Hilbert function spaces and the Nevanlinna-Pick problem on the polydisc

TL;DR

This paper develops a geometric method for constructing Nevanlinna-Pick problems on the polydisc with predetermined uniqueness sets, advancing understanding of function spaces and interpolation problems in several complex variables.

Contribution

It introduces a geometric construction technique for Nevanlinna-Pick problems with specified uniqueness sets and addresses a question about Hilbert function spaces posed by Agler and McCarthy.

Findings

Established a procedure for constructing Nevanlinna-Pick problems with given uniqueness sets

Proved a new result about Hilbert function spaces related to the problem

Partially answered a question posed by Agler and McCarthy

Abstract

In the paper `Distinguished Varieties,' Agler and McCarthy used Hilbert function spaces to study the uniqueness properties of the Nevanlinna-Pick problem on the bidisc. In this work we give a geometric procedure for constructing a Nevanlinna-Pick problems on D^n with a specified set of uniqueness. On the way to establishing this procedure, we prove a result about Hilbert function spaces and partially answer a question posed by Agler and McCarthy.