A note on polydegree $(n,1)$ rational inner functions, slice matrices, and singularities

TL;DR

This paper studies specific rational inner functions in the unit polydisk with polydegree (n,1), analyzing their singularities and compositions, and confirms a higher-dimensional version of a previously posed question.

Contribution

It demonstrates that under certain conditions, compositions of these functions preserve singularities and inner properties, extending understanding in multivariable complex analysis.

Findings

Compositions retain singularities under irreducibility conditions.

Quantitative control of properties of composed functions.

Affirmative answer to a higher-dimensional version of a known question.

Abstract

We analyze certain compositions of rational inner functions in the unit polydisk $\mathbb{D}^{d}$ with polydegree $(n,1)$, $n\in \mathbb{N}^{d-1}$, and isolated singularities in $\mathbb{T}^d$. Provided an irreducibility condition is met, such a composition is shown to be a rational inner function with singularities in precisely the same location as those of the initial function, and with quantitatively controlled properties. As an application, we answer a $d$-dimensional version of a question posed in \cite{BPS22} in the affirmative.