Kummert's approach to realization on the bidisk

We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has minimal unitary transfer realization. A slight modification the extends functions which are isometric on two-torus and we use this largely elementary new proof existence Agler decompositions for Schur variables. recent result Dritschel prove variable always have finite-dimensional contractive realizations. Finally, polynomial realizations built out special nilpotent linear combinations.