Proof of expanding algebraic functions by Puiseux series

  • Thread starter jackmell
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  • #1
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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
 
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  • #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.
 
  • #3
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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.
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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