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

## Authors

- First Online:

- Received:
- Revised:

DOI: 10.1007/s00229-010-0405-x

- Cite this article as:
- Gorai, S. manuscripta math. (2011) 135: 43. doi:10.1007/s00229-010-0405-x

## 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.