## Section: New Results

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

Participant : Alban Quadrat.

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.