# Branch points and branching number on Riemann surface

I am attempting to analyze the Riemann surface of the algebraic function $w=(\sqrt{z} - 1)^{1/4}$. To do this, I started out by writing as a polynomial, $P(w,z) = w^8 +2w^4 + 1 -z = 0$. Next, I want to identify the branch points. I understand these points to be the points where the inverse function theorem fails, so I $\dfrac{\partial P}{\partial w}=0,$ identify the values of $w$ satisfying this, and use them to determine some corresponding $z$'s. This gives the ordered pairs $(w,z) = (0,0), (1,3), (i,3), (-1,3), (-i,3)$. (Here I am just listing the pairs).

Next I would like to compute a branching number for each of these points. I understand that this amounts to determining the minimal number $l \in \mathbb N$ so that $\dfrac{\partial^l P}{\partial w^l}(w,z) \neq 0.$ The branching number is defined as $l-1$. Using this, it is easy to compute the branching numbers. The branching number for the first is $3$, and all the others have branching number $1$.

From here I want to compute the genus of the Riemann surface of this polynomial. To do this, I should expect the sum over all the branching numbers to be an even number, by considering Riemann-Hurwitz. However, their sum is currently $7$. This made me think I had missed a point, and was thinking about the point at infinity. However, I do not understand (or maybe it's obvious and I've just forgotten) how to determine the branching number of the point at infinity.

I am a bit unsure about even my work so far though, because it is a result in Schlag that the degree of a polynomial is equal to the number of sheets of the Riemann surface. This means that this surface has $8$ sheets. By Riemann-Hurwitz, we would expect something of the form $g_{RS} = 1+8(g-1)+\dfrac{\sum B(p)}{2}$, where $g$ is the genus of the Riemann sphere, which is $0$. In other words, $g_{RS} = -7 + \dfrac{\sum B(p)}{2}$, implying that the branching number of infinity must be very large, since genus must be at least $0$.

So my question is, did anything go wrong in this analysis? If so, where? Furthermore, how do I compute this branching number at infinity?

To find the branch points, we need to find the roots of the discriminant $$\Delta_w(P(z, w)) = \Delta_w(w^8 + 2 w^4 - z + 1) = (64 z)^4 (1 - z)^3.$$ This gives two branch points. Another way would be to solve $d z/d w = 0$ and then find the corresponding values of $z$.
At $z = 0$, we have $P(0, w) = (w^4 + 1)^2$. There are four double roots. Computing the derivatives wrt $w$ essentially does the same thing, finding the multiplicities of the roots.
At $z = 1$, $P(1, w) = w^4 (w^4 + 2)$. There is one root of the fourth order and four single roots.
To deal with $z = \infty$, we have to compute the Puiseux series. The expansion around infinity is determined by the terms $w^8 - z$, giving a ramification index of 8 (the branching number is 7).
Adding the branching numbers corresponding to each point $(z_0, w_0)$, we find the genus: $$g = \frac {4 (2 - 1) + (4 - 1) + (8 - 1)} 2 - 7 = 0.$$ Which is as expected, because $P(z, w)$ is linear in $z$ and thus has a trivial rational parametrization.