Don't understand why I'm being flagged for this response: Knowing the monodromy (ramified coverings) around each singular point enables us to easily compute the genus using the Riemann-Hurwitz formula:

$$G=1/2\sum^N (e_p-1)-D+1$$

where the set $\{e_p\}$ is the set of ramified coverings around all singular points including the point at infinity. Lets take the simplest example: $f(z,w)=w^2-z$ which is the square root function. This function of course fully-ramifies into a 2-cycle branch at the origin. But that means it also fully-ramifies at the point of infinity. So we have two ramifications: ((1,2),(1,2)). Plugging those into the expression we obtain: $1/2(1+1)-2+1=0$ which is of course the genus of any function $f(z,w)=w^n-z$.

And in the case of more complicated functions, we need to numerically integrate over all coverings in order to compute the ramified coverings: This integration is relatively simple to do hence the term "straight-forward".

Don't understand why I'm being flagged for this response: Knowing the monodromy (ramified coverings) around each singular point enables us to easily compute the genus using the Riemann-Hurwitz formula:

$$G=1/2\sum^N (e_p-1)-D+1$$

where the set $\{e_p\}$ is the set of ramified coverings around all singular points including the point at infinity. Lets take the simplest example: $f(z,w)=w^2-z$ which is the square root function. This function of course fully-ramifies into a 2-cycle branch at the origin. But that means it also fully-ramifies at the point of infinity. So we have two ramifications: ((1,2),(1,2)). Plugging those into the expression we obtain: $1/2(1+1)-2+1=0$ which is of course the genus of any function $f(z,w)=w^n-z$.

And in the case of more complicated functions, we need to numerically integrate over all coverings in order to compute the ramified coverings: This integration is relatively simple to do hence the term "straight-forward". And I asked if someone could check it because I don't have Maple and am not 100% sure it is correct.