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Singularities of rational inner functions in higher dimensions
Journal article   Open access   Peer reviewed

Singularities of rational inner functions in higher dimensions

Kelly Bickel, James Eldred Pascoe and Alan Sola
American journal of mathematics, v 144(4), pp 1115-1157
01 Aug 2022
DOI: https://doi.org/10.1353/ajm.2022.0025
url
https://arxiv.org/abs/1906.10913View
Submitted Open arXiv.org - Non-exclusive license to distribute

Abstract

Mathematics Physical Sciences Science & Technology
We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree (m, n, 1) RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of nontrivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

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