  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.
