Pure gaps on curves with many rational places

Abstract

We consider the algebraic curve defined by $y^m=f(x)$ where $m\ge 2$ 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.