Lu Qi-Keng’s Problem for Intersection of Two Complex Ellipsoids

Abstract

Lu Qi-Keng's problem for intersection of two complex ellipsoids Tomasz BeberokIn this paper We investigate the Lu Qi-Keng problem for intersection of two complex ellipsoids {z ∈ C 3 : |z 1 | 2 + |z 2 | q < 1, |z 1 | 2 + |z 3 | r < 1}.

The Bergman kernel: Explicit formulas, deflation, Lu Qi-Keng problem and Jacobi polynomials

“…The explicit formula of the Bergman kernel function for the domains D Using the same method as in [1] we will prove that it is also true for n = 3.…”

“…2 if polynomial G(ǫx, ǫy) does not satisfy the stability property (see [1], for details) for some 0 < ǫ < 1. By following Šiljak and Stipanović [7] we consider polynomial…”

“…In 2015 [1] the author of this paper computed the Bergman kernel for D q,r 1 = {z = (z 1 , w 1 , w 2 ) ∈ C 3 : |z 1 | 2 + |w 1 | q < 1, |z 1 | 2 + |w 2 | r < 1} explicitly. The goal of this paper is to extend the result in [1] to higher dimensional case, namely to D q,r n = {(z, w 1 , w 2 ) ∈ C n+2 : |z 1 | 2 + · · · + |z n | 2 + |w 1 | q < 1, |z 1 | 2 + · · · + |z n | 2 + |w 2 | r < 1}.…”

“…The goal of this paper is to extend the result in [1] to higher dimensional case, namely to D q,r n = {(z, w 1 , w 2 ) ∈ C n+2 : |z 1 | 2 + · · · + |z n | 2 + |w 1 | q < 1, |z 1 | 2 + · · · + |z n | 2 + |w 2 | r < 1}. This paper will be organized as follows.…”

Essential normality of Bergman modules over intersections of complex ellipsoids

“…Proof. (Compare [7,8,9].) Using polar coordinates z j = x j e √ −1θ j , w kl = y kl e √ −1ϕ kl , we have…”