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?



What element of $\pi_2(S^2)$ do rational functions represent?
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 $\max(\deg a,\deg b)$.


