Talk:Newton's identities

&lambda; or x?
It would help if the variable in the polynomial would be x instead of &lambda;. Perhaps the roots could be named r1, r2 etc and the coefficients c1, c2 etc. --MarSch 15:37, 30 August 2005 (UTC)


 * I specifically wanted to differentiate the variable $$\lambda$$ from the roots $$x_1, x_2, \dots x_n$$. I guess it's a matter of taste, since your suggested notations seemed ugly to me :-/  At least I did think about the notation.  Did I need to say that I don't want to change it?---CH  (talk) 04:18, 1 September 2005 (UTC)

Schur polynomial vandalism
Watch out, formulas like that are vulnerable to vandalism in which someone changes the powers to plausible looking but incorrect values. Indeed, an anonymous editor using the IP address 165.123.166.36 (registered to the University of Pennsylvania) has done just that, although this might possibly have been a well-intentioned edit. ---CH (talk) 17:05, 11 November 2005 (UTC)


 * Heh, except that in this case 165.123.166.36 was correct :-/ Oh, well, exception which proves the rule about anonymous editors.  I don't know how this expression got so messed up in the first place, but anyway, I think we all agree the current version is correct.---CH  (talk) 01:14, 13 November 2005 (UTC)

Coefficients in polynomial expansion
For the polynomial expansion to be coherent with the first form of Newton's identities given in the article, it should be written
 * $$ p(\lambda) = \prod_{\alpha=1}^n \left( \lambda - x_\alpha \right) = \sum_{j=0}^n (-1)^j a_j \lambda^{n-j}$$

where $$a_0 = 1$$. Gpicard 16:48, 12 April 2006 (UTC)


 * Done. Thanks. Zaslav 06:45, 30 March 2007 (UTC)

Students beware
I extensively rewrote the November 2005 version of this article and had been monitoring it for bad edits until a few months ago, but I am leaving the WP and am now abandoning this article to its fate.

Just wanted to provide notice that I am only responsible (in part) for the last version I edited; see User:Hillman/Archive. I emphatically do not vouch for anything you might see in more recent versions. In particular, I haven't verified the claim in the preceding section. There were good reasons why I chose the somewhat unconventional looking signs that I did in the Novemver 2005 version. Changing signs to "prettify" a formula like this is a good way to change truth into falsehood, although might be correct in what he says above, since I haven't had the heart to check.

Good luck to all students in your search for information, regardless!---CH 02:25, 1 July 2006 (UTC)

barnstar of sorts
Excellent article! I read it all the way through rather carefully today. (I'd give user:Hillman a barnstar on at user talk:Hillman but for the fact that he's gone into seclusion and blanked and protected his user- and talk-pages.) Michael Hardy 00:04, 29 January 2007 (UTC)

Appropriate contents
I deleted the section on "Enumerative combinatorics". It didn't have any content about Newton's identities. It was also vague on what it was about. I suggest that it might belong somewhere else, but only after improvement. Zaslav 21:46, 26 March 2007 (UTC)

The section "Relation with symmetric groups" is poorly written. How can a function of t s be defined without reference to quantities called t? Better explanation is needed. I am trying an improvement, keeping in mind that the contents should be related closely to Newton's identities. Zaslav 21:49, 26 March 2007 (UTC)
 * Changed my mind: I cannot fix this section, I have no idea what the purpose is.  Only the first few lines seem to have any connection with the symmetric group.  If anyone cares to, they should make sure the definitions are complete and the purpose is explained very carefully, and that the connection to Newton's identities is explained carefully.  I am going to delete most of it for now.  Zaslav 21:54, 26 March 2007 (UTC)

Computer science application
I'm not going to add this to the article myself because of my conflict of interest, but: Newton's identities are used to decode the data in a space-optimal streaming algorithm for maintaining sets of items subject to insertions and deletions of single items. The Bloom filter part is in a different algorithm for a similar problem, and is independent of the Newton identity part. —David Eppstein 15:21, 26 September 2007 (UTC)

What do Gröbner bases have to do with Newton's identities?
I am somewhat shocked by the prominent mention of Gröbner bases in this article. Why does there have to be a plug for them here? So they can be used to do computational problems in polynomial rings, granted. But Girard and Newton did not need to do that to find the identities of this article, and neither do we. One does not lecture on Gauss' elimination either on every occurrence of an inverse matrix. Can anybody justify what the mention of Gröbner bases adds to our understanding of these identities? If not, I am going to reduce their mention here to a very strict minimum. Marc van Leeuwen (talk) 08:58, 23 March 2008 (UTC)


 * You are right. This most likely has to do with the way mathematical articles on technical subjects are often written: someone (not an expert in the field) finds out an interesting topic X and starts an article/expands an existing article, including his motivation for being interested in X into the lead, even it is not at all the fundamental feature of X. Occasionally, such explanations are completely off the mark. I am fairly certain that this is what happened here: read the annotation for the Sturmfels' book under References. Just go ahead and correct. Arcfrk (talk) 05:30, 24 March 2008 (UTC)

Page is not suitable for general public
This page presents the subject in a semi-difficult way to understand. Efforts can be made so that non-Math PHD's understand what the heck it is that it's trying to say. Examples of usage will be of the greatest help. — Preceding unsigned comment added by 5.162.181.155 (talk) 08:01, 9 November 2013 (UTC)

I agree. I learnt these identities fifty years ago in a school text book. Recently I have found them in W.L.Ferrar's "Higher Algebra" (which is a sequel to "Higher Algebra for Schools"), first published in 1943 page 175.

Waring's formula?
In the subsection Expressing power sums in terms of elementary symmetric polynomials the formula expressing $p_{m}$ in terms of the $e_{i}$ is credited to Newton and Girard, but I have seen similar formulas referred to as "Waring's formula" in the mathematics literature. Both PlanetMath and MathWorld seem to do so. I don't know the history. Does anyone know what the story is? I noticed this when I saw a mention of "Waring's formula" in The Online Encyclopedia of Integer Sequences, and couldn't find anything about it in Wikipedia. Will Orrick (talk) 14:46, 9 June 2020 (UTC)


 * I don't know how closely 'Waring's formula' is related to 'Newton's (or Girard's) identities', but Waring, in math contexts, usually means Edward Waring, who came considerably after Newton, around the 1790s. So if they are essentially the same formulae, Waring would have no claim to priority, and would probably have known Newton's treatment.2A00:23C8:7906:1301:6810:70BD:8251:9EB1 (talk) 17:37, 17 October 2020 (UTC)
 * Thanks for your comment. For Wikipedia to be as widely useful as possible, readers need to be able to locate relevant articles by searching on commonly used names, even if those names are unjust, or historically inaccurate. So I guess my question is whether the term "Waring's formula" commonly refers to any of the formulas in this article and, if so, how that should be handled. Will Orrick (talk) 21:17, 17 October 2020 (UTC)

Simplifying the signs
The elementary symmetric polynomials $$e_k$$ appear in the context $$(-1)^k e_k$$. The substitution $$e_k=(-1)^k b_k$$ will simplify many formulas in the article.

$$ \prod_{k=1}^n (x-x_k) = \sum_{k=0}^n b_{n-k} x^k$$

Bo Jacoby (talk) 11:52, 12 September 2020 (UTC)