This chapter describes the basics for configuring Magma's powerful Gröbner basis machinery, which lies at the heart of computations with ideals and modules over multivariate polynomial rings. Later chapters will describe the many functions and operations available to the user for working with ideal and modules.
Gröbner bases were introduced by Bruno Buchberger [Buc65] and at the heart of the theory is the Buchberger algorithm which computes a Gröbner basis of an ideal starting from an arbitrary basis (generating set) of the ideal. The two books Ideals, Varieties and Algorithms [CLO96] and Gröbner Bases [BW93] have also inspired much of the design and presentation of ideals of multivariate polynomial rings in Magma.
Since V2.11 (May 2004), Magma also contains a highly optimized implementation of the Faugère F4 algorithm [Fau99], based on sparse linear algebra techniques, which usually performs dramatically better than the Buchberger algorithm (see [Ste04]).
Chapter MULTIVARIATE POLYNOMIAL RINGS deals with the basics of multivariate polynomial rings and their elements (for which there are very many functions), so it is recommended that that chapter be perused before reading this one.
Permutation and matrix groups have a natural action on multivariate polynomial rings. This leads to the subject of invariant rings of finite groups, which is covered in Chapter INVARIANT THEORY. See also the chapters on affine algebras (Chapter AFFINE ALGEBRAS) and on modules over affine algebras (Chapter MODULES OVER MULTIVARIATE RINGS), and the chapter on algebraically closed fields (Chapter ALGEBRAICALLY CLOSED FIELDS), which allows one to compute the variety of an ideal over the algebraic closure of the base field.