Hulls of subsets of the torus in \(\mathbb{C}^2\)
type: Item
instance of: scholarly article
title: Hulls of subsets of the torus in \(\mathbb{C}^2\)
zbMATH Open document ID: 0910.32013
publication date: 1998-09-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75303
review text: Let \(X\) be a compact subset of \(\mathbb{C}^2\). Its polynomial convex hull \(\widehat X\) is the set of points \(q\in\mathbb{C}^2\) such that for every polynomial \(P(z_1,z_2)\) one has \(| P(z) |\leq \max \{| P(z) |: z\in X\}\). The obvious reason for a point \(q\) to belong to a polynomial hull of \(X\) is if one could find a 1-dimensional analytic subvariety \(V\ni q\) with \(\partial V\subset X\) (maximum principle). The goal of the paper is the following: Theorem. There exists a compact subset \(X\) of the torus \(\mathbb{T}^2 =\{(z,w) \in\mathbb{C}^2: | z| =| w| =1\}\) such that \(\widehat X\setminus X\) is non-empty but contains no analytic subset of positive dimension. The construction of \(X\) is parallel to the construction of Wermer of a compact \(X\subset \{(z,w) \in\mathbb{C}^2: | z|=1, | w|\leq 1\}\) with similar properties, but is more tricky.
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32B15
zbMATH DE Number: 1199205
zbMATH Keywords: polynomial hull
zbMATH Keywords: analytic structure
zbMATH Keywords: torus
author: Herbert J. Alexander
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
OpenAlex ID: W2327020374
cites work: Continuing 1-dimensional analytic sets
cites work: Joint Quasitriangularity of Essentially Unitary Pairs
cites work: Courants positifs extremaux et conjecture de Hodge
cites work: Q4352976
cites work: Polynomial convexity, rational convexity, and currents
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cites work: Meromorphic mappings
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cites work: Polynomially convex hulls and analyticity
DOI: 10.5802/AIF.1639
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1199205