On the constant $D(q)$ defined by Homma

Abstract

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.