Bounded analytic functions on the polydisc

In the paper 'Distinguished Varieties' Agler and McCarthy proved several connections between the theory of bounded analytic functions on the bidisc and 1-dimensional algebraic varieties that exit the bidisc through the distinguished boundary. In this paper we extend several of their results to the theory of bounded analytic functions on the polydisc. We give sufficient conditions for a rational inner function on the polydisc to be uniquely determined in the Schur class of the polydisc by it's values on a finite set of points. This follows from giving sufficient conditions for a Pick problem on the polydisc to have a unique solution. We demonstrate that our results can be though of as a generalization to the polydisc of the Schwarz Lemma and Pick's Theorem on the disc. We establish our results by studying the Pick problem with Hilbert function space technique