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

Abstract

On a compact Riemann surface $X$ of genus $g$, one of the questions is the existence of meromorphic functions having poles at a point $P$ on $X$. One of the theorems is the Weierstrass gap theorem that determines a sequence of $g$ numbers $1 < n_k < 2g$, $1 \leq k \leq g$ for which a meromorphic function with the order with $n_k$ fails to exist at $P$. In this note, we give proof of the Weierstrass gap theorem in cohomology terminology. We see that an interesting combinatorial problem may be formed as a byproduct from the statement of the Weierstrass gap theorem.