Local ABC theorems for analytic functions

TL;DR

This paper extends the classical polynomial abc theorem to local settings for analytic functions on bounded domains, providing sharp estimates that generalize the original theorem.

Contribution

It introduces local abc-type theorems for analytic functions on bounded domains, generalizing the classical polynomial abc theorem with sharp estimates.

Findings

Established sharp local abc-type inequalities for analytic functions.

Generalized the global abc theorem to bounded domains via limiting arguments.

Provided bounds that are optimal for any domain considered.

Abstract

The classical $abc$ theorem for polynomials (often called Mason's theorem) deals with nontrivial polynomial solutions to the equation $a+b=c$. It provides a lower bound for the number of distinct zeros of the polynomial $abc$ in terms of $\deg{a}$, $\deg{b}$, and $\deg{c}$. We prove some "local" $abc$-type theorems for 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 they imply (a generalization of) the original "global" $abc$ theorem by a limiting argument.