An Application of Descriptive Set Theory to Complex Analysis

TL;DR

This paper proves that algebraic isomorphisms between certain rings of complex analytic functions are also topological, establishing a strong link between algebraic and topological structures in complex analysis.

Contribution

It demonstrates that any algebraic isomorphism from a Polish ring to a ring of holomorphic functions is necessarily a topological isomorphism, a novel result in the context of complex analysis and descriptive set theory.

Findings

Algebraic isomorphisms are topological for rings of holomorphic functions.

Certain rings of bounded and meromorphic functions cannot be Polish rings or fields.

A new general result about the structure of rings of complex analytic functions.

Abstract

The purpose of this paper is to prove a new general result about rings of complex analytic functions. Let $\Omega$ be an arbitrary nonempty open subset of the complex plane $\mathbb C$, $\mathcal{A}(\Omega)$ be the set of holomorphic functions on $\Omega$ viewed as a Polish ring (not a Polish algebra over $\mathbb C$) in the usual compact open topology, let $R$ be a Polish ring and let $\varphi : R \to \mathcal{A}(\Omega)$ be an abstract algebraic isomorphism. The main goal of this paper is to prove Theorem 36 that $\varphi$ is a topological isomorphism. A special result of Bers is an easy corollary. Two additional items supplement these results, viz., that $B(\mathbb{D})$, the abstract ring of bounded analytic functions on the unit disk, cannot be made into a Polish ring and that $\mathcal{M}(\Omega)$, the abstract field of meromorphic functions on $\Omega$, cannot be made into a Polish field.