##  [Newton's Identities](/newtons-identities-0) 

 Definition

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

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

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

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

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

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

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

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

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.