Ideal intersections in rings of partial differential operators

Abstract

An important step in solving linear differential equations in closed form is its factorization and generating the Loewy decomposition from it. For ordinary equations this is fairly straightforward because all operators involved generate principal ideals in the corresponding ring of operators. This is different for linear partial differential equations and the operators associated with them; its so-called non-unique factorizations have created some confusion in the past. Fundamental to Loewyʼs theory is the concept of a completely reducible operator; it is defined to be the left intersection of its irreducible constituents. Consequently a systematic investigation of the intersection ideals in the corresponding rings of differential operators is an indispensable requirement for generalizing Loewyʼs theory. The article at hand gives a thorough description of the possible intersection ideals of two first-order operators in two or three variables. Furthermore, it is shown how software provided on the website www.alltypes.de may be applied for solving concrete problems.