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.

Equations

$${\mathbb{D}}^2=\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1|<1,|z_2|<1\}$$

$$-\frac{3z_1{z}_2-z_1-z_2}{3-z_1-z_2}\text{and}-\frac{2z_1{z}_2-z_1-z_2}{2-z_1-z_2};$$

$$\varphi (z_1,z_2)=e^{i\alpha }{z}_1^M{z}_2^N\frac{\tilde{p}(z_1,z_2)}{p(z_1,z_2)}$$

$$\tilde{p}(z_1,z_2)=z_1^m{z}_2^n\overline{p\left(\frac{1}{{\overline{z}}_1},\frac{1}{{\overline{z}}_2}\right)}$$

$${\mathcal{Z}}_{\tilde{p}}=\{z\in {\mathbb{C}}^2:\tilde{p}(z)=0\},$$

$${\mathcal{Z}}_{\tilde{p}}^1={\mathcal{Z}}_{\tilde{p}}\cap \left(\overline{\mathbb{D}}\times \mathbb{T}\right)\text{and}{\mathcal{Z}}_{\tilde{p}}^2={\mathcal{Z}}_{\tilde{p}}\cap \left(\mathbb{T}\times \overline{\mathbb{D}}\right).$$

$$\mathrm{dist}\big{(}w_{k},\mathbb{T}^{2}\big{)}\leq C\ \mathrm{dist}\big{(}w_{k},\tau\big{)}^{K_{i}}\qquad\forall k\in\mathbb{N}.$$

$${\mathcal{C}}_{\lambda }=\{\zeta \in {\mathbb{T}}^2:\varphi (\zeta )=\lambda \},\text{for}\lambda \in \mathbb{T},$$

$$\varphi (z_1,z_2)=-\frac{2z_1{z}_2-z_1-z_2}{2-z_1-z_2}.$$

$${\mathcal{C}}_1^{\ast }(1,1)=\{(e^{i{t}_1},1)\}\cup \{(1,e^{i{t}_2})\},$$

$${\mathcal{C}}_{\lambda }=\{(z_1,{\psi }^{\lambda }(z_1)):z_1\in \mathbb{T}\}\text{where}{\psi }^{\lambda }(z_1)=\lambda \frac{1-\frac{1-\overline{\lambda }}{2}{z}_1}{z_1-\frac{1-\lambda }{2}},$$

$$|{\psi }^{\lambda }(z_1)-{\psi }^{\mu }(z_1)|\asymp |1-z_1{|}^2\text{as}{z}_1\to 1.$$

$$\frac{\partial \varphi }{\partial z_1}\in L^{\mathfrak{p}}({\mathbb{T}}^2)⟺K<\frac{1}{\mathfrak{p}-1}⟺\frac{\partial \varphi }{\partial z_2}\in L^{\mathfrak{p}}({\mathbb{T}}^2).$$

$$\varphi (z_1,z_2)=-\frac{4z_1^2{z}_2-z_1^2-3z_1{z}_2-z_1+z_2}{4-3z_1-z_2-z_1{z}_2+z_1^2}$$

$$4(z_1{z}_2-1)(z_1-1)=0$$

$${\mathcal{C}}_1^{\ast }(1,1)=\{(1,e^{i{t}_2}))\}\cup \{(e^{i{t}_1},e^{-i{t}_1})\}$$

$$z_2={\psi }^{\lambda }(z_1)=\frac{4\lambda -3\lambda z_1+\lambda z_1^2+z_1^2+z_1}{4z_1^2+\lambda z_1-3z_1+\lambda +1},z_1\in \mathbb{T}.$$

$$\phi (z_1,z_2)=-\frac{2z_1^2{z}_2-z_1-1}{2-z_1{z}_2-z_1^2{z}_2},$$

$$\begin{gathered} p(z_1,z_2)=-32+38z_2-10z_2^2+34z_1-32z_1{z}_2+2z_1{z}_2^2-30z_1^2+36z_1^2{z}_2 \\ -2z_1^2{z}_2^2+10z_1^3-8z_1^3{z}_2-6z_1^3{z}_2^2-6z_1^4+14z_1^4{z}_2-8z_1^4{z}_2^2 \end{gathered}$$

$$\begin{gathered} \tilde{p}(z_1,z_2)=-8+14z_2-6z_2^2-6z_1-8z_1{z}_2+10z_1{z}_2^2-2z_1^2+36z_1^2{z}_2-30z_1^2{z}_2^2 \\ +2z_1^3-32z_1^3{z}_2+34z_1^3{z}_2^2-10z_1^4+38z_1^4{z}_2-32z_1^4{z}_2^2 \end{gathered}$$

$$16=N(p,\tilde{p})=N_{{\mathbb{T}}^2}(p,\tilde{p})=14+2,$$

$$\varphi (z_1,z_2)=\eta z_1^M{z}_2^N\frac{\tilde{p}(z_1,z_2)}{p(z_1,z_2)},$$

$${\mathcal{Z}}_{\tilde{p}}:=\{(z_1,z_2)\in {\mathbb{C}}^2:\tilde{p}(z_1,z_2)=0\}$$

$${\mathcal{L}}_{\lambda }(\varphi ):=\left\{(z_1,z_2)\in {\mathbb{C}}^2:\tilde{p}(z_1,z_2)=\lambda p(z_1,z_2)\right\}.$$

$$\beta :\Pi \to \mathbb{D},\beta (w):=\frac{1+iw}{1-iw}\text{ and }{\beta }^{-1}:\mathbb{D}\to \Pi ,{\beta }^{-1}(z):=i\left[\frac{1-z}{1+z}\right].$$

$$z_1={\psi }_1^0(z_2),\dots ,z_1={\psi }_{L_0}^0(z_2),$$

$$q(w_1,w_2):=(1-i{w}_1{)}^m(1-i{w}_2{)}^n\tilde{p}\left(\beta (w_1),\beta (w_2)\right).$$

$$w_1={\Psi }_1^0\left(w_2^{\frac{1}{N_1}}\right),\dots ,w_1={\Psi }_L^0\left(w_2^{\frac{1}{N_L}}\right),$$

$$z_1=\beta \left({\Psi }_1^0\left({\beta }^{-1}(z_2{)}^{\frac{1}{N_1}}\right)\right),\dots ,z_1=\beta \left({\Psi }_L^0\left({\beta }^{-1}(z_2{)}^{\frac{1}{N_L}}\right)\right).$$

$$z_1={\psi }_1^0(z_2),\dots ,z_1={\psi }_{L_0}^0(z_2).$$

Full text

$\dagger$ Research supported in part by National Science Foundation DMS grant #1448846.

$\ddagger$ Research supported by National Science Foundation Mathematical Science Postdoctoral Research Fellowship DMS 1606260.

1. 1 Introduction and overview
2. 2 Level sets near singularities
3. 3 Contact order vs order of contact
4. 4 Equal contact orders
5. 5 Fine contact order vs fine order of contact
6. 6 Constructions of rational inner functions
7. 7 A zoo of rational inner functions

1. Introduction and overview

1.1. Introduction

A rational function of a complex variable can be described in terms of its zeros and poles, and the behavior of the function near these points is in principle easy to capture in terms of their integer orders. The exact location and nature of the zeros and poles of a one-variable rational function are decisive in many applications: for instance, critical points and poles determine much of the dynamical properties of a rational function in iteration theory, and the zeros and poles of rational functions in one variable govern the stability of associated systems in control theory. This latter fact, that the qualitative nature of a system is determined by the location of zeros of polynomials defining an associated rational function, leads to the important notion of a stable polynomial, one that has all roots outside the unit disk (or the left half-plane, depending on context).

When studying a rational function of several variables in a mathematical or engineering context, one is again led to consider points where numerator and denominator vanish, but now a new and subtle phenomenon manifests itself: simultaneous vanishing at a point does not necessarily lead to algebraic cancellation. Nevertheless, it may still happen that the rational function retains some smoothness and boundedness properties near a common zero of numerator and denominator, and this then leads to a rich geometric structure at this point.

This paper is devoted to a detailed study of singularities of a certain important class of rational functions in two variables. We work on the unit bidisk

[TABLE]

and are interested in zeros and singularities on the distinguished boundary of the bidisk, which we identify with the two-torus $\mathbb{T}^{2}=\mathbb{T}\times\mathbb{T}$, the Cartesian product of two copies of the unit circle $\mathbb{T}=\{z\in\mathbb{C}\colon|z|=1\}$. The distinguished boundary $\mathbb{T}^{2}$ supports the maximum modulus principle for the bidisk, and is determining for most of the function-theoretic questions we will address in this paper. A rational inner function (RIF) on the bidisk is a rational function $\phi\colon\mathbb{D}^{2}\to\mathbb{C}$ that is analytic and bounded in $\mathbb{D}^{2}$ and has $|\phi(\zeta)|=1$ for almost every $\zeta\in\mathbb{T}^{2}$. Examples of such functions are

[TABLE]

the first one is smooth on the closed bidisk $\overline{\mathbb{D}^{2}}$, but the second example exhibits what is known as a “non-essential singularity of the second kind” at $(1,1)$: the function has a non-tangential limit at $(1,1)$ but the vanishing polynomials $2z_{1}z_{2}-z_{1}-z_{2}$ and $2-z_{1}-z_{2}$ do not share a common factor.

The numerators and denominators in these examples can be obtained from each other by reflection in the unit circle. In fact, W. Rudin and E.L. Stout showed, [RS65] and [Rudin, Chapter 5], that all RIFs on the bidisk are of the form

[TABLE]

where $\alpha$ is a real number, $M$ and $N$ are non-negative integers, $p$ is a semi-stable polynomial, and the polynomial

[TABLE]

is the reflection of $p$. The pair of integers $(m,n)\in\mathbb{N}^{2}$ is referred to as the bidegree of $p$ and is given by the largest powers of $z_{1}$ and $z_{2}$ that appear in $p$. A polynomial $p\in\mathbb{C}[z_{1},z_{2}]$ is said to be semi-stable if it has no zeros in $\mathbb{D}^{2}$; it is stable (or strictly stable) if it is non-vanishing on the closed bidisk. For simplicity, we usually consider rational inner functions of the form $\phi=\frac{\tilde{p}}{p}$ in this paper; monomial factors do not materially affect our conclusions.

The study of rational inner functions and semi-stable polynomials has a rich tradition in complex analysis [KV79, McD87, AMcCY12, Kne10b], operator theory [AglMcC, BicKne13, BicLi17, GKVW17, BicGor18], algebraic geometry [AMcCS06, AMcCS08], and systems theory and engineering [Kum02, BSV05, GW06, Wag11]. We refer the reader to references provided in these papers for further work on these topics. Recently, Knese [Kne15] initiated the study of $L^{2}(\mathbb{T}^{2})$-integrability of rational functions of the form $q/p$, where $p$ is assumed semi-stable but not necessarily strictly stable. In [BPS18], the authors derived a concrete relationship between the numerical stability of a rational inner function $\phi$, as measured by the $L^{\mathfrak{p}}(\mathbb{T}^{2})$-integrability of $\frac{\partial\phi}{\partial z_{1}}$ and $\frac{\partial\phi}{\partial z_{2}}$, and “fine semi-stability” of its zero set, captured by contact orders at a singularity. These measure how fast the zero set of $\tilde{p}$ approaches $\mathbb{T}^{2}$ in relation to how the fast the zero set approaches the singularity, if one variable is restricted to $\mathbb{T}$. Informally, contact order can be defined for $\phi=\frac{\tilde{p}}{p}$ as follows. Setting

[TABLE]

we define the facial varieties

[TABLE]

The $z_{i}$-contact order of $\phi$ is given by the largest number $K_{i}$ such that there exists a sequence $\{w_{k}\}\subseteq\mathcal{Z}_{\tilde{p}}^{i}$ converging to a singular point $\tau\in\mathbb{T}^{2}$ of $\phi$ and a positive constant $C$ such that

[TABLE]

(The precise definition of contact order is given in [BPS18, Section 2] and Section 2 below.)

In this paper, we study the numerical stability of an RIF and the geometry of its zero set via level curves of the RIF restricted to the two-torus. This approach allows us to “visualize” the geometry of singularities of an RIF on $\mathbb{T}^{2}$ in a concrete and appealing way. More precisely, one of our main goals is to show how to divine “fine semi-stability”, that is, compute contact orders and related quantities by examining unimodular level curves

[TABLE]

and how they come together at singularities of $\phi$ on the two-torus. We show that such level curves are in fact smooth, in the sense that they can be parametrized by analytic functions. From this fact we are able to derive many properties of $\phi$ at its singularities, including for instance that its first partials enjoy the same $L^{\mathfrak{p}}(\mathbb{T}^{2})$-integrability properties. Using smoothness of level curves together with certain embedding constructions, we are further able to apply our results concerning rational inner functions to draw conclusions about how special varieties in $\mathbb{C}^{2}$ intersect the two-torus.

1.2. Overview

We proceed with an overview of the results contained in this paper: in what follows, these results are stated in a non-technical way, with references to precise versions in the body of the paper. Several of our theorems, which are valid for general rational inner functions, can be illustrated by examining the simple rational inner function

[TABLE]

The RIF $\phi$ has a single singularity at $(1,1)$ and a computation shows that $\phi(1,1)=1$, in the sense of non-tangential limits. By solving $-(2z_{1}z_{2}-z_{1}-z_{2})=2-z_{1}-z_{2}$ we find that the level curve of $\phi$ corresponding to this value coincides with the union of coordinate axes,

[TABLE]

and thus consists of smooth components with a transversal intersection. For $\lambda\in\mathbb{T}\setminus\{1\}$, the associated level curves of $\phi$ are smooth and are described by

[TABLE]

the reciprocal of a Möbius transformation of the disk. A plot of level curves, including the value curve, is provided in Figure 1. Here, and throughout, we identify the two-torus with $(-\pi,\pi]\times(-\pi,\pi]$ for computational purposes. Thus the point $(1,1)\in\mathbb{T}^{2}$ corresponds to $(0,0)$ in our plots.

We now observe that all level curves pass through the singularity at $(1,1)$ in the second and fourth quadrants, and any pair of level curves with the exception of $\mathcal{C}^{*}_{1}(1,1)$ touch to order $2$ at the origin: that is, for any pair $\lambda,\mu\in\mathbb{T}\setminus\{1\}$,

[TABLE]

The first fact illustrates what was called a Horn Lemma in [BPS18]: level curves of an RIF are highly constrained in the way they pass through singularities. A precise formulation is given in Lemma 2.13. We use the Horn Lemma to prove one of the main results of our paper, namely that smoothness of unimodular level curves holds for any RIF.

Theorem** (2.9).**

The components of each unimodular level curve $\mathcal{C}_{\lambda}$ of a rational inner function $\phi$ can be parametrized by analytic functions.

The fact that we may have to resolve a level curve into components is illustrated above by the splitting of $\mathcal{C}^{*}_{1}(1,1)$ into horizontal and vertical axes.

ln [BPS18], the contact orders of the rational inner function (1) with respect to both variables were computed, and were both found to be equal to $2$. In this paper, we show that contact order for a general RIF can be computed from unimodular level curves.

Theorem** (3.1).**

The $z_{1}$-contact order of $\phi$ at a singularity $\tau\in\mathbb{T}^{2}$ is determined by the maximal order of vanishing of $\psi^{\lambda}_{i}-\psi^{\mu}_{j}$ at $\tau$, where $z_{1}=\psi^{\mu}_{i}(z_{2})$ ($i=1,\ldots,m$) and $z_{1}=\psi^{\nu}_{j}(z_{2})$