## Section: Software and Platforms

### OreModules

Participants : Alban Quadrat [correspondent] , Daniel Robertz [Univ. Aachen] , Frédéric Chyzak [Inria Rocquencourt, Algorithms Project] .

The OreModules package [92] , based on the commercial Maple package Ore${}_{-}$algebra [93] , is dedicated to the study of linear multidimensional systems defined over certain Ore algebras of functional operators (e.g., ordinary or partial differential systems, time-delay systems, discrete systems) and their applications in mathematical systems theory, control theory and mathematical physics. OreModules is original because it combines the recent developments of the Gröbner bases over some noncommutative polynomial rings [100] , [102] and new algorithms of algebraic analysis in order to effectively check classical properties of module theory (e.g., existence of a non-trivial torsion submodule, torsion-freeness, reflexiveness, projectiveness, stably freeness, freeness), it gives their system-theoretical interpretations (existence of autonomous elements or successive parametrizations, existence of minimal/injective parametrizations or Bézout equations) [109] , [108] , [91] and it computes important tools of homological algebra (e.g., (minimal) free resolutions, split exact sequences, extension functors, projective or Krull dimensions, Hilbert power series). The abstract language of homological algebra used in the algebraic analysis approach carries over to the implementations in OreModules : up to the choice of the domain of functional operators which occurs in a given system, all algorithms are stated and implemented in sufficient generality such that linear systems defined over the Ore algebras developed in the Ore${}_{-}$algebra package are covered at the same time. Applications of the OreModules package to mathematical systems theory, control theory and mathematical physics are illustrated in a large library of examples. The binary of the package is freely available at http://wwwb.math.rwth-aachen.de/OreModules/ .

A Mathematica version of the OreModules package is in development. It is being developed by Maris Tõnso (Institute of Cybernetics, University of Tallinn), Thomas Cluzeau (ENSIL, University of Limoges) and A. Quadrat within the PHC Parrot project CASCAC. The Mathematica version of the OreModules package is based on the implementation of Gröbner bases over Ore algebras available in the Mathematica HolonomicFunctions package developed by Christoph Koutschan.