A New Algorithm for Gröbner Bases and a New Application

In my PhD thesis 1965 and the subsequent publication 1970 in aequationes mathematicae, I introduced the notion of Gröbner bases and proved a charaterization theorem on which an algorithm for constructing Gröbner bases can be based. The characterization theorem states that a set F of multivariate polynomials (over a field) is a Gröbner bases iff all S-polynomials of F reduce to 0 w.r.t. F. The algorithm for constructing a Gröbner basis G for any given F, proceeds by iterating the formation of S-polynomials and adding non-zero reduction results to the basis until no more nonzero results occur. The final basis is a Gröbner basis G for F. By an application of Dixon’s lemma it can be shown that this algorithm always terminates. In this approach to Gröbner bases, the notion of S-polynomials plays the crucial role: