Points on singular Frobenius nonclassical curves

TL;DR

This paper extends the understanding of Frobenius nonclassical curves by establishing a lower bound on the number of rational points for singular cases, generalizing previous results for nonsingular curves.

Contribution

It proves that the known point count formula for nonsingular curves also provides a lower bound for singular Frobenius nonclassical curves.

Findings

The number of rational points is at least d(q-d+2) for singular curves.

The lower bound matches the known count for nonsingular curves.

Singular curves can have fewer points, but not fewer than this bound.

Abstract

In 1990, Hefez and Voloch proved that the number of $F_q$-rational points on a nonsingular plane $q$-Frobenius nonclassical curve of degree $d$ is $N = d(q-d+2)$. We address these curves in the singular setting. In particular, we prove that $d(q-d + 2)$ is a lower bound on the number of $F_q$-rational points on such curves of degree $d$.