Rational inner functions on the polydisk -- a survey

TL;DR

This survey reviews the theory of rational inner functions on the polydisk, covering interpolation, sums of squares, and boundary behavior, with a focus on two variables and some discussion of higher dimensions.

Contribution

It provides a comprehensive overview of key results and open problems in the study of rational inner functions on the polydisk, especially in two variables.

Findings

Summarizes interpolation results for rational inner functions.

Discusses sums of squares formulas related to these functions.

Examines boundary behavior and higher-dimensional cases.

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.