On the Distribution of Zero Sets of Holomorphic Functions

Abstract

Let $M$ be a subharmonic function with Riesz measure ${\nu }_M$ in a domain $D$ in the $n$-dimensional complex Euclidean space ${\mathbb{C}}^n$, and let $f$ be a nonzero function that is holomorphic in $D$, vanishes on a set $\mathsf{Z}\subset D$, and satisfies $|f|\le \exp M$ on $D$. Then restrictions on the growth of ${\nu }_M$ near the boundary of $D$ imply certain restrictions on the area/volume of $\mathsf{Z}$. We give a quantitative study of this phenomenon in the subharmonic framework.

15 pages; in Russian; with corrections in normalizing factors and dividers