Domain of convergence of Puiseux series

[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 existence 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

 

[jvicens]

Hello Jack,

Thank you for your question. I can tell you that there are indeed general methods for computing the domain of convergence for Puiseux series representations of algebraic varieties. One such method is known as the Puiseux-Newton algorithm, which is a generalization of Newton's method for finding roots of polynomials. This algorithm can be used to compute the Puiseux series expansion of a given algebraic variety and determine its domain of convergence.

To apply this algorithm, one needs to first transform the given ideal into a monomial ideal, which is a polynomial ideal where all the terms have the same degree. This can be done by multiplying each term with a suitable power of a variable. Then, using the Puiseux-Newton algorithm, one can compute the Puiseux series expansion of the variety with respect to this monomial ideal. The algorithm also provides a criterion for the convergence of the series, which can be used to determine the domain of convergence.

In your example, the given ideal can be transformed into a monomial ideal by multiplying with z^2. Then, using the Puiseux-Newton algorithm, one can compute the Puiseux series expansion of the variety with respect to this monomial ideal, which will yield the desired result. However, it is important to note that in some cases, the algorithm may not be able to provide a symbolic expression for the coefficients a_n, in which case one may need to resort to numerical methods for computing the series.

I hope this helps answer your question. If you have any further inquiries, please don't hesitate to ask. Best of luck with your research!