# Power sum symmetric polynomial

In mathematics, specifically in commutative algebra, the **power sum symmetric polynomials** are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the *rationals,* but not over the *integers.*

## Definition

The power sum symmetric polynomial of degree *k* in [math]\displaystyle{ n }[/math] variables *x*_{1}, ..., *x*_{n}, written *p*_{k} for *k* = 0, 1, 2, ..., is the sum of all *k*th powers of the variables. Formally,

- [math]\displaystyle{ p_k (x_1, x_2, \dots,x_n) = \sum_{i=1}^n x_i^k \, . }[/math]

The first few of these polynomials are

- [math]\displaystyle{ p_0 (x_1, x_2, \dots,x_n) = 1 + 1 + \cdots + 1 = n \, , }[/math]
- [math]\displaystyle{ p_1 (x_1, x_2, \dots,x_n) = x_1 + x_2 + \cdots + x_n \, , }[/math]
- [math]\displaystyle{ p_2 (x_1, x_2, \dots,x_n) = x_1^2 + x_2^2 + \cdots + x_n^2 \, , }[/math]
- [math]\displaystyle{ p_3 (x_1, x_2, \dots,x_n) = x_1^3 + x_2^3 + \cdots + x_n^3 \, . }[/math]

Thus, for each nonnegative integer [math]\displaystyle{ k }[/math], there exists exactly one power sum symmetric polynomial of degree [math]\displaystyle{ k }[/math] in [math]\displaystyle{ n }[/math] variables.

The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring.

## Examples

The following lists the [math]\displaystyle{ n }[/math] power sum symmetric polynomials of positive degrees up to *n* for the first three positive values of [math]\displaystyle{ n. }[/math] In every case, [math]\displaystyle{ p_0 = n }[/math] is one of the polynomials. The list goes up to degree *n* because the power sum symmetric polynomials of degrees 1 to *n* are basic in the sense of the theorem stated below.

For *n* = 1:

- [math]\displaystyle{ p_1 = x_1\,. }[/math]

For *n* = 2:

- [math]\displaystyle{ p_1 = x_1 + x_2\,, }[/math]
- [math]\displaystyle{ p_2 = x_1^2 + x_2^2\,. }[/math]

For *n* = 3:

- [math]\displaystyle{ p_1 = x_1 + x_2 + x_3\,, }[/math]
- [math]\displaystyle{ p_2 = x_1^2 + x_2^2 + x_3^2\,, }[/math]
- [math]\displaystyle{ p_3 = x_1^3+x_2^3+x_3^3\,, }[/math]

## Properties

The set of power sum symmetric polynomials of degrees 1, 2, ..., *n* in *n* variables generates the ring of symmetric polynomials in *n* variables. More specifically:

**Theorem**. The ring of symmetric polynomials with rational coefficients equals the rational polynomial ring [math]\displaystyle{ \mathbb Q[p_1,\ldots,p_n]. }[/math] The same is true if the coefficients are taken in any field of characteristic 0.

However, this is not true if the coefficients must be integers. For example, for *n* = 2, the symmetric polynomial

- [math]\displaystyle{ P(x_1,x_2) = x_1^2 x_2 + x_1 x_2^2 + x_1x_2 }[/math]

has the expression

- [math]\displaystyle{ P(x_1,x_2) = \frac{p_1^3-p_1p_2}{2} + \frac{p_1^2-p_2}{2} \,, }[/math]

which involves fractions. According to the theorem this is the only way to represent [math]\displaystyle{ P(x_1,x_2) }[/math] in terms of *p*_{1} and *p*_{2}. Therefore, *P* does not belong to the integral polynomial ring [math]\displaystyle{ \mathbb Z[p_1,\ldots,p_n]. }[/math]
For another example, the elementary symmetric polynomials *e*_{k}, expressed as polynomials in the power sum polynomials, do not all have integral coefficients. For instance,

- [math]\displaystyle{ e_2 := \sum_{1 \leq i\lt j \leq n} x_ix_j = \frac{p_1^2-p_2}{2} \, . }[/math]

The theorem is also untrue if the field has characteristic different from 0. For example, if the field *F* has characteristic 2, then [math]\displaystyle{ p_2 = p_1^2 }[/math], so *p*_{1} and *p*_{2} cannot generate *e*_{2} = *x*_{1}*x*_{2}.

*Sketch of a partial proof of the theorem*: By Newton's identities the power sums are functions of the elementary symmetric polynomials; this is implied by the following recurrence relation, though the explicit function that gives the power sums in terms of the *e*_{j} is complicated:

- [math]\displaystyle{ p_n = \sum_{j=1}^n (-1)^{j-1} e_j p_{n-j} \,. }[/math]

Rewriting the same recurrence, one has the elementary symmetric polynomials in terms of the power sums (also implicitly, the explicit formula being complicated):

- [math]\displaystyle{ e_n = \frac{1}{n} \sum_{j=1}^n (-1)^{j-1} e_{n-j} p_j \,. }[/math]

This implies that the elementary polynomials are rational, though not integral, linear combinations of the power sum polynomials of degrees 1, ..., *n*.
Since the elementary symmetric polynomials are an algebraic basis for all symmetric polynomials with coefficients in a field, it follows that every symmetric polynomial in *n* variables is a polynomial function [math]\displaystyle{ f(p_1,\ldots,p_n) }[/math] of the power sum symmetric polynomials *p*_{1}, ..., *p*_{n}. That is, the ring of symmetric polynomials is contained in the ring generated by the power sums, [math]\displaystyle{ \mathbb Q[p_1,\ldots,p_n]. }[/math] Because every power sum polynomial is symmetric, the two rings are equal.

(This does not show how to prove the polynomial *f* is unique.)

For another system of symmetric polynomials with similar properties see complete homogeneous symmetric polynomials.

## References

- Ian G. Macdonald (1979),
*Symmetric Functions and Hall Polynomials*. Oxford Mathematical Monographs. Oxford: Clarendon Press. - Ian G. Macdonald (1995),
*Symmetric Functions and Hall Polynomials*, second ed. Oxford: Clarendon Press. ISBN:0-19-850450-0 (paperback, 1998). - Richard P. Stanley (1999),
*Enumerative Combinatorics*, Vol. 2. Cambridge: Cambridge University Press. ISBN:0-521-56069-1

## See also

Original source: https://en.wikipedia.org/wiki/Power sum symmetric polynomial.
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