An extended S-polynomial for computing Gröbner bases

Gröbner bases algorithm is an important symbolic method for solving polynomial equations, and S-polynomial is the crucial notion of Buchberger´s algorithm. In order to improve the efficiency of Buchberger´s algorithm, many researchers focused on polynomial reduction. Unlike the past approach, extended S-polynomial is introduced in this paper. It is obvious that S-polynomial is a special case of extended S-polynomial and it can derive out Gröbner bases also. However, compared with S-polynomial, it is distinctive that some polynomials with more lower degrees may be produced by extended S-polynomial ahead, this is very helpful to enhance the efficiency of the next polynomial reductions.