In the following, note that since ideals of a full polynomial ring P are regarded as subrings of P, the ring P itself is a valid ideal as well (the ideal containing 1).
The following basic constructions involve no Gröbner basis computation.
Given ideals I and J of the same polynomial ring P, return the sum of I and J, which is the ideal generated by the generators of I and those of J.
Given ideals I and J of the same polynomial ring P, return the product of I and J, which is the ideal generated by the products of the generators of I and those of J.
Given an ideal I of the polynomial ring P, and an integer k, return the k-th power of I.
Given an ideal I of a polynomial ring P over a field and an ideal J of P, such that J ⊂I, return the affine algebra I/J.
The following important basic operations on ideals involve Gröbner basis computation and use the standard algorithms as described in Chapter 1.8 of [GP02], for example, unless otherwise stated.
Given an ideal I of a polynomial ring P over a field K, return the dimension of P/I as a K-vector space. Note that this is quite different from the function Dimension below (which returns the Krull dimension of an ideal). If I is not of Krull dimension 0 then the vector space is infinite and Infinity is returned.
Given ideals I and J of the same polynomial ring P, return the colon ideal I:J (or ideal quotient of I by J), consisting of the polynomials f of P such that f * g is in I for all g in J.
Given an ideal I and an element f of a polynomial ring P, return the saturation (colon) ideal I:f^∞, consisting of the polynomials g of P such that there exists an i ≥1 with fi * g ∈I. An integer s with s ≥1 is also returned such that I:f^∞= I:fs. Note that if s is not needed, only one return value of the function should be expected which increases the efficiency enormously. Note also that this function is not equivalent to taking the ideal quotient of I by the ideal of P generated by f. It is in some ways a more natural operation mathematically, corresponding to taking the full inverse image of the localised ideal If under the localisation map P -> Pf, and can be faster than the I:f computation, if s is not required. In this case, the computation goes by the elimination of extra variable t from the ideal <I, 1 - f * t>.
Given an ideal I and an element f of a polynomial ring P, return the saturation (colon) ideal C=I:f^∞, and a polynomial g∈P such that C=I:< g > and g is of minimal degree. The irreducible factors of g will be a subset of the irreducible factors of f (and the corresponding multiplicities may be greater or lesser, depending on how often an irreducible factor divides the ideal I).
Given ideals I and J of some polynomial ring P, return the saturation (I:J^∞): that is, the ideal { f ∈P: exists n > 0, f Jn ⊆I}.
Given an ideal I of a polynomial ring P, return the saturation of I with respect to the irrelevant ideal of P -- that is, the ideal of all elements of P having positive degree.
Given an ideal I of a generic polynomial ring P, return P.
Given an ideal I, return the leading monomial ideal of I; that is, the ideal generated by all the leading monomials of I.
Given ideals I and J of the same polynomial ring P, return the intersection of I and J.
Given a set or sequence S of ideals of the same polynomial ring P, return the intersection of all the ideals of S.
Homogeneous: BoolElt Default: true
Given an ideal I of a polynomial ring P over a field, computes and returns a maximal regular sequence in I. The algorithm used is that of Eisenbud and Sturmfels ([ES94]) that tries to construct a regular sequence of fairly sparse polynomials. If parameter Homogeneous is true (the default), and I is a homogeneous ideal with respect to the variable weights, then the regular sequence constructed will also consist of homogeneous polynomials.
a: RngMPolElt Default: 1
a: RngMPolElt Default: 1
In each case P is a multivariate polynomial ring and I is an ideal of P. In the third case R is an affine quotient algebra of the form P/J. In the second case J is another ideal of P and we write R for the affine algebra P/J. In the first case, let R = P.The Rees algebra R(I) is the finitely-generated, graded polynomial algebra isomorphic to the algebra
R direct-sum I direct-sum I2 direct-sum I3 direct-sum ... where I gives the first graded part, I2 the second etc. and the multiplication is the obvious one. Here I is thought of as an ideal of R, rather than P for the second and third signatures. Proj of this algebra represents the blow-up of the affine scheme Spec(R) along the closed subscheme defined by I (see Chapter 2, Section 7 of [Har77]).
The function returns the Rees ideal, K, such that, if R1 is the generic polynomial ring of K, then R1/K is an affine algebra isomorphic to R(I)/<a - torsion>, where a is an element of P (or R in the third case) that gives a non-zero divisor in R and is 1 by default. In the first case, any such a remains a non-zero divisor in R(I), so is redundant. However, in the second and third cases, a can be specified to be not equal to 1 by use of the a parameter. Geometrically, dividing out by a-torsion gives the coordinate ring of the maximal closed subscheme of the blow-up that is flat over the generic point and the codimension one points defined by the vanishing of a, if these points are regular.
Given two ideals I and J of the same polynomial ring P, return whether I and J are equal. Involves the use of a Gröbner basis for each ideal.
Given two ideals I and J of the same polynomial ring P, return whether I and J are not equal. Involves the use of a Gröbner basis for each ideal.
Given two ideals I and J in the same polynomial ring P return whether I is not contained in J. Involves the use of a Gröbner basis for J.
Given two ideals I and J in the same polynomial ring P return whether I is contained in J. Involves the use of a Gröbner basis for J.
Given an ideal I of the polynomial ring P, return whether I is the zero ideal (contains zero alone).
Given an ideal I of the polynomial ring P, return whether I is proper; that is, whether I is strictly contained in P, or whether the Gröbner basis of I does not contain 1 alone.
Given an ideal I of the polynomial ring P, this function returns whether I is homogeneous with respect to the weights on the variables of P (i.e., whether I possesses a basis consisting of homogeneous polynomials alone). Checks whether the current basis of I consists of homogeneous polynomials and, if not and the current basis isn't Gröbner, then whether an easy Gröbner basis consists of homogeneous elements.
Given an ideal I of the polynomial ring P, return whether I is principal, and if so, return also a generator of I. This will be true if and only if an arbitrary Gröbner basis consists of a single (generating) element.
Given an ideal I of the polynomial ring P, return whether I is primary. An ideal I is primary if and only if for all ab ∈I, either a ∈I or bn ∈I for some n ≥1. The restrictions on I are the same as for the function PrimaryDecomposition---see the description of that function. In general, this function computes or retrieves the primary decomposition and checks whether it has a unique element.
Given an ideal I of the polynomial ring P, return whether I is prime. An ideal I is prime if and only if for all ab ∈I, either a ∈I or b ∈I. The restrictions on I are the same as for the function PrimaryDecomposition---see the description of that function. Again, this function computes the primary decomposition or uses the already stored one.
Given an ideal I of the polynomial ring P, return whether I is maximal. The restrictions on I are the same as for the function PrimaryDecomposition---see the description of that function. Checks first whether I is zero-dimensional (see below) and, if so, then checks whether it is prime. NB: given that I is of dimension 0, the prime/primary decomposition computation is relatively fast.
Given an ideal I of the polynomial ring P, return whether I is radical; that is, whether the radical of I is I itself. The restrictions on I are the same as for the function Radical---see the description of that function. The function computes the radical or uses the already stored one.
Given an ideal I of the polynomial ring P, defined over a field, return whether I is zero-dimensional (so the quotient of P by I has non-zero finite dimension as a vector space over the coefficient field -- see the section on dimension for further details). Note that the full polynomial ring P as an ideal of itself has dimension -1, so it is not zero-dimensional.
Given an ideal I of the polynomial ring P, return whether the monomial order of I is the grevlex order.
> P<x,y,z> := PolynomialRing(RationalField(), 3);
> I := ideal<P | x*y - 1, x^3*z^2 - y^2, x*z^3 - x - 1>;
> J := ideal<P | x*y - 1, x^2*z - y, x*z^3 - x - 1>;
> A := I * J;
> A;
Ideal of Polynomial ring of rank 3 over Rational Field
Order: Lexicographical
Variables: x, y, z
Basis:
[
x^2*y^2 - 2*x*y + 1,
x^3*y*z - x^2*z - x*y^2 + y,
x^2*y*z^3 - x^2*y - x*y - x*z^3 + x + 1,
x^4*y*z^2 - x^3*z^2 - x*y^3 + y^2,
x^5*z^3 - x^3*y*z^2 - x^2*y^2*z + y^3,
x^4*z^5 - x^4*z^2 - x^3*z^2 - x*y^2*z^3 + x*y^2 + y^2,
x^2*y*z^3 - x^2*y - x*y - x*z^3 + x + 1,
x^3*z^4 - x^3*z - x^2*z - x*y*z^3 + x*y + y,
x^2*z^6 - 2*x^2*z^3 + x^2 - 2*x*z^3 + 2*x + 1
]
> M := I meet J;
> M;
Ideal of Polynomial ring of rank 3 over Rational Field
Order: Lexicographical
Variables: x, y, z
Basis:
[
x^4 + x^3 - x*z^2 + z^12 - 4*z^9 + 6*z^6 - z^4 - 4*z^3 + z + 1,
x^5 + x^4 - x^2*z^2 + z^9 - 3*z^6 + 3*z^3 - z - 1,
x*z^3 - x - 1,
y - z^3 + 1
]
> A eq M;
true
> QuotientDimension(A);
24
> ColonIdeal(I, J);
Ideal of Polynomial ring of rank 3 over Rational Field
Order: Lexicographical
Variables: x, y, z
Inhomogeneous, Dimension 0
Basis:
[
x*y - 1,
x^3*z^2 - y^2,
x*z^3 - x - 1
]
Given a polynomial f from a polynomial ring P, together with an ideal I of P, return whether f is in I. The function computes the normal form of f relative to some Gröbner basis of I and checks if this is zero.
Given a polynomial f from a polynomial ring P, together with an ideal I of P, return whether f is not in I. As with in, this performs a normal form computation.
Given a polynomial f from a polynomial ring P, together with an ideal I of P, return whether f is in the radical of I. Note that using this function is much quicker in general than actually computing the radical of I. It uses the algorithm described in section 1.8.6 of [GP02].
Return the ideal generated by all first partial derivatives of the polynomial f.
> P<x, y, z> := PolynomialRing(RationalField(), 3); > I := ideal<P | (x + y)^3, (y - z)^2, y^2*z + z>; > NormalForm(y^2*z + z, I); 0 > NormalForm(x^3, I); -3*x^2*y - 3*x*z^4 - 6*x*z^2 + 1/2*z^3 + 3/2*z > NormalForm(z^4 + y^2, I); 2*z^4 + 2*z^2 > x + y in I; false > IsInRadical(x + y, I); true > IsInRadical((x + y)^2, I); true > IsInRadical(z, I); false > SPolynomial(x^4 + y - z, x^2 + y - z); -x^2*y + x^2*z + y - z