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Take a rational function of a single complex variable. View it as a continuous function from the Riemann sphere to itself. Is there a nice way to compute which element of $\pi_2(S^2)$ this corresponds to?

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up vote 16 down vote accepted

It's the cardinality of the preimage of a generic point, because generically the local degree of a complex analytic function is always +1. If the rational function is $a(x)/b(x)$, then the number of solutions in $x$ to $a(x)/b(x) = y$ for generic $y$ is $\max(\deg a,\deg b)$.

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Does this extends somehow to the quaternionic projective line? –  Mariano Suárez-Alvarez Oct 28 '11 at 15:51

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