  We study the ideal membership problem in $H^\infty$ on the unit disc. Thus,
given functions $f,f_1,\ldots,f_n$ in $H^\infty$, we seek sufficient conditions
on the size of $f$ in order for $f$ to belong to the ideal of $H^\infty$
generated by $f_1,\ldots,f_n$. We provide a different proof of a theorem of
Treil, which gives the sharpest known sufficient condition. To this end, we
solve a closely related problem in the Hilbert space $H^2$, which is equivalent
to the ideal membership problem by the Nevanlinna-Pick property of $H^2$.
