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No topological condition implies equality of polynomial and rational hulls


Author: Alexander J. Izzo
Journal: Proc. Amer. Math. Soc. 147 (2019), 5195-5207
MSC (2010): Primary 32E20, 32A65, 46J10, 46J15
DOI: https://doi.org/10.1090/proc/14628
Published electronically: July 10, 2019
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Abstract: It is shown that no purely topological condition implies the equality of the polynomial and rational hulls of a set: For any uncountable, compact subset $ K$ of a Euclidean space, there exists a set $ X$, in some $ \mathbb{C}^N$, that is homeomorphic to $ K$ and is rationally convex but not polynomially convex. In addition, it is shown that for the surfaces in $ \mathbb{C}^3$ constructed by Izzo and Stout, whose polynomial hulls are nontrivial but contain no analytic discs, the polynomial and rational hulls coincide, thereby answering a question of Gupta. Equality of polynomial and rational hulls is shown also for $ m$-dimensional manifolds ($ m\geq 2$) with polynomial hulls containing no analytic discs constructed by Izzo, Samuelsson Kalm, and Wold and by Arosio and Wold.


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Additional Information

Alexander J. Izzo
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email: aizzo@bgsu.edu

DOI: https://doi.org/10.1090/proc/14628
Received by editor(s): July 18, 2018
Received by editor(s) in revised form: February 26, 2019
Published electronically: July 10, 2019
Dedicated: Dedicated to Hari Bercovici
Communicated by: Harold P. Boas
Article copyright: © Copyright 2019 American Mathematical Society