Value sets of Dickson polynomials over finite fields

Abstract

Let F q denote the finite field of order q where q is a prime power. If a ∈ F q and d ≥ 1 is an integer, define the Dickson polynomial g d(x, a) = ∑ t=0 [ d 2 ] ( d (d−t) )( t d−t)(−a tx d−2t . Let { g d ( x, a) | x ∈ F q } denote the image or value set of the polynomial g d ( x, a). In this paper we determine the cardinality of the value set for the Dickson polynomial g d ( x, a) over the finite field F q .