On the polynomial convexity of the union of more than two totally-real planes in C^2

TL;DR

This paper investigates the local polynomial convexity at the origin of unions of multiple totally-real planes in a72, providing new sufficient conditions and extending known results for pairs of planes to more complex configurations.

Contribution

It introduces a novel sufficient condition for polynomial convexity of unions of multiple totally-real planes in a72, generalizing Weinstock's theorem and analyzing cases with three planes.

Findings

Established a sufficient condition for N=2 planes.

Provided an open condition for three planes.

Argued the optimality of the conditions for three planes.

Abstract

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.