A note on Diophantine systems involving three symmetric polynomials

TL;DR

This paper proves the existence of infinitely many integer triples satisfying certain symmetric polynomial Diophantine systems and extends recent results, also exploring systems involving sums of powers with specific equalities.

Contribution

It demonstrates the existence of infinitely many solutions to symmetric polynomial Diophantine systems and extends prior work by Zhang, Cai, and the author.

Findings

Infinitely many rational solutions exist for the specified systems.

Existence of at least k n-tuples with equal sums of powers for certain exponents.

Similar results established for different sets of power exponents.

Abstract

Let $\bar{X}_{n}=(x_{1},\ldots,x_{n})$ and $\sigma_{i}(\bar{X}_{n})=\sum x_{k_{1}}\ldots x_{k_{i}}$ be $i$-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers $a, b, c$ such that for each $1\leq i\leq n$ the system of Diophantine equations \begin{equation*} \sigma_{i}(\bar{X}_{2n})=a, \quad \sigma_{2n-i}(\bar{X}_{2n})=b, \quad \sigma_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each $k$ there are at least $k$ $n$-tuples of integers with the same sum of $i$-th powers for $i=1,2,3$. Similar result is proved for $i=1,2,4$ and $i=-1,1,2$.