Newton's identities relate power sums to what kind of polynomials?

Newton’s identities relate power sums to elementary symmetric polynomials. For variables x₁ through xₙ, a power sum is pₖ = x₁ᵏ + ··· + xₙᵏ, while an elementary symmetric polynomial adds products of distinct variables, such as x₁x₂ + x₁x₃ + x₂x₃.

The identities provide recursive formulas connecting these two families. For example, p₁ = e₁ and p₂ = e₁² − 2e₂. They therefore let mathematicians recover information about the coefficients of a polynomial from sums of powers of its roots, without solving for those roots individually.

Applied to a monic polynomial, the elementary symmetric polynomials correspond, with alternating signs, to its coefficients. The roots may be repeated, and the formulas still count them with their algebraic multiplicities.

The name is historical but not exclusive to Newton: Albert Girard obtained the first few related formulas in 1629. The identities now serve algebra, combinatorics, invariant theory, Galois theory, and matrix theory, where power sums of roots can be computed as traces of matrix powers.