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We use a special version of the Corona Theorem in several variables, validwhen all but one of the data functions are smooth, to generalize to thepolydisc and to the ball results obtained by El Fallah, Kellay and Seip aboutcyclicity of non vanishing bounded holomorphic functions in large enough Banachspaces of analytic functions determined either by weighted sums of powers ofTaylor coefficients or by radially weighted integrals of powers of the modulusof the function.

Cyclicity of non vanishing functions in the polydisc and in the ball