Sets of uniqueness for uniform limits of polynomials in several complex variables

TL;DR

This paper studies sets of uniqueness for uniform limits of polynomials in several complex variables, showing that positive measure sets are uniqueness sets under certain conditions, with extensions to different metrics.

Contribution

Introduces the concept of sets of uniqueness for polynomial limits in several complex variables and characterizes them based on measure and metric considerations.

Findings

Sets of positive measure are sets of uniqueness.

The converse that measure zero sets are not uniqueness sets does not hold.

Extends the analysis to the chordal metric on the Riemann sphere.

Abstract

We investigate the sets of uniform limits $A(\bar{B}_n)$, $A(\bar{D}^I)$ of polynomials on the closed unit ball $\bar{B}_n$ of $\mathbb{C}^n$ and on the cartesian product $\bar{D}^I$ where $I$ is an arbitrary set and $\bar{D}$ is the closed unit disc in $\mathbb{C}$. We introduce the notion of set of uniqueness for $A(\bar{D}^I)$ (respectively for $A(\bar{B}_n)$) for compact subsets $K$ of $T^I$ (respectively of $\partial \bar{B}_n$) where $T=\partial D$ is the unit circle. Our main result is that if $K$ has positive measure then $K$ is a set of uniqueness. The converse does not hold. Finally, we do a similar study when the uniform convergence is not meant with respect to the usual Euclidean metric in $\mathbb{C}$, but with respect to the chordal metric $\chi$ on $\mathbb{C} \cup \{\infty \}$.