Involutive Division Technique: Some Generalizations and Optimizations
Vladimir P. Gerdt
TL;DR
The paper addresses the problem of computing involutive bases, a specialized form of Gröbner bases, by introducing a broad class of involutive divisions induced by admissible orderings. It develops a formal framework with noetherian and constructive properties, and presents completion and basis algorithms (InvolutiveCompletion and MinimalInvolutiveBasis) that operate under varying main and completion orderings. Key contributions include the pair and monotonicity properties, which enable efficient recomputation and optimization, and the demonstration of finite involutive completions for constructive divisions. The work offers practical avenues for faster polynomial-ideal computations through dynamic division refinements and orderings, with potential improvements over standard Gröbner-basis methods in targeted scenarios.
Abstract
In this paper, in addition to the earlier introduced involutive divisions, we consider a new class of divisions induced by admissible monomial orderings. We prove that these divisions are noetherian and constructive. Thereby each of them allows one to compute an involutive Groebner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. We study dependence of involutive algorithms on the completion ordering. Based on properties of particular involutive divisions two computational optimizations are suggested. One of them consists in a special choice of the completion ordering. Another optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm.