Wronskians and deep zeros of holomorphic functions

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 ${\sum }_{j=0}^n{\lambda }_j{f}_j$ 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.

22 pages; to appear in J. Math. Pures Appl

References in corpus (4)

• Wronskians and linear independence
• Zeros of analytic functions, with or without multiplicities
• Factoring derivatives of functions in the Nevanlinna and Smirnov classes
• On the singular factor of a linear combination of holomorphic functions