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This chapter describes multivariate polynomial rings in Magma.
A multivariate polynomial ring in any number of variables n≥1
can be created over an arbitrary coefficient ring R, and we will
denote it by P=R[x1, ..., xn]. Certain functions,
however, will only apply for coefficient rings satisfying certain
conditions.
Magma contains a powerful system for computing with ideals of
multivariate polynomial rings. This is based on the construction
of Gröbner bases of such ideals. This chapter only deals
with polynomial rings and operations on their elements; see
Chapter GRÖBNER BASES for the details concerning ideals and
Gröbner bases.
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).
Let P be the polynomial ring R[x1, ..., xn] of rank n over a
ring R. A monomial (or power product) of P is a product of
powers of the variables of P, that is, an expression of the form
x1e1 ... xnen with ei ≥0 for 1 ≤i ≤n.
Multivariate polynomials in Magma are stored efficiently in distributive
form, using arrays of coefficient-monomial pairs, where the coefficient
is in the base ring R. The word `term' will always refer to a
coefficient multiplied by a monomial.
Various orders can be applied to the monomials, and these are
of great importance when dealing with Gröbner bases. A polynomial
ring in Magma may defined with a certain monomial order, but as this
does not affect the basic arithmetic operations in the polynomial
ring, these orders are not described here but in the chapter
dealing with Gröbner bases (see Section Representation and Monomial Orders).
Since V2.7 (June 2000), a new generalized monomial representation has
been developed, which uses differing byte sizes for monomials depending
on the size of the monomials encountered. Monomial overflow is
rigorously detected, and the system automatically extends the byte size
of the monomials in the background if possible. Thus there is no need
for the user to know beforehand the maximum degree which may occur,
and much memory is also saved for low- to medium-degree computations.
The total degree of any monomial may now be anything up to 230 - 1 =
1073741823.
It is possible but not advised to use distributive `multivariate' polynomials
in one single variable. (See Chapter UNIVARIATE POLYNOMIAL RINGS which devoted to
univariate polynomial rings.)
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