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Connecting branches of algebraic functions

  1. Dec 11, 2013 #1
    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
     
  2. jcsd
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