Density of polynomials in classes of functions on products of planar domains

We give sufficient conditions on planar domains for polynomials to be dense in the algebras A and A-infinity of the product of these domains, endowed with their natural topologies. We also characterize the uniform limits, with respect to the chordal metric, of polynomials on the product of the closures of these domains. The products may be finite products or infinite products, even uncountable.