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

TL;DR

This paper investigates Homma's constant $D(q)$, an analogue of the Ihara constant $A(q)$, by establishing new upper and lower bounds for its value over finite fields.

Contribution

The paper derives new bounds for Homma's constant $D(q)$, expanding understanding of its behavior without the nonsingularity constraint.

Findings

Established upper bounds for $D(q)$.

Established lower bounds for $D(q)$.

Enhanced understanding of the relationship between $D(q)$ and $A(q)$.

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.