On the number of solutions of equations of Dickson polynomials over finite fields
The authors consider the following extension of a diagonal equation over the finite field \(\mathrm{GF}(q)\) and study the number \(N_k\) of its solutions in \(\mathrm{GF}(q)^k\): \[ c_1D_{n_1}(x_1,a_1)+c_2D_{n_2}(x_2,a_2)+\ldots+c_kD_{n_k}(x_k,a_k)=c \] where \(c_i\in \mathrm{GF}(q)^*\), \(c\in \mathrm{GF}(q)\) and \(D_{n_i}(x_i,a_i)\) denotes the Dickson polynomial of degree \(n_i\) with parameter \(a_i\in \mathrm{GF}(q)\). They give a formula for the number \(N_1\) in terms of characters on \(\mathrm{GF}(q^2)\) and estimate \(N_k\) for general \(k\geq 2\). The result implies the existence of a solution for \(k>2\) if \(q\) is sufficiently large. The formula for \(N_1\) is used to re-prove a result on the size of the value set of Dickson polynomials given by \textit{W.-S. Chou}, \textit{J. Gomez-Calderon}, and \textit{G. L. Mullen} [J. Number Theory 30, 334--344 (1988; Zbl 0689.12012)].
scientific article; zbMATH DE number 5348117