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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.
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