On maximal plane curves of degree $3$ over $\mathbb{F}_4$, and Sziklai's example of degree $q-1$ over $\mathbb{F}_q$
DOI:
https://doi.org/10.13069/jacodesmath.v11i2.273Keywords:
Plane curve, Finite field, Rational point, Maximal curveAbstract
An elementary and self-contained argument for the complete determination 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.
Received: 21 January 2023 | Accepted: 14 June 2023Downloads
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