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 $\alpha\mapsto f(\alpha)$ 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.