24 - Gröbner I

Summary

Buchberger completed his thesis in 1965 and published his results in 1970. The next year, Gröbner quoted them in his notes of a course held by him in Turin and Milan in April–May 1971. There, in a section devoted to the determination of the primary components in the Lasker–Noether decomposition of an ideal, he concluded with the following remark:

OSSERVAZIONE: Riguardo ai calcoli che occorre eseguire per risolvere i problemi della teoria degli ideali negli anelli di polinomi, giova notare che, in linea di principio, tutti i calcoli si possono ridurre alla risoluzione di sistemi di equazioni lineari. Infatti basta risolvere il problema dato nei singoli spazi vettoriali P(^t)…In questo procedimento è lecito fermarsi ad un certo grado (finito) T che corresponde al grado massimo attinto dai polinomi che formano la base dell'ideale cercato.

Un criterio per determinare tale numero T è s tato indagato da B. BUCHBERGER (Aequationes mathematicae, Vol. 4, Fasc. 3, 1970, S. 377–388)

REMARK: With regard to the calculations needed to solve the problems in the theory of ideals of polynomial rings, it is helpful to remark that, in principle, all computations can be reduced to the resolution of systems of linear equations. In fact it is sufficient to solve the given problem in the single vector spacesP(^t) [the set of all polynomials of degree bounded by t] In this procedure it is sufficient to terminate at a fixed (finite) degree T corresponding to the maximal degree reached by the polynomials which are a basis of the required ideal.