The Bergman kernel function: Explicit formulas and zeroes. (English) Zbl 0919.32013

The authors discuss three basic principles for obtaining new Bergman kernel functions from old ones. The methods of deflation and inflation relate kernel functions of different dimensions, whereas the well known principle of folding relates kernel functions of domains in the same dimension. Using these techniques the authors show how to obtain the Bergman kernel function of some special domains with minimal computational effort. As an application of the explicit formulas, the authors show that the kernel function of the convex domain in \(\mathbb C^3\) defined by the inequality \(| z_1| + | z_2| + | z_3| <1\) does have zeroes. As a consequence, when \(n\geq 3\), there exists a smooth, bounded, strongly convex Reinhardt domain in \(\mathbb C^n\) whose Bergman kernel function has zeroes.
Reviewer: M.Stoll (Columbia)


32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
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