Involutive Bases of Polynomial Ideals
Vladimir P. Gerdt, Yuri A. Blinkov
TL;DR
This work develops a generalized involutive framework for polynomial ideals by introducing an involutive division $L$ that splits variable sets into multiplicative and non-multiplicative components relative to leading monomials. It defines $L$-reduction and $NF_L$, proves that an $L$-involutive basis is a (generally redundant) Gröbner basis, and presents an algorithmic construction with termination guarantees for noetherian divisions, implemented in Reduce for Pommaret-like behavior. Key contributions include formalizing noetherity, continuity, and constructivity of involutive divisions, designing InvolutiveCompletion and InvolutiveBasis algorithms with Buchberger's chain criterion, and demonstrating practical performance and connections to differential-algebraic analysis. The results extend involutive methods beyond the classical Pommaret approach, offering a robust, configurable framework for efficient ideal computation with potential applications in algebraic analysis and PDEs. All mathematical constructs are integrated to bridge involutive and Gröbner perspectives, enabling systematic generation of involutive bases when they exist and efficient handling of a broad class of polynomial ideals.
Abstract
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.