THE FACTORABLE CORE OF POLYNOMIALS OVER FINITE-FIELDS

For a polynomial f(x) over a finite field Fq, denote the polynomial f(y) − f(x) by φf(x, y). The polynomial φf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of φf which splits completely into linear factors over the algebraic closure of Fq is exactly φf. This illuminates an aspect of work of D.R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on value sets, including the exhibition of a class of permutation polynomials, are also mentioned. © 1990, Australian Mathematical Society. All rights reserved.