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

Abstract

AbstractWe analyze certain compositions of rational inner functions in the unit polydisk $$\mathbb {D}^{d}$$ D d with polydegree (n, 1), $$n\in \mathbb {N}^{d-1}$$ n ∈ N d - 1 , and isolated singularities in $$\mathbb {T}^d$$ 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 Bickel et al. (Am J Math 144: 1115–1157, 2022) in the affirmative.