Talk:Power sum symmetric polynomial

Newton's Identities

To replace a redirect to Newton's identities I am preparing a page on the power sum symmetric polynomials to go along with those on the elementary symmetric polynomials and the complete homogeneous symmetric polynomials as well as other types of bases of symmetric polynomials that knowledgeable WPians can prepare short pages about. The power sum polynomials are of interest for many reasons not closely related to their (aso important) connection with Newton's identities. It is not appropriate to slip them as a minor topic into Newton's identities. I ask anyone who disagrees to please think it over carefully and discuss the matter before cancelling this page. Thank you. Zaslav 23:52, 25 March 2007 (UTC)

Please overlook annoyed tone, which is the after-effect of a problem caused by some kind of stupid error of my own. Zaslav 20:57, 26 March 2007 (UTC)

I would be glad to see a complete proof that the p.s. polynomials form an algebraic basis for all symmetric polynomials over a field, written in here (similarly to elementary symmetric polynomials), to supplement the nice informal treatment at Newton's identities. Zaslav 01:08, 26 March 2007 (UTC)

Surely it only works over characteristic 0? E.g. in characteristic 2, x^2+y^2=(x+y)^2, so F[x+y,x^2+y^2]=F[x+y] with rank one while F[x+y,xy] has rank two. —David Eppstein 02:00, 28 March 2007 (UTC)

I don't doubt you're right. We need an expert to clean this up, but pending that, we do our best. Zaslav 18:55, 29 March 2007 (UTC)

Added your example to the article. Thanks. Zaslav 19:08, 29 March 2007 (UTC)

Working on this page

I'll probably do some work on this page soon. Currently the contents seems correct, but I can see several points to improve.

• There is too much stress on the fact that all symmetric polynomials can be expressed in terms of the first n power sums (if the base ring contains Q). In itself this is not extremely interesting; there are (many) other sequences of symmetric polynomials that do this, and even better (without requiring rational coefficients). The importance of the result is due in large part to the fact that many operations are computationally easier to describe on the basis of power sums than on other bases (for instance plethysm).
• The "sketch of a proof" could be simplified, and made to a complete proof both ways. Basically Newton's identities allow expression the first n power sums in the corresponding elementary symmetric polynomials, and vice versa if the rationals are available; this reduces the statement to the fundamental theorem of symmetric polynomials.
• It would seem good to say something about the failure of the theorem in finite characteristic: the Frobenius endomorphism makes any power sum whose degree is divisible by the characteristic algebraically dependent on previous ones, and consequently the sequence of power sums will also be missing elements in those degrees for generating all symmetric polynomials. In fact the mentioned algebraic dependence is extremely simple (given by a monomial), and this has arithmetic consequences for the coefficients of power sums expressed in elementary symmetric polynomials, but I'd like to see a reference discussing this before writing anything about it (to avoid WP:OR).
• Something should be said about $p_{0}$ which is unlike other power sums, and most other special kinds of symmetric polynomials, in that it depends on n in a way other than to control the number of monomials: its constant coefficient also depends on $n$; see symmetric function why this is relevant (many authors do not define $p_{0}$ at all, including those in the references of this article; in fact it would be nice to know if the definition of $p_{0}$, natural as it may seem, can be found in the literature at all).

I've already done quite a bit on related articles like symmetric polynomial, symmetric function and Newton's identities, so in part I will be picking up (or moving) stuff from there, trying to make things coherent (and not too redundant). Marc van Leeuwen (talk) 07:40, 20 April 2008 (UTC)

There is some algorithmic significance to these particular polynomials being able to recover the other symmetric polynomials (or the values from which the polynomials were derived) more than other sequences of polynomials, because power sums are easy to story and maintain: see my paper arXiv:0704.3313. Note also that this paper does not use fields that contain the rational numbers: for any finite subsequence of the power sum polynomials one can use an appropriate finite field.

However, I suspect the main significance of the power sum symmetric polynomials is statistical: they are the same as the moments of the data around zero, a fact that is neglected in the present article. So the fact that the power sum symmetric polynomials give complete information about their values is the same as the statistical fact that sufficiently many moments describe a data set accurately. —David Eppstein (talk) 15:12, 20 April 2008 (UTC)