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

TL;DR

This paper investigates the Diophantine equations $\sigma_k(x_1,\dots,x_n)=\sigma_n(x_1,\dots,x_n)$ for positive integers, showing that for every fixed $n$ and $1\le k<n$ there exists at least one solution, but only finitely many in number; in the notable case $k=n-2$, the authors prove that the number of solutions grows without bound as $n$ increases. The authors connect the problem to unit-fraction (Egyptian fraction) decompositions, Sylvester-type sequences, and majorization techniques to derive both upper and lower bounds on the count of solutions, introducing sequences $u_n$ and $v_n$ with explicit recurrences that control the size of components. They provide detailed analyses for the four key subcases $k=1,2,n-1,n-2$, yielding precise growth and bound results and constructing infinite families in certain regimes. The work links elementary symmetric polynomials with classical Diophantine and Egyptian-fraction problems, offering new bounds and growth phenomena that illuminate the structure of solutions to these symmetric-sum equations.

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.

Theorems & Definitions (32)

• Theorem 1
• Lemma 2
• Theorem 3
• Conjecture 4
• Remark 5
• Remark 6
• Definition 7
• Theorem 8
• Theorem 9
• Theorem 10