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## Section: New Results

### Equidimensional block-triangular representation of linear functional systems

In [30] , it is shown that every linear functional system (e.g., PD systems, differential time-delay systems, difference systems) is equivalent to a linear functional system defined by an upper block-triangular matrix of functional operators: each diagonal block is respectively formed by a generating set of the elements of the system satisfying a purely $i$-codimensional system. Hence, the system can be integrated in cascade by successively solving (inhomogeneous) $i$-codimensional linear functional systems to get a Monge parametrization of its solution space [120] . The results are based on an explicit construction of the grade/purity filtration of the module associated with the linear functional system. This new approach does not use complex Grothendieck spectral sequence arguments as is done in the literature of modern algebra [86] , [87] . To our knowledge, the algorithm obtained in [30] is the most efficient algorithm existing in the literature of non-commutative algebra. It was implemented in the PurityFiltration package developed in Maple (see Section  5.6 ) and in the homalg package of GAP 4 (see Section  5.7 ). Classes of overdetermined/underdetermined linear systems of partial differential equations which cannot be directly integrated by Maple can be solved using the PurityFiltration package.