The Lu Qi-Keng conjecture fails for strongly convex algebraic domains. (English) Zbl 0911.32037

A domain \(\Omega\) in \(\mathbb C^n\) is said to be Lu Qi-Keng if its Bergman kernel has no zeros in \(\Omega\times\Omega\). Boas, Fu and Straube seem to have proved (in an as yet unpublished paper) in a non-constructive way that there exist strongly convex real analytic domains failing to have this property for \(n\geq 3\). In this paper examples of failing strongly convex algebraic domains for \(n\geq 4\) are given. First they prove, using the explicit expression of the Bergman kernel established in a previous paper, that \(B_*=\{z\in \mathbb C^n: | z| ^2+| z\cdot z| \leq 1\}\) is not Lu Qi-Keng. Then \(B_*\) is approximated by an increasing sequence of strongly convex algebraic domains, and it follows that almost all of these must fail to be Lu Ki-Qeng.
Reviewer: J.Cnops (Gent)


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