Level curve portraits of rational inner functions

TL;DR

This paper studies the behavior of rational inner functions on the bidisk near boundary singularities using level sets, revealing their structure, stability, and methods for constructing functions with prescribed properties.

Contribution

It introduces a detailed analysis of level sets of rational inner functions, connecting their boundary behavior to zero sets and providing new construction methods.

Findings

Unimodular level sets can be parametrized with analytic curves.

Partial derivatives of rational inner functions share the same $L^p$-integrability.

Relations between intersection multiplicities and vanishing orders at singularities.

Abstract

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary $\mathbb{T}^2$ using level sets. We show that the unimodular level sets of a rational inner function $\phi$ can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of $\phi$. We apply these results to obtain a detailed description of the fine numerical stability of $\phi$: for instance, we show that $\frac{\partial \phi}{\partial z_1}$ and $\frac{\partial \phi}{\partial z_2}$ always possess the same $L^{\mathfrak{p}}$-integrability on $\mathbb{T}^2$, and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.