Distinguished varieties and the Nevanlinna-Pick interpolation problem on the symmetrized bidisk

Abstract

We study the uniqueness of the solutions of a solvable Pick interpolation problem in the symmetrized bidiskThe uniqueness set is the largest set in G where all the solutions to a solvable Pick problem coincide. For a solvable Pick problem in G, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in G. The algebraic variety is called the uniqueness variety. In the first main result of this paper, we show that if an N -point solvable Pick problem is such that it has no solutions of supremum norm (over G) less than one and that each of the (N − 1)-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the N -point problem contains a distinguished variety containing all the initial nodes. Here, a distinguished variety is an algebraic variety that intersects the domain G and exits through its distinguished boundary. The proof of the first main result requires a thorough understanding of distinguished varieties. Indeed, this article is as much a study of the uniqueness varieties as it is about distinguished varieties. We obtain a complete algebraic and geometric characterizations of distinguished varieties.

Rational penta-inner functions and the distinguished boundary of the pentablock

“…The symmetrized bidisc and the tetrablock have attracted a considerable amount of interest in recent years. For a greater exposition on these domains, see [1,4,7,9,12,14,17,21]. An equivalent definition of the symmetrized bidisc is given by…”

Determining sets for holomorphic functions on the symmetrized bidisk

“…Here, for a function f, we use the standard notation Z( f ) for the zero set of f. Note that if E is the uniqueness set for ( f , D), then for every z ∈ Ω/E, there exists a function g ∈ S(Ω) such that g = f on D but f (z) ≠ g(z). Remarkably, when D is a finite subset of G, then for any function f ∈ S(G), the uniqueness set for ( f , D) is an affine variety (see [6,25]). This is owing to the fact that every solvable Pick data in G always has a rational inner solution (see [3,25]).…”

“…Here, the distinguished boundary of a bounded domain Ω ⊂ C d is the Šilov boundary with respect to the algebra of complex-valued functions continuous on Ω and holomorphic in Ω. A special type of algebraic varieties has been prevalent in the study of uniqueness of the solutions of a Pick interpolation problem (see [6,[22][23][24][25]27]). We define it below.…”

“…Remarkably, when D is a finite subset of G, then for any function f ∈ S(G), the uniqueness set for (f, D) is an affine variety (see [16], [3]). This is owing to the fact that every solvable Pick data in G always has a rational inner solution (see [16], [5]). Also note that if f and g agree on D, then D is determining for (f, E) if and only if D is determining for (g, E) also.…”

“…An example of a distinguished variety with respect to G is {(2z, z 2 ) : z ∈ C}. We refer the readers to the papers [3,11,16,17,29] for results concerning these varieties and their connection to interpolation problems.…”