## On some Diophantine systems involving symmetric polynomials

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- by Maciej Ulas PDF
- Math. Comp.
**83**(2014), 1915-1930 Request permission

## Abstract:

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. 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.

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## Additional Information

**Maciej Ulas**- Affiliation: Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics, Łojasiewicza 6, 30 - 348 Kraków, Poland
- Email: maciej.ulas@uj.edu.pl
- Received by editor(s): August 27, 2012
- Received by editor(s) in revised form: November 12, 2012, and December 17, 2012
- Published electronically: October 22, 2013
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 1915-1930 - MSC (2010): Primary 11D25, 11G05
- DOI: https://doi.org/10.1090/S0025-5718-2013-02778-0
- MathSciNet review: 3194135