How many ways to compute Puiseux series?

  • Context: Graduate 
  • Thread starter Thread starter jackmell
  • Start date Start date
  • Tags Tags
    Series
Click For Summary
SUMMARY

There are three established methods for computing Puiseux expansions of algebraic functions: (1) Newton-polygon, (2) creating a differential equation for the algebraic function and solving it via power series, and (3) a matrix method. These methods were highlighted by Adrien Poteaux in his 2008 paper, indicating their recognition in the field. While these methods are widely acknowledged, the discussion reveals that they may not suffice for all problems related to Puiseux series, prompting further inquiry into additional techniques.

PREREQUISITES
  • Understanding of Puiseux series and their applications in algebraic geometry
  • Familiarity with Newton polygons and their role in algebraic expansions
  • Knowledge of differential equations and power series solutions
  • Basic concepts of matrix methods in mathematical computations
NEXT STEPS
  • Research advanced techniques for computing Puiseux series beyond the three mentioned methods
  • Explore the implications of Adrien Poteaux's work on Puiseux expansions
  • Learn about the application of matrix methods in algebraic function analysis
  • Investigate alternative computational methods in algebraic geometry
USEFUL FOR

Mathematicians, algebraic geometers, and researchers interested in the computation of Puiseux series and their applications in solving complex algebraic problems.

jackmell
Messages
1,806
Reaction score
54
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
 
Physics news on Phys.org
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:

Similar threads

Graduate Can Laurent-Puiseux series be computed for annular regions?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Problem expanding algebraic function via Newton polygon
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Laurent series for algebraic functions
  • · Replies 9 ·
Replies
9
Views
3K
Courses Advanced Math for Computational Material Science Engineering
  • · Replies 2 ·
Replies
2
Views
2K
Graduate How Can We Ensure Convergence in Function Approximations Beyond Taylor Series?
  • · Replies 23 ·
Replies
23
Views
4K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Basic Scheme and Fundamental Equations of String Theory
  • · Replies 26 ·
Replies
26
Views
4K
Graduate Which algebraic branches does a pochammer contour traverse?
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Make Intelligent Initial Guesses for Newton-Raphson Programmatically
  • · Replies 16 ·
Replies
16
Views
4K
Admissions I Need some advice about my physics undergraduate options
  • · Replies 11 ·
Replies
11
Views
3K