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

B. Krishna Das, Poornendu Kumar, Haripada Sau

TL;DR

This work analyzes Nevanlinna-Pick interpolation on the symmetrized bidisk $\mathbb{G}$, showing that the uniqueness set for solvable extremal data is an algebraic variety, the uniqueness variety, which contains a $\mathbb{G}$-distinguished variety canonically constructed from the data. It provides two independent proofs of this containment: one via pulling back to the bidisk and applying its distinguished-variety theory, and another intrinsic approach using admissible kernels and complete non-unitarity to produce a $\mathcal{W}$ with sheets through the interpolation nodes that force a fixed interpolation value. A central theme is the tight link between distinguished varieties and numerical contractions: $\mathcal{W}_F=\{(s,p): \det(F^*+pF-sI)=0\}$ is a distinguished variety iff $F$ is completely non-unitary, and every such distinguished variety arises this way. The paper also develops a finite-dimensional realization theory for rational inner functions on $\mathbb{G}$, and an admissible-kernel extension framework that allows controlled extension of interpolation data to distinguished varieties, bridging operator theory, complex geometry, and multivariable interpolation with concrete geometric descriptions of solution sets and examples illustrating the sharpness of the results.

Abstract

Starting with a solvable Nevanlinna-Pick interpolation problem with the initial data coming from the symmetrized bidisk, this paper studies the corresponding uniqueness set, i.e., the largest set in the domain where all solutions to the problem coincide. It is shown that the uniqueness set coincides with an algebraic variety in the domain. The algebraic variety - canonically constructed from the interpolation data - is called the uniqueness variety. It was shown that the uniqueness variety contains a distinguished variety which by definition is the zero set of a two-variable polynomial that intersects the domain and exits through its distinguished boundary. A complete algebraic and geometric characterizations of distinguished varieties are obtained in this paper.

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

TL;DR

This work analyzes Nevanlinna-Pick interpolation on the symmetrized bidisk , showing that the uniqueness set for solvable extremal data is an algebraic variety, the uniqueness variety, which contains a -distinguished variety canonically constructed from the data. It provides two independent proofs of this containment: one via pulling back to the bidisk and applying its distinguished-variety theory, and another intrinsic approach using admissible kernels and complete non-unitarity to produce a with sheets through the interpolation nodes that force a fixed interpolation value. A central theme is the tight link between distinguished varieties and numerical contractions: is a distinguished variety iff is completely non-unitary, and every such distinguished variety arises this way. The paper also develops a finite-dimensional realization theory for rational inner functions on , and an admissible-kernel extension framework that allows controlled extension of interpolation data to distinguished varieties, bridging operator theory, complex geometry, and multivariable interpolation with concrete geometric descriptions of solution sets and examples illustrating the sharpness of the results.

Abstract

Starting with a solvable Nevanlinna-Pick interpolation problem with the initial data coming from the symmetrized bidisk, this paper studies the corresponding uniqueness set, i.e., the largest set in the domain where all solutions to the problem coincide. It is shown that the uniqueness set coincides with an algebraic variety in the domain. The algebraic variety - canonically constructed from the interpolation data - is called the uniqueness variety. It was shown that the uniqueness variety contains a distinguished variety which by definition is the zero set of a two-variable polynomial that intersects the domain and exits through its distinguished boundary. A complete algebraic and geometric characterizations of distinguished varieties are obtained in this paper.

Paper Structure

This paper contains 8 sections, 24 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Distinguished varieties and numerical contractions
  3. Rational inner functions on the symmetrized bidisk
  4. Admissible extension of admissible kernels
  5. Distinguished varieties and extremal Pick problems -- proofs of Theorem \ref{['T:SandIntro']}
  6. The first proof
  7. The second proof
  8. Examples

Key Result

Theorem 1.2

There is always a rational solution to a solvable Pick problem in ${\mathbb{G}}$. Moreover, the rational solution can be obtained so that it is unimodular a.e. (with respect to the Lebesgue measure) on the symmetrized torus, i.e.,

Theorems & Definitions (49)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: See Theorem 5.1-9 in GustavsonRao
  • Proposition 2.2
  • Lemma 2.3
  • proof : Proof of (1)
  • Theorem 2.4: See Theorem 2.1 in ay-jga
  • ...and 39 more