Wronskians and deep zeros of holomorphic functions

Given linearly independent holomorphic functions $f_0,\mathrm{.}\mathrm{.}\mathrm{.},f_n$ on a planar domain $\mathrm{\Omega }$, let $\mathcal{E}$ be the set of those points $z\in \mathrm{\Omega }$ 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 $\mathrm{\Omega }$ (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.