There are no algebraically integrable ovals in even-dimensional spaces

2014/07/27 by V. A. Vassiliev, Vassiliev, Victor A.
Mathematics · #14D05 #20F55 #44A99 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1407.7221

openalex publication_date 2014/07/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that there are no bounded domains with smooth boundaries in even-dimensional Euclidean spaces, such that the volumes cut off from them by affine hyperplanes depend algebraically on these hyperplanes. For convex ovals in R², this is Lemma XXVIII from Newton's "Principia".

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