On the roots of the substitution Dickson polynomials (Q1607858)

Let \(\mathbb{F}_q\) denote the finite field of order \(q\) (\(q\) odd). For \(a\in \mathbb{F}_q\) and an integer \(d\geq 1\), the Dickson polynomial \(g_d(x,a)\) is defined as \[ g_d(x,a)= \mathop{{\sum_{t=0}}^{\lfloor\frac 12 d\rfloor}} \frac{d}{d-t} \binom {d-t}{t} (-a)^t x^{d-2t}. \] It is shown that under the composition of multi-valued functions, the set of the \(y\)-radical roots of \(g_d(x,a)- g_d(y,a)\) is generated by one of the roots.