Pure gaps on curves with many rational places

TL;DR

This paper extends the concept of pure gaps to c-gaps on algebraic curves over finite fields and provides criteria for identifying such gaps, with applications to curves with many rational places.

Contribution

It introduces the notion of c-gaps on curves and develops a criterion for their identification, advancing the understanding of gap structures on algebraic curves.

Findings

Established a criterion for c-gaps at rational places.

Constructed many families of pure gaps on specific curves.

Enhanced the understanding of rational place distributions.

Abstract

We consider the algebraic curve defined by $y^m = f(x)$ where $m \geq 2$ and $f(x)$ is a rational function over $\mathbb{F}_q$. We extend the concept of pure gap to {\bf c}-gap and obtain a criterion to decide when an $s$-tuple is a {\bf c}-gap at $s$ rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.