Newton's identities

In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.

Let x₁, …, x_n be variables, denote for k ≥ 1 by p_k(x₁, …, x_n) the k-th power sum:

and for k ≥ 0 denote by e_k(x₁, …, x_n) the elementary symmetric polynomial (that is, the sum of all distinct products of k distinct variables), so

Then Newton's identities can be stated as

valid for all n ≥ 1 and k ≥ 1.

Also, one has

for all k > n ≥ 1.

Concretely, one gets for the first few values of k:

The form and validity of these equations do not depend on the number n of variables (although the point where the left-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities in the ring of symmetric functions. In that ring one has

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