Gröbner bases and involutive bases
A V Astrelin, Oleg Golubitsky, Ε. V. Pankratiev · Algebra · 2012
Recently, constructive methods became widely used in commutative algebra. These methods are mainly based on the theory of Gröbner bases and involutive bases. Due to various applications, the investigations of effectiveness of constructing of the Gröbner bases are very urgent. The algorithm of constructing of the Gröbner bases is bases on considering of S-polynomials and applying of a normal simplificator. Usually, in order to refine the algorithm of the Gröbner bases construction, one uses more elaborate choice of S-polynomials. The influence of the normal simplificator on the effectiveness of the Gröbner bases construction is studied insufficiently. In the paper we try to establish a relation between the theory of Gröbner bases and the theory of involutive bases. It is known that any involutive basis contains as a subset the Gröbner basis of the ideal. It is shown that the involutive basis corresponds to the Gröbner basis with a fixed normal simplificator. Thus, the dependence of the effectiveness of the Gröbner basis construction on the chosen normal simplificator is emphasized. An attempt to describe the normal simplificators corresponding to involutive bases is fulfilled.