Nevanlinna-Pick Families and Singular Rational Varieties

2018/11/08 by Davidson, Kenneth R., Shamovich, Eli
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1811.03247

Abstract

The goal of this note is to apply ideas from commutative algebra (a.k.a. affine algebraic geometry) to the question of constrained Nevanlinna-Pick interpolation. More precisely, we consider subalgebras A ⊂ ℂ[z₁,…,z_d], such that the map from the affine space to the spectrum of A is an isomorphism except for finitely many points. Letting \mathfrakA be the weak-* closure of A in M_d -- the multiplier algebra of the Drury-Arveson space. We provide a parametrization for the Nevanlinna-Pick family of M_k(\mathfrakA) for k ≥ 1. In particular, when k=1 the parameter space for the Nevanlinna-Pick family is the Picard group of A.

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