Equidimensional triangularization of multidimensional linear systems

Based on the results obtained in [12] on the purity filtration of a finitely presented module associated with a multidimensional linear system, this paper aims at obtaining an equivalent block-triangular representation of the multidimensional linear system defined by equidimensional diagonal blocks. The multidimensional linear system can then be integrated in cascade by solving equidimensional homogeneous linear systems. Many multidimensional linear systems defined by under/overdetermined linear systems of partial differential equations can be explicitly solved by means of the PURITYFILTRATION and AbelianSystems packages, but cannot be computed by classical computer algebra systems such as Maple. The results developed in this paper generalize those obtained in the literature on Monge parametrizations and on the classification of autonomous elements by their codimensions.