Newton-Girard Formulas

The Newton-Girard formulas are identities relating the elementary symmetric polynomials to the power sums

(1)

The identities are given by

(2)

for each , with . The first few identities, taking for , are

			(3)
			(4)
			(5)
			(6)

If , ..., are the roots, counted with multiplicity, of the monic polynomial

(7)

then the higher power sums satisfy the recurrence relation

(8)

for . This follows by multiplying by and summing over .

For example, for the roots of , repeated use of these identities gives

(9)

Expressions for higher power sums in terms of coefficients can also be obtained by polynomial reduction modulo the relations in Vieta's formulas (Trott 2006).

Conversely, for a fixed positive integer , the first power sums determine , ..., recursively. They therefore determine a unique monic polynomial of polynomial degree and its multiset of roots in , counted with multiplicity.