发表机构
Kanagawa University; Gyeongsang National University（神奈川大学; 庆尚国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完成 Hermitian-相对曲线有理点全为拐点的分类，并分类含非拐点有理点的曲线，枚举并证明所有有理点与拐点数量组合均可实现。
AI 中文摘要
在文献[6]中，我们引入了 Hermitian-相对曲线的概念，即由 $(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A (x,y,z)^t =0$ 定义的平面曲线，其中 $A \in GL(3, \mathbb{F}_q)$。在研究了它们的基本性质之后，我们对具有两个或更多有理拐点的曲线进行了分类。在本文中，我们首先通过证明存在恰好具有一个有理点的曲线，完成了对所有有理点均为拐点的曲线的分类。作为延续，我们随后对至少具有一个非拐点有理点的 Hermitian-相对曲线进行分类。我们根据由有理点数和拐点数组成的有序对来对这些曲线进行分类。最后，我们枚举所有可能的有序对，并证明每个有序对都能由一条合适的曲线实现。
英文摘要
In [6], we introduced the notion of a Hermitian-relative curve, which is a plane curve defined by $(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A (x,y,z)^t =0$ with $A \in GL(3, \mathbb{F}_q)$. After investigating their basic properties, we classified those curves with two or more rational inflexions. In this paper, we first complete the classification of curves whose rational points are all inflexions by demonstrating the existence of curves with exactly one rational point. As a continuation, we then classify Hermitian-relative curves that possess at least one rational point which is not an inflexion. We categorize these curves according to the ordered pair consisting of the number of rational points and the number of inflexions. Finally, we enumerate all possible pairs and prove that each pair is realized by a suitable curve.
Comments33 pages
