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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).
Subsections
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.
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).
IdealQuotient(I, J) : RngMPol, RngMPol -> RngMPol
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.
IdealQuotient(I, f) : RngMPol, RngMPolElt -> RngMPol, RngIntElt
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.
Saturation(I, f) : RngMPol, RngMPolElt -> RngMPol, RngMPolElt
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 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.
Given ideals I and J of some polynomial ring P, return the saturation
(I:J^∞): that is, the ideal
{ f ∈P: exists n > 0, fn J ⊆I}.
Given an ideal I of a polynomial ring P and an element x of P, return
the saturation of I with respect to the principal ideal generated by x.
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 two ideals I and J of the same polynomial ring P,
return whether I and J are equal.
Given two ideals I and J of the same polynomial ring P,
return whether I and J are not equal.
Given two ideals I and J in the same polynomial ring P
return whether I is not contained in J.
Given two ideals I and J in the same polynomial ring P
return whether I is contained in 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 Groebner
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).
Given an ideal I of the polynomial ring P, return whether I is principal,
and if so, return also a generator of I.
IsPrimary(I) : RngMPol -> BoolElt
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.
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.
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.
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.
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
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.
We construct some ideals in Q[x, y, z] and perform basic arithmetic on them.
> 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.
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.
Return the ideal generated by all first partial derivatives
of the polynomial f.
Given a polynomial f from a polynomial ring P, together with an ideal
I of P, return the unique normal form of f with respect to
(the Gröbner basis of) I. The normal form of f is zero if and
only if f is in I.
Given a polynomial f from a polynomial ring P, together with a set
or sequence S of polynomials from P, return a normal form g of f
with respect to S. (This is not unique in general. If the normal
form of f is zero then f is in the ideal generated by S, but the
converse is false in general. In fact, the normal form is unique if
and only if S forms a Gröbner basis.) If S is a sequence, one
may also assign a second return value C which gives the coordinates of
the reduction, so that C[i].S[i] is subtracted from f for
each i to yield g.
Given a polynomial f from a polynomial ring P, together with an ideal
I of P, return whether f is not in I.
Given elements f and g from a polynomial ring P, return the S-polynomial
of f and g.
We demonstrate the element operations with respect to an ideal
of Q[x, y, z].
> 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
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