Optimal plane curves of degree $q-1$ over a finite field

TL;DR

This paper classifies all degree $q-1$ plane curves over finite fields that reach the Sziklai bound for rational points, showing they are projectively equivalent to specific binomial curves with a sum condition.

Contribution

It provides a complete classification of extremal degree $q-1$ plane curves over finite fields that attain the Sziklai bound, linking them to explicit binomial forms.

Findings

Curves attaining the Sziklai bound are projectively equivalent to specific binomial curves.

The classification completes the understanding of extremal curves for the Sziklai bound.

The results also relate to Frobenius classical extremal curves of degree $q-1$.

Abstract

Let $q\geq 5$ be a prime power. In this note, we prove that if a plane curve $\mathcal{X}$ of degree $q - 1$ defined over $\mathbb{F}_q$ without $\mathbb{F}_q$-linear components attains the Sziklai upper bound $(d-1)q+1 = (q - 1)^2$ for the number of its $\mathbb{F}_q$-rational points, then $\mathcal{X}$ is projectively equivalent over $\mathbb{F}_q$ to the curve $ \mathcal{C}_{(\alpha,\beta,\gamma)} : \alpha X^{q - 1} + \beta Y^{q - 1} + \gamma Z^{q - 1} = 0$ for some $\alpha, \beta, \gamma \in \mathbb{F}_q^{*}$ such that $\alpha + \beta + \gamma = 0$. This completes the classification of curves that are extremal with respect to the Sziklai bound. Also, since the Sziklai bound is equal to the St\"ohr-Voloch's bound for plane curves of degree $q - 1$, our main result classifies the $\mathbb{F}_q$-Frobenius classical extremal plane curves of degree $q - 1$.