On algebraically integrable domains in Euclidean spaces

2017/05/17 by Mark Agranovsky, Agranovsky, Mark
Mathematics · #44A12 #51M99 #Algebraic and Geometric Analysis #Analytic and geometric function theory #FOS: Mathematics #Holomorphic and Operator Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1705.06063

openalex publication_date 2017/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let D be a bounded domain D in \mathbb Rⁿ with infinitely smooth boundary and n is odd. We prove that if the volume cut off from the domain by a hyperplane is an algebraic function of the hyperplane, free of real singular points, then the domain is an ellipsoid. This partially answers a question of V.I. Arnold: whether odd-dimensional ellipsoids are the only algebraically integrable domains?

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