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In the paper `Distinguished Varieties,' Agler and McCarthy used Hilbertfunction spaces to study the uniqueness properties of the Nevanlinna-Pickproblem on the bidisc. In this work we give a geometric procedure forconstructing a Nevanlinna-Pick problems on D^n with a specified set ofuniqueness. On the way to establishing this procedure, we prove a result aboutHilbert function spaces and partially answer a question posed by Agler andMcCarthy.

Hilbert function spaces and the Nevanlinna-Pick problem on the polydisc