Rational inner functions in the Schur-Agler class of the polydisk

TL;DR

This paper investigates the class of rational inner functions on the polydisk that admit transfer function realizations, focusing on their minimality and differences from the two-variable case.

Contribution

It characterizes the Schur-Agler class of rational inner functions in multiple variables and explores the limitations of minimal representations beyond two variables.

Findings

Two-variable rational inner functions have transfer function realizations.

Such realizations do not extend straightforwardly to three or more variables.

Minimality of representations is more complex in higher dimensions.

Abstract

Every two variable rational inner function on the bidisk has a special representation called a transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a transfer function realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might suggest.