On maximal plane curves of degree $3$ over $\mathbb{F}_4$,and Sziklai's example of degree $q-1$ over $\mathbb{F}_q$

TL;DR

This paper classifies maximal degree 3 plane curves over 4, revealing two equivalent curves up to projective transformation, and discusses Sziklai's degree q-1 examples over q, complementing existing theorems.

Contribution

It provides a complete classification of maximal degree 3 curves over q, including Sziklai's examples, extending the characterization of Hermitian curves.

Findings

Two maximal degree 3 curves over q are projectively equivalent over q.

The classification complements existing theorems on Hermitian curves.

Sziklai's degree q-1 examples are characterized within this classification.

Abstract

The classification of maximal plane curves of degree $3$ over $\mathbb{F}_4$ will be given, which complements Hirschfeld-Storme-Thas-Voloch's theorem on a characterization of Hermitian curves in $\mathbb{P}^2$. This complementary part should be understood as the classification of Sziklai's example of maximal plane curves of degree $q-1$ over $\mathbb{F}_q$. Although two maximal plane curves of degree $3$ over $\mathbb{F}_4$ up to projective equivalence over $\mathbb{F}_4$ appear, they are birationally equivalent over $\mathbb{F}_4$ each other.