  We study similarity classes of commuting row contractions annihilated by what
we call higher order vanishing ideals of interpolating sequences. Our main
result exhibits a Jordan-type direct sum decomposition for these row
contractions. We illustrate how the family of ideals to which our theorem
applies is very rich, especially in several variables. We also give two
applications of the main result. First, we obtain a purely operator theoretic
characterization of interpolating sequences. Second, we classify certain
classes of cyclic commuting row contractions up to quasi-similarity in terms of
their annihilating ideals. This refines some of our recent work on the topic.
We show how this classification is sharp: in general quasi-similarity cannot be
improved to similarity. The obstruction to doing so is the existence, or lack
thereof, of norm-controlled similarities between commuting tuples of nilpotent
matrices, and we investigate this question in detail.
