In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a symmetric polynomial if for any permutation of the subscripts one has .
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every symmetric polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial.
Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomials play important roles alongside the elementary ones. The resulting structures, and in particular the ring of symmetric functions, are of great importance in combinatorics and in representation theory.
Examples
The following polynomials in two variables X1 and X2 are symmetric:
as is the following polynomial in three variables X1, X2, X3:
There are many ways to make specific symmetric polynomials in any number of variables (see the various types below). An example of a somewhat different flavor is
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant).
On the other hand, the polynomial in two variables
is not symmetric, since if one exchanges and one gets a different polynomial, . Similarly in three variables
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric:
Applications
Galois theory
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field. These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots. Moreover the fundamental theorem of symmetric polynomials implies that a polynomial function f of the n roots can be expressed as (another) polynomial function of the coefficients of the polynomial determined by the roots if and only if f is given by a symmetric polynomial.
This yields the approach to solving polynomial equations by inverting this map, "breaking" the symmetry – given the coefficients of the polynomial (the elementary symmetric polynomials in the roots), how can one recover the roots? This leads to studying solutions of polynomials using the permutation group of the roots, originally in the form of Lagrange resolvents, later developed in Galois theory.
Relation with the roots of a monic univariate polynomial
Consider a monic polynomial in t of degree n
with coefficients ai in some field K. There exist n roots x1,…,xn of P in some possibly larger field (for instance if K is the field of real numbers, the roots will exist in the field of complex numbers); some of the roots might be equal, but the fact that one has all roots is expressed by the relation
By comparing coefficients one finds that
These are in fact just instances of Vieta's formulas. They show that all coefficients of the polynomial are given in terms of the roots by a symmetric polynomial expression: although for a given polynomial P there may be qualitative differences between the roots (like lying in the base field K or not, being simple or multiple roots), none of this affects the way the roots occur in these expressions.