Wronskians and deep zeros of holomorphic functions

TL;DR

This paper investigates the rarity of high-multiplicity zeros in linear combinations of holomorphic functions, linking deep zeros to the zeros of the Wronskian and extending the analysis to various function spaces.

Contribution

It establishes that deep zeros are rare by connecting them to the zeros of the Wronskian and explores similar phenomena in different function spaces with boundary conditions.

Findings

Deep zeros correspond to zeros of the Wronskian.

Deep zeros form a discrete set in the domain.

The study extends to various function spaces with boundary smallness conditions.

Abstract

Given linearly independent holomorphic functions $f_0,...,f_n$ on a planar domain $\Omega$, let $\mathcal E$ be the set of those points $z\in\Omega$ where a nontrivial linear combination $\sum_{j=0}^n\lambda_jf_j$ may have a zero of multiplicity greater than $n$, once the coefficients $\lambda_j=\lambda_j(z)$ are chosen appropriately. An elementary argument involving the Wronskian $W$ of the $f_j$'s shows that $\mathcal E$ is a discrete subset of $\Omega$ (and is actually the zero set of $W$); thus "deep" zeros are rare. We elaborate on this by studying similar phenomena in various function spaces on the unit disk, with more sophisticated boundary smallness conditions playing the role of deep zeros.