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Finite Fields and Their Applications
Article . 2023 . Peer-reviewed
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https://doi.org/10.48550/arxiv...
Article . 2022
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Optimal plane curves of degree q − 1 over a finite field

Authors: Walteir de Paula Ferreira; Pietro Speziali;

Optimal plane curves of degree q − 1 over a finite field

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 $q - 1$.

Keywords

Algebra and Number Theory, Applied Mathematics, General Engineering, Theoretical Computer Science, Mathematics - Algebraic Geometry, FOS: Mathematics, Algebraic Geometry (math.AG), 114H37, 14H05

21 references, page 1 of 3

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[4] M. Datta, Maximum number of Fq-rational points on nonsingular threefolds in P4. Finite Fields and Their Applications, 59 (2019), 86-96.

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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