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How to compute monodromy of a particular algebraic function

  1. Nov 19, 2013 #1
    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
  2. jcsd
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