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

TL;DR

This paper investigates when elementary symmetric polynomials of positive integers are equal, proving existence and finiteness of solutions for certain cases and showing infinite solutions as the number of variables grows.

Contribution

It establishes new results on the existence, finiteness, and infinitude of solutions to symmetric polynomial equalities involving positive integers.

Findings

Solutions exist for the equality of the nth and kth elementary symmetric polynomials when k < n.

The number of solutions for the equality of the nth and (n-2)th elementary symmetric polynomials tends to infinity as n increases.

Finitely many solutions exist for the equality of the nth and kth elementary symmetric polynomials with k < n.

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.