Section: New Results

Algorithmic study of linear functional systems

Participants : Alban Quadrat [correspondent] , Thomas Cluzeau [ENSIL, Univ. Limoges] .

In [77] , [76] , [97] , 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 [129] [89] . 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 involved spectral sequence arguments as is done in the literature of modern algebra [102] , [103] . To our knowledge, the algorithm obtained in [77] , [97] 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  5.7 ) and in the homalg package of GAP 4 (see Section  5.8 ).

Given a linear multidimensional system (e.g., ordinary/partial differential systems, differential time-delay systems, difference systems), Serre's reduction aims at finding an equivalent linear multidimensional system which contains fewer equations and fewer unknowns. Finding Serre's reduction of a linear multidimensional system can generally simplify the study of structural properties and of different numerical analysis issues, and it can sometimes help solving the linear multidimensional system in closed form. In [17] , Serre's reduction problem is studied for underdetermined linear systems of partial differential equations with either polynomial, formal power series or analytic coefficients and with holonomic adjoints in the sense of algebraic analysis [102] , [103] . These linear partial differential systems are proved to be equivalent to a linear partial differential equation. In particular, an analytic linear ordinary differential system with at least one input is equivalent to a single ordinary differential equation. In the case of polynomial coefficients, we give an algorithm which computes the corresponding linear partial differential equation.

In [45] , we give a complete constructive form of the classical Fitting's lemma in module theory which studies the relation between equivalences of linear systems and isomorphisms of their associated finitely presented modules. The corresponding algorithms were implemented in the OreMorphisms package (see Section  5.5 ).