Exercise 13, § 4

Soit $n\in \mathbf{N}$.

a) A subset V of $\mathbf{C}^n$ is polynomially convex if and only if, for every $x\in \mathbf{C}^n-$ V, there exists an entire function $f:\mathbf{C}^n\rightarrow \mathbf{C}$ such that

$|f(x)|>$ sup$_{y\in V}|f(y)|$.

b) Let $K\subset \mathbf{R}^n$ be a compact subset. Then K is polynomially convex in $\mathbf{C}^n$.

c) Let V be a compact polynomially convex subset of $\mathbf{C}^n$ and $f\in$ $\mathscr{O}(V)$. Then the graph of $f$ is polynomially convex in $\mathbf{C}^{n+1}$. (Use the Oka–Weil theorem).