The Lu Qi-Keng conjecture fails for strongly convex algebraic domains
type: Item
instance of: scholarly article
title: The Lu Qi-Keng conjecture fails for strongly convex algebraic domains
zbMATH Open document ID: 0911.32037
author: Peter Pflug
author: El-Hassan Youssfi
published in: Archiv der Mathematik
publication date: 1999-04-27T00:00:00Z
review text: 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 Boas, Fu and Straube seem to have proved (in an as yet unpublished paper) in a non-constructive way that there exist strongly convex real analytic domains failing to have this property for \(n\geq 3\).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.
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 1214006
zbMATH Keywords: zeros of Bergman kernels
Wikidata QID: Q123219249
reviewed by: J. Cnops
MaRDI profile type: Publication
full work available at URL: s000130050259
OpenAlex ID: W2014872705
DOI: 10.1007/S000130050259
description: scientific article; zbMATH DE number 1214006