# How to compute monodromy of a particular algebraic function

1. Nov 19, 2013

### jackmell

Given the algebraic function:

$$w=z^{p/q}(1-z)^{r/s},\quad (p,q,r,s)\in\mathbb{Z}\backslash\{0\}$$

and I choose a particular determination to begin tracking around the singular point $z=1$ starting at the point $z=1/2$ along the contour $z=1+1/2 e^{it}$, is there a way to compute other than by exhaustive search, which determination I land on when I return to the starting point?

Can we even predict the branching geometry for this particular function? Will I encounter a fully-ramified geometry around this singular point or will it separate into a predictable set of cycles or will the cycling geometry be a function of the parameters $p,q,r, s$ that can be computed for any values of the parameters other than by actually tracking around the singular point and determining the monodromy numerically?

Here's one I'm working with:

$1-z^{15}(1-z)^{28}w^{20}=0$

corresponding to $p/q=-3/4$ and $r/s=-7/5$. This function ramifies into four 5-cycle branches over $z=1$. I believe so anyway. Anyone reading this want to confirm that? Suppose I next choose one of the determinations for one of those cycles. Now, without directly computing the argument change around the contour, can I predict or compute as a function of the $(p,q,r,s)$ parameters, which determination of the branch I'll land on when I go once around the circular contour about this singular point?

Last edited: Nov 19, 2013