Michael E. Zieve:
Bivariate factorizations via Galois theory, with application to exceptional polynomials,
J. Algebra 210 (1998), 670–689. MR 99m:12001

(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[XY].  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 MüllerCohen–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.

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