Notes on Gröbner bases
A self-contained exposition of the theory of Gröbner bases and its applications, mainly based on work of Buchberger. A Gröbner basis is a basis for an ideal with properties that allow a lot of computational problems to be solved that involve ideals. The authors advocates the constructive point of view within algebraic geometry. They remark that algorithmic algebraic geometry involves the search for a variety of bounds. The authors first recall the basic facts needed from polynomial and ideal theory, then they present the definition of a Gröbner basis using normal forms. In subsequent sections, the normal form algorithm is presented, together with results on bounds for normal form algorithms. Next, they present some alternative definitions of Gröbner bases (e.g. using Church-Rosser relations, \(NF_ G(f)\) and \(NF_ G(S(f,g)))\). The authors also present the basic algorithm of Buchberger which transforms a finite set of polynomials into a Gröbner basis. This algorithm is simple but nondeterministic as it calls the normal form algorithms. Gröbner bases can in addition be turned into a unique object if one restricts oneself to reduced Gröbner bases. Moreover, reduced Gröbner bases can be determined in an algorithmic way. Finally a few applications are presented: algorithms to reduce modulo an ideal, to test membership of an ideal, to test congruence modulo an ideal and to test whether two ideal are equal or contained in each other. There are also some applications in residue class ring theory and the theory of solving systems of polynomial equations.
scientific article; zbMATH DE number 4123853
scientific article