A New Family of Exceptional Polynomials in Characteristic Two

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 7→ f(α) induces a bijection on an infinite algebraic extension of k. Our polynomials have degree 2e−1(2e − 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.