Rational curves with many rational points over a finite field

TL;DR

This paper investigates a specific rational plane curve over a finite field that attains the maximum number of rational points allowed by the Aubry-Perret bound, exploring its structure and generalizations.

Contribution

It introduces a particular rational curve over a finite field that reaches the Aubry-Perret bound and analyzes its rational point configuration and generalizations.

Findings

Curve attains Aubry-Perret bound for rational points

Detailed configuration of rational points on the curve

Generalizations of the curve are presented

Abstract

We study a particular plane curve over a finite field whose normalization is of genus 0. The number of rational points of this curve achieves the Aubry-Perret bound for rational curves. The configuration of its rational points and a generalization of the curve are also presented.