2011/08/29 by Sushil Gorai, Gorai, Sushil
Computer Science · Engineering · Mathematics · #32E20 #46J10 #Advanced Topics in Algebra #Complex Variables (math.CV) #FOS: Mathematics #Matrix Theory and Algorithms #graph theory and CDMA systems #math.CV #msc:32E20 #msc:46J10
paper · pdf · doi:10.48550/arxiv.1108.5625
19 pages
openalex publication_date 2011/08/29 · arxiv created 2011/08/30 · arxiv updated 2011/08/31 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this paper we shall discuss local polynomial convexity at the origin of the union of finitely many totally-real planes through 0 ∈ℂ2. The planes, say P0,..., PN, satisfy a mild transversality condition that enables us to view them in Weinstock normal form, i.e. P0=ℝ2 and Pj=M(Aj):=(Aj+i\mathbbI)ℝ2, j=1,...,N, where each Aj is a 2× 2 matrix with real entries. Weinstock has solved the problem completely for N=1 (in fact, for pairs of transverse, maximally totally-real subspaces in ℂn ∀ n≥ 2). Using a characterization of simultaneous triangularizability of 2× 2 matrices over the reals, given by Florentino, we deduce a sufficient condition for local polynomial convexity of the union of the above planes at 0∈ ℂ2. Weinstock's theorem for ℂ2 occurs as a special case of our result. The picture is much clearer when N=2. For three totally-real planes, we shall provide an open condition for local polynomial convexity of the union. We shall also argue the optimality (in an appropriate sense) of the conditions in this case.