The Lu Qi-Keng conjecture fails for strongly convex algebraic domains

Abstract.

In this note, we give an example of an algebraic strongly convex $S^{1} \cdot \hbox {SO}(n, \Bbb R)$-invariant domain $\Omega$ in ${\Bbb C}^n, n\geq 4,$ such that $\Omega$ is not Lu Qi-Keng.