We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability ...of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree $(m,n,1)$ RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of nontrivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.
Motivated by classical notions of partial convexity, biconvexity, and bilinear matrix inequalities, we investigate the theory of free sets that are defined by (low degree) noncommutative matrix ...polynomials with constrained terms. Given a tuple of symmetric polynomials
Γ
, a free set
K
is called
Γ
-convex if for all
X
∈
K
and isometries
V
satisfying
V
∗
Γ
(
X
)
V
=
Γ
(
V
∗
X
V
)
, we have
V
∗
X
V
∈
K
.
We establish an Effros–Winkler Hahn–Banach separation theorem for
Γ
-convex sets; they are delineated by linear pencils in the coordinates of
Γ
and the variables
x
.
Motivated by classical notions of bilinear matrix inequalities (BMIs) and partial convexity, this article investigates partial convexity for noncommutative functions. It is shown that noncommutative ...rational functions that are partially convex admit novel butterfly-type realizations that necessitate square roots. A strengthening of partial convexity arising in connection with BMIs--xy-convexity--is also considered. A characterization of xy-convex polynomials is given. Keywords: Partial convexity, biconvexity, bilinear matrix inequality (BMI), noncommutative rational function, noncommutative polynomial, realization theory.
We provide an effective single-matrix criterion, in terms of what we call the elementary Pick matrix, for the solvability of the noncommutative Nevanlinna-Pick interpolation problem in the row ball, ...and provide some applications. In particular we show that the so-called “column-row property” fails for the free semigroup algebras, in stark contrast to the analogous commutative case. Additional applications of the elementary Pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting. Finally we present some numerical results related to the failure of the column-row property.
A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as ...normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem is given: a positive semidefinite univariate trace polynomial is a quotient of sums of products of squares and traces of squares of trace polynomials.
We analyze the singularities of rational inner functions (RIFs) on the unit bidisk and study both when these functions belong to Dirichlet‐type spaces and when their partial derivatives belong to ...Hardy spaces. We characterize derivative Hp membership purely in terms of contact order, a measure of the rate at which the zero set of an RIF approaches the distinguished boundary of the bidisk. We also show that derivatives of RIFs with singularities fail to be in Hp for p⩾32 and that higher non‐tangential regularity of an RIF paradoxically reduces the Hp integrability of its derivative. We derive inclusion results for Dirichlet‐type spaces from derivative inclusion for Hp. Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of RIFs fails to belong to the unweighted Dirichlet space.
Given a polynomial \(p\) with no zeros in the polydisk, or equivalently the poly-upper half-plane, we study the problem of determining the ideal of polynomials \(q\) with the property that the ...rational function \(q/p\) is bounded near a boundary zero of \(p\). We give a complete description of this ideal of numerators in the case where the zero set of \(p\) is smooth and satisfies a non-degeneracy condition. In three variables, we give a description of the ideal in terms of an integral closure when \(p\) 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 paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators. These invariant skew fields are always finitely ...generated, which contrasts with the free algebra case. For abelian groups or solvable groups G with a well-behaved representation theory it is shown that the invariant skew fields are free on |G|(d−1)+1 generators. Finally, positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented.
Call a noncommutative (nc) rational function r regular if it has no singularities, that is, r(X) is defined for all tuples of self‐adjoint matrices X. In this paper, regular nc rational functions r ...are characterized via the properties of their (minimal size) linear systems realizations r=b∗L−1c. It is shown that r is regular if and only if L=A0+∑jAjxj is free elliptic. Roughly speaking, a linear pencil L is free elliptic if, after a finite sequence of basis changes and restrictions, the real part of A0 is positive definite and the other Aj are skew‐adjoint. The second main result is a solution to an nc version of Hilbert's 17th problem: a positive regular nc rational function is a sum of squares.
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