Asymmetric approach to computation of Gröbner bases

A new approach to Buchberger's algorithm based on the use of essential multiplications and nonmultiplicative prolongations instead of traditional $S$-polynomials is described. In the framework of this approach, both Buchberger's algorithm for computing Gröbner bases and Gerdt–Blinkov algorithm for computing involutive bases obtain a unified form of description. The new approach is based on consideration of the process of determining an $S$-polynomial as a process of constructing a nonmultiplicative prolongation of a polynomial and its subsequent reducing with respect to an essential multiplication. An advantage of the method is that some “redundant” $S$-pairs are automatically excluded from consideration.