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?

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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?

Keyon Fitzgerald · Accepted Answer

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 · Accepted Answer

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&#039;s book on &quot;Iteration of Rational functions&quot;.

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

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