  In this paper we consider an algorithmic technique more general than that
proposed by Zharkov and Blinkov for the involutive analysis of polynomial
ideals. It is based on a new concept of involutive monomial division which is
defined for a monomial set. Such a division provides for each monomial the
self-consistent separation of the whole set of variables into two disjoint
subsets. They are called multiplicative and non-multiplicative. Given an
admissible ordering, this separation is applied to polynomials in terms of
their leading monomials. As special cases of the separation we consider those
introduced by Janet, Thomas and Pommaret for the purpose of algebraic analysis
of partial differential equations. Given involutive division, we define an
involutive reduction and an involutive normal form. Then we introduce, in terms
of the latter, the concept of involutivity for polynomial systems. We prove
that an involutive system is a special, generally redundant, form of a Groebner
basis. An algorithm for construction of involutive bases is proposed. It is
shown that involutive divisions satisfying certain conditions, for example,
those of Janet and Thomas, provide an algorithmic construction of an involutive
basis for any polynomial ideal. Some optimization in computation of involutive
bases is also analyzed. In particular, we incorporate Buchberger's chain
criterion to avoid unnecessary reductions. The implementation for Pommaret
division has been done in Reduce.
