The \(*\)-product of domains in several complex variables. (English) Zbl 1531.32003
Summary: In this article, we investigate the problem of computing the \(*\)-product of domains in \(\mathbb{C}^N\). Assuming that \(0\in G\subset \mathbb{C}^N\) is an arbitrary Runge domain and \(0\in D\subset \mathbb{C}^N\) is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a geometric relation between \(D*G\) and another domain in \(\mathbb{C}^N\) which is ’extremal’ (in an appropriate sense) with respect to a special coefficient multiplier dependent only on the dimension \(N\). Next, for \(N = 2\), we derive a characterization of the latter domain expressed in terms of planar geometry. These two results, when combined together, give a formula which allows to calculate \(D*G\) for two-dimensional domains \(D\) and \(G\) satisfying the outlined assumptions.
MSC:
| 32A05 | Power series, series of functions of several complex variables |
| 32D15 | Continuation of analytic objects in several complex variables |
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