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Variation of fibers and polynomially convex hulls. (English) Zbl 0683.32010
Let \(Y=\{(z,w)\in {\mathbb{C}}^ 2:\) \(| w-\alpha (z)| \leq R(z)\), \(| z| =1\}\), where \(\alpha\) (z) and R(z) are continuous and satisfy \(| \alpha (z)| <R(z)\). Denote the fiber above z, \(| z| \leq 1\), of the polynomial hull \(\hat Y\) of Y by \(\hat Y(z)\). The author shows that \(\hat Y(z)\) is a disc that contains zero, and that there is a real valued solution of \(\partial^ 2u/\partial z\partial \bar z=e^{2u}\), \(| z| \leq 1\), from which the radius and the center of \(\hat Y(z)\) can be computed by a simple formula.
Reviewer: H.Röhrl

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