# Vanishing Bergman kernels on the disk

## TL;DR

This paper investigates the zeroes of Bergman kernels on the disk, introduces a method to generate kernels with finitely many zeroes, and proves that radial-weighted kernels cannot have infinitely many zeroes, with brief discussion on Segal-Bargmann spaces.

## Contribution

It provides a novel method for constructing Bergman kernels with a finite number of zeroes and establishes a limitation for radial-weighted kernels on the disk.

## Key findings

- Bergman kernels can be generated with arbitrarily many zeroes, but finitely many.
- Radial weights on the disk produce kernels with only finitely many zeroes.
- Brief discussion on zeroes in Segal-Bargmann spaces.

## Abstract

We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal-Bargmann spaces of the complex plane are briefly discussed.