Bounded meromorphic functions on the complex 2-disc

TL;DR

This paper characterizes bounded holomorphic functions on the complex 2-disc that extend meromorphically beyond, solving a conjecture using algebraic geometry tools like Zariski's theorem.

Contribution

It provides a complete description of such functions and proves a conjecture by applying classical algebraic geometry to complex analysis.

Findings

Characterization of bounded meromorphic functions on the 2-disc

Resolution of the conjecture by Bickel et al.

Application of Zariski's theorem in complex analysis

Abstract

We describe bounded, holomorphic functions on the complex 2-disc, that admit meromorphic extension to a larger 2-disc. This solves a conjecture of Bickel, Knese, Pascoe and Sola. The key technical ingredient is an old theorem of Zariski about integrally closed ideals in 2-dimensional regular rings. v2: Remark 15 is added.