Extreme points and saturated polynomials

Abstract

We consider the problem of characterizing the extreme points of the set of analytic functions f on the bidisk with positive real part and f (0) = 1. If one restricts to those f whose Cayley transform is a rational inner function, one gets a more tractable problem. We construct families of such f that are extreme points and conjecture that these are all such extreme points. These extreme points are constructed from polynomials dubbed T 2 -saturated, which roughly speaking means they have no zeros in the bidisk and as many zeros as possible on the boundary without having infinitely many zeros.

Clark Measures for Rational Inner Functions

“…In [32], Rudin posed the question "What are the extreme points of P 2 (or equivalently, of M (P 2 ))?" While this question is still open, a number of interesting examples and related results (often in the n-variable situation) have been proved by Forelli [15], Knese [22], and Mc-Donald [25,26,27,28]. As the Clark measures σ α are trivially in M (P 2 ) when φ(0) = 0, it makes sense to consider our investigations in the context of Rudin's question and these subsequent results.…”

Local theory of stable polynomials and bounded rational functions of several variables

“…One basic conclusion of this is that if x n → (0, 0) in R 2 and f (x n ) → s ∈ R then eventually the points x n are trapped in the region (31) for s 1 = s − δ, s 2 = s + δ for δ > 0. Another conclusion is that the closure of f (D 2 r ∩ H 2 ) contains R for all r > 0 since all of the level sets of f pass through (0, 0) and points on these level sets can be perturbed to points of H 2 .…”

“…Since V α is unitary for α ∈ T, we also see that D is a rank one perturbation of a unitary. This is presented in Section 9 of [31].…”

“…Now, since p(z) + wq(z) = 0 for z ∈ D 2 and w ∈ D we get a more general family of perturbations of stable polynomials than simply p + w p by considering p + wq. Further interest in perturbations of stable polynomials comes from a problem studied in [31] of characterizing the extreme points of real rational Pick functions. Rational Schur functions on D 2 possess contractive transfer function realizations.…”

An elementary method to compute the algebra generated by some given matrices and its dimension

“…In early 2017, an implementation of the method was made in Mathematica by Igor Klep to find a counterexample to an unpublished conjecture of Knese about extreme points of the cone of rational inner Herglotz functions, as in acknowledged in Section 10 of [7]. The Mathematica code has been used modified and shared for other problems by Klep and others.…”