# Domain of convergence of Puiseux series

1. Jan 7, 2012

### jackmell

May I ask if it's appropriate to ask in this sub-forum, are there any general methods for computing the domain of convergence for Puiseux series representations of algebraic varieties? I'm unable to find anything on the matter.

For example, suppose I'm given the ideal in $\mathbb{C}[z,w]$:

$$a_n(z)w^n+\cdots+a_0(z)$$

and I proceed to compute a Puiseux-series representation of (a component of) the variety. Is there nothing I can do to determine (symbolically) the domain of convergence for this series? I'm aware and familiar with Newton-Puiseux's theorem which guarantees the existance of such series but says nothing (quantitative) about computing their region of convergence.

Ok, here's an example: Consider the ideal $f(z,w)=(w-1)(w-2)^2(w-3)^3-z$ in $\mathbb{C}[z,w]$, prove (or disprove) that a double-valued (analytically-continued) manifold of the variety can be represented by the Puiseux expansion below convergent in the proposed (not proven) region.

$$w(z)=\sum_{n=0}^{\infty} a_n z^{n/2},\quad 0\leq|z|<\left|\frac{-587+145\sqrt{13}}{1458}\right|$$

So I'm only conjecturing the region of convergence is as above. Is there a way to prove this rigorously without having a symbolically-computable expression for the coefficients $a_n$?

Thanks,
Jack

Last edited: Jan 7, 2012