Singularities of rational inner functions in higher dimensions

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

“…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 [9] in the affirmative.…”

“…A series of recent papers with Bickel and Pascoe [7,8,9]; Bickel, Knese, and Pascoe [10]; and Tully-Doyle [21] deal with aspects of RIF theory that are particular to dimensions d ≥ 2. Namely, unlike in one dimension, RIFs in two or more variables can have singularities on the d-torus, arising at points ζ ∈ T d where p(ζ) = 0 and p(ζ) = 0 vanish without having common factors that cancel out.…”

“…Namely, unlike in one dimension, RIFs in two or more variables can have singularities on the d-torus, arising at points ζ ∈ T d where p(ζ) = 0 and p(ζ) = 0 vanish without having common factors that cancel out. A d-dimensional example (see [15,Section 5] and [9,Example 2.5]) is given by…”

“…One would like to describe RIF singularities in detail, and there are different ways of doing this. The papers [7,8,9], as well as [10], investigate for which p ≥ 1 the partial derivative of a RIF has ∂φ ∂z d ∈ L p (T d ). Roughly speaking, the smaller the maximal p for which integrability holds, the stronger the singularity of φ; for the example (1.2), the maximal integrability index is p = 1 2 (d + 1); see [9] and [10] for comprehensive discussions.…”

“…The papers [7,8,9], as well as [10], investigate for which p ≥ 1 the partial derivative of a RIF has ∂φ ∂z d ∈ L p (T d ). Roughly speaking, the smaller the maximal p for which integrability holds, the stronger the singularity of φ; for the example (1.2), the maximal integrability index is p = 1 2 (d + 1); see [9] and [10] for comprehensive discussions. The paper [7] and the work of Bergqvist [5] also consider other notions of derivative integrability corresponding to norms of Dirichlet type.…”