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This chapter describes the Magma functionality for ideals over polynomial
rings. For the basics on multivariate polynomial rings and their elements,
see Chapter MULTIVARIATE POLYNOMIAL RINGS. Most of the significant operations with ideals
construct or utilise a previously-constructed Gröbner basis. The monomial
ordering used for this basis can greatly affect the speed and memory usage
of these operations. This ordering is attached to the polynomial ring in
which the ideals are created. For information on Gröbner bases and the
creation of polynomial rings with specified orders, see Chapter GRÖBNER BASES.
That chapter also tells the user how to compute and return a Gröbner basis,
or just to compute it internally for later use in the operations described below,
with many additional configuration parameters to optimise the computation.
Users may ignore the issue when creating the ambient polynomial rings by
allowing Magma to make default choices. It is, however, highly recommended
that users who wish to work with complicated ideals thoroughly acquaint
themselves with the options available. Magma has an extremely powerful
Gröbner basis engine and often makes sophisticated choices internally
of alternative monomial orders for particular computations. Ultimately,
however, the user may significantly speed up his work by a judicious choice
of order. We note here that the default order is the lexicographical
one, a total elimination order well suited to finding solutions of zero-dimensional
systems of polynomial equations but tending to produce very large bases that
can take much time and memory to compute. For homogeneous ideals of rings
with the standard weighting (all variables have weight one), the grevlex order
is usually the best in practice and there is theoretical justification for
this. In the case that the ring has a different weighting and the ideal is
homogeneous with respect to that, the weighted grevlex order is the best choice.
In any case, the EasyIdeal and EasyBasis intrinsics
of the Gröbner basis chapter return to the user a basis for an internally
chosen good order and these "easy" bases are used in many internal functions
if a basis with respect to the polynomial ring order has not already been
computed and stored.
The functions and operations described here cover a wide range of commutative
algebra functionality. This includes sums and intersections, colon ideals and
saturations, elimination, radicals and primary decompositions, Noether
normalisations and computation of Hilbert polynomials and Hilbert series.
Related chapters including other polynomial ring functionality relying
on Gröbner bases are the chapter on invariant rings of finite group
actions, Chapter INVARIANT THEORY, and the chapters on affine
algebras (Chapter AFFINE ALGEBRAS) and on modules over affine algebras (Chapter
MODULES OVER MULTIVARIATE RINGS). The chapter on algebraically closed fields
(Chapter ALGEBRAICALLY CLOSED FIELDS) describes functions that allows one to compute the variety
of an ideal over the algebraic closure of the base field. And, of course,
the Algebraic Geometry component of Magma and parts of the Arithmetic Geometry
are built upon the commutative algebra here.
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