Ideal intersections in rings of partial differential operators
The factorization of linear ordinary differential operators and the corresponding Loewy decomposition is one of the most important tools in solving linear ordinary differential equations in closed form. For linear partial differential operators a corresponding theory is essentially missing, and the present paper is a first step to overcome this deficiency. The author aims at a complete understanding of the left intersection ideals in the respective ring of differential operators and how to obtain a proper analog to the completely reducible components of Loewy. The left intersection ideals for first-order operators in two or three variables over the coefficient ring of rational functions are studied in detail. Also, a complete list of possible types of intersection and sum ideals is given including the corresponding ideals of constraints in the coefficient ring. As expected, the number of possible types is highly increasing with the number of variables. Several figures are provided which are helpful for understanding the branching into the respective cases.
scientific article; zbMATH DE number 5907658