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.
Categories: math.NT