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Let $\mathcal{X}$ be a projective, irreducible, nonsingular algebraic curveover the finite field $\mathbb{F}_q$ with $q$ elements and let$|\mathcal{X}(\mathbb{F}_q)|$ and $g(\mathcal X)$ be its number of rationalpoints and genus respectively. The Ihara constant $A(q)$ has been intensivelystudied 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 $\mathcal X$ goesto infinity. In 2012 Homma defined an analogue $D(q)$ of $A(q)$, where thenonsingularity of $\mathcal X$ is dropped and $g(\mathcal X)$ is replaced withthe degree of $\mathcal X$. We will call $D(q)$ Homma's constant. In thispaper, upper and lower bounds for the value of $D(q)$ are found.

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