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

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.