. Motivated by studying boundary singularities of rational functions in two variables that are analytic on a domain, we investigate local integrability on R2 near (0, 0) of rational functions with denominator non-vanishing in the bi-upper half-plane but with an isolated zero (with respect to R2) at the origin. we give a necessary and sufficient test for membership in a local Lp(R2) space and we give a complete description of all numerators Q such that Q/P is locally in a given Lp space. As applications, we prove that every bounded rational function on the bidisk has partial derivatives belonging to L1 on the two-torus. In addition, we give a new proof of a conjecture started by Bickel, Knese, Pascoe, and Sola [Ann. Polon. Math. 133 (2024), pp. 95-169] and completed by Kollar [Period. Math. Hungar. 88 (2024), pp. 1-7] characterizing the ideal of Q such that Q/P is locally bounded. A larger takeaway from this work is that a local model for stable polynomials we employ is a flexible tool and may be of use for other local questions about stable polynomials.

We study polynomials with no zeros on the unit ball in complex Euclidean space with a view toward characterizing when a rational function is bounded on the ball. We give a complete local description of such polynomials in two variables near a boundary zero. In higher dimensions, we give a partial characterization of a simple boundary zero. Several applications are given including boundedness of rational functions with boundary singularities and constructions of examples with prescribed local properties.

Corrigendum to “Three radii associated to Schur functions on the polydisk”

We give an elementary proof that the Agler norm of a function is determined by its norm on commuting tuples of nilpotent matrices. The proof is a variation on a standard cone separation argument. The topic is closely related to the Eschmeier-Patton-Putinar formulation of Carath & eacute;odory-Fej & eacute;r interpolation.

Given a polynomial p$p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials q$q$ with the property that the rational function q/p$q/p$ is bounded near a boundary zero of p$p$. We give a complete description of this ideal of numerators in the case where the zero set of p$p$ is smooth and satisfies a nondegeneracy condition. We also give a description of the ideal in terms of an integral closure when p$p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.

We prove several results about functions which preserve the Schur-Agler class under Hadamard or coefficient-wise product. First, functions which preserve the Schur class necessarily preserve the Schur-Agler class. Second, “moments” of certain commuting operator tuples form coefficients of Schur-Agler class preservers. Finally, any preserver of the full matrix Schur-Agler class must have coefficients given by moments of commuting operator tuples. We also point out that any counterexample to the multivariable von Neumann inequality can be used to derive a non-trivial Agler class preserver.

We define the Schur-Agler class in infinite variables to consist of functions whose restrictions to finite dimensional polydisks belong to the Schur-Agler class. We show that a natural generalization of an Agler decomposition holds and the functions possess transfer function realizations that allow us to extend the functions to the unit ball of $\ell^\infty$. We also give a Pick interpolation type theorem which displays a subtle difference with finitely many variables. Finally, we make a brief connection to Dirichlet series derived from the Schur-Agler class in infinite variables via the Bohr correspondence.

Given a polynomial p $p$ with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials q $q$ with the property that the rational function q / p $q/p$ is bounded near a boundary zero of p $p$ . We give a complete description of this ideal of numerators in the case where the zero set of p $p$ is smooth and satisfies a nondegeneracy condition. We also give a description of the ideal in terms of an integral closure when p $p$ has an isolated zero on the distinguished boundary. Constructions of multivariate stable polynomials are presented to illustrate sharpness of our results and necessity of our assumptions.

. This article examines three radii associated to bounded analytic functions on the polydisk: the well-known Bohr radius, the Bohr-Agler radius, and the Schur-Agler radius. We prove explicit upper and lower bounds for the Bohr-Agler radius, an explicit lower bound for the Schur-Agler radius, and an asymptotic upper bound for the Schur-Agler radius. The Bohr-Agler radius obeys the same (known) asymptotic as the Bohr radius while we show the Schur-Agler radius is roughly of the same growth as the Bohr radius. As a corollary, we bound the Bohr radius on the bidisk below by 0.3006. Finally, we improve some estimates of P. G. Dixon on Agler norms of homogeneous polynomials using some modern inequalities.

We provide detailed local descriptions of stable polynomials in terms of their homogeneous decompositions, Puiseux expansions, and transfer function realizations. We use this theory to first prove that bounded rational functions on the polydisk possess nontangential limits at every boundary point. We relate higher non-tangential regularity and distinguished boundary behavior of bounded rational functions to geometric properties of the zero sets of stable polynomials via our local descriptions. For a fixed stable polynomial p, we analyze the ideal of numerators q such that q/p is bounded on the bi-upper half plane. We completely characterize this ideal in several geometrically interesting situations including smooth points, double points, and ordinary multiple points of p. Finally, we analyze integrability properties of bounded rational functions and their derivatives on the bidisk.

Rational inner functions are a generalization of finite Blaschke products to several variables. In this article we survey a variety of results about rational inner functions related to interpolation, sums of squares formulas, and boundary behavior. We mostly focus on two variables however in the final section we discuss higher dimensions.

We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization.A slight modification of the approach extends to rational functions which are isometric on the two-torus and we use this to give a largely elementary new proof of the existence of Agler decompositions for every matrix-valued Schur function in two variables.We use a recent result of Dritschel to prove two variable matrix-valued rational Schur functions always have finite-dimensional contractive transfer function realizations.Finally, we prove that two variable matrix-valued polynomial inner functions have transfer function realizations built out of special nilpotent linear combinations.

Motivated by the discovery of large-scale ionized clouds around AGN host galaxies, and particularly the large fraction of those which are consistent with photoionized gaseous tidal debris, we have searched for [O III] emission over wide fields around a set of Seyfert galaxies previously mapped in H I, many of which show extended gas features. The detection threshold was designed to reach mean emission-line surface brightness 10 times fainter than seen in such AGN-ionized clouds as Hanny's Voorwerp, so that similar structures at larger distances (and ages) could be detected. Of 24 Seyfert galaxies, we find one extended emission feature, a discrete cloud projected 12 kpc from the center of Mkn 1 and spanning a transverse extent of 8 kpc. Optical spectroscopy of the Mkn 1 cloud confirms its redshift association with the Mkn 1- NGC 451 galaxy pair, shows it to closely match the kinematics of nearby H I, and reveals emission-line ratios requiring photoionization by the AGN at roughly the direct observed luminosity of the nucleus. Given the small fraction of H I features with detected [O III] emission, we constrain the typical opening angle of ionization cones in Seyfert galaxies to be of order 20 deg, if active episodes are long compared to the light-travel times involved. An appendix presents a derivation of an analytical expression for the probability of intersection of a cone with randomly oriented arcs, approximating the geometry of H I clouds and tails exposed to ionization cones. For the entire sample, the full opening angle of bicones must be <20 deg if the AGN are continuously bright for scales longer than the light-travel times across the H I structures. Since many ionization cones are observed to be much broader than this, our low detection fraction may add to evidence for the ubiquity of strong variations in AGN luminosity on scales 10,000-100,000 years.

A short and simple proof of necessity in the McCullough-Quiggin characterization of positive semi-definite kernels with the complete Pick property is presented.

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 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.

A classical inequality of Szász bounds polynomials with no zeros in the upper half plane entirely in terms of their first few coefficients. Borcea–Brändén generalized this result to several variables as a piece of their characterization of linear maps on polynomials preserving stability. In this paper, we use determinantal representations to prove Szász type inequalities in two variables and then prove that one can use the two variable inequality to prove an inequality for several variables.

Consider the Dirichlet-type space on the bidisk consisting of holomorphic functions f(z_1,z_2):=∑_k,l≥0a_klz_1^kz_2^l such that ∑_^k,l≥0(k+1)^α_1(l+1)^α_2|a_kl|^2<∞. Here the parameters α 1 , α 2 are arbitrary real numbers. We characterize the polynomials that are cyclic for the shift operators on this space. More precisely, we show that, given an irreducible polynomial p ( z 1 , z 2 ) depending on both z 1 and z 2 and having no zeros in the bidisk: if α 1 + α 2 ≤ 1, then p is cyclic; if α 1 + α 2 > 1 and minα 1 , α 2 ≤ 1, then p is cyclic if and only if it has finitely many zeros in the two-torus 𝕋^2 ; if minα 1 , α 2 > 1, then p is cyclic if and only if it has no zeros in 𝕋^2 .

We prove a generalization of the Hermitian version of the Helton-Vinnikov determinantal representation of hyperbolic polynomials to the class of semi-hyperbolic polynomials, a strictly larger class, as shown by an example. We also prove that certain hyperbolic polynomials affine in two out of four variables divide a determinantal polynomial. The proofs are based on work related to polynomials with no zeros on the bidisk and tridisk.

We study two questions. When does a function belong to the union of Lebesgue spaces and when does a function have an $A_1$ majorant? We show these questions are fundamentally related. For functions restricted to a fixed cube we prove that the following are equivalent: a function belongs to $L^p$ for some $p>1$; the function has an $A_1$ majorant; for any $p>1$ the function belongs to $L^p_w$ for some $A_p$ weight $w$. We also examine the case of functions defined on ${\mathbb R}^n$ and give characterizations of the union of $L^p_w$ over $w$ in $A_p$ and when a function has an $A_1$ majorant on all of ${\mathbb R}^n$.