Let f be a meromorphic (equivalently, rational) function on the Riemann sphere with k pr

aniawnua

aniawnua

Answered question

2022-05-24

Let f be a meromorphic (equivalently, rational) function on the Riemann sphere with k prescribed poles. What is the number of critical points (or critical values) of such function f? I believe the answer is 2 k 2, but why?

Answer & Explanation

Keyon Fitzgerald

Keyon Fitzgerald

Beginner2022-05-25Added 10 answers

If you know some geometry or the Riemann-Hurwitz formula,
f is a degree k map from P 1 ( C ) to P 1 ( C ), both surfaces have genus 0.
Then the formula says that 2 0 2 = k ( 2 0 2 ) + n where n is the number of critical points (in the generic case where they are all distinct so they make simple branch points)
From this you get n = 2 k 2.
meindwrhc

meindwrhc

Beginner2022-05-26Added 3 answers

For a generic degree d rational map f = P / Q under the conditions that (a) it is reduced, (b) that P and Q have same degree, (c) Q has simple zeros only and (d) that is not a critical point, then the number of critical points is indeed 2 d 2. You may see this from f = ( P Q Q P ) / Q 2 and use the above requirements to see that the degree of the numerator is 2 d 2 and that it has no zeros in commun with Q. When one of the above mentioned conditions fail various other things may happen. You may look in Beardon's book on "Iteration of Rational functions".

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