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

We investigate the sets of uniform limits A(Bn), A(DI) of polynomials on the closed unit ball Bn of Cn and on the cartesian product DI where I is an arbitrary set, maybe finite, infinite denumerable or non-denumerable and D is the closed unit disc in C. The class A(DI) contains exactly all functions f:DI→C continuous with respect to the product topology on DI and separately holomorphic. We consider sets of uniqueness for A(DI) (respectively for A(Bn)) to be compact subsets K of TI (respectively of ∂Bn) where T=∂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 C, but with respect to the chordal metric χ on C∪{∞}. © 2015 Elsevier Inc..