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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.

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

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