  We give a solution to Pick's interpolation problem on the unit polydisc in
$\mathbb{C}^n$, $n\geq 2$, by characterizing all interpolation data that admit
a $\mathbb{D}$-valued interpolant, in terms of a family of positive-definite
kernels parametrized by a class of polynomials. This uses a duality approach
that has been associated with Pick interpolation, together with some
approximation theory. Furthermore, we use duality methods to understand the set
of points on the $n$-torus at which the boundary values of a given solution to
an extremal interpolation problem are not unimodular.
