ON THE NUMBER OF SOLUTIONS OF EQUATIONS OF DICKSON POLYNOMIALS OVER FINITE FIELDS

Abstract

Let $k, n_1, \dots, n_k$ be fixed positive integers, $c_1, \dots, c_k \in GF(q)^*$, and $a_1, \dots, a_k, c \in GF(q)$. We study the number of solutions in $GF(q)$ of the equation $c_1D_{n_1}(x_1, a_1) + c_2D_{n_2}(x_2, a_2) + \cdots + c_kD_{n_k}(x_k, a_k) = c$, where each $D_{n_i}(x_i, a_i)$, $1 \leq i \leq k$, is the Dickson polynomial of degree $n_i$ with parameter $a_i$. 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].