Distribution of power residues over shifted subfields and maximal cliques in generalized Paley graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929608094384128 |
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| author | Martin, Greg Yip, Chi Hoi |
| author_facet | Martin, Greg Yip, Chi Hoi |
| contents | 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 $\mathbb{F}_{q^d}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_04312 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of power residues over shifted subfields and maximal cliques in generalized Paley graphs Martin, Greg Yip, Chi Hoi Number Theory Combinatorics 11T24, 11B30, 05C25 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 $\mathbb{F}_{q^d}$. 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. |
| title | Distribution of power residues over shifted subfields and maximal cliques in generalized Paley graphs |
| topic | Number Theory Combinatorics 11T24, 11B30, 05C25 |
| url | https://arxiv.org/abs/2403.04312 |