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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.

Traces of analytic uniform algebras on subvarieties and test collections