Minimal area of zero sets in tube domains of C 2

Abstract

Let Ω be a convex domain in C 2 symmetric with respect to the origin and let f(z, w) run over the class of analytic functions on Ω which vanish at the origin. What is the minimum of the areas of the zero sets Z(f) in Ω? If Ω is a ball or a four-dimensional cube with edges parallel to the axes, the minimum area is attained only if f is a suitable linear function (Lelong-Rutishauser, Katsnelson-Ronkin). In the present paper it is shown that the latter is the case also if Ω is a tube domain T D = {( z, w) ϵ C 2|( x, u) ϵD}. Thus, somewhat surprisingly, the minimum area of a zero set Z( f) through the origin in a convex symmetric tube domain T D is precisely twice the area of the base D. The special case where D is a circular disc had been treated earlier by Alexander and Osserman.