# Zero Sets for Spaces of Analytic Functions

## TL;DR

This paper proves that Gaussian analytic functions almost surely do not share zeros with functions in certain weighted spaces, confirming longstanding conjectures and extending results on zero sets in Bergman and Bargmann-Fock spaces.

## Contribution

It establishes that under mild conditions, Gaussian analytic functions almost surely do not vanish where functions in specific weighted spaces do, confirming conjectures of Shapiro and Zhu.

## Key findings

- Confirmed Shapiro's conjecture for Bergman spaces.
- Resolved Zhu's question for Bargmann-Fock spaces.
- Extended results to unions of zero sets in these spaces.

## Abstract

We show that under mild conditions, a Gaussian analytic function $\boldsymbol F$ that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock space has the property that a.s. no non-zero function in that space vanishes where $\boldsymbol F$ does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann-Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro (1979) on Bergman spaces and allowing us to strengthen a result of Zhu (1993) on Bargmann-Fock spaces.