(The published version is available online.)
We give a general method for factoring Rg := g(X) - g(Y), where g is a polynomial over a field K. Our approach often works when g varies over an infinite family of polynomials.
Factorizations of Rg are especially important when g is an exceptional polynomial, which by definition means that the scalar multiples of X - Y are the only absolutely irreducible factors of Rg in K[X, Y]. Exceptional polynomials arise in various investigations in case K is finite, since in that case a polynomial g is exceptional if and only if the map a → g(a) induces a bijection on infinitely many finite extensions of K. We apply our method to factor Rg for each g in the infinite family of exceptional polynomials discovered recently by Lenstra and Zieve. We also apply our method to factor Rg in case g is one of the Müller–Cohen–Matthews exceptional polynomials; in this case these factorizations had been discovered previously by more complicated methods.
Additional comment from September 2008: The results of this paper were used in the subsequent paper by Guralnick and Zieve which, together with a paper by Guralnick, Rosenberg and Zieve, yields a complete classification of indecomposable exceptional polynomials of non-prime power degree.
Michael Zieve: home page publication list