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In this note, we give an example of a strongly convex algebraic complete Reinhardt domain which is not Lu Qi-Keng in Cn for any n ≥ 3. 0. Introduction and statement of the main result Let us first recall that a domain Ω in C is said to be Lu Qi-Keng if its Bergman kernel function KΩ(z, w) is non zero at any point (z, w) ∈ Ω × Ω. The question of finding smooth pseudoconvex domains in C whose Bergman kernel function has zeros was raised by Lu Qi-Keng [Lu]. It is well-known [SY] that when n = 1, then a smooth domain Ω in C is Lu Qi-Keng if and only if it is simply connected. For a complete history of this subject, see [Bo1], [Bo2]. In higher dimensions, Boas, Fu and Straube [BFS] proved that, for any n ≥ 3, there exists a strongly convex real analytic domain in C which is not Lu Qi-Keng. Their example is deduced by exhaustion from the Hartogs triangle in C. Recently, P. Pflug and E. H. Youssfi [PY] proved that the minimal balls defined in [HP] are non Lu Qi-Keng for any n ≥ 4. As a consequence, they gave concrete strongly convex algebraic and S · SO(n,R)-invariant non Lu Qi-Keng balls in C for any n ≥ 4, but these balls are not Reinhardt. In this note, we shall give concrete strongly convex algebraic complete Reinhardt domains which are non Lu Qi-Keng in C for any n ≥ 3. This completes the works in [PY] and [BFS]. Consider, for a positive integer n, a non-negative integer m and a real number a ∈ ]0, 1], the domain Ωa,k,n,m := { z = (z1, z2, . . . , zn+m) ∈ C : Na,k(z) < 1 } , where Na,k(z) :=  ∑ 1,... , n+1∈{−1,+1} α 1,... , n+1(z) + a α(z)  1 2k

The Lu Qi-Keng conjecture fails for strongly convex algebraic complete Reinhardt domains in $\mathbb {C}^n\ (n \ge 3)$