Involutive directions and new involutive divisions

In this paper, we propose the concept of involutive direction as a vector representation for the concept of involutive division proposed by Gerdt and his co-workers. With this representation, most of the properties of involutive divisions such as Noetherity, Artinity, and constructivity, can be greatly simplified. A new algorithm to compute the involutive completion is also given. Based on the vector representation, two new types of involutive divisions are found and proved to be Noetherian, Artinian, and constructive. These new divisions may lead to new methods of finding integrability conditions of partial differential equations and computing Gröbner bases of polynomial ideals.