Joachim Apel
The object of this paper is to enlighten the relationship between the two classical theories of passive complete orthonomic systems of PDEs at the one hand side and Gr\"obner bases of finitely generated modules over polynomial rings at the other hand side. The link between both types of canonical forms are the Riquier bases (also called {\lq}involutive bases{\rq} in the literature) which are both, a particular type of Gr\"obner bases which carry some additional structure and a natural translation of the notion of passive complete orthonomic systems of PDEs into the language of polynomial modules.
We will point out some desirable applications which a ``good'' notion of Riquier bases could provide. Unfortunately, these desires turn out to collide and we will discuss the problem of finding a reasonable compromise.