A new family of exceptional polynomials in characteristic two
Abstract
We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. The strategy of our proof is to identify the curves that can arise as the Galois closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification in Galois covers of curves, as well as the computation of the automorphism groups of all curves in a certain 2-parameter family.
Keywords
Cite
@article{arxiv.0707.1837,
title = {A new family of exceptional polynomials in characteristic two},
author = {Robert M. Guralnick and Joel E. Rosenberg and Michael E. Zieve},
journal= {arXiv preprint arXiv:0707.1837},
year = {2013}
}
Comments
30 pages; changed notation, fixed some minor errors, added Corollary 1.3, and removed the perfectness hypothesis in Theorem 1.2