The set of pure gaps at several rational places in function fields

In this work, we explore the use of maximal elements in generalized Weierstrass semigroups and their relationship with pure gaps, extending the results in Castellanos et al. [J Pure Appl Algebra 228(4):107513, 2024]. We provide a method to completely determine the set of pure gaps at several rational places in a function field F over a finite field, where the periods of certain places are the same, and determine its cardinality. As an example, we calculate the cardinality and provide a simple, explicit description of the set of pure gaps at several rational places distinct from the infinity place on Kummer extensions, offering a different characterization from that presented by Hu and Yang [Des Codes Cryptogr 86(1):211–230, 2018]. Furthermore, we present some applications in coding theory and AG codes with good parameters.