Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Locally polynomially integrable surfaces and finite stationary phase
  expansions   Let $M$ be a strictly convex smooth connected hypersurface in $\mathbb R^n$
and $\widehat{M}$ its convex hull. We say that $M$ is locally polynomially
integrable if the $(n-1)-$ dimensional volumes of the sections of $\widehat M$
by hyperplanes, sufficiently close to the tangent hyperplanes to $M,$ depend
polynomially on the distance of the hyperplanes to the origin. It is
conjectured that only quadrics in odd dimensional spaces possess such a
property. The main result of this article partially confirms the conjecture.
The study of integrable domains and surfaces is motivated by a conjecture of
V.I. Arnold about algebraically integrable domains. The result and the proof
are related to study oscillating integrals for which the asymptotic stationary
phase expansions consist of finite number of terms.

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The categories of this paper are: stat
32
The categories of this paper are: atom-ph
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn