I would like to have a simple proof for the following result:


Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined). There is a natural extension to a map $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$, considering $\mathbb{S}^2=\mathbb{C}\cup\infty$ the compactification of the complex numbers (by taking limits, perhaps some analysis is needed). **Then $\text{deg}(\bar{f})$(the topological degree of maps between spheres)$=\max\{\text{deg}(p),\text{deg}(q)\}$**.


Any idea is welcome.



Degree of a rational function
The fundamental theorem of algebra tells us that the number of solutions to $f(z) = a$ is the maximum of the degrees of the numerator and denominator.


