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The polynomial equation approach to the design of optimal discrete-time multivariable regulators is studied. Kucera's solution to the problem is rederived in the light of work concerning the role, and the necessity, of the various matrix diophantine equations which appear. In particular, a constructive derivation of the pair of coupled bilateral matrix diophantine equations, which the optimal controller matrices satisfy, is given. The conditions are reviewed under which only one of the bilateral equations may be used to calculate the unique optimal controller. As a simpler alternative, the conditions under which a single unilateral equation (related to the original couple of bilateral equations) may be used are also reviewed. Finally, guided by these theoretical results, a formal design procedure is given that exploits the advantages of using these simplifications.