Janet's approach to presentations and resolutions for polynomials and linear pdes

Janet's algorithm to create normal forms for systems of linear pdes is outlined and used as a tool to construct resolutions for finitely generated modules over polynomial rings over fields as well as over rings of linear differential operators with coefficients in a differential field. The main result is that a Janet basis for a module allows to read off a Janet basis for the syzygy module. Two concepts are introduced: The generalized Hilbert series allowing to read off a basis (over the ground field) of the modules, once the Janet basis is constructed, and the Janet graph, containing all the relevant information connected to the Janet basis. In the context of pdes, the generalized Hilbert series enumerates the free Taylor coefficients for power series solutions. Rather than presenting Janet's algorithm as a powerful computational tool competing successfully with more commonly known Grobner basis techniques, it is used here to prove theoretical results.