Carath\'{e}odory interpolation on the non-commutative polydisk
  The Carath\'{e}odory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carath\'{e}odory--Fej\'{e}r problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carath\'{e}odory (resp., Carath\'{e}odory--Fej\'{e}r) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $\Lambda$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $\Lambda$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$. 
Dmitry S. Kalyuzhny\u{\i}-Verbovetzki\u{\i}