  Given a complex domain $\Omega$ and analytic functions
$\varphi_1,\ldots,\varphi_n : \Omega \to \mathbb{D}$, we give geometric
conditions for $H^\infty(\Omega)$ to be generated by functions of the form $g
\circ \varphi_k$, $g \in H^\infty(\mathbb{D})$. We apply these results to the
extension of bounded functions on an analytic one-dimensional complex
subvariety of the polydisk $\mathbb{D}^n$ to functions in the Schur-Agler
algebra of $\mathbb{D}^n$, with an estimate on the norm of the extension. Our
proofs use some extension of the techniques of separation of singularities by
Havin, Nersessian and Ortega-Cerd\'a.
