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- W4299441517 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 $1leq ileq n$ 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 $i=-1,1,2$." @default.
- W4299441517 created "2022-10-02" @default.
- W4299441517 creator A5005743799 @default.
- W4299441517 date "2013-05-27" @default.
- W4299441517 modified "2023-10-18" @default.
- W4299441517 title "A note on Diophantine systems involving three symmetric polynomials" @default.
- W4299441517 doi "https://doi.org/10.48550/arxiv.1305.6241" @default.
- W4299441517 hasPublicationYear "2013" @default.
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