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Rational inner functions on the polydisk -- a survey

Greg Knese

TL;DR

This survey analyzes rational inner functions on the polydisk, focusing on the two-variable setting while outlining higher-dimensional challenges. It presents a canonical representation $f=\frac{\tilde{p}}{p}$ with a Pfister-type refinement $q(z)=a z^{m}\tilde{p}(z)$ that underpins many constructions and boundary phenomena, including boundary singularities exemplified by simple RIFs. The paper connects interpolation (Pick, Agler) to sums-of-squares identities and determinantal representations, linking function theory with multivariable operator theory and real algebraic geometry, and it discusses the Schur–Agler class and its limits in higher dimensions. It also develops a detailed local theory of boundary behavior via Puiseux expansions, Horn regions, and contact order, and surveys the state of knowledge and open problems for three or more variables, highlighting the rich interplay between analysis, geometry, and operator theory in several complex variables.

Abstract

Rational inner functions are a generalization of finite Blaschke products to several variables. In this article we survey a variety of results about rational inner functions related to interpolation, sums of squares formulas, and boundary behavior. We mostly focus on two variables however in the final section we discuss higher dimensions.

Rational inner functions on the polydisk -- a survey

TL;DR

This survey analyzes rational inner functions on the polydisk, focusing on the two-variable setting while outlining higher-dimensional challenges. It presents a canonical representation with a Pfister-type refinement that underpins many constructions and boundary phenomena, including boundary singularities exemplified by simple RIFs. The paper connects interpolation (Pick, Agler) to sums-of-squares identities and determinantal representations, linking function theory with multivariable operator theory and real algebraic geometry, and it discusses the Schur–Agler class and its limits in higher dimensions. It also develops a detailed local theory of boundary behavior via Puiseux expansions, Horn regions, and contact order, and surveys the state of knowledge and open problems for three or more variables, highlighting the rich interplay between analysis, geometry, and operator theory in several complex variables.

Abstract

Rational inner functions are a generalization of finite Blaschke products to several variables. In this article we survey a variety of results about rational inner functions related to interpolation, sums of squares formulas, and boundary behavior. We mostly focus on two variables however in the final section we discuss higher dimensions.
Paper Structure (7 sections, 18 theorems, 70 equations, 1 figure)

This paper contains 7 sections, 18 theorems, 70 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic facts
  3. Interpolation
  4. Sums of squares
  5. Boundary behavior
  6. Higher dimensions
  7. Acknowledgements

Key Result

theorem 1

Given $p,q \in \mathbb{C}[z_1,\dots, z_d]$ with no common factors, where $p(z) \ne 0$ for $z\in \mathbb{D}^d$, the function $f = q/p$ is a rational inner function if and only if $q(z)$ is of the form where $a\in \mathbb{T}$, $m \in \mathbb{Z}_{\geq 0}^d$ is a $d$-tuple of non-negative integers.

Figures (1)

  • Figure 1: Various level sets for a rational inner function

Theorems & Definitions (26)

  • definition 1
  • theorem 1
  • lemma 1
  • proof
  • proof : Proof of Theorem \ref{['RIFchar']}
  • proposition 1
  • proof
  • proposition 2
  • lemma 2
  • proof
  • ...and 16 more