Spaces with polynomial hulls that contain no analytic discs

TL;DR

The paper constructs specific compact sets in complex Euclidean spaces with nontrivial polynomially convex hulls that contain no analytic discs, answering longstanding questions in complex analysis.

Contribution

It introduces new notions of polynomial and rational convexity, generalizes key results, and constructs explicit examples of sets with desired convexity properties.

Findings

Existence of a Cantor set in ${f C}^3$ with a nontrivial polynomially convex hull containing no analytic discs.

Construction of arcs and curves in ${f C}^4$ with similar properties.

Every uncountable, compact subspace can be embedded into some ${f C}^N$ with a polynomially convex hull containing no analytic discs.

Abstract

Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ with a nontrivial polynomially convex hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and curves in ${\mathbb C}^4$ with nontrivial polynomially convex hulls that contain no analytic discs. This answers a question raised a few years ago by Bercovici 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 polynomially convex 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.