On gap sets in arbitrary Kummer extensions of $K(x)$

TL;DR

This paper provides an explicit description of the gap set at totally ramified places in Kummer extensions of function fields, revealing structural properties of the Weierstrass semigroup and generalizing known formulas for Weierstrass weights.

Contribution

It offers a compact, explicit characterization of gap sets and Weierstrass semigroups in Kummer extensions, including symmetry conditions and asymptotic weight formulas.

Findings

Explicit description of gap sets at ramified places

Determination of generating sets for Weierstrass semigroups

Generalization of Towse's asymptotic weight formula

Abstract

Let $K$ be an algebraically closed field, and let $F/K(x)$ be a Kummer extension of function fields of genus $g$. We provide a compact and explicit description of the gap set $G(Q)$ at any totally ramified place $Q$ of the extension $F/K(x)$. As a consequence, we deduce structural properties of the Weierstrass semigroup $H(Q)$; in particular, we determine a generating set for $H(Q)$, and we characterize its symmetry in certain cases. We also generalize a formula due to Towse that describes the asymptotic behavior of the sum of the Weierstrass weights at all totally ramified places of the extension $F/K(x)$ relative to $g^3-g$.