Local Polynomial Rings and Ideals

Creation of Local Polynomial Rings and Accessing their Monomial Orders

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.

LocalPolynomialRing(K, n): Rng, RngIntElt -> RngMPolLoc

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.

LocalPolynomialRing(K, n, order): Rng, RngIntElt, MonStgElt, ... -> RngMPolLoc

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.

MonomialOrder(R): RngMPolLoc -> Tup

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.

MonomialOrderWeightVectors(R): RngMPol -> [ [ FldRatElt ] ]

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(R): RngMPol -> RngMPolLoc

Localization(I): RngMPol -> RngMPolLoc

Given a (global) multivariate polynomial ring \(R=K[x_1, \ldots, x_n]\) (or an ideal \(I\) of such an \(R\)), return the localization \(K[x_1, \ldots, x_n]_{\langle x_1, \ldots, x_n\rangle}\) 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.

Example: Local Polynomial Rings with Different Orders (ex-ab71f5)

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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

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Creation of Ideals and Accessing their Bases

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 \(f\).

IdealWithFixedBasis(B): [ RngMPolLocElt ] -> RngMPolLoc

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.

Basis(I): RngMPolLoc -> [ RngMPolLocElt ]

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.

BasisElement(I, i): RngMPolLoc, RngIntElt -> RngMPolLocElt

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].