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Given a complex domain   Ω   and analytic functionsφ  1   ,  …  ,   φ  n   :  Ω  →  D   , we find geometric conditions allowing us to conclude thatH  ∞    (  Ω  )is generated by functions of the form    g  ∘   φ  k    ,    g  ∈   H  ∞    (  D  )    . This is then applied to the construction of an extension of bounded holomorphic functions on an analytic one‐dimensional complex subvariety of the polydiskD   n    to functions in the Schur–Agler algebra ofD   n   , with an estimate on the norm of the extension. Our proofs use an extension of the technique of separation of singularities by Havin, Nersessian and Ortega‐Cerdá.