Affine \(n\) space over a ring#

sage.schemes.affine.affine_space.AffineSpace(n, R=None, names=None, ambient_projective_space=None, default_embedding_index=None)#

Return affine space of dimension n over the ring R.

EXAMPLES:

The dimension and ring can be given in either order:

sage: AffineSpace(3, QQ, 'x')
Affine Space of dimension 3 over Rational Field
sage: AffineSpace(5, QQ, 'x')
Affine Space of dimension 5 over Rational Field
sage: A = AffineSpace(2, QQ, names='XY'); A
Affine Space of dimension 2 over Rational Field
sage: A.coordinate_ring()
Multivariate Polynomial Ring in X, Y over Rational Field

Use the divide operator for base extension:

sage: AffineSpace(5, names='x')/GF(17)
Affine Space of dimension 5 over Finite Field of size 17

The default base ring is \(\ZZ\):

sage: AffineSpace(5, names='x')
Affine Space of dimension 5 over Integer Ring

There is also an affine space associated to each polynomial ring:

sage: R = GF(7)['x, y, z']
sage: A = AffineSpace(R); A
Affine Space of dimension 3 over Finite Field of size 7
sage: A.coordinate_ring() is R
True
class sage.schemes.affine.affine_space.AffineSpace_field(n, R, names, ambient_projective_space, default_embedding_index)#

Bases: AffineSpace_generic

curve(F)#

Return a curve defined by F in this affine space.

INPUT:

  • F – a polynomial, or a list or tuple of polynomials in the coordinate ring of this affine space

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: A.curve([y - x^4, z - y^5])                                           # needs sage.schemes
Affine Curve over Rational Field defined by -x^4 + y, -y^5 + z
line_through(p, q)#

Return the line through p and q.

INPUT:

  • p, q – distinct rational points of the affine space

EXAMPLES:

sage: # needs sage.libs.singular sage.schemes
sage: A3.<x,y,z> = AffineSpace(3, QQ)
sage: p1 = A3(1, 2, 3)
sage: p2 = A3(4, 5, 6)
sage: L = A3.line_through(p1, p2); L
Affine Curve over Rational Field defined by -1/6*x + 1/6*y - 1/6,
  -1/6*x + 1/6*z - 1/3, -1/6*y + 1/6*z - 1/6, -1/6*x + 1/3*y - 1/6*z
sage: L(p1)
(1, 2, 3)
sage: L(p2)
(4, 5, 6)
sage: A3.line_through(p1, p1)
Traceback (most recent call last):
...
ValueError: not distinct points
points_of_bounded_height(**kwds)#

Return an iterator of the points in this affine space of absolute height of at most the given bound.

Bound check is strict for the rational field. Requires this space to be affine space over a number field. Uses the Doyle-Krumm algorithm 4 (algorithm 5 for imaginary quadratic) for computing algebraic numbers up to a given height [DK2013].

The algorithm requires floating point arithmetic, so the user is allowed to specify the precision for such calculations. Additionally, due to floating point issues, points slightly larger than the bound may be returned. This can be controlled by lowering the tolerance.

INPUT:

kwds:

  • bound - a real number

  • tolerance - a rational number in (0,1] used in doyle-krumm algorithm-4

  • precision - the precision to use for computing the elements of bounded height of number fields

OUTPUT:

  • an iterator of points in self

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: list(A.points_of_bounded_height(bound=3))
[(0, 0), (1, 0), (-1, 0), (1/2, 0), (-1/2, 0), (2, 0), (-2, 0), (0, 1),
 (1, 1), (-1, 1), (1/2, 1), (-1/2, 1), (2, 1), (-2, 1), (0, -1), (1, -1),
 (-1, -1), (1/2, -1), (-1/2, -1), (2, -1), (-2, -1), (0, 1/2), (1, 1/2),
 (-1, 1/2), (1/2, 1/2), (-1/2, 1/2), (2, 1/2), (-2, 1/2), (0, -1/2), (1, -1/2),
 (-1, -1/2), (1/2, -1/2), (-1/2, -1/2), (2, -1/2), (-2, -1/2), (0, 2), (1, 2),
 (-1, 2), (1/2, 2), (-1/2, 2), (2, 2), (-2, 2), (0, -2), (1, -2), (-1, -2),
 (1/2, -2), (-1/2, -2), (2, -2), (-2, -2)]

sage: u = QQ['u'].0
sage: A.<x,y> = AffineSpace(NumberField(u^2 - 2, 'v'), 2)                   # needs sage.rings.number_field
sage: len(list(A.points_of_bounded_height(bound=2, tolerance=0.1)))         # needs sage.rings.number_field
529
translation(p, q=None)#

Return the automorphism of the affine space translating p to the origin.

If q is given, the automorphism translates p to q.

INPUT:

  • p – a rational point

  • q – (default: None) a rational point

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: p = A(1,2,3)
sage: q = A(4,5,6)
sage: A.translation(p, q)
Scheme endomorphism of Affine Space of dimension 3 over Rational Field
  Defn: Defined on coordinates by sending (x, y, z) to
        (x + 3, y + 3, z + 3)
sage: phi = A.translation(p)
sage: psi = A.translation(A.origin(), q)
sage: psi * phi
Scheme endomorphism of Affine Space of dimension 3 over Rational Field
  Defn: Defined on coordinates by sending (x, y, z) to
        (x + 3, y + 3, z + 3)
sage: psi * phi == A.translation(p, q)
True
weil_restriction()#

Compute the Weil restriction of this affine space over some extension field.

If the field is a finite field, then this computes the Weil restriction to the prime subfield.

OUTPUT: Affine space of dimension d * self.dimension_relative() over the base field of self.base_ring().

EXAMPLES:

sage: # needs sage.rings.number_field
sage: R.<x> = QQ[]
sage: K.<w> = NumberField(x^5 - 2)
sage: AK.<x,y> = AffineSpace(K, 2)
sage: AK.weil_restriction()
Affine Space of dimension 10 over Rational Field
sage: R.<x> = K[]
sage: L.<v> = K.extension(x^2 + 1)
sage: AL.<x,y> = AffineSpace(L, 2)
sage: AL.weil_restriction()
Affine Space of dimension 4 over Number Field in w
 with defining polynomial x^5 - 2
class sage.schemes.affine.affine_space.AffineSpace_finite_field(n, R, names, ambient_projective_space, default_embedding_index)#

Bases: AffineSpace_field

class sage.schemes.affine.affine_space.AffineSpace_generic(n, R, names, ambient_projective_space, default_embedding_index)#

Bases: AmbientSpace, AffineScheme

Affine space of dimension \(n\) over the ring \(R\).

EXAMPLES:

sage: X.<x,y,z> = AffineSpace(3, QQ)
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.base_ring()
Rational Field
sage: X.category()
Category of schemes over Rational Field
sage: X.structure_morphism()
Scheme morphism:
  From: Affine Space of dimension 3 over Rational Field
  To:   Spectrum of Rational Field
  Defn: Structure map

Loading and saving:

sage: loads(X.dumps()) == X
True

We create several other examples of affine spaces:

sage: AffineSpace(5, PolynomialRing(QQ, 'z'), 'Z')
Affine Space of dimension 5 over Univariate Polynomial Ring in z over Rational Field

sage: AffineSpace(RealField(), 3, 'Z')                                          # needs sage.rings.real_mpfr
Affine Space of dimension 3 over Real Field with 53 bits of precision

sage: AffineSpace(Qp(7), 2, 'x')                                                # needs sage.rings.padics
Affine Space of dimension 2 over 7-adic Field with capped relative precision 20

Even 0-dimensional affine spaces are supported:

sage: AffineSpace(0)
Affine Space of dimension 0 over Integer Ring
change_ring(R)#

Return an affine space over ring R and otherwise the same as this space.

INPUT:

  • R – commutative ring or morphism.

OUTPUT: An affine space over R.

Note

There is no need to have any relation between R and the base ring of this space, if you want to have such a relation, use self.base_extend(R) instead.

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(3, ZZ)
sage: AQ = A.change_ring(QQ); AQ
Affine Space of dimension 3 over Rational Field
sage: AQ.change_ring(GF(5))
Affine Space of dimension 3 over Finite Field of size 5

sage: K.<w> = QuadraticField(5)                                             # needs sage.rings.number_field
sage: A = AffineSpace(K, 2, 't')                                            # needs sage.rings.number_field
sage: A.change_ring(K.embeddings(CC)[1])                                    # needs sage.rings.number_field
Affine Space of dimension 2 over Complex Field with 53 bits of precision
chebyshev_polynomial(n, kind='first', monic=False)#

Generate an endomorphism of this affine line by a Chebyshev polynomial.

Chebyshev polynomials are a sequence of recursively defined orthogonal polynomials. Chebyshev of the first kind are defined as \(T_0(x) = 1\), \(T_1(x) = x\), and \(T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\). Chebyshev of the second kind are defined as \(U_0(x) = 1\), \(U_1(x) = 2x\), and \(U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)\).

INPUT:

  • n – a non-negative integer.

  • kind – first or second specifying which kind of chebyshev the user would like to generate. Defaults to first.

  • monic – True or False specifying if the polynomial defining the system should be monic or not. Defaults to False.

OUTPUT: DynamicalSystem_affine

EXAMPLES:

sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(5, 'first')                                    # needs sage.schemes
Dynamical System of Affine Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x) to (16*x^5 - 20*x^3 + 5*x)

sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 'second')                                   # needs sage.schemes
Dynamical System of Affine Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x) to (8*x^3 - 4*x)

sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(3, 2)                                          # needs sage.schemes
Traceback (most recent call last):
...
ValueError: keyword 'kind' must have a value of either 'first' or 'second'

sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(-4, 'second')
Traceback (most recent call last):
...
ValueError: first parameter 'n' must be a non-negative integer

sage: A = AffineSpace(QQ, 2, 'x')
sage: A.chebyshev_polynomial(2)
Traceback (most recent call last):
...
TypeError: affine space must be of dimension 1

sage: A.<x> = AffineSpace(QQ, 1)
sage: A.chebyshev_polynomial(7, monic=True)                                 # needs sage.schemes
Dynamical System of Affine Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x) to (x^7 - 7*x^5 + 14*x^3 - 7*x)

sage: F.<t> = FunctionField(QQ)
sage: A.<x> = AffineSpace(F, 1)
sage: A.chebyshev_polynomial(4, monic=True)                                 # needs sage.schemes
Dynamical System of Affine Space of dimension 1
over Rational function field in t over Rational Field
  Defn: Defined on coordinates by sending (x) to (x^4 + (-4)*x^2 + 2)
coordinate_ring()#

Return the coordinate ring of this scheme, if defined.

EXAMPLES:

sage: R = AffineSpace(2, GF(9,'alpha'), 'z').coordinate_ring(); R           # needs sage.rings.finite_rings
Multivariate Polynomial Ring in z0, z1 over Finite Field in alpha of size 3^2
sage: AffineSpace(3, R, 'x').coordinate_ring()                              # needs sage.rings.finite_rings
Multivariate Polynomial Ring in x0, x1, x2 over Multivariate Polynomial Ring
in z0, z1 over Finite Field in alpha of size 3^2
ngens()#

Return the number of generators of self, i.e. the number of variables in the coordinate ring of self.

EXAMPLES:

sage: AffineSpace(3, QQ).ngens()
3
sage: AffineSpace(7, ZZ).ngens()
7
origin()#

Return the rational point at the origin of this affine space.

EXAMPLES:

sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: A.origin()
(0, 0, 0)
sage: _ == A(0,0,0)
True
projective_embedding(i=None, PP=None)#

Return a morphism from this space into an ambient projective space of the same dimension.

INPUT:

  • i – integer (default: dimension of self = last coordinate) determines which projective embedding to compute. The embedding is that which has a 1 in the i-th coordinate, numbered from 0.

  • PP – (default: None) ambient projective space, i.e., codomain of morphism; this is constructed if it is not given.

EXAMPLES:

sage: AA = AffineSpace(2, QQ, 'x')
sage: pi = AA.projective_embedding(0); pi
Scheme morphism:
  From: Affine Space of dimension 2 over Rational Field
  To:   Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x0, x1) to (1 : x0 : x1)
sage: z = AA(3, 4)
sage: pi(z)
(1/4 : 3/4 : 1)
sage: pi(AA(0,2))
(1/2 : 0 : 1)
sage: pi = AA.projective_embedding(1); pi
Scheme morphism:
  From: Affine Space of dimension 2 over Rational Field
  To:   Projective Space of dimension 2 over Rational Field
  Defn: Defined on coordinates by sending (x0, x1) to (x0 : 1 : x1)
sage: pi(z)
(3/4 : 1/4 : 1)
sage: pi = AA.projective_embedding(2)
sage: pi(z)
(3 : 4 : 1)

sage: A.<x,y> = AffineSpace(ZZ, 2)
sage: A.projective_embedding(2).codomain().affine_patch(2) == A
True
rational_points(F=None)#

Return the list of F-rational points on the affine space self, where F is a given finite field, or the base ring of self.

EXAMPLES:

sage: A = AffineSpace(1, GF(3))
sage: A.rational_points()
[(0), (1), (2)]
sage: A.rational_points(GF(3^2, 'b'))                                       # needs sage.rings.finite_rings
[(0), (b), (b + 1), (2*b + 1), (2), (2*b), (2*b + 2), (b + 2), (1)]

sage: AffineSpace(2, ZZ).rational_points(GF(2))
[(0, 0), (0, 1), (1, 0), (1, 1)]
subscheme(X, **kwds)#

Return the closed subscheme defined by X.

INPUT:

  • X - a list or tuple of equations.

EXAMPLES:

sage: A.<x,y> = AffineSpace(QQ, 2)
sage: X = A.subscheme([x, y^2, x*y^2]); X
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
  x,
  y^2,
  x*y^2

sage: # needs sage.libs.singular
sage: X.defining_polynomials ()
(x, y^2, x*y^2)
sage: I = X.defining_ideal(); I
Ideal (x, y^2, x*y^2) of Multivariate Polynomial Ring in x, y over Rational Field
sage: I.groebner_basis()
[y^2, x]
sage: X.dimension()
0
sage: X.base_ring()
Rational Field
sage: X.base_scheme()
Spectrum of Rational Field
sage: X.structure_morphism()
Scheme morphism:
  From: Closed subscheme of Affine Space of dimension 2 over Rational Field
        defined by: x, y^2, x*y^2
  To:   Spectrum of Rational Field
  Defn: Structure map
sage: X.dimension()
0
sage.schemes.affine.affine_space.is_AffineSpace(x)#

Return True if x is an affine space.

EXAMPLES:

sage: from sage.schemes.affine.affine_space import is_AffineSpace
sage: is_AffineSpace(AffineSpace(5, names='x'))
True
sage: is_AffineSpace(AffineSpace(5, GF(9, 'alpha'), names='x'))                 # needs sage.rings.finite_rings
True
sage: is_AffineSpace(Spec(ZZ))
False