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Proof of expanding algebraic functions by Puiseux series

  1. Nov 27, 2011 #1
    Hi,

    I'm told that a basic result in algebraic geometry is that all algebraic functions [itex]w(z)[/itex] of one variable given by the equation:

    [tex]f(w,z)=a_n(z)w^n+a_{n-1}(z)w^{n-1}+\cdots+a_0(z)=0[/tex]

    can be written in terms of fractional power series of the form:

    [tex]w(z)=\sum_{n=-\infty}^{\infty}a_n \left(z^{1/d}\right)^n[/tex]

    Might someone here suggest an accessible reference that goes over the proof or perhaps take a moment to explain some of the basic ideas behind the proof to me?

    Thanks,
    Jack
     
    Last edited: Nov 27, 2011
  2. jcsd
  3. Nov 27, 2011 #2

    Hurkyl

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    My gut feeling is just set d=n! (or to the lcm of 1, 2, .., n), plug in, and show that you can solve for the coefficients.
     
  4. Nov 27, 2011 #3
    Afraid I don't follow that Hurkyl. My main question is how do I show the existence of such a power series representation and then show that it actually converges to the function in the specified domain.
     
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