关于Homma定义的常数$D(q)$
On the constant $D(q)$ defined by Homma
摘要：
设$\mathcal{X}$是具有$q$元素的有限域$\mathbb{F}_q$上的一条射影的、不可约的、非奇异的代数曲线，$|\mathcal{X}(\mathbb{F}_q)|$和$g(\mathcal X)$分别是它的有理点数和亏格数. Ihara常数$A(q)$在过去的几十年中得到了深入的研究，它被定义为当$\mathcal X$的亏格趋于无穷大时$|\mathcal{X}(\mathbb{F}_q)|/g(\mathcal X)$的极限上界。2012年，Homma定义了$A(q)$的类似物$D(q)$，其中$\mathcal X$的非奇异性被丢弃，$g(\mathcal X)$被$\mathcal X$的次数取代。我们称之为$D(q)$ Homma常数。本文给出了$D(q)$值的上、下界。
英文摘要：
Let $\mathcal{X}$ be a projective, irreducible, nonsingular algebraic curve over the finite field $\mathbb{F}_q$ with $q$ elements and let $|\mathcal{X}(\mathbb{F}_q)|$ and $g(\mathcal X)$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ has been intensively studied during the last decades, and it is defined as the limit superior of $|\mathcal{X}(\mathbb{F}_q)|/g(\mathcal X)$ as the genus of $\mathcal X$ goes to infinity. In 2012 Homma defined an analogue $D(q)$ of $A(q)$, where the nonsingularity of $\mathcal X$ is dropped and $g(\mathcal X)$ is replaced with the degree of $\mathcal X$. We will call $D(q)$ Homma's constant. In this paper, upper and lower bounds for the value of $D(q)$ are found.
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