(Both the published version and the arXiv version are available online.)

Let *k* be a field of characteristic *p*≥0,
and let *g* be a polynomial over *k* which is
indecomposable (not a composition of lower-degree polynomials over *k*).
Guralnick and Saxl
showed that, if *g* decomposes over an extension of *k*,
then the degree of *g* is either a power of *p* or
21 or 55. We begin by determining all possibilities with the latter degrees.
It turns out that there exists such a polynomial of degree 21 if and only if
*p*=7 and *k* contains nonsquares, in which
case the polynomials can be presented explicitly. A similar conclusion holds
for degree 55.

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}*(

We solve these two problems as consequences of a more general problem.
Let *k* be a field of characteristic *p*>0,
let *q* be a power of *p*, and let *t* be
transcendental over *k*. We determine all polynomials *g*
in *k*[*X*] of degree *q*(*q*-1)/2
for which *g*(*X*)-*t*
has simple roots and its Galois group over *k*(*t*)
has a transitive normal subgroup isomorphic to PSL(2,*q*).
These include the polynomials discussed in the previous two paragraphs.
More generally,
Guralnick and Saxl have
produced a list of groups satisfying
the known conditions necessary for occurring as the Galois group of
*g*(*X*)-*t*
over *K*(*t*), where *g* is a polynomial
over an algebraically closed field *K*. Our result determines all such
polynomials for one of the main families of groups on the Guralnick–Saxl list; this complements
work of
Abhyankar,
who has exhibited polynomials realizing some of the other
families of groups on the list.

Our strategy for proving these results is as follows. Let *g* be
a polynomial of degree *q*(*q*-1)/2 over an algebraically
closed field *K* of characteristic *p*, where *q*
is a power of *p*. Let Ω be
the splitting field of *g*(*X*)-*t* over
*K*(*t*), and suppose that
*G* := Gal(Ω/*K*(*t*))
normalizes PSL(2,*q*). Then *G* is contained in
the automorphism group PΓL(2,*q*) of PSL(2,*q*).
We use the classification of subgroups of PΓL(2,*q*), together
with Hilbert's different formula, to determine all possibilities for the inertia
and higher ramification groups in Ω/*K*(*t*)
that are consistent with *K*(*x*) having genus zero.
For each such possibility, we determine all candidates for Ω: this is the
most difficult step, and involves various ingredients. We then determine
the automorphism group of each of these (infinitely many) candidates for Ω,
and use invariant theory to compute the subfields of Ω corresponding to
*K*(*t*) and *K*(*x*), and
finally to compute the polynomial *g*. This solves the geometric part of
our problem; to solve the original arithmetic problems, we use the factorizations
of *g*(*X*)-*g*(*Y*)
determined previously by
Cohen–Matthews
and
Zieve.

Our results should be viewed in the context of
Abhyankar's conjecture
(the Raynaud/Harbater theorem). Given a smooth projective irreducible curve *C*
over an algebraically closed field of characteristic *p*, and a finite set
*S* of points on *C*, this conjecture describes the groups occurring
as Galois groups of covers of *C* whose branch locus is contained in
*S*. However, this work does not address the problem of determining the
dimension of the moduli space of covers having prescribed Galois group and prescribed
ramification data, let alone the problem of constructing the covers explicitly.
The *p*=0 case of Abhyankar's conjecture follows from
Riemann's
existence theorem; however, even in that situation, the
proof is nonconstructive. We have constructed all polynomial covers of
**P**^{1} realizing certain permutation representations of
PSL(2,*q*). In future work we hope to extend this to other classes
of groups, which will provide data towards a conjecture on the dimension of the
moduli space of covers having prescribed Galois group and prescribed ramification.

*Michael Zieve*:
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