Toral algebraic sets and function theory on polydisks Jim Agler John E. McCarthy Mark Stankus Article

DOI :
10.1007/BF02922130

Cite this article as: Agler, J., McCarthy, J.E. & Stankus, M. J Geom Anal (2006) 16: 551. doi:10.1007/BF02922130
Abstract A toral algebraic set A is an algebraic set in ℂ^{n} whose intersection with T^{n} 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 .

Math Subject Classifications 14J70 32A65

Key Words and Phrases Toral variety inner function H ^{∞} Pick interpolation Communicated by John Erik Fornæss

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© Mathematica Josephina, Inc. 2006

Authors and Affiliations Jim Agler John E. McCarthy Mark Stankus 1. U.C. San Diego La Jolla 2. Washington University St. Louis 3. California Polytechnic State University San Luis Obispo