Nevanlinna-Pick families and singular rational varieties (Q2220153)

Originally, the Nevanlinna-Pick interpolation problem was stated and solved for analytic functions from the disk to the disk. Replacing the domain of bounded analytic interpolation by a multiply connected domain turned out to be quite challenging, cf.\ [\textit{M. B. Abrahamse}, Mich. Math. J. 26, 195--205 (1979; Zbl 0442.30034)]. Concepts of the geometry of algebraic curves naturally entered there into full light. Exploring more complicated domains of definition for bounded analytic interpolation, such as singular affine curves, revealed a whole new phenomenology, at the interface between algebraic geometry and operator algebras. The article under review treats with high authority such a situation: affine rational curves carrying a Hilbert space metric induced by Drury-Arveson inner product. The main results align to and clarify a variety of known theorems proved by several authors over the last decade [\textit{M. A. Dritschel} et al., Oper. Matrices 10, No. 4, 829--861 (2016; Zbl 1387.47008); \textit{M. Raghupathi}, J. Math. Anal. Appl. 355, No. 1, 258--276 (2009; Zbl 1221.30083)]. For the entire collection see [Zbl 1455.47001].