INVOLUTIVE BASES OF POLYNOMIAL IDEALS

In this paper we consider an algorithmic technique more general than t hat proposed by Zharkov and Blinkov for the involutive analysis of pol ynomial ideals. It is based on a new concept of involutive monomial di vision which is defined for a monomial set. Such a division provides f or each monomial the self-consistent separation of the whole set of va riables into two disjoint subsets. They are called multiplicative and non-multiplicative. Given an admissible ordering, this separation is a pplied 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 differen tial equations. Given involutive division, we define an involutive red uction and an involutive normal form. Then we introduce, in terms of t he latter, the concept of involutivity for polynomial systems. We prov e that an involutive system is a special, generally redundant, form of a Grobner basis. An algorithm for construction of involutive bases is proposed. It is shown that involutive divisions satisfying certain co nditions, for trample, those of Janet and Thomas, provide an algorithm ic construction of an involutive basis for any polynomial ideal. Some optimization in computation of involutive bases is also analyzed. In p articular, we incorporate Buchberger's chain criterion to avoid unnece ssary reductions. The implementation for Pommaret division has been do ne in Reduce. (C) 1998 IMACS/Elsevier Science B.V.