Abstract
Let M be a strictly convex smooth connected hypersurface in ℝn and \(\widehat{M}\) its convex hull. We say that M is locally polynomially integrable if for every point a ∈ M the (n − 1)-dimensional volume of the cross-section of \(\widehat{M}\) by a parallel translation of the tangent hyperplane at a to a small distance t depends polynomially on t. 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 studying oscillating integrals for which the asymptotic stationary phase expansions consist of a finite number of terms.
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To Larry Zalcman, with friendship and appreciation
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Agranovsky, M. Locally polynomially integrable surfaces and finite stationary phase expansions. JAMA 141, 23–47 (2020). https://doi.org/10.1007/s11854-020-0124-5
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DOI: https://doi.org/10.1007/s11854-020-0124-5