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