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≥2 and q ≡ 1 d, we show that for each positive integer m such that rad(m) rad(d), there are maximal cliques of size approximately $q/m$ in the d-Paley graph defined on Fqᵈ. 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.