Abstract. We give a new proof on the disk that a Pick problem can be solved by a rational function that is unimodular on the unit circle and for which the number of poles inside the disk is no more than the number of non-positive eigenvalues of the Pick matrix. We use this method to find rational solutions to Pick problems on the bidisk.

AMS Subject Classification (1991): 30E05, 32E30

Keyword(s): Nevanlinna--Pick interpolation, Takagi problem, rational interpolation, degenerate Pick problem

Received April 13, 2012, and in revised form December 12, 2012. (Registered under 61/2012.)