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. Youriens (talk) 11:56, 10 October 2021 (UTC)

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). Youriens (talk) 11:56, 10 October 2021 (UTC)

References

I do not think this is an improvement. Power series are familiar to every single-variable calculus student (at least in the US), whereas Laurent series are almost never met (at least under that name) until later. Puiseux series need not be evaluated / evaluable (formal Puiseux series are a perfectly reasonable object), and the technicalities thereof hardly seem lead-worthy. Compared with the trivial matter of what to call the variable, these are much larger issues. (I wouldn't object to changing the variable to x, though.) --JBL (talk) 19:08, 10 October 2021 (UTC)

Puiseux series are distinguished by their fractional powers. That is a quintessential feature of the series quite different from power series with integer exponents. Is it unreasonable to expect readers to have some background in Complex Variables/Analysis to have any real success in understanding what these series really represent? Seems to me Calculus is not quite enough. I wrote Laurent series because there is a very close connection between Laurent's Theorem and Puiseux series, an important one I think, and should be explained somewhere in the article and just mentioning it in the lead is a first exposure. I specifically gave a real example of evaluating the series, using a specific root and explaining the conjugate series, to relax the reader as many are intimidated by multi-valued functions and giving a simple practical example they can understand gives them confidence to proceed further. Perhaps I could omit the term "conjugate" as that may be a bit too technical for the lead but I do feel explaining how to practically evaluate the series sometime soon in the discussion is helpful. Youriens (talk) 12:39, 11 October 2021 (UTC)

The article introduction is not for expressing the quintessence of a subject, but for summarizing the article and providing information that allows readers to decide whether they want read the article further. When technically possible, the introduction must also recall what is the subject of the article to readers who have heard of it and do not remember exactly the definition. I agree with JBL that the current version better fills these requirement than yours. In particular, as everyone who knows what a Laurent series knows also what a power series is, "power series" must be preferred to "Laurent series" in the first paragraph, even if it is worth to mention them later in the introduction, as it is done now.

As two editors prefer the current version, please discuss (if needed) the current version rather than yours. D.Lazard (talk) 13:29, 11 October 2021 (UTC)

Is it unreasonable to expect readers to have some background in Complex Variables/Analysis to have any real success in understanding what these series really represent? You titled this section "Some suggestions for appealing to a wider audience". Appealing to as wide an audience as possible is a great idea; insisting that the audience must share your particular background and perspective is the opposite of that. It's quite clear that students who have taken one year of calculus should in principle be able to understand what is a Puiseux series, and even the content of Puiseux's theorem, and therefore we should aspire to introduce those basic ideas using terminology that would not unnecessarily exclude those potential readers. Speaking personally, I have no training in complex analysis (I mean, I took some classes as an undergraduate nearly 2 decades ago that did not mention Puiseux series, as far as I recall), but I occasionally come across (formal) Puiseux series in my work, so your description of the audience excludes even some PhD-holding professional research mathematicians who might be interested in the article content. --JBL (talk) 14:26, 11 October 2021 (UTC)

Copy that. I see some changes have already been made which I feel is an improvement. However, I feel one statement can be improved further.

Original statement: (possibly zero excluded, in the case of a solution that tends to the infinity at when x tends to 0).

Recommended change: "(zero excluded in the case of expansion about a pole). For example $y(x)$ given by the expression $f(x,y)=1+(2+x)y+xy^{2}+xy^{3}=0$ has a pole at the origin so the Puiseux series for at least one branch contains a ${\frac {1}{x^{p}}}$ term so is undefined at zero." Youriens (talk) 14:57, 11 October 2021 (UTC)

This does not belong to the lead, but will be worth to include in the section § Newton–Puiseux theorem, when this section will be correctly written. D.Lazard (talk) 15:11, 12 October 2021 (UTC)

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. Youriens (talk) 11:56, 10 October 2021 (UTC)

It seems that the only difference with the current version is that Puiseux expansion are considered in the neighborhood of any point. As series Puiseux in $x-x_{0}$ are not defined before, it seem better to keep the case $x_{0}\neq 0$ outside the lead. However, the current formulation can (and should) be improved. D.Lazard (talk) 13:54, 10 October 2021 (UTC)

Ok, how about just state the expansion can be at any point but is frequently done at the origin. I think presenting the expansion at any point in the lead while perhaps focusing initially at the origin does not add unnecessary complexity and prepares the reader for a more comprehensive understanding. Consider Runge's Theorem which studies expansions centered at infinity. Casually and briefly mentioning this Theorem in the discussion for example gives a concrete application of Puiseux series and may interest the reader to pursue this further.

Another example is studying the branching geometry at singular points other than zero important for studying contour integration over these functions. Consider the pochhammer contour for the beta function. This is a difficult concept to understand without a clear understanding of the underlying geometry and Puiseux series can help with that. The problem of course is most everyone is intimidated by the fractional powers and more so when they are combined into an extremely complex object like the algebraic curve of f(x,y)=0. But this can be mitigated if the principles are presented in an intuitive and easy to follow manner by anyone interested in Complex Analysis. This I can do as I hope I did with the edit suggestions I made of the first paragraph above.Youriens (talk) 16:02, 10 October 2021 (UTC)

I agree that Puiseux series in $x-x_{0}$ are important. But here we are in the lead, and is it standard, in an article introduction, to avoid, as possible, technical details (see MOS:LEAD and MOS:MATH#Article introduction). This suggest to detail Puiseux expansions outside the origin in a specific section that could be a first subsection of § Algebraic curves. By the way, this section begins with two errors: by excluding branches with a vertical tangent, and, in the second paragraph by identifying a branch with a point (incidently, it is not here that normalization must be considered, as it suffices to define, as usually, a branch as an analytic curve). D.Lazard (talk) 16:35, 10 October 2021 (UTC)

Ok, thanks for that reference. I do not see where branches with vertical tangents are being excluded. My example of $1+x^{1/3}+\cdots$ has a branch point at the origin with a vertical tangent. Also I do not see where a branch is being identified with a point. My example refers implicitly to a 3-cycle branch defined locally by three single-valued Puiseux series. Can you explain a bit further please?Youriens (talk) 12:39, 11 October 2021 (UTC)

OK for the vertical tangent. But this shows that the section is very badly written, as it may confuse readers who well know the subject. The sentence "let us define the branches of X at p to be the points ..." is definitively an identification of branches with some points. This could make sense if the language of schemes would be assumed, but this is not the case, and it must not be the case if a wide audience is desired. D.Lazard (talk) 13:48, 11 October 2021 (UTC)

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)

This sounds to me like a good thing to have in the article, and a bad thing to have in the lead of the article. --JBL (talk) 19:08, 10 October 2021 (UTC)

Completeness property

Hello everyone, it seems confusing the way the completeness of Puiseux series is explained. Would someone (who is confortable enough with this subject) please explain better why the completeness is claimed for Puiseux series, then contradicted in the generalization section ?

Best regards. 79.95.86.167 (talk) 17:29, 27 November 2021 (UTC)

 Fixed. Good point; thanks for pointing it. D.Lazard (talk) 18:33, 27 November 2021 (UTC)

pdfs with registration?

Did anyone check the links? For the last link (Algebraic Curves, R.J. Walker) I thought that I downloaded the book several days ago. But now when I open it, I see that it leads to a site where I should download it. The url type says that I need to register (that's why I didn't remove this link) - first I didn't notice that.

Do people check these links or are they just scam / fishing / fraud?

I looked up Help:URL and Help:Link, but could not find any security policy in this case (and what if the link just changes - should we register ourselves to check that it is safe?..)

Yaroslav Nikitenko (talk) 23:12, 11 August 2022 (UTC)