We study the uniqueness of the solutions of a solvable Pick interpolation problem in the symmetrized bidisk \(\begin{aligned} {\mathbb {G}}=\{(z_1+z_2,z_1z_2): z_1,z_2\in {\mathbb {D}}\}. \end{aligned}\) The uniqueness set is the largest set in \({\mathbb {G}}\) where all the solutions to a solvable Pick problem coincide. There is a canonical construction of an algebraic variety, which coincides with the uniqueness set in \({\mathbb {G}}\) . The algebraic variety is called the uniqueness variety. A solvable Pick problem is called extremal if it has no solutions of supremum norm (over \({\mathbb {G}}\) ) less than one. We show that if an N-point extremal Pick problem is such that none of the \((N-1)\) -point sub-problems is extremal, then the uniqueness variety contains a distinguished variety that contains all the initial nodes. Here, a distinguished variety is an algebraic variety that intersects the domain \({\mathbb {G}}\) 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 Nevanlinna-Pick interpolation problem as it is about distinguished varieties. We obtain complete algebraic and geometric characterizations of distinguished varieties solving an unsettled problem left open by Pal and Shalit, J. Funct. Anal. 2014.
