Bruno Buchberger and the world of Groebner bases

摘要

In 1965, Bruno Buchberger submitted his remarkable dissertation (Buchberger, 1965) at the University of Innsbruck in Austria. Given the stature now accorded to the work, it was met initially with a curious absence of fanfare. In fact, Buchberger had completely solved the problem given to him by his advisor Wolfgang Grobner (1899-1980), namely to build An algorithm for finding the basis elements of the residue class ring of a zero-dimensional polynomial ideal (German title "Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal"), and he coined the term "Grobner bases" in Buchberger (1976) as a tribute to his mentor. At that time only few mathematicians were interested in the algorithmic solution of algebraic problems, and the dawning field of computer science did not feel a strong need for deeper algebra. Despite its modest title, the algorithm presented in Buchberger's thesis (now duly referred to as "Buchberger's algorithm") is actually developed for ideals of any dimension. In addition to the standard ingredients of Grobner bases theory (e.g. the leading term ideal, S-polynomials, uniqueness properties, ideal membership), first applications in algebraic geometry become visible: characterizing nontrivial and zero-dimensional ideals, computation of Hilbert functions, and deriving the multiplication table for the residue class ring. The thesis is available in an English translation in the JSC Special Issue (Buchberger, 2006); the interested reader is strongly advised to read Buchberger's own comments in the editorial of Buchberger (2006). It should also be emphasized that the original algorithm was implemented on the university's state-of-the-art machine, a ZUSE Z23, complete with examples successfully computed by the implementation. The original magnetic drum of the Z23 can now be seen in a showcase at the Kepler university.