No topological condition implies equality of polynomial and rational\n hulls

2018/06/15 by Alexander J. Izzo, Izzo, Alexander J.
Mathematics · #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1806.06160

openalex publication_date 2018/06/15 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

It is shown that no purely topological condition implies the equality of the\npolynomial and rational hulls of a set: For any compact subset K of a\nEuclidean space, there exists a set X, in some mathbb C^N, that is\nhomeomorphic to K and is rationally convex but not polynomially convex. In\naddition, it is shown that for the surfaces in mathbb C³ constructed by\nIzzo and Stout, whose polynomial hulls are nontrivial but contain no analytic\ndiscs, the polynomial and rational hulls coincide, thereby answering a question\nof Gupta. Equality of polynomial and rational hulls is shown also for\nm-dimensional manifolds (m\≥ 2) with polynomial hulls containing no\nanalytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and\nWold.\n

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