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Polynomial hulls of arcs and curves. (English) Zbl 1475.32007

The author studies polynomial hulls of arcs and simple closed curves, where an arc is a space homeomorphic to the closed unit interval and a simple closed curve is a space homeomorphic to the unit circle. He proves that there exist arcs and simple closed curves in \(\mathbb C^3\) with nontrivial polynomial hulls that contain no analytic discs. In any bounded connected Runge domain in \(\mathbb C^N\), \(N\ge 2\), he proves the existence of polynomially convex arcs and polynomially convex simple closed curves of almost full \(2N\)-dimensional Lebesgue measure through any given point. Proofs rely on the existence of appropriate Cantor sets.

MSC:

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
32A38 Algebras of holomorphic functions of several complex variables
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
46J10 Banach algebras of continuous functions, function algebras
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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References:

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