Postprint version. Published in Journal of Geometric Analysis, Volume 16, Issue 4, December 1, 2006, pages 551-562. Copyright © 2006 Springer. The original publication is available at http://www.springerlink.com/content/3374962337rg2m18.
A toral algebraic set A is an algebraic set in Cn whose intersection with Tn is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.