  In the paper `Distinguished Varieties,' Agler and McCarthy used Hilbert
function spaces to study the uniqueness properties of the Nevanlinna-Pick
problem on the bidisc. In this work we give a geometric procedure for
constructing a Nevanlinna-Pick problems on D^n with a specified set of
uniqueness. On the way to establishing this procedure, we prove a result about
Hilbert function spaces and partially answer a question posed by Agler and
McCarthy.
