Involutive Division Techniques: Some Generalizations and Optimizations

A new class of involutive divisions induced by certain orderings of monomials is considered. It is proved that these divisions are Noetherian and constructive. Therefore, each of them allows one to compute an involutive Gröbner basis of a polynomial ideal by sequentially examining multiplicative reductions of nonmultiplicative prolongations. The dependence of involutive algorithms on the completion ordering is studied. Based on the properties of particular involutive divisions, two computational optimizations are suggested. One of them consists of a special choice of the completion ordering. The other optimization is related to recomputing multiplicative and nonmultiplicative variables in the course of the algorithm. Bibliography: 17 titles.

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