On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

Abstract

Let $\mathcal{G}$ be the projective plane curve defined over ${\mathbb{F}}_q$ given by $$a{X}^n{Y}^n-X^n{Z}^n-Y^n{Z}^n+b{Z}^{2n}=0,$$ where $ab\notin \{0,1\}$, and for each $s\in \{2,\dots ,n-1\}$, let ${\mathcal{D}}_s^{P_1,P_2}$ be the base-point-free linear series cut out on $\mathcal{G}$ by the linear system of all curves of degree $s$ passing through the singular points $P_1=(1:0:0)$ and $P_2=(0:1:0)$ of $\mathcal{G}$. The present work determines an upper bound for the number $N_q(\mathcal{G})$ of ${\mathbb{F}}_q$-rational points on the nonsingular model of $\mathcal{G}$ in cases where ${\mathcal{D}}_s^{P_1,P_2}$ is ${\mathbb{F}}_q$-Frobenius classical. As a consequence, when ${\mathbb{F}}_q$ is a prime field, the bound obtained for $N_q(\mathcal{G})$ improves in several cases the known bounds for the number $n_P$ of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point $P$ distinct from its vertices.