Polynomial Hulls of Smooth Discs: A Survey

Abstract

We remark that as a consequence of Taylor’s Theorem, polynomially convex sets are the compact sets that are holomorphically convex relative to the ambient space C. The problem of deciding whether a compact set in C is polynomially convex or not is a fundamental and difficult problem in complex analysis. We note that polynomial convexity is a global condition. A closed subset E ⊂ C is called locally polynomially convex at z ∈ E if there exists r > 0 such that E ∩ clos B(z, r) is polynomially convex, where