Fixed Points of the Dickson Polynomials of the Second Kind

Abstract

The permutation behavior of Dickson polynomials of the first kind has been extensively studied, while such behavior for Dickson polynomials of the second kind is less known. Necessary and sufficient conditions for a polynomial of the second kind to be a permutation over some finite fields have been established by Cohen, Matthew, and Henderson. We introduce a new way to define these polynomials and determine the number of their fixed points.