Polynomially convex set

Remarks

Always $K \subseteq \hat K$, and $\hat K$ is compact. It coincides with the holomorphically convex hull of $K$ in $D = \C^n$, because entire functions are uniform limits of their Taylor polynomials on compact sets.

Example: a compact convex set $K$ is polynomially convex. If $z \notin K$, there is a complex linear function $\ell$ with $\operatorname{Re} \ell(z) > \max_K \operatorname{Re} \ell$. Then $f = e^{\ell}$ satisfies $\abs{f(z)} > \max_K \abs{f}$, and the Taylor polynomials of $f$ converge to $f$ uniformly on $K \cup \set{z}$, so some polynomial $p$ has $\abs{p(z)} > \max_K \abs{p}$.

Non-example: the unit circle $K = \set{\abs{z} = 1}$ in $\C$. By the maximum principle, $\abs{p(0)} \le \max_K \abs{p}$ for every polynomial $p$, so $0 \in \hat K$ but $0 \notin K$.

For $n = 1$, a compact set $K \subseteq \C$ is polynomially convex exactly when $\C \setminus K$ is connected.