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

In this paper we discuss local polynomial convexity at the origin of the union of finitely many totally-real planes through 0 ∈ C 2 . The planes, say P 0 , … , P N , satisfy a mild transversality condition that enables us to view them in Weinstockʹs normal form, i.e., P 0 = R 2 and P j = M ( A j ) : = ( A j + i I ) R 2 , j = 1 , … , N , where each A j is a 2 × 2 matrix with real entries. Weinstock has solved the problem completely for pairs of transverse, maximally totally-real subspaces in C n ∀ 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 ∈ C 2 . Weinstockʹs theorem for C 2 occurs as a special case of our result.