# Power Sum Polynomials

April 24, 2020

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### Abstract

This article provides a formalisation of the symmetric multivariate polynomials known as power sum polynomials. These are of the form pn(X1,…, Xk) = X1n + … + Xkn. A formal proof of the Girard–Newton Theorem is also given. This theorem relates the power sum polynomials to the elementary symmetric polynomials sk in the form of a recurrence relation (-1)k k sk = ∑i∈[0,k) (-1)i si pk-i .

As an application, this is then used to solve a generalised form of a puzzle given as an exercise in Dummit and Foote's Abstract Algebra: For k complex unknowns x1, …, xk, define pj := x1j + … + xkj. Then for each vector a ∈ ℂk, show that there is exactly one solution to the system p1 = a1, …, pk = ak up to permutation of the xi and determine the value of pi for i>k.