The techniques of predictive control are based on building a prediction about the future plant behaviour. One way to obtain such a prediction requires solving a polynomial diophantine equation (PDE). There are different methods to solve such an equation, which are good for design purposes but present some restrictions when studying the behaviour of solutions and the performance of the methods in which they are applied. In this article an analytical solution for the PDE is developed, which constitutes a new analysis tool for predictive methods. Also, first results are briefly described on the application of this solution to the analysis of the generalised predictive control method.