The use of Gröbner basis computations for treating systems of polynomial equations has become an important tool in many areas. This paper introduces of the concept of standard bases (a generalization of Gröbner bases) and the application to some problems from algebraic geometry. The examples are presented as SINGULAR commands. A general introduction to Gröbner bases can be found in the textbook [CLO], an introduction to syzygies in [E] and [St1].
SINGULAR is a computer algebra system for computing information about singularities, for use in algebraic geometry. The basic algorithm in SINGULAR is a general standard basis algorithm for general monomial orderings(see [GG]). This includes wellorderings (Buchberger algorithm ([B1],[B2]) and tangent cone orderings (Mora algorithm ([M1],[MPT])) as special cases: It is able to work with non-homogeneous and homogeneous input and also to compute in the localization of the polynomial ring in 0. For a complete description of SINGULAR see [Si].