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 {mod}\ d), we show that for each positive integer m such that \operatorname {rad}(m) \mid \operatorname {rad}(d), there are maximal cliques of size approximately q/m in the d-Paley graph defined on \mathbb {F}_{q^d}. We also confirm a conjecture of Goryainov, Shalaginov, and Yip on the maximality of certain cliques in generalized Paley graphs, as well as an analogous conjecture of Goryainov for Peisert graphs. [ABSTRACT FROM AUTHOR]