# Proof of expanding algebraic functions by Puiseux series

1. Nov 27, 2011

### jackmell

Hi,

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

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

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

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

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. Nov 27, 2011

### Hurkyl

Staff Emeritus
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. Nov 27, 2011

### jackmell

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.