The three-point Pick-Nevanlinna interpolation problem on the polydisc

We give a necessary and sufficient condition for the existence of a holomorphic interpolant, for prescribed three-point Pick-Nevanlinna data. This is equivalent, due to results of Agler-McCarthy and Knese, to the existence of an interpolating rational inner function on the unit polydisc. Characterizing the latter reduces the search for a three-point interpolant to finding a single rational inner function that satisfies a certain positivity condition and arises from a polynomial of a very special form. This in turn relies on a pair of results on the factorization of rational inner functions