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The polynomial hull of a set of finite linear measure in \({\mathbb{C}}^ n\). (English) Zbl 0615.32009
From the author’s introduction: ”For a compact subset X of \({\mathbb{C}}^ n\) denote the polynomially convex hull by \(\hat X.\) In the author’s paper in Ann. J. Math. 93, 65-74 (1971; Zbl 0221.32011), it was shown that if \(\Gamma\) is a compact connected subset of \({\mathbb{C}}^ n\) of finite linear measure, then \({\hat \Gamma}\setminus \Gamma\) is a (possibly empty) pure one-dimensional analytic subset of \({\mathbb{C}}^ n\setminus \Gamma\). The proof built upon previous work by Wermer, Bishop and Stolzenberg on the hulls of curves. The hypothesis that \(\Gamma\) be connected, or more generally, that \(\Gamma\) be contained in a connected set of finite linear measure, was required by the technique used, and it remained an open question whether it was needed. Our main result is that connectedness is indeed required.
Theorem. There exists a compact subset \(\Gamma\) of \({\mathbb{C}}^ 2\) of finite linear measure such that \({\hat \Gamma}\setminus \Gamma\) is not a one-dimensional analytic subset of \({\mathbb{C}}^ 2\setminus \Gamma.\)
The set \(\Gamma\) which we shall construct is not highly pathological. It is a countable union of real analytic simple closed curves. While \({\hat \Gamma}\setminus \Gamma\) is not an analytic subset of \({\mathbb{C}}^ 2\setminus \Gamma\) in the Theorem, it is however a countable union of analytic sets. The general structure of \(\hat X\setminus X\) when X has finite linear measure (and is disconnected) remains open.”
Reviewer: E.Bedford

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32C25 Analytic subsets and submanifolds
Zbl 0221.32011
Full Text: DOI
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