Value sets of Dickson polynomials over Galois rings

Abstract

The Galois ring GR( p n , m) is a finite extension of the ring of integers modulo p n . We consider Dickson polynomials g d ( x, a) over Galois rings. In particular, we find an estimate of the cardinality of the value set { g d ( x, a)∥ ∈ GR( p n , m)} and determine the cardinality of the preimage of g d ( x, a) in certain cases.