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arXiv · 2601.14057

On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity

Sándor Z. Kiss·Csaba Sándor·Maciej Zakarczemny
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Abstract

We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many solutions. Furthermore, we consider the equality of the values of the $n$th and $(n-2)$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. In particular, we show that the number of solutions of this equation tends to infinity if $n$ tends to infinity.

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Sándor Z. Kiss, Csaba Sándor, Maciej Zakarczemny. 2026-01-20. On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity. https://arxiv.org/abs/2601.14057

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