Polynomially convex set

Summary

A compact set $K \subseteq \C^n$ is polynomially convex if every point outside $K$ can be separated from $K$ by a polynomial. This means that some polynomial is larger in absolute value at the point than everywhere on $K$. They are the natural sets for polynomial approximation. By the Oka–Weil theorem, every function holomorphic near such a set is a uniform limit of polynomials on it. In $\C$ they are the compact sets with connected complement. For $n \ge 2$ the condition is much subtler, and it is not determined by the topology of $K$ alone.