Zeros of analytic functions, with or without multiplicities

Abstract

The classical Mason-Stothers theorem deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound on the number of distinct zeros of the polynomial $abc$ in terms of the degrees of $a$, $b$ and $c$. We extend this to general analytic functions living on a reasonable bounded domain $\Omega \subset \mathbb{C}$, rather than on the whole of $\mathbb{C}$. The estimates obtained are sharp, for any $\Omega$, and a generalization of the original result on polynomials can be recovered from them by a limiting argument.

This is a retitled and slightly revised version of my paper arXiv:1004.3591

Cited by in corpus (5)

• On the singular factor of a linear combination of holomorphic functions
• Two problems on coinvariant subspaces of the shift operator
• ABC-type estimates via Garsia-type norms
• Mason's theorem with a difference radical
• Wronskians and deep zeros of holomorphic functions