Complete set of Pure Gaps in Function Fields

Abstract

In this work, we provide a way to completely determine the set of pure gaps $G_0(P_1,P_2)$ at two rational places $P_1,P_2$ in a function field $F$ over a finite field ${\mathbb{F}}_q$, and its cardinality. Furthermore, we given a bound for the cardinality of the set $G_0(P_1,P_2)$ which is better, in some cases, than the generic bound given by Homma and Kim. As a consequence, we completely determine the set of pure gaps and its cardinality for two families of function fields: the $GK$ function field and Kummer extensions.

22 pages