On generalized Weierstrass Semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$
Alonso S. Castellanos, Erik A. R. Mendoza, Guilherme Tizziotti
TL;DR
This work develops a unified framework for generalized Weierstrass semigroups in arbitrary Kummer extensions of $\mathbb{F}_q(x)$ by leveraging discrepancy methods to describe the absolute and relative maximal elements $\widehat{\Gamma}({\bf Q})$ and $\widehat{\Lambda}({\bf Q})$. The authors provide an explicit parametrization of these maximal sets and deduce the minimal generating sets $\Gamma({\bf Q})$ and $\Lambda({\bf Q})$ for the Weierstrass semigroup, enabling gap and pure-gap analysis in a broad class of function fields. They then apply the general theory to concrete families, including maximal curves $\mathcal{X}_{a,b,n,s}$, $\mathcal{Y}_{n,s}$, curves of the form $y^m=f(x)$ with separable $f$, and the Beelen-Montanucci curve, delivering explicit formulas for the maximal-element sets in each case. By relaxing conditions such as inclusion of the point at infinity or gcd constraints, the results generalize and unify previous contributions in the theory of Weierstrass semigroups, with potential implications for algebraic-geometry codes and gap analysis in new families of curves.
Abstract
In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of function field $\mathbb{F}_q(x)$. We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit descriptions of the sets of absolute and relative maximal elements within these semigroups are provided. Additionally, we apply our results to function fields of the maximal curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$, which cannot be covered by the Hermitian curve, and the Beelen-Montanucci curve. Our results generalize and unify several earlier contributions in the theory of Weierstrass semigroups, providing new perspectives on the relationship between these semigroups and function fields.