Local geometry of zero sets of holomorphic functions near the torus
type: Item
instance of: scholarly article
title: Local geometry of zero sets of holomorphic functions near the torus
zbMATH Open document ID: 1153.14002
author: Jim Agler
author: John E. McCarthy
author: Mark Stankus
published in: The New York Journal of Mathematics
publication date: 2008-12-03T00:00:00Z
full work available at URL: https://eudml.org/doc/129524
full work available at URL: 14-24.html
review text: Let \(\mathbb D\) denote the unit disk in the complex plane, \(\mathbb T\) be a unit circle, \(\mathbb E\) be the complement of the closed unit disk in \(\mathbb C\) and let \(A(\mathbb D^n)\) denote the polydisk algebra, the algebra of functions that are continuous on the closure of \(\mathbb D^n\) and holomorphic on the interior. When studying function theory on the polydisk \(\mathbb D^n\), it is often useful to focus on the torus \(\mathbb T^n\), which is the distinguished boundary of \(\mathbb D^n\). In several important ways, the behavior of a function in \(A(\mathbb D^n)\) is controlled by its behavior on \(\mathbb T^n\): not only is \(\mathbb T^n\) a set of uniqueness, but every function in the algebra attains its maximum modulus on \(\mathbb T^n\). We say that a variety (an irreducible algebraic set) \(V\) is toral if its intersection with \(\mathbb T^n\) is fat enough to be a determining set for holomorphic functions on \(V\). Otherwise we call the variety atoral. A polynomial in \(\mathbb C[z_1,\dots,z_n]\) is toral (respectively, atoral) if the zero set of every irreducible factor is toral (respectively, atoral). Earlier the authors shown that knowing an algebraic set is toral has interesting consequences in function theory and operator theory. In this paper, they study the localized versions of this and other related geometric properties of zero sets. Let \(f\) be a holomorphic function defined on an open subset \(G\) of \(\mathbb C^n\) and suppose \(\tau\in\mathbb T^n\cap G\). We say \(f\) is locally toral at \(\tau\) if \(f(\tau) = 0\) and, for every neighborhood \(U\) of \(\tau \in G\), there is a neighborhood \(V\) of \(\tau \in U\) such that \(Z_f\cap V\cap \mathbb T^n\) is determining for \(Z_f\cap V\). An irreducible holomorphic function may no longer be irreducible if its domain is restricted to a smaller set, however, there is a well-defined notion of locally irreducible at a point. If \(f\) is locally irreducible at \(\tau\), then we say \(f\)…
Mathematics Subject Classification ID: 14-06
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 32B99
Mathematics Subject Classification ID: 14J99
Mathematics Subject Classification ID: 32A38
zbMATH DE Number: 5377300
zbMATH Keywords: polydisk algebra of holomorphic functions
zbMATH Keywords: local geometry of analytic sets near torus
zbMATH Keywords: global geometry of algebraic sets
zbMATH Keywords: toral polynomial
zbMATH Keywords: locally toral
zbMATH Keywords: locally atoral
zbMATH Keywords: locally sidal
zbMATH Keywords: locally asidal
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5377300