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**Which algebraic branches does a pochhammer contour traverse?**

Hi,

Got another question about pochhammer continuation of beta function in regards to algebraic function cycles:

Which cycles of a function does the pochhammer contour traverse? Take for example the function:

$$w=z^{4/5}(1-z)^{3/7}$$

That corresponds to the expresson:

$$f(z,w)=z^{28}(1-z)^{15}-w^{35}=0$$

We have 35 coverings ramifying at the origin into seven 5-cycle branches. Suppose I pin one of the horizontal legs of the pochhammer contour over one determination of one of those seven branches. Which determination and which branches will the remainder of the contour traverse? Can this be predicted for any algebraic function? Will it remain on only one branch or will it traverse two three or four different branches? I don't know.

I suppose I could expand the function into it's Puiseux series, pin the contour at a reference point, say ##z=1/10##, track it around, selecting the four values at the reference point, compare those values with each series at the reference point, and compare the results. That is however a completely empirical and brute-force approach and I was wondering if there is a more algebraic way of doing this.

Ok thanks,

Jack

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