Power Sum Symmetric Polynomial - Properties

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 The same is true if the coefficients are taken in any field whose characteristic is 0.

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

has the expression

which involves fractions. According to the theorem this is the only way to represent in terms of p₁ and p₂. Therefore, P does not belong to the integral polynomial ring 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,

The theorem is also untrue if the field has characteristic different from 0. For example, if the field F has characteristic 2, then, so p₁ and p₂ cannot generate e₂ = x₁x₂.

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 (see Newton's identities):

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

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 of the power sum symmetric polynomials p₁, ..., p_n. That is, the ring of symmetric polynomials is contained in the ring generated by the power sums, 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.