Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rational points on certain families of symmetric equations   We generalize the work of Dem'janenko and Silverman for the Fermat quartics,
effectively determining the rational points on the curves
$x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic
Jacobians are at most one. As an application, we explicitly describe
$X_d(\mathbb{Q})$ for certain $d\geq3$, where $X_d: T_d(x)+T_d(y)=1$ and $T_d$
is the monic Chebychev polynomial of degree $d$. Moreover, we show how this
later problem relates to orbit intersection problems in dynamics. Finally, we
construct a new family of genus $3$ curves which break the Hasse principle,
assuming the parity conjecture, by specifying our results to quadratic twists
of $x^4-4x^2-4y^2+y^4=-6$.

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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph