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Polynomial hulls with convex fibers. (English) Zbl 0538.32011
Let Y be a compact set in $${\mathbb{C}}^ n$$ and denote by $$\hat Y$$ the polynomial hull of Y, i.e. the set of those points in $${\mathbb{C}}^ n$$ at which the modulus of each polynomial on $${\mathbb{C}}^ n$$ is dominated by its maximum over Y. Except in special cases it is not known how to calculate the points of $$\hat Y$$, given Y. In this paper a class of sets Y in $${\mathbb{C}}^ 2$$ is presented for which a satisfactory description of $$\hat Y$$ is possible. For each $$\lambda$$ in $${\mathbb{C}}$$ one puts $$Y_{\lambda}=\{w|(\lambda,w)\in Y\}$$ and $${\hat Y}_{\lambda}$$ is defined similarly. $$\Gamma$$ denotes the unit circle. The first result is the following: assume that $$Y\subset \{(\lambda,w)| \lambda \in \Gamma \}$$ and that $$Y_{\lambda}$$ is convex for each $$\lambda\in \Gamma$$. Then $$\hat Y$$ consists of Y together with the union of the graphs $$w=\phi(\lambda)$$ of all bounded analytic functions on $$| \lambda |<1$$ with $$\phi(\lambda)\in Y_{\lambda}$$ for almost all $$\lambda$$ in $$\Gamma$$. In the particular case that each $$Y_{\lambda}$$, $$\lambda\in \Gamma$$, is a disk: $$Y_{\lambda}=\{w| | w- \alpha(\lambda)| \leq R(\lambda)\}$$, where $$\alpha$$ is a given complex-valued function continuous on $$\Gamma$$ and $$R>0$$ and $$\in C^ 2(\Gamma)$$, the following is shown: Assuming that $$| \alpha(\lambda)| \leq R(\lambda)$$, $$\lambda\in \Gamma$$, and that there is a b, $$| b|<1$$, such that $${\hat Y}_ b$$ contains more than one point, there exist analytic functions A,B,C,D on $$| \lambda |<1$$ such that $$\hat Y\cap \{| \lambda |<1\}=\{(\lambda,w)| |(A(\lambda)w+C(\lambda))(B(\lambda)w+D(\lambda))^{-1}| \leq 1,| \lambda |<1\}.$$ A,B,C,D are calculated from $$\alpha$$ and R. The method of proof makes key use of work of J. Garnett [”Bounded analytic functions” (1981; Zbl 0469.30024); Chapter 4].
Reviewer: Reviewer (Berlin)

##### MSC:
 32D99 Analytic continuation 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 32E05 Holomorphically convex complex spaces, reduction theory
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##### References:
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