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On the polynomial hull of a graph. (English) Zbl 0798.32026
Let $$G \subset \mathbb{R}^ 3$$ be a bounded strictly convex domain and let $$\varphi : \partial G \to \mathbb{R}$$ be a continuous function. Put $$\Gamma : = \{(z,w) \in \mathbb{C}^ 2:(z, \text{Re} w) \in G$$, $$\text{Im} w = \varphi (z, \text{Re} w)\}$$ and let $$\widehat \Gamma$$ denote the polynomial hull of $$\Gamma$$.
The author studies the structure of the set $$\widehat \Gamma \backslash \Gamma$$. The main result is the following theorem: There exists a family $$(D_ \alpha)_{\alpha \in A} \subset \mathbb{C}^ 2$$ of pairwise disjoint holomorphic discs such that $$\widehat \Gamma \backslash \Gamma = \bigcup_{\alpha \in A}D_ \alpha$$ and for each $$\alpha \in A$$ there exist a simply connected domain $$\Omega_ \alpha \subset \mathbb{C}$$ and a function $$f_ \alpha : \overline \Omega_ \alpha \to \mathbb{C}$$ with the following properties:
$$f_ \alpha \in {\mathcal C} (\overline \Omega_ \alpha) \cap {\mathcal O} (\Omega_ \alpha)$$,
$$D_ \alpha = \{(z,f_ \alpha (z)) : z \in \Omega_ \alpha\}$$,
$$\Gamma \supset \{(z,f_ \alpha (z)) : z \in \partial \Omega_ \alpha\}$$,
$$\mathbb{C} \backslash \overline \Omega_ \alpha$$ is connected,
if the set $$\partial \Omega_ \alpha \backslash \partial \overline \Omega_ \alpha$$ is nonempty, then it has a noncountable number of connected components.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations
##### Keywords:
bounded strictly convex domain; polynomial hull
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