A note on meromorphic functions on a compact Riemann surface having poles at a single point

TL;DR

This paper provides a proof of the Weierstrass gap theorem for meromorphic functions on compact Riemann surfaces, explores related combinatorial problems, and discusses implications for Weierstrass points.

Contribution

It offers a new proof of the Weierstrass gap theorem using cohomology dimensions and introduces a related combinatorial problem.

Findings

Proof of the Weierstrass gap theorem using cohomology groups

Identification of a combinatorial problem related to the theorem

Insights into the existence of meromorphic functions at Weierstrass points

Abstract

The Riemann -Rock theorem plays a central role in the theory of Riemann surfaces with applications to several branches in Mathematics and Physics. Suppose $X$ ia a compact Riemann surface of genus $g$ and $P \in X$. By the Riemann-Roch theorem there exists a meromorphic function on $X$ having a pole at $P$ and is holomorphic in $X \setminus \{P\}$. The Weierstrass gap theorem gives more information on the order of the pole at $P$. It determines a sequence of $g$ distinct numbers $1 < n_k < 2g$, $1 \leq k \leq g$ for which a meromorphic function with the order $n_k$, fails to exist at $P$ and it can be obtained again as an application of Riemann-Roch theorem. In this note, we give proof of the Weierstrass gap theorem, using the dimensions of the cohomology groups and find an interesting combinatorial problem, which may be seen as a byproduct from the statement of the Weierstrass gap theorem. A short note is given at the end on Weierstrass points where a meromorphic function with lower order pole $\leq g$ exists and obtain some consequences of Weierstrass gap theorem.