A note on Diophantine systems involving three symmetric polynomials

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 $i=-1,1,2$.Comment: to appear in J. Number Theor

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.Comment: to appear in Math. Com