# Symmetric polynomial

A polynomial $f$ with coefficients in a field or a commutative associative ring $K$ with a unit, which is a symmetric function in its variables, that is, is invariant under all permutations of the variables:

$\label{symm} f(x_1,\ldots,x_n) = f(\pi(x_1),\ldots,\pi(x_n)).$

The symmetric polynomials form the algebra $S(x_1,\ldots,x_n)$ over $K$.

The most important examples of symmetric polynomials are the elementary symmetric polynomials

$s_k(x_1,\ldots,x_n) = \sum_{1 \leq i_1 < \ldots < i_k \leq n} x_{i_1} \ldots x_{i_k}$

and the power sums

$p_k(x_1,\ldots,x_n) = x_1^k + \ldots + x_n^k.$

The latter can be expressed in terms of elementary symmetric polynomials by recurrence formulas, called Newton's formulas:

\begin{aligned} p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots + (-1)^{k-1} p_1 s_{k-1}+ (-1)^{k} k s_{k} &= 0 \quad &\text{if }1 \leq k \leq n, \\ p_k - p_{k-1} s_1 + p_{k-2} s_2 + \ldots + (-1)^{n-1} p_{k-n+1} s_{n-1}+ (-1)^{n} p_{k-n} s_{n} &= 0 \quad &\text{if }k > n. \end{aligned}

For the elementary symmetric polynomials $s_1,\ldots,s_k$ ($1 \leq k \leq n$) of the roots of an arbitrary polynomial in one variable with leading coefficient 1, $x^n+\alpha_1 x^{n-1}+ \ldots + \alpha_n$, one has $\alpha_k=(-1)^k s_k$ (see Viète theorem).

The fundamental theorem on symmetric polynomials: Every symmetric polynomial is a polynomial in the elementary symmetric polynomials, and this representation is unique. In other words, the elementary symmetric polynomials are a set of free generators for the algebra $S(x_1,\ldots,x_n)$. If the field has characteristic 0, then the polynomials $p_1,\ldots,p_n$ also form a set of free generators of this algebra.

A skew-symmetric, or alternating, polynomial is a polynomial $f(x_1,\ldots,x_n)$ satisfying the relation \ref{symm} if $\pi$ is even and the relation

$f(x_1,\ldots,x_n) = -f(\pi(x_1),\ldots,\pi(x_n))$

if $\pi$ is odd. Any skew-symmetric polynomial can be written [Pr, Thm. 3.1.2] in the form $\Delta_n g$, where $g$ is a symmetric polynomial and

$\Delta_n = \prod_{i<j} (x_i-x_j).$

This representation is not unique, in view of the relation $\Delta_n^2=\operatorname{Dis}(s_1,\ldots,s_n)$.

How to Cite This Entry:
Symmetric polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_polynomial&oldid=39117
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article