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Locally polynomially integrable surfaces and finite stationary phase expansions
Journal d'Analyse Mathématique ( IF 1 ) Pub Date : 2020-09-01 , DOI: 10.1007/s11854-020-0124-5
Mark Agranovsky

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.

中文翻译:

局部多项式可积曲面和有限固定相扩展

令 $M$ 是 $\mathbb R^n$ 和 $\widehat{M}$ 其凸包中的严格凸光滑连通超曲面。我们说 $M$ 是局部多项式可积的,如果 $\widehat M$ 的截面的 $(n-1)-$ 维体积通过超平面,足够接近到 $M 的切超平面,$ 多项式依赖于距离超平面到原点。据推测,只有奇维空间中的二次曲线才具有这样的性质。本文的主要结果部分证实了这一猜想。可积域和曲面的研究受到 VI Arnold 关于代数可积域的猜想的启发。结果和证明与研究振荡积分有关,其中渐近固定相扩展由有限项组成。
更新日期:2020-09-01
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