The number of critical points of a rational function on the Riemann sphere

Question

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 $2k-2$, but why?

Answer 1

If you know some geometry or the Riemann-Hurwitz formula,

$f$ is a degree $k$ map from $\Bbb P^1(\Bbb C)$ to $\Bbb P^1(\Bbb C)$, 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 $n = 2k-2$.

Answer 2

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 $\infty$ is not a critical point, then the number of critical points is indeed $2d-2$. 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 $2d-2$ 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".