Plane-filling curves of small degree over finite fields

TL;DR

This paper investigates the existence and properties of smooth plane-filling curves over finite fields with degrees higher than the known minimum, expanding understanding of their geometric and algebraic structure.

Contribution

It extends the study of plane-filling curves to degrees greater than the minimal, providing new insights into their construction and characteristics.

Findings

Minimum degree of smooth plane-filling curves is q+2.

Existence of smooth plane-filling curves of degree q+3 and higher.

Structural properties of higher-degree plane-filling curves.

Abstract

A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth plane-filling curve is $q+2$. We study smooth plane-filling curves of degree $q+3$ and higher.