Transition systems have been intensively applied to the modelling of complex systems. Their safety properties can be verified using model-checking procedures by iterative computation of fixed points. The approach has to face two main difficulties: the complexity of computations on the data domain and the termination of the iterative algorithm. In many cases an analysis of the transition system can be exploited in order to speed up the calculus. Metatransitions are upper approximations of transition relations: they lead in one step to a superset of the states occuring on an infinite trajectory. Using polynomials we compute metatransitions for linear transition systems. We apply this method to a train controller.