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] except for scalar multiples of X — Y; when k is finite, this condition is equivalent to saying that the map α ↦ f(α) induces a bijection on an infinite algebraic extension of k. Our polynomials have degree 2 e―1 (2 e ― 1), where e > 1 is odd. We also prove that this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic.