## Section: Software and Platforms

### Stafford

Participants : Alban Quadrat [correspondent] , Daniel Robertz [Univ. Aachen] .

The Stafford package of OreModules [92] contains an implementation of two constructive versions of Stafford's famous but difficult theorem [124] stating that every ideal over the Weyl algebra ${A}_{n}\left(k\right)$ (resp., ${B}_{n}\left(k\right)$) of partial differential operators with polynomial (resp., rational) coefficients over a field $k$ of characteristic 0 (e.g., $k=\mathbb{Q}$, $\mathbb{R}$) can be generated by two generators. Based on this implementation and algorithmic results developed in [119] by the authors of the package, two algorithms which compute bases of free modules over the Weyl algebras ${A}_{n}\left(\mathbb{Q}\right)$ and ${B}_{n}\left(\mathbb{Q}\right)$ have been implemented. The rest of Stafford's results developed in [124] have recently been made constructive in [121] (e.g., computation of unimodular elements, decomposition of modules, Serre's splitting-off theorem, Stafford's reduction, Bass' cancellation theorem, minimal number of generators) and implemented in the Stafford package. The development of the Stafford package was motivated by applications to linear systems of partial differential equations with polynomial or rational coefficients (e.g., computation of injective parametrization, Monge problem, differential flatness, the reduction and decomposition problems and Serre's reduction problem). To our knowledge, the Stafford package is the only implementation of Stafford's theorems nowadays available. The binary of the package is freely available at http://wwwb.math.rwth-aachen.de/OreModules/ .