Some observations concerning polynomial convexity

Abstract

In this paper we discuss a couple of observations related to polynomial convexity. More precisely, (i) We observe that the union of finitely many disjoint closed balls with centres in ${\cup }_{\theta \in [0,\pi /2]}{e}^{i\theta }V$ is polynomially convex, where $V$ is a Lagrangian subspace of ${\mathbb{C}}^n$. (ii) We show that any compact subset $K$ of $\{(z,w)\in {\mathbb{C}}^2:q(w)=\overline{p(z)}\}$, where $p$ and $q$ are two non-constant holomorphic polynomials in one variable, is polynomially convex and $\mathcal{P}(K)=\mathcal{C}(K)$.

9 pages