In this work, using maximal elements in generalized Weierstrass semigroups and its relationship with pure gaps, we extend the results in \cite{CMT2024} and provide a way to completely determine the set of pure gaps at several rational places in an arbitrary function field $F$ over a finite field and its cardinality. As an example, we determine the cardinality and a simple explicit description of the set of pure gaps at several rational places distinct to the infinity place on Kummer extensions, which is a different characterization from that presented by Hu and Yang in \cite{HY2018}. Furthermore, we present some applications in coding theory and AG codes with good parameters.

翻译：本研究利用广义Weierstrass半群中的极大元素及其与纯缺口的关系，扩展了文献\cite{CMT2024}的结果，提出了一种完整确定有限域上任意函数域$F$中多个有理点处纯缺口集合及其基数的方法。作为实例，我们确定了库默尔扩张中不同于无穷远点的多个有理点处纯缺口集合的基数及其简洁显式描述——这一刻画不同于胡和杨在\cite{HY2018}中的表述。此外，我们展示了该结果在编码理论及具有良好参数的代数几何码中的若干应用。