论文标题
有限字段上的最佳平面曲线$ q-1 $ $ q-1 $
Optimal plane curves of degree $q-1$ over a finite field
论文作者
论文摘要
令$ q \ geq 5 $为主要功率。在此注释中，我们证明，如果一架平面曲线$ \ Mathcal $ Q -Q -1 $ $ Q -1 $在$ \ Mathbb {f} _Q $上定义的，而无需$ \ MATHBB {f} _Q $ -linear -lineareartents AD AD AD AD SZIKIKLAI上限$（D -1）Q+（D -1）Q+1 =（d -1）Q+1 =（Q -1） $ \ mathbb {f} _q $ - rational点，然后$ \ mathcal {x} $在$ \ mathbb {f} _q $上等于curve $ \ mathcal {c} _ { - 1} = 0 $对于某些$α，β，γ\ in \ mathbb {f} _q^{*} $，使得$α+β+γ= 0 $。这完成了相对于sziklai界限的极端曲线的分类。 另外，由于Sziklai界限等于Stöhr -voloch的限制为$ q -1 $的平面曲线，因此我们的主要结果对$ \ mathbb {f} _Q $ -FROBENIUS经典的极端平面曲线$ q -1 $。
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}_{(α,β,γ)} : αX^{q - 1} + βY^{q - 1} + γZ^{q - 1} = 0$ for some $α, β, γ\in \mathbb{F}_q^{*}$ such that $α+ β+ γ= 0$. This completes the classification of curves that are extremal with respect to the Sziklai bound. Also, since the Sziklai bound is equal to the Stöhr-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 $q - 1$.
