantly larger than its genus.
A key step in our proof is the computation of the automorphism groups of curves of
the form *v*^{q}+*v*=*h*(*w*),
with *h* varying over a two-parameter family of rational functions.
Our method for this is rather general, and applies to many families of rational
functions *h*.

*Michael Zieve*:
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