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

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.