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.
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.
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.
Now one may change the point of view, by taking the roots rather than the coefficients as basic parameters for describing P, and considering them as indeterminates rather than as constants in an appropriate field; the coefficients ai then become just the particular symmetric polynomials given by the above equations.
An example of such relations are Newton's identities, which express the sum of any fixed power of the roots in terms of the elementary symmetric polynomials.
Powers and products of elementary symmetric polynomials work out to rather complicated expressions.
If one seeks basic additive building blocks for symmetric polynomials, a more natural choice is to take those symmetric polynomials that contain only one type of monomial, with only those copies required to obtain symmetry.
To do this it suffices to separate the different types of monomial occurring in P. In particular if P has integer coefficients, then so will the linear combination.
The Newton identities provide an explicit method to do this; it involves division by integers up to n, which explains the rational coefficients.
More precisely: For example, for n = 2, the relevant complete homogeneous symmetric polynomials are h1(X1, X2) = X1 + X2 and h2(X1, X2) = X12 + X1X2 + X22.
The first polynomial in the list of examples above can then be written as As in the case of power sums, the given statement applies in particular to the complete homogeneous symmetric polynomials beyond hn(X1, ..., Xn), allowing them to be expressed in terms of the ones up to that point; again the resulting identities become invalid when the number of variables is increased.
An important aspect of complete homogeneous symmetric polynomials is their relation to elementary symmetric polynomials, which can be expressed as the identities Since e0(X1, ..., Xn) and h0(X1, ..., Xn) are both equal to 1, one can isolate either the first or the last term of these summations; the former gives a set of equations that allows one to recursively express the successive complete homogeneous symmetric polynomials in terms of the elementary symmetric polynomials, and the latter gives a set of equations that allows doing the inverse.
They are however not as easy to describe as the other kinds of special symmetric polynomials; see the main article for details.
They are also important in combinatorics, where they are mostly studied through the ring of symmetric functions, which avoids having to carry around a fixed number of variables all the time.