On the polynomial convexity of the union of two maximal totally real subspaces of \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: On the polynomial convexity of the union of two maximal totally real subspaces of \(\mathbb C^n\)
zbMATH Open document ID: 0628.32015
DOI: 10.1007/BF01457016
author: Barnet M. Weinstock
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164453
review text: An example due to Burns has shown that the union of two totally real planes which meet only at \(0\) in \({\mathbb C}^ 2\) need not have a polynomially convex intersection with the unit ball. This paper contains a complete description of those pairs of totally real \(n\)-dimensional subspaces of \({\mathbb C}^ n\) which meet only at \(0\) whose union has the property that every compact subset is polynomially convex. After a change of coordinates one can reduce to the case that the pair is \(({\mathbb R}^ n,(A+i){\mathbb R}^ n)\) where \(A\) is a real \(n\times n\) matrix such that \((A+i)\) is non-singular. The paper \(({\mathbb R}^ n,(A+i){\mathbb R}^ n)\) has the stated property if and only if \(A\) has no eigenvalue of the form \(t_i\), where \(t>1\). In this case, every continuous function on \({\mathbb R}^ n\cup (A+i){\mathbb R}^ n=X\) can be approximated uniformly on compact sets by polynomials. If \(A\) has \(k\) eigenvalues of the form \(t_i\), \(t>1,\) and if each such eigenvalue is of simple type then there is a \(k\)-parameter family of polyannular regions in \({\mathbb C}^ n\) whose distinguished boundaries lie in \(X\) such that every continuous function on \(X\) which extends analytically to each polyannulus is the uniform limit on compact subsets of \(X\) of polynomials.
Mathematics Subject Classification ID: 32E20
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32E30
Mathematics Subject Classification ID: 32D15
zbMATH DE Number: 4021578
zbMATH Keywords: union of totally real \(n\)-dimensional subspaces
zbMATH Keywords: polynomially convex compact subsets
zbMATH Keywords: polynomial approximation
MaRDI profile type: Publication
cites work: Q4774586
cites work: Totally real imbeddings and the universal covering spaces of domains of holomorphy: Some examples
cites work: Q4104417
OpenAlex ID: W2093993763
description: scientific article; zbMATH DE number 4021578