Lu Qi-Kengʹs problem on some complex ellipsoids

In this note, we give an improvement on the Bergman kernel for the domain { ( w , z ) ∈ C m × C n : ‖ w ‖ m 2 p + ‖ z ‖ n 2 < 1 } . As an application, we describe how the zeroes of the kernel depend on the defining parameters p , m , n . We also consider the domain { { ( w , z ) ∈ C 2 : | w | 4 + | z | 4 < 1 } .