A new family of exceptional polynomials in characteristic two

TL;DR

This paper introduces a new family of exceptional polynomials over fields of characteristic two, completing their classification for degrees not a power of the characteristic, using advanced Galois theory and curve automorphism analysis.

Contribution

It constructs explicit new exceptional polynomials of specific degrees and completes their classification, expanding understanding of their structure and properties.

Findings

Identified the form of curves arising as Galois closures of exceptional polynomials.

Proved these polynomials have no nontrivial absolutely irreducible factors in K[x,y].

Computed automorphism groups of related algebraic curves.

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.