Passive complete orthonomic systems of PDEs and involutive bases of polynomial modules

Abstract

The objective of this article is to enlighten the relationship between the two classical theories of passive complete orthonomic systems of PDEs on the one hand and Gröbner bases of finitely generated modules over polynomial rings on the other hand. The link between both types of canonical forms are the involutive bases which are both, a particular type of Gröbner bases which carry some additional structure and a natural translation of the notion of passive complete orthonomic systems of linear PDEs with constant coefficients into the language of polynomial modules. We will point out some desirable applications which a "good" notion of involutive bases could provide. Unfortunately, these desires turn out to collide and we will discuss the problem of finding a reasonable compromise. © Springer-Verlag Berlin Heidelberg 2003.