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?
1 Answer
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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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2$\begingroup$ Does this extends somehow to the quaternionic projective line? $\endgroup$ Oct 28, 2011 at 15:51