Hulls of subsets of the torus in \(\mathbb{C}^2\)
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.
scientific article; zbMATH DE number 1199205
scientific article