The polyphase representation with respect to sampling lattices in multidimensional (M-D) multirate signal processing allows us to identify perfect reconstruction (PR) filter banks with unimodular Laurent polynomial matrices, and various problems in the design and analysis of invertible MD multirate systems can be algebraically formulated with the aid of this representation. While the resulting algebraic problems can be solved in one dimension (1-D) by the Euclidean Division Algorithm, we show that Gröbner bases offers an effective solution to them in the M-D case.