Introduction

This chapter describes ideals of multivariate polynomial rings. Magma contains a powerful system for computing with such ideals which is based on the construction of Gröbner bases of such ideals. 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.

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 RINGS OF FINITE GROUPS. See also the chapters on affine algebras (Chapter AFFINE ALGEBRAS) and on modules over affine algebras (Chapter MODULES OVER AFFINE ALGEBRAS), 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.