Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $i=-1,1,2$.

23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an