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

Abstract

We analyze certain compositions of rational inner functions in the unit polydisk mathbb {D}^{d} with polydegree (n, 1), nin 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 Bickel et al. (Am J Math 144: 1115–1157, 2022) in the affirmative.