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.