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 Post subject: On Zeros And PolesPosted: Thu Feb 04, 2016 10:53 am
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Joined: Tue Nov 10, 2015 8:25 pm
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Let $f$ be a meromorphic function on a (connected) Riemann Surface $X$. Show that the zeros and the poles of $f$ are isolated points.

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 Post subject: Re: On Zeros And PolesPosted: Mon Mar 14, 2016 3:49 pm
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Joined: Tue Nov 10, 2015 8:25 pm
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Let $X$ be a Riemann surface and let $f$ be a meromorphic function on $X$. Then $f$ can be considered as a holomorphic map $f \ \colon X \longrightarrow \hat{\mathbb{C}}$ which is not identically equal to $\infty$. But for holomorphic maps between Riemann Surfaces the Identity Theorem holds.
• If the set of zeros of $f$ contained a limit point, then, by the Identity Theorem, $f$ should be equal to $0$. But we have assumed that $f$ is not constant.
• If the set of poles of $f$ contained a limit point, then, by the Identity Theorem, $f$ should be equal to $\infty$. But we have excluded that case by definition of a meromorphic function.
Hence, the zeros and the poles of $f$ are isolated points.

Remark: Using the Identity Theorem we can also prove that the set of ramification points of a proper, non-constant, holomorphic map between Riemann Surfaces consists only of isolated points.

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