On connection between constructive involutive divisions and monomial orderings

Abstract

This work considers the basic issues of the theory of involutive divisions, namely, the property of constructivity which assures the existence of minimal involutive basis. The work deals with class of ≻-divisions which possess many good properties of Janet division and can be considered as its analogs for orderings different from the lexicographic one. Various criteria of constructivity and non-constructivity are given in the paper for these divisions in terms of admissible monomial orderings ≻. It is proven that Janet division has the advantage in the minimal involutive basis size of the class of ≻-divisions for which x1≻ x2 ≻ ... ≻ x n holds. Also examples of new involutive divisions which can be better than Janet division in minimal involutive basis size for some ideals are given. © Springer-Verlag Berlin Heidelberg 2006.