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 ₀ S ₁ ∩ T ₀ S 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 ${dim_\mathbb{R}(T_0S_1\cap T_0S_2)=1}$, 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.