  We use a special version of the Corona Theorem in several variables, valid
when all but one of the data functions are smooth, to generalize to the
polydisc and to the ball results obtained by El Fallah, Kellay and Seip about
cyclicity of non vanishing bounded holomorphic functions in large enough Banach
spaces of analytic functions determined either by weighted sums of powers of
Taylor coefficients or by radially weighted integrals of powers of the modulus
of the function.
