, Volume 135, Issue 1-2, pp 43-62
Date: 11 Nov 2010

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

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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 1T 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.