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Local polynomial rings are created from
a coefficient field, the number of variables, and a monomial order.
If no order is specified,
the monomial order is taken to be the local lexicographical order.
Create a local polynomial ring in n>0 variables over the field K.
The local lexicographical ordering
on the monomials is used for this default construction.
LocalPolynomialAlgebra(K, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPolLoc
Create a local polynomial ring in n>0 variables over the
ring R with the given order order on the monomials. See the
above section on local monomial orders for the valid values for the
argument order.
LocalPolynomialRing(K, n, T) : Rng, RngIntElt, Tup -> RngMPolLoc
Create a local polynomial ring in n>0 variables over the
field K with the order given by the tuple T on the monomials.
T must be a tuple whose components match the valid arguments
for the monomial orders in Section Elements and Local Monomial Orders.
Such a tuple is also returned by the next function.
Given a local polynomial ring R (or an ideal thereof), return a description
of the monomial order of R. This is returned as a tuple which
matches the relevant arguments listed for each possible order in
Section Elements and Local Monomial Orders, so may be passed as the
third argument to the function LocalPolynomialRing above.
Given a polynomial ring R of rank n (or an ideal thereof),
return the weight vectors of the underlying monomial order as a
sequence of n sequences of n rationals. See, for example,
[CLO98, p. 153] for more information.
Localization(I) : RngMPol -> RngMPolLoc
Given a (global) multivariate polynomial ring R=K[x1, ..., xn]
(or an ideal I of such an R), return the localization
K[x1, ..., xn]< x1, ..., xn > of R
(or the ideal of the localization of R which corresponds to I).
The print names for the variables of R are carried over.
We show how one can construct local polynomial rings with different orders.
Note the order on the monomials for elements of the rings.
> K := RationalField();
> R<x,y,z> := LocalPolynomialRing(K, 3);
> R;
Localization of Polynomial Ring of rank 3 over Rational Field
Order: Local Lexicographical
Variables: x, y, z
> MonomialOrder(R);
<"llex">
> MonomialOrderWeightVectors(R);
[
[ 0, 0, -1 ],
[ 0, -1, 0 ],
[ -1, 0, 0 ]
]
> 1 + x + y + z + x^7 + x^8*y^7 + y^5 + z^10;
1 + x + x^7 + y + y^5 + x^8*y^7 + z + z^10
> R<x,y,z> := LocalPolynomialRing(K, 3, "lgrevlex");
> R;
Localization of Polynomial Ring of rank 3 over Rational Field
Order: Local Graded Reverse Lexicographical
Variables: x, y, z
> MonomialOrder(R);
<"lgrevlex">
> MonomialOrderWeightVectors(R);
[
[ -1, -1, -1 ],
[ -1, -1, 0 ],
[ -1, 0, 0 ]
]
> 1 + x + y + z + x^7 + x^8*y^7 + y^5 + z^10;
1 + z + y + x + y^5 + x^7 + z^10 + x^8*y^7
As for global polynomial rings, within the general context of
ideals of local polynomial rings, the term "basis" will refer
to an ordered sequence of polynomials which generate an ideal.
(Thus a basis can contain duplicates and zero elements so is not like
a basis of a vector space.)
ideal<R | L> : RngMPolLoc, List -> RngMPolLoc
Given a local polynomial ring R,
return the ideal of R generated by the elements of R specified by
the list L. Each term of the list L must be an expression defining
an object of one of the following types:
- (a)
- An element of R;
- (b)
- A set or sequence of elements of R;
- (c)
- An ideal of R;
- (d)
- A set or sequence of ideals of R.
Ideal(B) : [ RngMPolLocElt ] -> RngMPolLoc
Ideal(B) : { RngMPolLocElt } -> RngMPolLoc
Given a set or sequence B of polynomials from a local polynomial ring R,
return the ideal of R generated by the elements of B with
the given basis B. This is equivalent to the above ideal
constructor, but is more convenient when one simply has a set or sequence
of polynomials.
Ideal(f) : RngMPolLocElt -> RngMPolLoc
Given a polynomial f from a local polynomial ring R,
return the principal ideal of R generated by f.
Given a sequence B of polynomials from a local polynomial ring R,
return the ideal of R generated by the elements of B with the given
fixed basis B. When the function Coordinates is called, its
result will be with respect to the entries of B instead of the Gröbner
basis of I.
WARNING: this function should only be used
when it is desired to express polynomials of the ideal in terms of the
elements of B, as the computation of the Gröbner basis in this case
is very expensive, so it should be avoided if these expressions are
not wanted.
Given an ideal I, return the current basis of I. This will
be the standard basis of I if it is computed; otherwise it
will be the original basis.
Given an ideal I together with an integer i, return the i-th element
of the current basis of I. This the same as Basis(I)[i].
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