# Algebraic dependence of meromorphic functions on a compact Riemann surface

I am trying to show given two meromorphic functions $$f,g$$ on a compact Riemann surface $$X$$ that there exists a non-zero polynomial $$P \in \mathbb{C}[X,Y]$$ such that $$P(f,g) = 0$$.

I have seen this question Meromorphic Function in a Compact Riemann Surface however I didn't find the discussion very useful.

Firstly, why should I expect such a result to be true?

Also, how can I go about proving this?

One thing I noticed is a polynomial in $$f,g$$ can be written as $$(1,f,...,f^r)A(1,g,...,g^r)^t$$ for some matrix of coefficients $$A$$, but I did not find a way to use this.

• If you cannot solve this, take a look for instance at the end of chapter 7 of R.Narasimhan "Compact Riemann surfaces". – Moishe Kohan Jan 12 at 17:01