Toral Algebraic Sets and Function Theory on Polydisks

2005/08/17 by Jim Agler, John McCarthy, Agler, Jim +3 · 1 citation
Mathematics · #14J70 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:14J70

paper · pdf · doi:10.48550/arxiv.math/0508331

arxiv created 2005/08/17 · arxiv updated 2009/12/01

Abstract

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.

Cited by

• Pairs of Clark Unitary Operators on the Bidisk and their Taylor Joint Spectra

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