Level curve portraits of rational inner functions

Abstract

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary T 2 using level sets. We show that the unimodular level sets of a rational inner function can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of. We apply these results to obtain a detailed description of the fine numerical stability of : for instance, we show that @ @z 1 and @ @z 2 always possess the same L p-integrability on 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.

Optimal approximants and orthogonal polynomials in several variables

“…. This polynomial is the denominator of a rational inner function in the bidisk constructed in [11], and hence it follows that b has no zeros in the bidisk and, in particular, is a cyclic vector in the Bergman space A 2 (D), viz. [9].…”

“…However, b does have a single boundary zero at (1, 1) ∈ T 2 . In fact, as is explained in [11], there are general methods for constructing rational inner functions in two variables having boundary singularities with prescribed properties. The denominator polynomials of such rational inner functions often exhibit some interesting features, which this led us to consider them when searching for counterexamples.…”

Dynamics of low-degree rational inner skew-products on $\mathbb{T}^2$

“…However, if a normalized φ has singular points, then φ * (ζ) is not typically continuous on T 2 . See [3,28,8,9] for more background material on RIFs.…”

Rational inner functions and their Dirichlet type norms

“…In [8], Knese gives an algebraic classification of the ideal of polynomials p such that q/p lies in L 2 (T 2 ) for a fixed polynomial q, and furthermore proves that rational inner functions have non-tangential limits everywhere on the n−torus. In [3] and [5], a thorough analysis of the unimodular level sets and singular sets of RIFs gives a fairly complete understanding of the L p integrability on T 2 of the partial derivatives of RIFs in 2 variables. For instance it is shown that ∂ z 1 φ and ∂ z 2 φ have the same integrability properties.…”

“…However, it is shown in [4] that a lot of important properties are lost in higher dimensions. For example, in two variables, the singular sets are isolated points, and the unimodular level sets passing through these singularities are curves parametrizable by analytic functions (See [5]). In dimension d ≥ 3 this is no longer true; the singular set is only guaranteed to be contained in an algebraic set of dimension n − 2; so for example in dimension 3 the singular set can be a union of points and curves (see Prop.…”