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

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 $\mathbb{C}\cup \{\infty \}$.

This paper is based on the master thesis of the first named author to be presented at the University of Athens, under the direction of the second named author (17 pages)