Let f be a rational function on a compact connected Riemann surface X. The rational function f induc

Arslan Lyons

Arslan Lyons

Answered question

2022-02-15

Let f be a rational function on a compact connected Riemann surface X. The rational function f induces a holomorphic map f:XR1(C).
Let x be a point on the Riemann sphere P1(C). ow can I check that if b is a branch point of f by looking at the derivative of f?
How does this work when X=P1(C)?

Answer & Explanation

Alexandra Haynes

Alexandra Haynes

Beginner2022-02-16Added 10 answers

If bϵX and f(b) 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), replace f with 1f.

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