Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Product Ranks of the $3\times 3$ Determinant and Permanent   We show that the product rank of the $3\times 3$ determinant is $5$, and the
product rank of the $3\times 3$ permanent is $4$. As a corollary, we obtain
that the tensor ranks of the $3 \times 3$ determinant and permanent are $5$ and
$4$, respectively. We show moreover that the border product rank of the $n
\times n$ permanent is larger than $n$ for any $n\geq 3$.

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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg