# On the Distribution of Zero Sets of Holomorphic Functions. IV. A Criterion

## TL;DR

This paper provides a comprehensive criterion for when a sequence of points in a domain can be the zero set of a holomorphic function bounded by an exponential of a difference of subharmonic functions.

## Contribution

It establishes necessary and sufficient conditions for zero sequences of holomorphic functions with exponential growth constraints in complex domains.

## Key findings

- Characterization of zero sets in terms of subharmonic functions.
- Complete description of zero sequence conditions.
- Application to classes of holomorphic functions with growth bounds.

## Abstract

Let $D$ be a proper domain in the extended complex plane ${\mathbb C}_{\infty}:={\mathbb C}\cup \{\infty\}$, $M=M_+-M_-\not\equiv \pm \infty$ be a difference of non-trivial subharmonic functions $M_{\pm}\not\equiv \mp \infty$ on $D$, $\text{Hol}(D,M)$ be the class of holomorphic function $f$ on $D$ satisfying $|f|\leq \text{const}_f\exp M$ om $D$, ${\sf Z}\subset D$ be a sequence of points in $D$ without limits points in $D$. We give a complete description of the conditions under which the sequence $\sf Z$ is a sequence of all zeros for some nonzero function $f\in \text{Hol}(D,M)$.