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Polynomial hulls in \({\mathbb{C}}^ 2\) and quasicircles. (English) Zbl 0703.32007
For compact \(X\subset {\mathbb{C}}^ n\) denote by \(\hat X\) its polynomial hull. The main result of the paper is:
Theorem 1.1. Let \(X\subset \partial D\times {\mathbb{C}}\) be a compact set such that for every \(\zeta\in \partial D\), \(X(\zeta)=\{w\in {\mathbb{C}}:\) (\(\zeta\),w)\(\in X\}\) is a simply connected continuum; here D is a unit disc in \({\mathbb{C}}\). Assume that \(\hat X\setminus X\) is nonempty. Then \(\hat X\setminus X\) is equal to the union of the graphs of all \(H^{\infty}(D)\) functions whose cluster values at \(\zeta\in \partial D\) belong to X(\(\zeta\)). (Below, an analytic disc will always denote such a graph). Furthermore,
(a) the relative boundary of \(\hat X\cap D\times {\mathbb{C}}\), denoted by S, is covered by analytic disks, and every two distinct analytic discs contained in S are disjoint;
(b) every analytic disc contained in \(\hat X\) and interesting S must be fully contained in S;
(c) for every \(z\in D\) the fiber \(Y(z)=\{w\in {\mathbb{C}}:\) (z,w)\(\in \hat X\}\), of \(\hat X,\) is a simply connected continuum and its topological boundary (in \({\mathbb{C}})\) is equal to the fiber of S, i.e. \(S(z)=\{w\in {\mathbb{C}}:\) (z,w)\(\in S\}\).
Reviewer: S.M.Ivashkovich

MSC:
32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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