On some Diophantine systems involving symmetric polynomials

Let $\sigma_{i}(x_{1},\ldots, x_{n})=\sum_{1\leq k_{1}<k_{2}<\ldots <k_{i}\leq n}x_{k_{1}}\ldots x_{k_{i}}$ be the $i$-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai \cite{ZC,ZC2}. More precisely, we prove that for each $n\geq 4$ and rational numbers $a, b$ with $ab\neq 0$, the system of diophantine equations \begin{equation*} \sigma_{1}(x_{1},\ldots, x_{n})=a, \quad \sigma_{n}(x_{1},\ldots, x_{n})=b, \end{equation*} has infinitely many solutions depending on $n-3$ free parameters. A similar result is proved for the system \begin{equation*} \sigma_{i}(x_{1},\ldots, x_{n})=a, \quad \sigma_{n}(x_{1},\ldots, x_{n})=b, \end{equation*} with $n\geq 4$ and $2\leq i< n$. Here, $a, b$ are rational numbers with $b\neq 0$. We also give some results concerning the general system of the form \begin{equation*} \sigma_{i}(x_{1},\ldots, x_{n})=a, \quad \sigma_{j}(x_{1},\ldots, x_{n})=b, \end{equation*} with suitably chosen rational values of $a, b$ and $i<j<n$. Finally, we present some remarks on the systems involving three different symmetric polynomials.