2.3 Power sums and elementary symmetric polynomials

Let \(\alpha _1, \ldots , \alpha _N\) be a collection of real or complex numbers (possibly with repetitions). The power sums are defined by

\begin{equation} p_j = \sum _{i=1}^N \alpha _i^j, \quad j \geq 0. \end{equation}

The elementary symmetric polynomials are the coefficients appearing in the factorization

\begin{equation} \prod _{i=1}^N (t - \alpha _i) = t^N - e_1 t^{N-1} + e_2 t^{N-2} - \cdots + (-1)^N e_N, \end{equation}

where

\begin{equation} e_k = \sum _{1 \leq i_1 {\lt} \cdots {\lt} i_k \leq N} \alpha _{i_1} \cdots \alpha _{i_k}. \end{equation}

Theorem 2.3.1 Newton’s identities

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The power sums and elementary symmetric polynomials are related by:

\begin{equation} k \cdot e_k = \sum _{i=1}^k (-1)^{i-1} e_{k-i} p_i, \quad k = 1, 2, \ldots , N, \end{equation}

where we set \(e_0 = 1\).

(See ../docs/ChebyshevCircles/Proofs/NewtonIdentities.htmlAPI Documentation | ../paper/chebyshev_circles.pdfPaper Theorem 2.3)

An immediate consequence is that the elementary symmetric polynomials (and hence the coefficients of the monic polynomial with roots \(\alpha _1, \ldots , \alpha _N\)) are determined by the power sums.

Corollary 2.3.2

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Let \(\alpha _1, \ldots , \alpha _N\) and \(\beta _1, \ldots , \beta _N\) be two collections of numbers. If

\begin{equation} \sum _{i=1}^N \alpha _i^j = \sum _{i=1}^N \beta _i^j \quad \text{for all } j = 1, 2, \ldots , N, \end{equation}

then the monic polynomials with roots \(\{ \alpha _i\} \) and \(\{ \beta _i\} \) have identical coefficients of degree \(k \geq 1\). They may differ in the constant term if \(\sum \alpha _i \neq \sum \beta _i\) (i.e., if \(p_0\) differs).

Proof ▶

By Theorem 2.3.1, if \(p_j(\alpha ) = p_j(\beta )\) for \(j = 1, \ldots , N\), then \(e_k(\alpha ) = e_k(\beta )\) for \(k = 1, \ldots , N\). By Vieta’s formulas, the coefficient of \(x^{N-k}\) in the monic polynomial \(\prod (x - \alpha _i)\) is \((-1)^k e_k\). Thus the coefficients match for \(k \geq 1\). The constant term \((-1)^N e_N\) involves the product of all roots, which may differ.