Exploring Polynomial Convexity Of Certain Classes Of Sets

Abstract

Let K be a compact subset of Cn . The polynomially convex hull of K is defined as The compact set K is said to be polynomially convex if = K. A closed subset is said to be locally polynomially convex at if there exists a closed ball centred at z such that is polynomially convex. The aim of this thesis is to derive easily checkable conditions to detect polynomial convexity in certain classes of sets in This thesis begins with the basic question: Let S1 and S2 be two smooth, totally real surfaces in C2 that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is locally polynomially convex at the origin? If then it is a folk result that the answer is, “Yes.” We discuss an obstruction to the presumed proof, and use a different approach to provide a proof. When dimR it turns out that the positioning of the complexification of controls the outcome in many situations. In general, however, local polynomial convexity of also depends on the degeneracy of the contact of T0Sj with We establish a result showing this. Next, we consider a generalization of Weinstock’s theorem for more than two totally real planes in C2 . Using a characterization, recently found by Florentino, for simultaneous triangularizability over R of real matrices, we present a sufficient condition for local polynomial convexity at of union of finitely many totally real planes is C2 . The next result is motivated by an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc generated by z and h — where h is a nowhereholomorphic harmonic function on D that is continuous up to ∂D — equals . The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h+ R, where R is a nonharmonic perturbation whose Laplacian is “small” in a certain sense. Ideas developed for the latter result, especially the role of plurisubharmonicity, lead us to our final result: a characterization for compact patches of smooth, totallyreal graphs in to be polynomially convex.