A collection of equalities connecting the power sums of the roots of a polynomial (s_k = sum r_i^k) with the polynomial's coefficients expressed as elementary symmetric polynomials; they give recursive formulae between power sums and elementary symmetric functions.

Principle

Symmetric polynomials in the roots can be expressed in two natural bases — power sums and elementary symmetric polynomials — and Newton's identities provide explicit linear relations (with combinatorial coefficients) that convert between these bases.

Demonstration

For a quadratic polynomial x^2 − sx + p with roots r1, r2, the identities give s_1 = r1 + r2 = s and s_2 = r1^2 + r2^2 = s^2 − 2p; these express power sums s_k in terms of coefficients s and p or vice versa.

Misapplication

Dividing by integers k appearing in the relations without checking the ground ring's characteristic leads to invalid steps in fields of characteristic dividing k; also misusing identities on infinite root sets without convergence considerations is incorrect.

Consequence

Enable computation of polynomial coefficients from power sums and the recovery of symmetric invariants from moment-like data; they are fundamental in elimination theory, computation of Newton sums, and relations among symmetric functions.

Reversal

The inverse conversion — expressing power sums in terms of elementary symmetric polynomials — is given by the same family of recursive relations applied in the other direction; neither direction is inherently primary but both are interdependent.

Boundary

Valid for polynomials over rings where the combinatorial integer coefficients make sense; over rings whose characteristic divides some integer coefficients (e.g., k) special care or modified statements are required, and analytic issues arise for infinite multisets of roots.

Semantic Tension

Related to Viète's formulas, which give direct expressions of coefficients as elementary symmetric polynomials of roots; Newton's identities bridge Viète's elementary symmetric perspective with power-sum (moment) information, sometimes causing confusion about which representation is more natural.

Synthesis

Newton's Identities are the algebraic bridge between power sums and elementary symmetric polynomials: they provide recursive formulae that translate moment-like root data into coefficient data and vice versa, subject to arithmetic constraints of the base ring.