Talk:Puiseux series

Counterexample for completeness of Puiseux series

For example, the series $\sum _{k=2}^{\infty }w^{k+{\frac {1}{k}}}$ does not converge. This is to explain why I just deleted the completeness claim from the main page. --Thomas Bliem (talk) 01:42, 2 June 2010 (UTC)

This is not a Puiseux series, the denominators in the exponents being unbounded. D.Lazard (talk) 12:23, 30 October 2012 (UTC)

A suggestion on how to start making this article more widely accessible

I was the one who placed the “too technical” template. I was redirected to this page from “Puiseux theorem,” to find out that the article only gives the “modern,” very algebraic statements. But more elementary theory does exist, and the discussion of it should precede the more “modern” formulations. I may rewrite the introductory sections of the article myself, but in the meantime, here is a suggestion on how to proceed.

A good place to find a contemporary presentation of the elementary theory seems to be the book Singular points of plane curves by C. T. C. Wall. The first sentence of Chapter 2 in that book says:

The theorem of Puiseux states that a polynomial equation $f(x,y)=0$ has a solution in which $y$ is expressed as a power series in fractional powers of $x$.

The book proceeds to “give several versions of this theorem, of increasing sharpness.” I think this article should follow that example. Reuqr (talk) 16:53, 7 April 2011 (UTC)

I believe that I have solved, for the lead, the "too technical" issue. I have left the tag, because some work is yet needed in the body of the article. In particular the use of Newton polygon to expand as Puiseux series the solutions of a bivariate equation should be explained and illustrated by examples. D.Lazard (talk) 12:29, 30 October 2012 (UTC)

I am also concerned with the excessive focus on the abstract algebra properties of the series and the omission of Puiseux's theorem, which I think more readers would be interested in.--Jasper Deng (talk) 06:24, 24 November 2015 (UTC)

Wrong statement?

I am confused by this statement (in "Analytic convergence").

When $K=\mathbb {C}$, i.e. the field of complex numbers, the Puiseux expansions defined above are convergent in the sense that for a given choice of n-th root of x, they converge for small enough $|x|$

Indeed this is wrong even for formal power series.128.178.14.87 (talk) 09:53, 30 July 2015 (UTC)

This sentence belongs to a subsection of section "Puiseux expansion of algebraic curves and functions", and thus concern only Puiseux expansion of algebraic curves. In this case, the statement is true. Similarly, the divergent Taylor expansion are always Taylor expansions of transcendental functions. I have edited the article to clarify this point. D.Lazard (talk) 10:42, 30 July 2015 (UTC)

Some suggestions for appealing to a wider audience

I am hesitant to go in and change the article however Puiseux series are intimidating to most everyone who encounters them and beginning the discussion here with an abstract description only frightens them further. Better to slowly ease them in with an undergraduate introduction to appeal to a much more wide audience, then follow with the more advanced description.

First paragraph about Puiseux series should be improved

The "T" only distracts from an already difficult subject. Use x. Also, would be more clear if written as:

In mathematics, Puiseux series are a generalization of Laurent series that allow for fractional exponents. They were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850. For example, the series $g(x)=1+x^{1/3}+x^{2/3}+x^{5/3}+\cdots$ is a Puiseux series in x. The entire series is evaluated using a chosen cube root of $x$. And therefore this series actually represents three (single-valued) series which make up a single conjugate class of Puiseux series. The other two series of this class can be obtained by conjugation or most often computed by the usual method of Newton Polygon (see below for more information).

References

Second paragraph in my opinion is not worded correctly

Recommend:

Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation $P(x,y)=0$, its solutions in y, viewed as functions of x, can be expanded as Puiseux series about a point $x_{0}$ that are convergent in some neighbourhood of $x_{0}$. In other words, every branch of an algebraic curve may be locally (in terms of x) described by a Puiseux series.

Next follow with a specific example

$F(x,y)=1+(x+x^{2})y+(3x^{2}+x^{3})y^{3}$ or at least one which fully ramifies to better exhibit a conjugate class and include either links to the Newton Polygon algorithm or a short discussion. Next show some actual calculations using the series, especially convergence examples. Include a plot or two of the beautiful underlying geometry.

I can do this work which would in my opinion break a long-standing ice jam with this subject and attract further interest and discovery about this very beautiful subject so ask everyone associated with this topic to allow me to propose changes for your review. Youriens (talk) 11:56, 10 October 2021 (UTC)