Newton's identities

Newton's identities relate the power sums of a set of variables to their elementary symmetric polynomials, allowing either family to be computed from the other.

Power sum: The sum of the same positive integer power of each variable in a collection. These sums are one side of each identity.
Symmetric polynomial: A polynomial unchanged when its variables are permuted. Both families in Newton's identities are symmetric polynomials.
Faddeev–LeVerrier algorithm: A matrix algorithm that computes characteristic-polynomial coefficients from traces of matrix powers. It applies Newton's identities to obtain coefficients without finding eigenvalues.
Isaac Newton: An English mathematician and natural philosopher who lived from 1642 to 1727. The identities bear Newton's name and appeared in his work on equations.
Elementary symmetric polynomial: The sum of all products formed by choosing a fixed number of distinct variables. These polynomials are the other side of each identity.
Binomial coefficient: The number of ways to choose a fixed-size subset from a finite set. Equivalent identity conventions use binomial coefficients to normalize sums.
Matrix trace: The sum of the diagonal entries of a square matrix. Traces of powers equal power sums of eigenvalues, supplying the algorithm's inputs.
Girard–Newton identities: An alternate name for relations between power sums and elementary symmetric polynomials. The paired name recognizes Albert Girard's earlier contributions to these relations.
Polynomial roots: Values at which a polynomial evaluates to zero. The identities recover information about roots from their symmetric sums.
Polynomial coefficient: A scalar multiplying a power of the variable in a polynomial expression. Elementary symmetric polynomials appear as signed coefficients of a monic root polynomial.

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