Newton's identities for power sums

Remark — Computing power sums: Newton's identities

Having defined $p_k = r_1^k+r_2^k+\cdots+r_n^k$ (the sum of the $k$-th powers of the roots), Newton’s recurrences hold: $p_1 = \sigma_1, \quad p_2 = \sigma_1 p_1 - 2\sigma_2, \quad p_3 = \sigma_1 p_2 - \sigma_2 p_1 + 3\sigma_3,\ \ldots$ They allow one to compute $\sum r_i^k$ without knowing the roots: extremely useful in olympiad problems.

Newton’s identities connect the power sums $p_k$ to the elementary symmetric functions $\sigma_k$ (the same as in Vieta’s relations). For the historical origin of these recurrences, as a bridge between seventeenth-century symbolic algebra and the modern theory of symmetric functions, see Stillwell.

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Topics: Complex numbers
Concepts: Newton’s identities · Vieta’s relations
Skills: Using formulae
People: Isaac Newton