Computing Invariants of Multidimensional Linear Systems on an Abstract Homological Level

Methods from homological algebra [16] play a more and more important role in the study of multidimensional linear systems [15], [14], [6]. The use of modules allows an algebraic treatment of linear systems which is independent of their presentations by systems of equations. The type of linear system (ordinary/partial differential equations, timedelay systems, discrete systems. . . ) is encoded in the (noncommutative) ring of (differential, shift, . . . ) operators over which the modules are defined. In this framework, homological algebra gives very general information about the structural properties of linear systems. Homological algebra is a natural extension of the theory of modules over rings. The category of modules and their homomorphisms is replaced by the category of chain complexes and their chain maps. A module is represented by any of its resolutions. The module is then recovered as the only nontrivial homology of the resolution. The notions of derived functors and their homologies, connecting homomorphism and the resulting long exact homology sequences play a central role in homological algebra. The MAPLE-package homalg [1], [2] provides a way to deal with these powerful notions. The package is abstract in the sense that it is independent of any specific ring arithmetic. If one specifies a ring in which one can solve the ideal membership problem and compute syzygies, the above homological algebra constructions over that ring become accessible using homalg. In this paper we introduce the package homalg and present several applications of homalg to the study of multidimensional linear systems using available MAPLE-packages which provide the ring arithmetics, e.g. OREMODULES [4], [5] and JANET [3], [13]. Keywords— Homological algebra, multidimensional linear systems, SMITH normal form, JACOBSON normal form, extension modules, computer algebra.