Newton's identities

In mathematics, specifically algebra, the Newton identities combine two fundamental types of symmetric polynomials in n number of variables, the elementary symmetric polynomials

And the power sums

These identities are generally attributed to considerations of Isaac Newton in 1666, but they can be found even at Albert Girard in 1629. Applications of these identities can be found in the Galois theory, invariant theory, group theory, combinatorics, but also outside mathematics, for example, in general relativity theory.

Derivation by means of formal power series

Let T be the variable in the ring of formal power series. Then, analogous to the set group of Vieta,

Since the polynomial p ( t ) has a constant coefficient 1, it is invertible in the ring of formal power series. For the logarithmic derivative is obtained

The quotient on the right side also exist as a formal power series, they arise as a geometric series. Thus applies

This can then be converted to

By comparison of the same magnitude of a T on both sides of the equation system for determining the results of the elementary symmetric polynomial power series and vice versa,

These relationships can be run by the Division of formal power series in p '( T) / p ​​( t ) to solve for the power sums, it is

Conversely, that the quotient of derivation and function, the derivative of the logarithm is thus valid for integration and exponentieren, resulting in the following relationships by comparing coefficients.