THE RESULTS OF THE PAPER:

You might be most interested in the problem related to coding theory, that is describing explicitly those curves that have the property of exactly  qt+1 points over infinitely many of the finite fields Fqt. This property is complementary to that of curves that achieve the extremal bound for rational points over finite fields. Often the same curve works for both problems – although for different values of t.

We use the classification of finite simple groups and covering theory in positive characteristic to do much more than solve  Carlitz's conjecture (1966). An exceptional polynomial f over a finite field Fq is a polynomial that is a permutation polynomial on infinitely many finite extensions of Fq. Carlitz's conjecture says f must be have odd degree (if q is odd). It is immediate that you can reduce to considering the case that f is indecomposable over Fq.

What we do is figure out the nature of the monodromy groups of exceptional polynomials in positive characteristic. The method for the original Schur conjecture shows quickly that the ramification over the infinite prime is wild (the prime characteristic divides the order of the inertia group). In particular, much more than their just having odd degrees, except in the case the characteristic is 2 or 3, those degrees actually are powers of the characteristic.

The more precise results are as follows. Recall an affine group – in what appears below – is a semidirect product of a vector space V over Fp and a subgroup of the general linear group GL(V) acting on V. In our results, pu = q, for some integer u.
TWO KEY IDEAS IN THE PAPER:

Total, but wild, ramification: There is a key connection between geometry of covers and group theory. As in characteristic 0 when dealing with polynomial covers, f, the total ramification over ∞ is the key to much computation. There, and in characteristics prime to the degree, the inertia group I is cyclic of order the degree.

In all cases the investigation is to draw conclusions on the arithmetic Gf^ (and geometric Gf) monodromy group of the cover. In dealing with polynomials over finite fields, if the degree is divisible by the characteristic, the statement we use for total ramification is that G(Tf, 1). I as a set equals Gf. That is, the cosets by elements in  Iof the stabilizer of an integer in the representation, G(Tf, 1), cover the group. Group theorists call this a factorization of the representation.

Division of group theory labor: Once I showed Guralnick and Saxl this factorization property, they were prepared on the basis of a number of papers extant in the literature to divide the territory of the classification of finite simple groups, so as to identify possible factorizations of groups that could be arithmetic monodromy of an exceptional polynomial.

This is based on the reduction to a primitive Gf, since the tool for applying the classification is the Aschbaker-O'nan-Scott theorem that "plugs in" simple groups to get primitive permutation representations. The category of affine groups remains separate in this association.

BEYOND THE PAPER:

Almost immediately on publication of the paper, P. Müller, then S. Cohen, and then H. Lenstra and M. Zieve produced polynomials realizing those exceptional characteristic 2 and 3 cases. These then, are the only exceptional (indecomposable) polynomial covers known with nonsolvable monodromy groups.

We don't, however, know which affine groups appear as the geometric monodromy group of exceptional polynomials outside the characteristic 2 and 3 cases.

Corollary 14.2 holds also for an indecomposable exceptional cover having (at least) one totally ramified place over a rational point of the base. Its arithmetic monodromy group is an affine group (if p > 2 or 3). Again, if q is odd these covers are of odd degree.

The methods allow considering covers XP1 – generalized exceptional covers – that include exceptional polynomials as a special case. These covers have this property. Over each Fqt  point of the projective line  P1, there is exactly one Fqt  point of X for infinitely many t. Thus, X has a rare diophantine property when X has genus greater than 0. It has exactly  qt+1 points in Fqt  for infinitely many t. This gives exceptional covers a special place in the theory of counting rational points on curves over finite fields explicitly.

INDECOMPOSABILITY STATEMENT:

Recall this says a polynomial indecomposable over K is indecomposable over the algebraic closure of K. It holds when the characteristic p of K doesn't divide the degree of the polynomial. Most counterexamples are of p power degree, but one (due to Peter Mueller) has p exactly dividing degree of f.

Many applications outside exceptional polynomials benefit from the Indecomposability Statement. When it holds, it reduces problems on polynomial values to the case where the geometric monodromy group is primitive. Indeed, some of technical group theory in [FGS] appears because we can't easily make such a reduction.