Involutive Division Technique: Some Generalizations and Optimizations

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.Comment: 19 page

Involutive Bases Algorithm Incorporating F5 Criterion

Faugere's F5 algorithm is the fastest known algorithm to compute Groebner bases. It has a signature-based and an incremental structure that allow to apply the F5 criterion for deletion of unnecessary reductions. In this paper, we present an involutive completion algorithm which outputs a minimal involutive basis. Our completion algorithm has a nonincremental structure and in addition to the involutive form of Buchberger's criteria it applies the F5 criterion whenever this criterion is applicable in the course of completion to involution. In doing so, we use the G2V form of the F5 criterion developed by Gao, Guan and Volny IV. To compare the proposed algorithm, via a set of benchmarks, with the Gerdt-Blinkov involutive algorithm (which does not apply the F5 criterion) we use implementations of both algorithms done on the same platform in Maple.Comment: 24 pages, 2 figure

Algorithmic Thomas Decomposition of Algebraic and Differential Systems

In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and non-vanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of a previous paper and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376