In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
Mathematical statement
Formulation in terms of symmetric polynomials Let x1, ..., xn be variables, denote for k ≥ 1 by pk(x1, ..., xn) the k-th power sum:
p k ( x 1 , … , x n ) = ∑ i = 1 n x i k = x 1 k + ⋯ + x n k , {\displaystyle p_{k}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}^{k}=x_{1}^{k}+\cdots +x_{n}^{k},}
and for k ≥ 0 denote by ek(x1, ..., xn) the elementary symmetric polynomial (that is, the sum of all distinct products of k distinct variables), so
… excerpt ends here. Continue reading the full article.
