Mathematics – Complex Variables
Scientific paper
2011-08-29
Mathematics
Complex Variables
19 pages
Scientific paper
In this paper we shall discuss local polynomial convexity at the origin of the union of finitely many totally-real planes through $0 \in\mathbb{C}^2$. The planes, say $P_0,..., P_N$, satisfy a mild transversality condition that enables us to view them in Weinstock normal form, i.e. $P_0=\mathbb{R}^2$ and $P_j=M(A_j):=(A_j+i\mathbb{I})\mathbb{R}^2$, $j=1,...,N$, where each $A_j$ is a $2\times 2$ matrix with real entries. Weinstock has solved the problem completely for N=1 (in fact, for pairs of transverse, maximally totally-real subspaces in $\mathbb{C}^n\, \forall n\geq 2$). Using a characterization of simultaneous triangularizability of $2\times 2$ matrices over the reals, given by Florentino, we deduce a sufficient condition for local polynomial convexity of the union of the above planes at $0\in \mathbb{C}^2$. Weinstock's theorem for $\mathbb{C}^2$ 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.
No associations
LandOfFree
On the polynomial convexity of the union of more than two totally-real planes in C^2 does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the polynomial convexity of the union of more than two totally-real planes in C^2, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the polynomial convexity of the union of more than two totally-real planes in C^2 will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-126801