# How many ways to compute Puiseux series?

• jackmell
In summary, there are currently three published methods for computing Puiseux expansions of algebraic functions: the Newton-polygon method, creating a differential equation and solving it via power series, and a matrix method. However, these methods may not be sufficient to solve all problems. Two years ago, Adrien Poteaux stated that there are only three known methods for computing Puiseux series, but it is uncertain if he is the expert in this area. The reference to Walker's work was also obtained.
jackmell
How many ways in the entire world are there for computing Puiseux expansions of algebraic functions? I know of three published methods:

(1) Newton-polygon

(2) Create a differential equation for the algebraic function then solve it via power series

(3) Some matrix method I'm not too familiar with.

Is that all?

Let me know guys if I'm missing something.

Thank you,
Jack

it's a big world. do you have a problem these do not suffice to solve? that would make a more limited question.

mathwonk said:
it's a big world. do you have a problem these do not suffice to solve? that would make a more limited question.

Unfortunately I do have a problem those three do not in themselves suffice to solve. However, if those are the only methods available then my problem is solved.

However two years ago, Adrien Poteaux in http://www.lifl.fr/~poteaux/fichiers/JSC_ISSAC08.pdf
stated, "we know of three methods to compute Puiseux series" and it looks like he's the expert in the matter although I'm not sure. Perhaps I should go with that. I did e-mail him but he never replied.

By the way, I did obtain a copy of Walker. Thank you for that reference. :)

Last edited:

## 1. What is a Puiseux series and how is it computed?

A Puiseux series is a mathematical series used to represent functions that have a singularity, or discontinuity, at a certain point. It is computed by expanding the function around the singularity using fractional powers instead of just integer powers.

## 2. How many ways are there to compute a Puiseux series?

There are several methods for computing a Puiseux series, including the Newton-Puiseux algorithm, the substitution method, and the elimination method. Each method may be more suitable for certain types of functions or singularities.

## 3. Can Puiseux series be computed for any function?

Yes, Puiseux series can be computed for any function that has a singularity at a certain point. However, the series may not always converge, in which case it is not a valid representation of the function.

## 4. How does the number of terms in a Puiseux series affect its accuracy?

The more terms included in a Puiseux series, the more accurate the representation of the function will be. However, including too many terms can also lead to computational errors and loss of accuracy.

## 5. Are there any applications for Puiseux series in real-world problems?

Yes, Puiseux series are commonly used in physics and engineering to model functions with singularities, such as in fluid dynamics and electrical circuits. They also have applications in computer graphics and numerical analysis.

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