The Carlitz-lenstra-wan Conjecture on Expectional Polynomials: an Elementary Version

We give a proof, following an argument of H.W. Lenstra, of the conjecture of L. Carlitz (1966) as generalized by D. Wan (1993). This says, there are no exceptional polynomials of degree n over Fq if (n, q− 1) > 1. Fried, Guralnick and Saxl proved a much stronger result, showing that primitive exceptional polynomials have monodromy groups with degrees either a power of the characteristic (and the monodromy group is affine), or they are cyclic, dihedral (from Tchebychev polynomials) or when the characteristic is p = 2 or 3 the monodromy group is PSL2(p) with a odd. In the original paper we didn’t realize the community wouldn’t recognize that the elementary Lenstra-Wan statement follows from [FGS93] – which was written before that statement was formulated. From [FGS93] – generalizing results from the proof of the Schur conjecture – a brief argument concludes the Wan conjecture by giving a strong characterization of the values of q for which an indecomposable polynomial of degree n (not a power of p) can be exceptional over Fq . By contrast, the Lenstra-Wan statement captures little of the content of [FGS93].