Weierstrass points on Kummer extensions

TL;DR

This paper investigates Weierstrass points on Kummer extensions, providing necessary and sufficient conditions for gaps at ramified places, thereby extending previous results in algebraic geometry.

Contribution

It offers a comprehensive characterization of Weierstrass gaps at ramified places in Kummer extensions, advancing the understanding of their algebraic structure.

Findings

Necessary and sufficient conditions for Weierstrass gaps at ramified places

Extension of previous results on Weierstrass points

Deeper insight into the structure of Kummer extensions

Abstract

For Kummer extensions $y^m=f(x)$, we discuss conditions for an integer be a Weierstrass gap at a place $P$. In the case of totally ramified places, the conditions will be necessary and sufficient. As a consequence, we extend independent results of several authors.