Abstract. We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in L 1 (C) whose associated Bergman kernels have infinitely many zeroes is exhibited.

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Under a mild technical assumption, we prove a necessary and sufficient condition for a totally real compact set in C n to be rationally convex. This generalizes a classical result of Duval-Sibony.

Under a mild technical assumption, we prove a necessary and sufficient condition for a totally real compact set in to be rationally convex. This generalises a classical result of Duval–Sibony.