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Distribution of power residues over shifted subfields and maximal cliques in generalized Paley graphs (2403.04312v2)
Published 7 Mar 2024 in math.NT and math.CO
Abstract: 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 $d\ge2$ and $q \equiv 1 \pmod d$, 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 $\mathbb{F}_{qd}$. 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.
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