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In the seventies, the study of transfer matrices of linear time-invariant systems led to the development of the polynomial approach (Kailath, Rosenbrock, et al.). In this approach, univariate polynomial matrices play a central role (e.g., Hermite and Smith normal forms, Bezout equations etc.). When generalizing linear systems given by ordinary differential/difference equations to differential time-delay systems, systems with parameters, 2-D or 3-D filters and circuits, one had to face the case of systems described by means of matrices with entries in multivariate polynomial rings. These new systems were called 2-D or 3-D and, more generally, multidimensional systems (Bose, Lin, Pommaret, Oberst, Youla, Wood, et al.). For such systems, no normal forms such as Hermite and Smith forms exist. In order to handle these problems, the concept of Grobner bases (developed by Buchberger) was introduced in multidimensional systems theory. In many ways, the computation of these bases can be seen as an extension of both Gaussian elimination and the Euclidean algorithm. Grobner bases have been implemented in nearly all modern computer algebra systems. Another difficulty with multidimensional systems is the lack of a canonical first order representation. Several models (Roesser, Fornasini-Marchesini, reduced Spencer form etc.) have been studied extensively. The geometric approach to systems and control (Basile,Marro, Wonham, et al.) has recently been extended to these classes of multidimensional systems yielding promising results. The reduction to first order representations (realization problem) is of great importance to many applications of multidimensional systems theory such as uncertainty modeling for robust control. Special computer algebra packages have been developed for the effective manipulation of linear fractional transformations. In his pioneering work, Kalman developed a module-theoretic approach to the realization problem. Independently, the Japanese school of Sato, Kashiwara, and Kawai proposed an

Special issue on symbolic methods in multidimensional systems theory