# Singularities of rational inner functions in higher dimensions

## TL;DR

This paper investigates the complex boundary behavior of rational inner functions in three or more dimensions, revealing that higher-dimensional cases exhibit more complex and less predictable properties than the well-understood two-variable scenario.

## Contribution

It provides a detailed analysis of the boundary and singularity behavior of rational inner functions in higher dimensions, extending understanding beyond the two-variable case.

## Key findings

- Boundary behavior becomes more complex in higher dimensions.
- Unimodular level sets reveal information about singularities.
- Higher dimensions lose some of the regularity properties seen in two-variable cases.

## Abstract

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.