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

## TL;DR

This paper establishes bounds on the number of rational points on certain algebraic curves over finite fields, improving known bounds for chords of polygons inscribed in a hyperbola when specific linear series are Frobenius classical.

## Contribution

It provides new upper bounds for rational points on generalized Fermat curves, linking algebraic geometry with combinatorial properties of polygons inscribed in hyperbolas.

## Key findings

- Derived upper bounds for rational points on the curves.
- Improved bounds for the number of chords passing through a point in polygons.
- Established conditions under which the bounds are tighter.

## Abstract

Let $\mathcal{G}$ be the projective plane curve defined over $\mathbb{F}_q$ given by $$aX^nY^n-X^nZ^n-Y^nZ^n+bZ^{2n}=0,$$ where $ab\notin\{0,1\}$, and for each $s\in\{2,\ldots,n-1\}$, 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.