Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Stable Equivalence and Cancellation Problems   Let $K$ be an arbitrary field of characteristic 0, and $\Aff^n$ the
$n$-dimensional affine space over $K$. A well-known cancellation problem asks,
given two algebraic varieties $V_1, V_2 \subseteq \Aff^n$ with isomorphic
cylinders $V_1 \times \Aff^1$ and $V_2 \times \Aff^1$, whether $V_1$ and $V_2$
themselves are isomorphic.
  In this paper, we focus on a related problem: given two varieties with
equivalent (under an automorphism of $\Aff^{n+1}$) cylinders $V_1 \times
\Aff^1$ and $V_2 \times \Aff^1$, are $V_1$ and $V_2$ equivalent under an
automorphism of $\Aff^n$? We call this stable equivalence problem. We show that
the answer is positive for any two curves $V_1, V_2 \subseteq \Aff^2$.
  For an arbitrary $n \ge 2$, we consider a special, arguably the most
important, case of both problems, where one of the varieties is a hyperplane.
We show that a positive solution of the stable equivalence problem in this case
implies a positive solution of the cancellation problem.

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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio