Some observations concerning polynomial convexity

TL;DR

This paper explores specific conditions under which unions of certain geometric objects and particular subsets in complex space are polynomially convex, contributing new insights to complex analysis and polynomial convexity theory.

Contribution

It demonstrates polynomial convexity of unions of disjoint balls with centers in a Lagrangian subspace and of certain subsets defined by polynomial relations, advancing understanding in polynomial convexity.

Findings

Union of disjoint balls with centers in a Lagrangian subspace is polynomially convex.

Certain subsets defined by polynomial relations are polynomially convex.

Polynomial and continuous functions coincide on these subsets.

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 $\mathscr{P}(K)=\mathscr{C}(K)$.