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

We investigate the sets of uniform limits A(B‾n), A(D‾I) of polynomials on the closed unit ball B‾n of CnCn and on the cartesian product D‾I where I   is an arbitrary set, maybe finite, infinite denumerable or non-denumerable and D‾ is the closed unit disc in CC. The class A(D‾I) contains exactly all functions f:D‾I→C continuous with respect to the product topology on D‾I and separately holomorphic. We consider sets of uniqueness for A(D‾I) (respectively for A(B‾n)) to be compact subsets K   of TITI (respectively of ∂B‾n) where T=∂DT=∂D is the unit circle. 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 CC, but with respect to the chordal metric χ   on C∪{∞}C∪{∞}.