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.
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.
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.
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.
> 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.)
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.
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.
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].