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Let $f$ be a rational function on a compact connected Riemann surface $X$. The rational function $f$ induces a holomorphic map $\overline{f}:X\to \mathbf{P}^1(\mathbf{C})$.

Let $x$ be a point on the Riemann sphere $\mathbf{P}^1(\mathbf{C})$. How can I check that if $b$ is a branch point of $\overline{f}$ by looking at the derivative of $f$?

How does this work when $X=\mathbf{P}^1(\mathbf{C})$?

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If $b\in X$ and $f(b)\neq\infty$ then $b$ is a branch point iff $f'(b)=0$ (derivative wrt. an arbitrary local coordinate; the ramification index is the maximal $k$ s.t. $f^{(k)}(b)=0$ (the number of branches meeting at $b$ is $k+1$)). If $f(b)=\infty$, replace $f$ with $1/f$.

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Just a minor nitpick on terminology: If $b$ is in $X$, one should call it a ramification point right? The branch locus being the image of the ramification locus. –  Harry Apr 2 '12 at 19:10
    
@Harry: you are of course right –  user8268 Apr 2 '12 at 19:12
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