On the polynomial convexity of the union of two maximal totally real subspaces of \(\mathbb C^n\)
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.
union of totally real \(n\)-dimensional subspaces
scientific article; zbMATH DE number 4021578