Local polynomial convexity of the union of two totally real surfaces at their intersection

Abstract

We consider the following question: Let $S_1$ and $S_2$ be two smooth, totally-real surfaces in $\mathbb{C}^2$ that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is $S_1 \cup S_2$ locally polynomially convex at the origin? If $T_0S_1 \cap T_0S_2=\{0\}$, then it is a folk result that the answer is yes. We discuss an obstruction to the presumed proof, and provide a different approach. When dimension of $T_0S_1 \cap T_0S_2$ over the field of real numbers is 1, we present a geometric condition under which no consistent answer to the above question exists. We then discuss conditions under which we can expect local polynomial convexity.