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

TL;DR

This paper explores conditions for the uniqueness of solutions to the Nevanlinna-Pick problem on the polydisc, building on previous geometric methods and employing algebraic geometry tools like Bezout's theorem.

Contribution

It proposes a conjecture on necessary and sufficient conditions for solution uniqueness and verifies it in three special cases using advanced geometric techniques.

Findings

Conjecture on solution uniqueness for Nevanlinna-Pick problems on the polydisc.

Verification of the conjecture in three specific cases.

Application of Bezout's theorem to analyze problem solutions.

Abstract

In \cite{ds_hfs}, a geometric procedure for constructing a Nevanlinna-Pick problem on $\D^n$ with a specified set of uniqueness was established. In this sequel we conjecture a necessary and a sufficient condition for a Nevanlinna-Pick problem on $\DT$ to have a unique solution. We use the results of \cite{ds_hfs} and Bezout's theorem to establish three special cases of this conjecture.