Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Extremal Positive Semidefinite Forms of Cubic Homogeneous Polynomials
  of Three Variables   Let $\mathcal{H}_{3,d}$ be the vector space of homogeneous three variable
polynomials of degree $d$, and $\mathcal{P}_{3,d}^+$ be the set of all elements
$f \in \mathcal{H}_{3,d}$ such that$f(x,y,z) \geq 0$ for all $x \geq 0$, $y
\geq 0$, $z \geq 0$. In this article, we determine all extremal elements of
$\mathcal{P}_{3,3}^+$. We prove that if $f \in \mathcal{P}_{3,3}^+$ is an
irreducible extremal element, then the zero locus $V_{\mathbb{C}}(f)$ in
$\mathbb{P}_{\mathbb{C}}^2$ is a rational curve whose singularity is an acnode
in the interior of $\mathbb{P}_+^2$ or a cusp on an edge of $\mathbb{P}_+^2$.
We also prove that if $f \in \mathcal{P}_{3,3}^+$ is an extremal element, then
$f(x^2,y^2,z^2)$ is an extremal element of $\mathcal{P}_{3,6}$, where
$\mathcal{P}_{3,d}$ is the set of all the elements $f \in \mathcal{H}_{3,d}$
such that $f(x,y,z) \geq 0$ for all $x$, $y$, $z \in \mathbb{R}$.

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The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
12
The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn