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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:
 3.2e+21 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
##### Keywords:
fiber of polynomial hull
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