Newton's identities

Statement

Let $R$ be a commutative ring and let $n \ge 1$. Let $e_k$ be the elementary symmetric polynomials and $p_k$ the power sums in $x_1, \dots, x_n$, with $e_0 = 1$ and $e_k = 0$ for $k > n$. Then for every $k \ge 1$, $$k \, e_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i} \, p_i.$$

Remarks

The first cases are $e_1 = p_1$, $2 e_2 = e_1 p_1 - p_2$ and $3 e_3 = e_2 p_1 - e_1 p_2 + p_3$. For $k > n$, the left side is $0$, and the identity expresses $p_k$ through $p_{k-1}, \dots, p_{k-n}$. Applied to the roots of a polynomial, the identities compute the power sums of the roots from the coefficients. Applied to the eigenvalues of a matrix $A$, they relate the traces of the powers of $A$ to the coefficients of its characteristic polynomial.

History

Isaac Newton found the identities around 1666. Albert Girard had already found the first four formulas in 1629.

Explanations

L1 human

Newton's identities convert between two ways of describing the roots of a polynomial: the sums of their powers, and the coefficients. Each identity gives the next quantity from the earlier ones.