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\title{Power Sum Polynomials\\ and the Girard--Newton Theorem}
\author{Manuel Eberl}
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\begin{abstract}
This article provides a formalisation of the symmetric multivariate polynomials known as \emph{power sum polynomials}. These are of the form $p_n(X_1,\ldots, X_k) = X_1 ^ n + \ldots + X_k ^ n$. A formal proof of the Girard--Newton Theorem is also given. This theorem relates the power sum polynomials to the elementary symmetric polynomials $s_k$ in the form of a recurrence relation $(-1)^k k s_k = \sum_{i=0}^{k-1} (-1)^i s_i p_{k-i}$\ .

As an application, this is then used to solve a generalised form of a puzzle given as an exercise in Dummit and Foote's \emph{Abstract Algebra}: For $k$ complex unknowns $x_1, \ldots, x_k$, define $p_j := x_1^j + \ldots + x_k^j$. Then for each vector $a\in\mathbb{C}^k$, show that there is exactly one solution to the system $p_1 = a_1, \ldots, p_k = a_k$ up to permutation of the $x_i$ and determine the value of $p_i$ for $i>k$.
\end{abstract}

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