Rings of Partial Differential Operators

Abstract

In the ring of ordinary differential operators all ideals are principal. Consequently, the relation between an individual operator and the ideal that is generated by it is straightforward. The situation is different in rings of partial differential operators where in general ideals may have any number of generators, and only a Janet basis provides a unique representation. Therefore a more algebraic language is appropriate for dealing with partial differential operators and the ideals or modules they generate. It is introduced in the first section of this chapter. Subsequently it is applied for discussing certain properties of ideals in those rings of partial differential operators that are applied in later parts of this monograph. General references for this chapter are the books by Kolchin [37] or van der Put and Singer [71], or the article by Buium and Cassidy [8].