Gröbner bases computation by triangularizing Macaulay matrices

Abstract

In my PhD thesis 1965 and the subsequent publication 1970 in aequationes mathematicae, I introduced the notion of Gröbner bases and proved a characterization theorem for Gröbner bases on which an algorithm for constructing Gröbner bases can be based. The main idea for the theorem and the algorithm was the notion of "S-polynomials". Most of the subsequent work on the algorithmic theory of Gröbner bases, including the implementation of the Gröbner bases technology in mathematical software systems like Mathematica, Maple, etc. was based on this approach.

In the early eighties, I proposed a completely different strategy for computing Gröbner bases proceeding by the following three steps:

1. Produce the set of all multiples $u.f$ of the polynomials $f$ in the initial basis $F$ by all power products $u$ (which we call "generalized Sylvester matrix" or "Macaulay matrix" of $F$).

2. Triangularize this matrix.

3. Take the "contour" in the diagonal of the matrix, i.e. the set of all those polynomials in the diagonal whose leading power product is not a multiple of the leading power product of any other polynomial in the diagonal.

It is easy to prove that the above procedure yields a Gröbner basis if one starts with the infinite matrix of all shifts $u.f$. However, of course, this is not an algorithm. So I posed the question whether one can give an a-priori bound on $D$ so that, if one puts all shifts $u.f$ with degree of $u$ smaller than $D$ to the initial matrix, the matrix resulting by triangularization and contour formation will be a Gröbner basis for $F$. Over the years, several people tried to find such a bound but only recently (2014) my PhD student Manuela Wiesinger-Widi was able to establish such a bound combining known bounds by G. Hermann and T. W. Dubé for the ideal membership and the Gröbner bases degree problem, respectively.