# Connecting branches of algebraic functions

1. Dec 11, 2013

### jackmell

Consider the algebraic function, $w(z)$ given by

$$f(z,w)=z^4(1-z)^3-w^{12}=0$$

where I have shown elsewhere $w(z)$ ramifies over the origin into three 4-cycle branches and four 3-cycle branches over z=1. Now choose one determination of say one of the 4-cycle branches over the point $z=1/2$ which I'll label as $w(4,1,1)$, that is, determination 1 on the first 4-cycle branch. Now, between the two singular points, these coverings are of course analytically continuous with one another in some way I don't understand so that $w(4,1,1)$ is analytically continuous with some determination of one of the 3-cycle branches about z=1. I'll label the branch and determination over z=1 that $w(4,1,1)$ is analytically continuous with as $w(3,k,j)$ since I don't know either the branch or the determination on that branch.

And therefore, I can do this sort of thing with all 12 coverings. That is, associate $w(4,n,m)$ with $w(3,k,j)$.

Now, is there any way to figure out the values of $n,m,k,j$ other than by brute-force numerical analysis of the data?

Ok thanks,
Jack