Polynomial hull ($\hat{K}=\{z:|p(z)|\le {\sup }_K|p|\text{ for every polynomial }p\}$)

For a compact set $K\subset {\mathbb{C}}^n$, its polynomial hull is the set displayed, where $p$ ranges over all complex multivariate polynomials. It contains $K$. In one complex dimension, bounded complementary components are filled in: the maximum modulus principle bounds every polynomial inside those components by its boundary values. Polynomial approximation and separation distinguish points outside the hull.

For a compact set $K\subset \mathbb{C}$, its polynomial hull is $\hat{K}=\{z:|P(z)|\le {\sup }_K|P|\text{ for every polynomial }P\}$. In one complex variable this is $K$ together with all bounded components of its complement. The maximum modulus principle proves inclusion of each filled hole. Conversely, polynomial approximation of an exterior pole by the Runge approximation theorem separates an exterior point from the filled compact set. The filled set has connected complement, which makes it the natural set for polynomial approximation of functions whose poles lie in the unbounded complementary component.