Abstract
Extensions of the notions of polynomial and rational hull are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set in \({\mathbb {C}}^3\) with a nontrivial polynomial hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and simple closed curves in \({\mathbb {C}}^4\) with nontrivial polynomial hulls that contain no analytic discs. This answers a question raised by Bercovici in 2014 and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace X of \({\mathbb {C}}^N\), for some N, in such a way as to have a nontrivial polynomial hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, X can be chosen so as to have the stronger property that P(X) has a dense set of invertible elements.
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Communicated by Ngaiming Mok.
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Izzo, A.J. Spaces with polynomial hulls that contain no analytic discs. Math. Ann. 378, 829–852 (2020). https://doi.org/10.1007/s00208-019-01838-z
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DOI: https://doi.org/10.1007/s00208-019-01838-z