  We give a characterization for the existence of a holomorphic interpolant on
the unit polydisc $\mathbb{D}^n,$ $n\geq 2,$ for prescribed three-point
Pick--Nevanlinna data. One of the key steps is a characterization for the
existence of an interpolant that is a rational inner function on
$\mathbb{D}^n.$ The latter reduces the search for a three-point interpolant to
finding a single rational inner function that satisfies a type of positivity
condition and arises from a polynomial of a very special form. This in turn
relies on a pair of results, which are of independent interest, on the
factorization of rational inner functions.
