Distribution of power residues over shifted subfields and maximal cliques in generalized Paley graphs

We derive an asymptotic formula for the number of solutions in a given subfield to certain system of equations over finite fields. As an application, we construct new families of maximal cliques in generalized Paley graphs. Given integers $d\ge 2$ and $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}d)$, we show that for each positive integer $m$ such that $\mathrm{rad}(m)\mid \mathrm{rad}(d)$, there are maximal cliques of size approximately $q\mathrm{/}m$ in the $d$-Paley graph defined on ${\mathbb{F}}_{q^d}$. We also confirm a conjecture of Goryainov, Shalaginov, and the second author on the maximality of certain cliques in generalized Paley graphs, as well as an analogous conjecture of Goryainov for Peisert graphs.