A note on polynomial convexity of the union of finitely many totally-real planes in C2

In this paper we 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's normal form, i.e., P0=R2 and Pj=M(Aj):=(Aj+iI)R2, j=1,…,N, where each Aj is a 2×2 matrix with real entries. Weinstock has solved the problem completely 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.