Zeros of analytic functions, with or without multiplicities

TL;DR

This paper extends the Mason-Stothers theorem from polynomials to general analytic functions on bounded domains, providing sharp estimates on zeros and their multiplicities, generalizing classical results with broad applicability.

Contribution

It generalizes the Mason-Stothers theorem to analytic functions on bounded domains, offering sharp bounds on zeros and multiplicities, beyond polynomial cases.

Findings

Sharp bounds on zeros of analytic functions on bounded domains

Generalization of Mason-Stothers theorem to broader class of functions

Recovery of polynomial results via limiting arguments

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.