The Carlitz-Lenstra-Wan conjecture on Expectional Polynomials

The accomplishments toward the classification of exceptional polynomials over a finite field Fq are explained briefly for the layman in [FGS]. Recall the meaning of exceptional:  The polynomial (or rational function) f(x) maps one-one on the finite field Fqt for infinitely many t. To know such a polynomial includes knowing at least the following:
  1. inertia data for the cover f(x): P1 → P1;
  2. inertia group data for the Galois closure ˆf: ˆX → P1 associated to f; and
  3. the arithmetic ˆGf and geometric group Gf of the cover ˆf
Exceptionality and what you get from items #1-#3: Would you want to know the coefficients of f? Many think that is a worthy goal. Yet, you would find those aren't especially helpful for deciphering the data #1–#3. The phrase "geometric group of the cover" refers to the Galois closure over the algebraic closure of Fq, with "arithmetic" refers to that Galois closure over Fq. Part of the theory of exceptional covers, the generalization of MacCluer's Theorem, characterizes exceptionality from knowing both ˆGf and Gf (and the permutation representation of them both that comes from the cover f).

That characterization:  The fiber product of f with itself, minus the diagonal, has no absolutely irreducible Fq components. This is akin to how (xr-1)/(x-1) has no absolutely irreducible irreducible factors over Fq, with r prime, unless r -1 divides q-1. Indeed, that gives the exact condition for xr to be exceptional over Fq.

A full discussion and history of that result is in [§3, FGS] with a later generalization to pr-exceptional covers (including Davenport pairs) in Principle 3.1 of  exceptTowYFFTA_519.pdf. The phrase "inertia data," even for a polynomial cover, is well-understood in terms of higher ramification groups. Yet, for a non-Galois cover, there is little literature that can could make computing the genus of such a cover from inertia data efficient. The first half of fr-mez.pdf was developed explicitly for that purpose.

The conclusion: Items #1-#3 allow you to detect an exceptional polynomial. The monodromy method, as applied to such problems as Schur's and Davenport's conjecture, used general group and arithmetic techniques on data of #1-#3 to characterize those very few, precious, finite groups (pairs of geometric and arithmetic monodromy groups) that could possibly produce polynomials contributing to their conjectures.

Further, though harder, in practice the method is often reversible. Given items #1-#3 group theoretically, often a fulfilling cover can be found producing this data. An historical instance of that, for exceptional polynomial covers, appears below. Note: For exceptional covers the arithmetic monodromy group always contains the geometric monodromy as a proper subgroup. The highest results in the monodromy method (as in the second half of fr-mez.pdf ) contribute to general problems of this Inverse Galois kind.

Comparison of the Carlitz-Wan-Lenstra Conjecture with the results of [FGS]: A polynomial f has a degree, n. That simple point makes characterizing exceptional polynomials much easier (see the relation to Serre's Open Image Theorem in exceptTowYFFTA_519.html) than rational functions. You detect polynomials from total ramification over the point at ∞ on P1. So, knowing that degree is a partial piece of #1 data.

The Carlitz-Wan-Lenstra Conjecture: For an exceptional f, n should be prime to q - 1. 

So, if for example, n is a power of the prime characteristic, this conjecture would give no further data what-so-ever about such an f.

[FGS] displayed all the data of #1-#3 that could possibly be attached to an exceptional polynomial, when n was not a power of that characteristic. Further, the paper produced all possible exceptional polynomials under those conditions – including solving the oldest unsolved problem in my career, Dixon's 1890 conjecture – except for a small, interesting collection. [FGS] showed these special polynomials could have geometric monodromy groups PSL2(pa) with a an odd integer, over the finite fields of (the same) characteristic p=2 and 3. Again, proving that the monodromy method works, this #1–#3 data was sufficient for Cohen-Matthews, Lenstra-Zieve and Mueller to find realizing polynomials soon after. exceptTowYFFTA_519.pdf recounts this in §6.4.3 in stating the following problem: Find the cryptographic period as a Poincaré series of such exceptional polynomials .

The case where an exceptional f has degree n=pu, p the characteristic of Fq: Then, excluding all these explicitly known cases, [FGS] showed the geometric monodromy group to be an affine group in a precise way:   ˆGf is the semi-direct product of a p' subgroup, H, of the general linear group GLu(Z/p) and Z/pu.  Further, when H is cyclic, [Thm. 11.1, FGS] characterizes them all. sch-carlitz.html continues this affine group topic. It also reminds of the known exceptional polynomials, and it describes the role of the finite group classification in the whole matter.