On the number of solutions of equations of Dickson polynomials over finite fields

Let k, n(1), ... , n(k) be fixed positive integers, C-1, ... , C-k is an element of GF(q)*, and a(1), ... , a(k), c is an element of GF(q). We study the number of solutions in GF(q) of the equation c(1)D(n1) (x(1), a(1)) + C2Dn2 (x(2), a(2)) + ... + CkDnk (x(k), a(k)) = c, where each D-ni (x(i), a(i)), 1 <= i <= k, is the Dickson polynomial of degree n(i) with parameter ai. We also employ the results of the k = 1 case to recover the cardinality of preimages and images of Dickson polynomials obtained earlier by Chou, Gomez-Calderon and Mullen [1].