Harmony of Gröbner bases and the modern industrial society

Bruno Buchberger, who was a student of Wolfgang Gröbner, introduced an algorithm to compute Gröbner bases in his dissertation in 1966. By virtue of his algorithm, a rich area, so called, "Computational Algebra" was born. Just at that time, Heisuke Hironaka introduced "standard bases" which is similar to Gröbner bases in his big paper on resolutions of singularities. In a word, a Gröbner basis is a set of generators of an ideal of a polynomial ring which possesses "nice properties," and we can compute it with starting from a system of generators of the ideal by using Buchberger algorithm. One of its "nice properties" is that it is useful to solve a system of equations. For example, in this picture, we regard the solutions of the upper system of equations as zeros of the ideal generated by 3 polynomials (with ignoring = 0 ), and compute a Gröbner basis with starting from this system of generators. It then turns out that the lower 3 polynomials form a Gröbner basis. Although both systems have the same set of solutions, to solve the lower system is much easier than to solve the upper system.