I am currently struggling with understanding how to find the branch points and behavior of covering of implicitly defined irreducible functions $P(x,y)$. An arbitrary example I am working on is to find the branch points and branching of $$P(x,y) = x^4 -4xy +3y^2$$ As I understand, branch points are at points $(x_0,y_0)$ where $P(x_0,y_0) = \partial x P(x_0,y_0) = 0$
So, in this case, the branch points would be given by the system: $$x^3 = y$$ $$x^4 -4x^4 +3x^6 = 0$$
And thus we have $$x^3 = y$$ $$3x^3(x^2-1)=0$$ So the branch points would be (0,0), (1,1), (-1, -1).
Now, I am interested in the branch behavior, and how the Riemann sheets are distributed. From what I understand, the degree of $x$ in $P(x,y)$ is the number of Riemann sheets at each branch point $y_0$. How would I determine the branch behavior?
I have gathered from an example that you find all zeroes of $P(x,y_0)$, where $y_0$ is a branch point, and the order of the zero at each $y_0$ is the number of sheets at that point. I don't completely understand why/if this is true.
Moreover, I have seen an example of Riemann sheets "grouping", as in $P(x,y) = w^6 - (z-3)^3(z+2)^2$. (Figuring out the sheet behavior was not explained) Are the sheets determined in the same "find roots and multiplicities" way?
I suppose my principal question here is, how can you figure out "branching behavior" and Riemann covers for a general polynomial? I cannot find any resources on it, so any information would be greatly appreciated!