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.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]a_n(z)w^n+\cdots+a_0(z)[/tex]

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 [itex]f(z,w)=(w-1)(w-2)^2(w-3)^3-z[/itex] in [itex]\mathbb{C}[z,w][/itex], 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.

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

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 [itex]a_n[/itex]?

Thanks,

Jack

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# Domain of convergence of Puiseux series

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