A proof of the Weierstraß gap theorem not using the Riemann-Roch formula (Q2299127)

Let \(X\) be a compact Riemann surface with genus \(g\ge 1\). The Weierstrass gap theorem states that for each \(P\in X\), there are exactly \(g\) integers, \(n_j(P)\), with \[ 1=n_1(P)< n_2(P) < \cdots < n_g(P) \le 2g-1 \] such that there is no meromorphic function on \(X\) which is holomorphic on \(X-\{P\}\) and has a pole of order \(n_j(P)\) at \(P\). The proof of the theorem is an application of the Riemann-Roch Theorem. The authors propose an alternative combinatorial proof that makes use of ideas from modular function algebras, integral bases and Puiseux series.