def create_key_and_extra_args(self, X, Y, category=None, base=ZZ,
check=True):
"""
Create a key that uniquely determines the Hom-set.
INPUT:
- ``X`` -- a scheme. The domain of the morphisms.
- ``Y`` -- a scheme. The codomain of the morphisms.
- ``category`` -- a category for the Hom-sets (default: schemes over
given base).
- ``base`` -- a scheme or a ring. The base scheme of domain
and codomain schemes. If a ring is specified, the spectrum
of that ring will be used as base scheme.
- ``check`` -- boolean (default: ``True``).
EXAMPLES::
sage: A2 = AffineSpace(QQ,2)
sage: A3 = AffineSpace(QQ,3)
sage: A3.Hom(A2) # indirect doctest
Set of morphisms
From: Affine Space of dimension 3 over Rational Field
To: Affine Space of dimension 2 over Rational Field
sage: from sage.schemes.generic.homset import SchemeHomsetFactory
sage: SHOMfactory = SchemeHomsetFactory('test')
sage: key, extra = SHOMfactory.create_key_and_extra_args(A3,A2,check=False)
sage: key
(..., ..., Category of schemes over Integer Ring)
sage: extra
{'Y': Affine Space of dimension 2 over Rational Field,
'X': Affine Space of dimension 3 over Rational Field,
'base_ring': Integer Ring, 'check': False}
"""
if not is_Scheme(X) and is_CommutativeRing(X):
X = Spec(X)
if not is_Scheme(Y) and is_CommutativeRing(Y):
Y = Spec(Y)
if is_Spec(base):
base_spec = base
base_ring = base.coordinate_ring()
elif is_CommutativeRing(base):
base_spec = Spec(base)
base_ring = base
else:
raise ValueError(
'The base must be a commutative ring or its spectrum.')
if not category:
from sage.categories.schemes import Schemes
category = Schemes(base_spec)
key = tuple([id(X), id(Y), category])
extra = {'X':X, 'Y':Y, 'base_ring':base_ring, 'check':check}
return key, extra
def create_key_and_extra_args(self, X, Y, category=None, base=ZZ,
check=True):
"""
Create a key that uniquely determines the Hom-set.
INPUT:
- ``X`` -- a scheme. The domain of the morphisms.
- ``Y`` -- a scheme. The codomain of the morphisms.
- ``category`` -- a category for the Hom-sets (default: schemes over
given base).
- ``base`` -- a scheme or a ring. The base scheme of domain
and codomain schemes. If a ring is specified, the spectrum
of that ring will be used as base scheme.
- ``check`` -- boolean (default: ``True``).
EXAMPLES::
sage: A2 = AffineSpace(QQ,2)
sage: A3 = AffineSpace(QQ,3)
sage: A3.Hom(A2) # indirect doctest
Set of morphisms
From: Affine Space of dimension 3 over Rational Field
To: Affine Space of dimension 2 over Rational Field
sage: from sage.schemes.generic.homset import SchemeHomsetFactory
sage: SHOMfactory = SchemeHomsetFactory('test')
sage: key, extra = SHOMfactory.create_key_and_extra_args(A3,A2,check=False)
sage: key
(..., ..., Category of schemes over Integer Ring)
sage: extra
{'Y': Affine Space of dimension 2 over Rational Field,
'X': Affine Space of dimension 3 over Rational Field,
'base_ring': Integer Ring, 'check': False}
"""
if not is_Scheme(X) and is_CommutativeRing(X):
X = Spec(X)
if not is_Scheme(Y) and is_CommutativeRing(Y):
Y = Spec(Y)
if is_Spec(base):
base_spec = base
base_ring = base.coordinate_ring()
elif is_CommutativeRing(base):
base_spec = Spec(base)
base_ring = base
else:
raise ValueError(
'The base must be a commutative ring or its spectrum.')
if not category:
from sage.categories.schemes import Schemes
category = Schemes(base_spec)
key = tuple([id(X), id(Y), category])
extra = {'X':X, 'Y':Y, 'base_ring':base_ring, 'check':check}
return key, extra
def __init__(self, X, coordinates, check=True):
r"""
See :class:`SchemeMorphism_point_toric_field` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
"""
# Convert scheme to its set of points over the base ring
if is_Scheme(X):
X = X(X.base_ring())
super(SchemeMorphism_point_toric_field, self).__init__(X)
if check:
# Verify that there are the right number of coords
# Why is it not done in the parent?
if is_SchemeMorphism(coordinates):
coordinates = list(coordinates)
if not isinstance(coordinates, (list, tuple)):
raise TypeError("coordinates must be a scheme point, list, "
"or tuple. Got %s" % coordinates)
d = X.codomain().ambient_space().ngens()
if len(coordinates) != d:
raise ValueError("there must be %d coordinates! Got only %d: "
"%s" % (d, len(coordinates), coordinates))
# Make sure the coordinates all lie in the appropriate ring
coordinates = Sequence(coordinates, X.value_ring())
# Verify that the point satisfies the equations of X.
X.codomain()._check_satisfies_equations(coordinates)
self._coords = coordinates
def __init__(self, X, coordinates, check=True):
r"""
See :class:`SchemeMorphism_point_toric_field` for documentation.
TESTS::
sage: fan = FaceFan(lattice_polytope.octahedron(2))
sage: P1xP1 = ToricVariety(fan)
sage: P1xP1(1,2,3,4)
[1 : 2 : 3 : 4]
"""
# Convert scheme to its set of points over the base ring
if is_Scheme(X):
X = X(X.base_ring())
super(SchemeMorphism_point_toric_field, self).__init__(X)
if check:
# Verify that there are the right number of coords
# Why is it not done in the parent?
if is_SchemeMorphism(coordinates):
coordinates = list(coordinates)
if not isinstance(coordinates, (list, tuple)):
raise TypeError("coordinates must be a scheme point, list, "
"or tuple. Got %s" % coordinates)
d = X.codomain().ambient_space().ngens()
if len(coordinates) != d:
raise ValueError("there must be %d coordinates! Got only %d: "
"%s" % (d, len(coordinates), coordinates))
# Make sure the coordinates all lie in the appropriate ring
coordinates = Sequence(coordinates, X.value_ring())
# Verify that the point satisfies the equations of X.
X.codomain()._check_satisfies_equations(coordinates)
self._coords = coordinates
def enum_affine_number_field(X, B):
"""
Enumerates affine points on scheme ``X`` defined over a number field. Simply checks all of the
points of absolute height up to ``B`` and adds those that are on the scheme to the list.
INPUT:
- ``X`` - a scheme defined over a number field.
- ``B`` - a real number.
OUTPUT:
- a list containing the affine points of ``X`` of absolute height up to ``B``,
sorted.
EXAMPLES::
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 2, 'v')
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: X = A.subscheme([y^2 - x])
sage: enum_affine_number_field(X(K), 4)
[(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v), (0, 0, v), (0, 0, 1),
(1, -1, -1), (1, -1, -v), (1, -1, -1/2*v), (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1),
(1, 1, -1), (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = AffineSpace(K, 2)
sage: X=A.subscheme(x-y)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: enum_affine_number_field(X, 3)
[(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2), (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2),
(1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]
"""
from sage.schemes.affine.affine_space import is_AffineSpace
if (is_Scheme(X)):
if (not is_AffineSpace(X.ambient_space())):
raise TypeError(
"ambient space must be affine space over a number field")
X = X(X.base_ring())
else:
if (not is_AffineSpace(X.codomain().ambient_space())):
raise TypeError(
"codomain must be affine space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(B):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def Jacobian(X, **kwds):
"""
Return the Jacobian.
INPUT:
- ``X`` -- polynomial, algebraic variety, or anything else that
has a Jacobian elliptic curve.
- ``kwds`` -- optional keyword arguments.
The input ``X`` can be one of the following:
* A polynomial, see :func:`Jacobian_of_equation` for details.
* A curve, see :func:`Jacobian_of_curve` for details.
EXAMPLES::
sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(C, morphism=True)
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
u^3 + v^3 + w^3
To: Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
Defn: Defined on coordinates by sending (u : v : w) to
(-u^4*v^4*w - u^4*v*w^4 - u*v^4*w^4 :
1/2*u^6*v^3 - 1/2*u^3*v^6 - 1/2*u^6*w^3 + 1/2*v^6*w^3 + 1/2*u^3*w^6 - 1/2*v^3*w^6 :
u^3*v^3*w^3)
"""
try:
return X.jacobian(**kwds)
except AttributeError:
pass
morphism = kwds.pop('morphism', False)
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
if is_MPolynomial(X):
if morphism:
from sage.schemes.curves.constructor import Curve
return Jacobian_of_equation(X, curve=Curve(X), **kwds)
else:
return Jacobian_of_equation(X, **kwds)
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(X) and X.dimension() == 1:
return Jacobian_of_curve(X, morphism=morphism, **kwds)
def Jacobian(X, **kwds):
"""
Return the Jacobian.
INPUT:
- ``X`` -- polynomial, algebraic variety, or anything else that
has a Jacobian elliptic curve.
- ``kwds`` -- optional keyword arguments.
The input ``X`` can be one of the following:
* A polynomial, see :func:`Jacobian_of_equation` for details.
* A curve, see :func:`Jacobian_of_curve` for details.
EXAMPLES::
sage: R.<u,v,w> = QQ[]
sage: Jacobian(u^3+v^3+w^3)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: C = Curve(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: P2.<u,v,w> = ProjectiveSpace(2, QQ)
sage: C = P2.subscheme(u^3+v^3+w^3)
sage: Jacobian(C)
Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
sage: Jacobian(C, morphism=True)
Scheme morphism:
From: Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
u^3 + v^3 + w^3
To: Elliptic Curve defined by y^2 = x^3 - 27/4 over Rational Field
Defn: Defined on coordinates by sending (u : v : w) to
(-u^4*v^4*w - u^4*v*w^4 - u*v^4*w^4 :
1/2*u^6*v^3 - 1/2*u^3*v^6 - 1/2*u^6*w^3 + 1/2*v^6*w^3 + 1/2*u^3*w^6 - 1/2*v^3*w^6 :
u^3*v^3*w^3)
"""
try:
return X.jacobian(**kwds)
except AttributeError:
pass
morphism = kwds.pop('morphism', False)
from sage.rings.polynomial.multi_polynomial_element import is_MPolynomial
if is_MPolynomial(X):
if morphism:
from sage.schemes.curves.constructor import Curve
return Jacobian_of_equation(X, curve=Curve(X), **kwds)
else:
return Jacobian_of_equation(X, **kwds)
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(X) and X.dimension() == 1:
return Jacobian_of_curve(X, morphism=morphism, **kwds)
def enum_product_projective_finite_field(X):
r"""
Enumerates projective points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the projective points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: PP.<x,y,z,w> = ProductProjectiveSpaces([1, 1], GF(3))
sage: from sage.schemes.product_projective.rational_point import \
enum_product_projective_finite_field
sage: enum_product_projective_finite_field(PP)
[(0 : 1 , 0 : 1), (0 : 1 , 1 : 0), (0 : 1 , 1 : 1),
(0 : 1 , 2 : 1), (1 : 0 , 0 : 1), (1 : 0 , 1 : 0),
(1 : 0 , 1 : 1), (1 : 0 , 2 : 1), (1 : 1 , 0 : 1),
(1 : 1 , 1 : 0), (1 : 1 , 1 : 1), (1 : 1 , 2 : 1),
(2 : 1 , 0 : 1), (2 : 1 , 1 : 0), (2 : 1 , 1 : 1),
(2 : 1 , 2 : 1)]
::
sage: PP.<x0,x1,x2,x3> = ProductProjectiveSpaces([1, 1], GF(17))
sage: X = PP.subscheme([x0^2 + 2*x1^2])
sage: from sage.schemes.product_projective.rational_point import \
enum_product_projective_finite_field
sage: len(enum_product_projective_finite_field(X))
36
"""
if (is_Scheme(X)):
if (not is_ProductProjectiveSpaces(X.ambient_space())):
raise TypeError(
"ambient space must be product of projective space over the rational field"
)
X = X(X.base_ring())
else:
if (not is_ProductProjectiveSpaces(X.codomain().ambient_space())):
raise TypeError(
"codomain must be product of projective space over the rational field"
)
R = X.codomain().ambient_space()
pts = []
for P in R.rational_points():
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def enum_affine_number_field(X, B):
"""
Enumerates affine points on scheme ``X`` defined over a number field. Simply checks all of the
points of absolute height up to ``B`` and adds those that are on the scheme to the list.
INPUT:
- ``X`` - a scheme defined over a number field.
- ``B`` - a real number.
OUTPUT:
- a list containing the affine points of ``X`` of absolute height up to ``B``,
sorted.
EXAMPLES::
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 2, 'v')
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: X = A.subscheme([y^2 - x])
sage: enum_affine_number_field(X(K), 4)
[(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v), (0, 0, v), (0, 0, 1),
(1, -1, -1), (1, -1, -v), (1, -1, -1/2*v), (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1),
(1, 1, -1), (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = AffineSpace(K, 2)
sage: X=A.subscheme(x-y)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: enum_affine_number_field(X, 3)
[(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2), (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2),
(1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]
"""
from sage.schemes.affine.affine_space import is_AffineSpace
if(is_Scheme(X)):
if (not is_AffineSpace(X.ambient_space())):
raise TypeError("ambient space must be affine space over a number field")
X = X(X.base_ring())
else:
if (not is_AffineSpace(X.codomain().ambient_space())):
raise TypeError("codomain must be affine space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(B):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def enum_projective_number_field(X,B):
"""
Enumerates projective points on scheme ``X`` defined over a number field. Simply checks all of the
points of absolute height of at most ``B`` and adds those that are on the scheme to the list.
INPUT:
- ``X`` - a scheme defined over a number field
- ``B`` - a real number
OUTPUT:
- a list containing the projective points of ``X`` of absolute height up to ``B``,
sorted.
EXAMPLES::
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^3 - 5,'v')
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: X = P.subscheme([x - y])
sage: enum_projective_number_field(X(K),125)
[(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
(1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3,'v')
sage: A.<x,y> = ProjectiveSpace(K,1)
sage: X=A.subscheme(x-y)
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: enum_projective_number_field(X,3)
[(1 : 1)]
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if(is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError("Ambient space must be projective space over a number field")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError("Codomain must be projective space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(B):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def __init__(self, C):
"""
Initialize.
TESTS::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^3 + y^3 + z^3)
sage: J = Jacobian_generic(C); J
Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: type(J)
<class 'sage.schemes.jacobians.abstract_jacobian.Jacobian_generic_with_category'>
Note: this is an abstract parent, so we skip element tests::
sage: TestSuite(J).run(skip =["_test_an_element",\
"_test_elements",\
"_test_elements_eq_reflexive",\
"_test_elements_eq_symmetric",\
"_test_elements_eq_transitive",\
"_test_elements_neq",\
"_test_some_elements"])
::
sage: Jacobian_generic(ZZ)
Traceback (most recent call last):
...
TypeError: Argument (=Integer Ring) must be a scheme.
sage: Jacobian_generic(P2)
Traceback (most recent call last):
...
ValueError: C (=Projective Space of dimension 2 over Rational Field)
must have dimension 1.
::
sage: P2.<x, y, z> = ProjectiveSpace(Zmod(6), 2)
sage: C = Curve(x + y + z, P2)
sage: Jacobian_generic(C)
Traceback (most recent call last):
...
TypeError: C (=Projective Plane Curve over Ring of integers modulo 6
defined by x + y + z) must be defined over a field.
"""
if not is_Scheme(C):
raise TypeError("Argument (=%s) must be a scheme." % C)
if C.base_ring() not in _Fields:
raise TypeError("C (=%s) must be defined over a field." % C)
if C.dimension() != 1:
raise ValueError("C (=%s) must have dimension 1." % C)
self.__curve = C
Scheme.__init__(self, C.base_scheme())
def __call__(self, x):
"""
Call syntax for Spec.
INPUT/OUTPUT:
The argument ``x`` must be one of the following:
- a prime ideal of the coordinate ring; the output will
be the corresponding point of X
- an element (or list of elements) of the coordinate ring
which generates a prime ideal; the output will be the
corresponding point of X
- a ring or a scheme S; the output will be the set X(S) of
S-valued points on X
EXAMPLES::
sage: S = Spec(ZZ)
sage: P = S(3); P
Point on Spectrum of Integer Ring defined by the Principal ideal (3) of Integer Ring
sage: type(P)
<class 'sage.schemes.generic.point.SchemeTopologicalPoint_prime_ideal'>
sage: S(ZZ.ideal(next_prime(1000000)))
Point on Spectrum of Integer Ring defined by the Principal ideal (1000003) of Integer Ring
sage: R.<x, y, z> = QQ[]
sage: S = Spec(R)
sage: P = S(R.ideal(x, y, z)); P
Point on Spectrum of Multivariate Polynomial Ring
in x, y, z over Rational Field defined by the Ideal (x, y, z)
of Multivariate Polynomial Ring in x, y, z over Rational Field
This indicates the fix of :trac:`12734`::
sage: S = Spec(ZZ)
sage: S(ZZ)
Set of rational points of Spectrum of Integer Ring
sage: S(S)
Set of rational points of Spectrum of Integer Ring
"""
if is_CommutativeRing(x):
return self.point_homset(x)
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x.Hom(self)
return SchemeTopologicalPoint_prime_ideal(self, x)
def __init__(self, C):
"""
TESTS::
sage: from sage.schemes.jacobians.abstract_jacobian import Jacobian_generic
sage: P2.<x, y, z> = ProjectiveSpace(QQ, 2)
sage: C = Curve(x^3 + y^3 + z^3)
sage: J = Jacobian_generic(C); J
Jacobian of Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3
sage: type(J)
<class 'sage.schemes.jacobians.abstract_jacobian.Jacobian_generic_with_category'>
Note: this is an abstract parent, so we skip element tests::
sage: TestSuite(J).run(skip =["_test_an_element",\
"_test_elements",\
"_test_elements_eq_reflexive",\
"_test_elements_eq_symmetric",\
"_test_elements_eq_transitive",\
"_test_elements_neq",\
"_test_some_elements"])
::
sage: Jacobian_generic(ZZ)
Traceback (most recent call last):
...
TypeError: Argument (=Integer Ring) must be a scheme.
sage: Jacobian_generic(P2)
Traceback (most recent call last):
...
ValueError: C (=Projective Space of dimension 2 over Rational Field) must have dimension 1.
sage: P2.<x, y, z> = ProjectiveSpace(Zmod(6), 2)
sage: C = Curve(x + y + z)
sage: Jacobian_generic(C)
Traceback (most recent call last):
...
TypeError: C (=Projective Plane Curve over Ring of integers modulo 6 defined by x + y + z) must be defined over a field.
"""
if not is_Scheme(C):
raise TypeError("Argument (=%s) must be a scheme."%C)
if C.base_ring() not in _Fields:
raise TypeError("C (=%s) must be defined over a field."%C)
if C.dimension() != 1:
raise ValueError("C (=%s) must have dimension 1."%C)
self.__curve = C
Scheme.__init__(self, C.base_scheme())
def Schemes(X=None):
"""
Construct a category of schemes.
EXAMPLES::
sage: Schemes()
Category of Schemes
sage: Schemes(Spec(ZZ))
Category of schemes over Integer Ring
sage: Schemes(ZZ)
Category of schemes over Integer Ring
"""
if X is None:
return Schemes_abstract()
from sage.schemes.generic.scheme import is_Scheme
if not is_Scheme(X):
X = Schemes()(X)
return Schemes_over_base(X)
def __classcall_private__(cls, X = None):
"""
Implement the dispatching ``Schemes(ZZ)`` -> ``Schemes_over_base``.
EXAMPLES::
sage: Schemes()
Category of schemes
sage: Schemes(Spec(ZZ))
Category of schemes over Integer Ring
sage: Schemes(ZZ)
Category of schemes over Integer Ring
"""
if X is not None:
from sage.schemes.generic.scheme import is_Scheme
if not is_Scheme(X):
X = Schemes()(X)
return Schemes_over_base(X)
else:
return super(Schemes, cls).__classcall__(cls)
def __classcall_private__(cls, X=None):
"""
Implement the dispatching ``Schemes(ZZ)`` -> ``Schemes_over_base``.
EXAMPLES::
sage: Schemes()
Category of schemes
sage: Schemes(Spec(ZZ))
Category of schemes over Integer Ring
sage: Schemes(ZZ)
Category of schemes over Integer Ring
"""
if X is not None:
from sage.schemes.generic.scheme import is_Scheme
if not is_Scheme(X):
X = Schemes()(X)
return Schemes_over_base(X)
else:
return super(Schemes, cls).__classcall__(cls)
def enum_affine_rational_field(X, B):
"""
Enumerates affine rational points on scheme ``X`` (defined over `\QQ`) up
to bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the affine points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(3,QQ)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
sage: enum_affine_rational_field(A(QQ),1)
[(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]
::
sage: A.<w,x,y,z> = AffineSpace(4,QQ)
sage: S = A.subscheme([x^2-y*z+3,w^3+z+y^2])
sage: enum_affine_rational_field(S(QQ),2)
[]
sage: enum_affine_rational_field(S(QQ),3)
[(-2, 0, -3, -1)]
::
sage: A.<x,y> = AffineSpace(2,QQ)
sage: C = Curve(x^2+y-x)
sage: enum_affine_rational_field(C,10)
[(-2, -6), (-1, -2), (0, 0), (1, 0), (2, -2), (3, -6)]
AUTHORS:
- David R. Kohel <*****@*****.**>: original version.
- Charlie Turner (06-2010): small adjustments.
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
if X.value_ring() is ZZ:
Q = [1]
else: # rational field
Q = range(1, B + 1)
R = [0] + [s * k for k in range(1, B + 1) for s in [1, -1]]
pts = []
P = [0] * n
m = ZZ(0)
try:
pts.append(X(P))
except TypeError:
pass
iters = [iter(R) for _ in range(n)]
for it in iters:
it.next()
i = 0
while i < n:
try:
a = ZZ(iters[i].next())
except StopIteration:
iters[i] = iter(R) # reset
P[i] = iters[i].next() # reset P[i] to 0 and increment
i += 1
continue
m = m.gcd(a)
P[i] = a
for b in Q:
if m.gcd(b) == 1:
try:
pts.append(X([num / b for num in P]))
except TypeError:
pass
i = 0
m = ZZ(0)
pts.sort()
return pts
def enum_projective_rational_field(X, B):
r"""
Enumerates projective, rational points on scheme ``X`` of height up to
bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme.
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the projective points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: P.<X,Y,Z> = ProjectiveSpace(2, QQ)
sage: C = P.subscheme([X+Y-Z])
sage: from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
sage: enum_projective_rational_field(C(QQ), 6)
[(-5 : 6 : 1), (-4 : 5 : 1), (-3 : 4 : 1), (-2 : 3 : 1),
(-3/2 : 5/2 : 1), (-1 : 1 : 0), (-1 : 2 : 1), (-2/3 : 5/3 : 1),
(-1/2 : 3/2 : 1), (-1/3 : 4/3 : 1), (-1/4 : 5/4 : 1),
(-1/5 : 6/5 : 1), (0 : 1 : 1), (1/6 : 5/6 : 1), (1/5 : 4/5 : 1),
(1/4 : 3/4 : 1), (1/3 : 2/3 : 1), (2/5 : 3/5 : 1), (1/2 : 1/2 : 1),
(3/5 : 2/5 : 1), (2/3 : 1/3 : 1), (3/4 : 1/4 : 1), (4/5 : 1/5 : 1),
(5/6 : 1/6 : 1), (1 : 0 : 1), (6/5 : -1/5 : 1), (5/4 : -1/4 : 1),
(4/3 : -1/3 : 1), (3/2 : -1/2 : 1), (5/3 : -2/3 : 1), (2 : -1 : 1),
(5/2 : -3/2 : 1), (3 : -2 : 1), (4 : -3 : 1), (5 : -4 : 1),
(6 : -5 : 1)]
sage: enum_projective_rational_field(C,6) == enum_projective_rational_field(C(QQ),6)
True
::
sage: P3.<W,X,Y,Z> = ProjectiveSpace(3, QQ)
sage: enum_projective_rational_field(P3, 1)
[(-1 : -1 : -1 : 1), (-1 : -1 : 0 : 1), (-1 : -1 : 1 : 0), (-1 : -1 : 1 : 1),
(-1 : 0 : -1 : 1), (-1 : 0 : 0 : 1), (-1 : 0 : 1 : 0), (-1 : 0 : 1 : 1),
(-1 : 1 : -1 : 1), (-1 : 1 : 0 : 0), (-1 : 1 : 0 : 1), (-1 : 1 : 1 : 0),
(-1 : 1 : 1 : 1), (0 : -1 : -1 : 1), (0 : -1 : 0 : 1), (0 : -1 : 1 : 0),
(0 : -1 : 1 : 1), (0 : 0 : -1 : 1), (0 : 0 : 0 : 1), (0 : 0 : 1 : 0),
(0 : 0 : 1 : 1), (0 : 1 : -1 : 1), (0 : 1 : 0 : 0), (0 : 1 : 0 : 1),
(0 : 1 : 1 : 0), (0 : 1 : 1 : 1), (1 : -1 : -1 : 1), (1 : -1 : 0 : 1),
(1 : -1 : 1 : 0), (1 : -1 : 1 : 1), (1 : 0 : -1 : 1), (1 : 0 : 0 : 0),
(1 : 0 : 0 : 1), (1 : 0 : 1 : 0), (1 : 0 : 1 : 1), (1 : 1 : -1 : 1),
(1 : 1 : 0 : 0), (1 : 1 : 0 : 1), (1 : 1 : 1 : 0), (1 : 1 : 1 : 1)]
ALGORITHM:
We just check all possible projective points in correct dimension
of projective space to see if they lie on ``X``.
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if (is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError(
"ambient space must be projective space over the rational field"
)
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError(
"codomain must be projective space over the rational field")
n = X.codomain().ambient_space().ngens()
zero = (0, ) * n
pts = []
for c in cartesian_product_iterator([srange(-B, B + 1) for _ in range(n)]):
if gcd(c) == 1 and c > zero:
try:
pts.append(X(c))
except TypeError:
pass
pts.sort()
return pts
def enum_projective_finite_field(X):
"""
Enumerates projective points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the projective points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(53)
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
sage: len(enum_projective_finite_field(P(F)))
2863
sage: 53^2+53+1
2863
::
sage: F = GF(9,'a')
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: C = Curve(X^3-Y^3+Z^2*Y)
sage: enum_projective_finite_field(C(F))
[(0 : 0 : 1), (0 : 1 : 1), (0 : 2 : 1), (1 : 1 : 0), (a + 1 : 2*a : 1),
(a + 1 : 2*a + 1 : 1), (a + 1 : 2*a + 2 : 1), (2*a + 2 : a : 1),
(2*a + 2 : a + 1 : 1), (2*a + 2 : a + 2 : 1)]
::
sage: F = GF(5)
sage: P2F.<X,Y,Z> = ProjectiveSpace(2,F)
sage: enum_projective_finite_field(P2F)
[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 2 : 1), (0 : 3 : 1), (0 : 4 : 1),
(1 : 0 : 0), (1 : 0 : 1), (1 : 1 : 0), (1 : 1 : 1), (1 : 2 : 1), (1 : 3 : 1),
(1 : 4 : 1), (2 : 0 : 1), (2 : 1 : 0), (2 : 1 : 1), (2 : 2 : 1), (2 : 3 : 1),
(2 : 4 : 1), (3 : 0 : 1), (3 : 1 : 0), (3 : 1 : 1), (3 : 2 : 1), (3 : 3 : 1),
(3 : 4 : 1), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1),
(4 : 4 : 1)]
ALGORITHM:
Checks all points in projective space to see if they lie on ``X``.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010).
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if (is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError(
"ambient space must be projective space over a finite")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError(
"codomain must be projective space over a finite field")
n = X.codomain().ambient_space().ngens() - 1
F = X.value_ring()
pts = []
for k in range(n + 1):
for c in cartesian_product_iterator([F for _ in range(k)]):
try:
pts.append(X(list(c) + [1] + [0] * (n - k)))
except TypeError:
pass
pts.sort()
return pts
def enum_projective_number_field(X, B, prec=53):
"""
Enumerates projective points on scheme ``X`` defined over a number field.
Simply checks all of the points of absolute height of at most ``B``
and adds those that are on the scheme to the list.
INPUT:
- ``X`` - a scheme defined over a number field.
- ``B`` - a real number.
- ``prec`` - the precision to use for computing the elements of bounded height of number fields.
OUTPUT:
- a list containing the projective points of ``X`` of absolute height up to ``B``,
sorted.
.. WARNING::
In the current implementation, the output of the [Doyle-Krumm] algorithm
for elements of bounded height cannot be guaranteed to be correct due to
the necessity of floating point computations. In some cases, the default
53-bit precision is considerably lower than would be required for the
algorithm to generate correct output.
EXAMPLES::
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^3 - 5,'v')
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: X = P.subscheme([x - y])
sage: enum_projective_number_field(X(K), 5^(1/3), prec=2^10)
[(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
(1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = ProjectiveSpace(K,1)
sage: X = A.subscheme(x-y)
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: enum_projective_number_field(X, 2)
[(1 : 1)]
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if (is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError(
"ambient space must be projective space over a number field")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError(
"codomain must be projective space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(B, prec):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
sage: phi = ZZ.hom(QQ); phi
Natural morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Natural morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Natural morphism:
From: Integer Ring
To: Rational Field
"""
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x
from sage.schemes.generic.morphism import is_SchemeMorphism
if is_SchemeMorphism(x):
return x
from sage.rings.ring import CommutativeRing
from sage.schemes.generic.spec import Spec
from sage.categories.map import Map
from sage.categories.all import Rings
if isinstance(x, CommutativeRing):
return Spec(x)
elif isinstance(x, Map) and x.category_for().is_subcategory(Rings()):
# x is a morphism of Rings
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))
def enum_projective_number_field(X, **kwds):
"""
Enumerates projective points on scheme ``X`` defined over a number field.
Simply checks all of the points of absolute height of at most ``B``
and adds those that are on the scheme to the list.
This algorithm computes 2 lists: L containing elements x in `K` such that
H_k(x) <= B, and a list L' containing elements x in `K` that, due to
floating point issues,
may be slightly larger then the bound. This can be controlled
by lowering the tolerance.
ALGORITHM:
This is an implementation of the revised algorithm (Algorithm 4) in
[DK2013]_. Algorithm 5 is used for imaginary quadratic fields.
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:
- a list containing the projective points of ``X`` of absolute height up to ``B``,
sorted.
EXAMPLES::
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^3 - 5,'v')
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: X = P.subscheme([x - y])
sage: enum_projective_number_field(X(K), bound=RR(5^(1/3)), prec=2^10)
[(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
(1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = ProjectiveSpace(K,1)
sage: X = A.subscheme(x-y)
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: enum_projective_number_field(X, bound=2)
[(1 : 1)]
"""
B = kwds.pop('bound')
tol = kwds.pop('tolerance', 1e-2)
prec = kwds.pop('precision', 53)
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if is_Scheme(X):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError(
"ambient space must be projective space over a number field")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError(
"codomain must be projective space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(bound=B, tolerance=tol,
precision=prec):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def enum_affine_number_field(X, **kwds):
"""
Enumerates affine points on scheme ``X`` defined over a number field. Simply checks all of the
points of absolute height up to ``B`` and adds those that are on the scheme to the list.
This algorithm computes 2 lists: L containing elements x in `K` such that
H_k(x) <= B, and a list L' containing elements x in `K` that, due to
floating point issues,
may be slightly larger then the bound. This can be controlled
by lowering the tolerance.
ALGORITHM:
This is an implementation of the revised algorithm (Algorithm 4) in
[Doyle-Krumm]_. Algorithm 5 is used for imaginary quadratic fields.
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:
- a list containing the affine points of ``X`` of absolute height up to ``B``,
sorted.
EXAMPLES::
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 2, 'v')
sage: A.<x,y,z> = AffineSpace(K, 3)
sage: X = A.subscheme([y^2 - x])
sage: enum_affine_number_field(X(K), bound=2**0.5)
[(0, 0, -1), (0, 0, -v), (0, 0, -1/2*v), (0, 0, 0), (0, 0, 1/2*v), (0, 0, v), (0, 0, 1),
(1, -1, -1), (1, -1, -v), (1, -1, -1/2*v), (1, -1, 0), (1, -1, 1/2*v), (1, -1, v), (1, -1, 1),
(1, 1, -1), (1, 1, -v), (1, 1, -1/2*v), (1, 1, 0), (1, 1, 1/2*v), (1, 1, v), (1, 1, 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = AffineSpace(K, 2)
sage: X=A.subscheme(x-y)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_number_field
sage: enum_affine_number_field(X, bound=3**0.25)
[(-1, -1), (-1/2*v - 1/2, -1/2*v - 1/2), (1/2*v - 1/2, 1/2*v - 1/2), (0, 0), (-1/2*v + 1/2, -1/2*v + 1/2),
(1/2*v + 1/2, 1/2*v + 1/2), (1, 1)]
"""
B = kwds.pop('bound')
tol = kwds.pop('tolerance', 1e-2)
prec = kwds.pop('precision', 53)
from sage.schemes.affine.affine_space import is_AffineSpace
if is_Scheme(X):
if not is_AffineSpace(X.ambient_space()):
raise TypeError(
"ambient space must be affine space over a number field")
X = X(X.base_ring())
elif not is_AffineSpace(X.codomain().ambient_space()):
raise TypeError("codomain must be affine space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(bound=B, tolerance=tol,
precision=prec):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
sage: phi = ZZ.hom(QQ); phi
Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
"""
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x
from sage.schemes.generic.morphism import is_SchemeMorphism
if is_SchemeMorphism(x):
return x
from sage.rings.morphism import is_RingHomomorphism
from sage.rings.ring import CommutativeRing
from sage.schemes.generic.spec import Spec
if isinstance(x, CommutativeRing):
return Spec(x)
elif is_RingHomomorphism(x):
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError("No way to create an object or morphism in %s from %s"%(self, x))
def enum_product_projective_rational_field(X, B):
r"""
Enumerate projective, rational points on scheme ``X`` of height up to
bound ``B``.
INPUT:
- ``X`` -- a scheme or set of abstract rational points of a scheme
- ``B`` -- a positive integer bound
OUTPUT:
- a list containing the product projective points of ``X`` of height up
to ``B``, sorted.
EXAMPLES::
sage: PP.<x0,x1,x2,x3,x4> = ProductProjectiveSpaces([1, 2], QQ)
sage: from sage.schemes.product_projective.rational_point import \
enum_product_projective_rational_field
sage: enum_product_projective_rational_field(PP,1)
[(-1 : 1 , -1 : -1 : 1), (-1 : 1 , -1 : 0 : 1), (-1 : 1 , -1 : 1 : 0),
(-1 : 1 , -1 : 1 : 1), (-1 : 1 , 0 : -1 : 1), (-1 : 1 , 0 : 0 : 1),
(-1 : 1 , 0 : 1 : 0), (-1 : 1 , 0 : 1 : 1), (-1 : 1 , 1 : -1 : 1),
(-1 : 1 , 1 : 0 : 0), (-1 : 1 , 1 : 0 : 1), (-1 : 1 , 1 : 1 : 0),
(-1 : 1 , 1 : 1 : 1), (0 : 1 , -1 : -1 : 1), (0 : 1 , -1 : 0 : 1),
(0 : 1 , -1 : 1 : 0), (0 : 1 , -1 : 1 : 1), (0 : 1 , 0 : -1 : 1),
(0 : 1 , 0 : 0 : 1), (0 : 1 , 0 : 1 : 0), (0 : 1 , 0 : 1 : 1),
(0 : 1 , 1 : -1 : 1), (0 : 1 , 1 : 0 : 0), (0 : 1 , 1 : 0 : 1),
(0 : 1 , 1 : 1 : 0), (0 : 1 , 1 : 1 : 1), (1 : 0 , -1 : -1 : 1),
(1 : 0 , -1 : 0 : 1), (1 : 0 , -1 : 1 : 0), (1 : 0 , -1 : 1 : 1),
(1 : 0 , 0 : -1 : 1), (1 : 0 , 0 : 0 : 1), (1 : 0 , 0 : 1 : 0),
(1 : 0 , 0 : 1 : 1), (1 : 0 , 1 : -1 : 1), (1 : 0 , 1 : 0 : 0),
(1 : 0 , 1 : 0 : 1), (1 : 0 , 1 : 1 : 0), (1 : 0 , 1 : 1 : 1),
(1 : 1 , -1 : -1 : 1), (1 : 1 , -1 : 0 : 1), (1 : 1 , -1 : 1 : 0),
(1 : 1 , -1 : 1 : 1), (1 : 1 , 0 : -1 : 1), (1 : 1 , 0 : 0 : 1),
(1 : 1 , 0 : 1 : 0), (1 : 1 , 0 : 1 : 1), (1 : 1 , 1 : -1 : 1),
(1 : 1 , 1 : 0 : 0), (1 : 1 , 1 : 0 : 1), (1 : 1 , 1 : 1 : 0),
(1 : 1 , 1 : 1 : 1)]
::
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces([2,1], QQ)
sage: X = PP.subscheme([x^2 + x*y + y*z, u*u-v*u])
sage: from sage.schemes.product_projective.rational_point import \
enum_product_projective_rational_field
sage: enum_product_projective_rational_field(X,4)
[(-2 : 4 : 1 , 0 : 1), (-2 : 4 : 1 , 1 : 1), (-1 : 1 : 0 , 0 : 1),
(-1 : 1 : 0 , 1 : 1), (-2/3 : -4/3 : 1 , 0 : 1), (-2/3 : -4/3 : 1 , 1 : 1),
(-1/2 : -1/2 : 1 , 0 : 1), (-1/2 : -1/2 : 1 , 1 : 1),
(0 : 0 : 1 , 0 : 1), (0 : 0 : 1 , 1 : 1), (0 : 1 : 0 , 0 : 1),
(0 : 1 : 0 , 1 : 1), (1 : -1/2 : 1 , 0 : 1), (1 : -1/2 : 1 , 1 : 1)]
"""
if(is_Scheme(X)):
if (not is_ProductProjectiveSpaces(X.ambient_space())):
raise TypeError("ambient space must be product of projective space over the rational field")
X = X(X.base_ring())
else:
if (not is_ProductProjectiveSpaces(X.codomain().ambient_space())):
raise TypeError("codomain must be product of projective space over the rational field")
R = X.codomain().ambient_space()
m = R.num_components()
iters = [ R[i].points_of_bounded_height(bound=B) for i in range(m) ]
dim = [R[i].dimension_relative() + 1 for i in range(m)]
dim_prefix = [0, dim[0]] # prefixes dim list
for i in range(1, len(dim)):
dim_prefix.append(dim_prefix[i] + dim[i])
pts = []
P = []
for i in range(m):
pt = next(iters[i])
for j in range(dim[i]):
P.append(pt[j]) # initial value of P
try: # add the initial point
pts.append(X(P))
except TypeError:
pass
i = 0
while i < m:
try:
pt = next(iters[i])
for j in range(dim[i]):
P[dim_prefix[i] + j] = pt[j]
try:
pts.append(X(P))
except TypeError:
pass
i = 0
except StopIteration:
iters[i] = R[i].points_of_bounded_height(bound=B)
pt = next(iters[i]) # reset
for j in range(dim[i]):
P[dim_prefix[i] + j] = pt[j]
i += 1
pts.sort()
return pts
def enum_affine_finite_field(X):
r"""
Enumerates affine points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the affine points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(7)
sage: A.<w,x,y,z> = AffineSpace(4, F)
sage: C = A.subscheme([w^2+x+4, y*z*x-6, z*y+w*x])
sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
sage: enum_affine_finite_field(C(F))
[]
sage: C = A.subscheme([w^2+x+4, y*z*x-6])
sage: enum_affine_finite_field(C(F))
[(0, 3, 1, 2), (0, 3, 2, 1), (0, 3, 3, 3), (0, 3, 4, 4), (0, 3, 5, 6),
(0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
(1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
(2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
(3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
(4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
(5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
(5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
(6, 2, 5, 2), (6, 2, 6, 4)]
::
sage: A.<x,y,z> = AffineSpace(3, GF(3))
sage: S = A.subscheme(x+y)
sage: enum_affine_finite_field(S)
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2),
(2, 1, 0), (2, 1, 1), (2, 1, 2)]
ALGORITHM:
Checks all points in affine space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
from sage.schemes.affine.affine_space import is_AffineSpace
if is_Scheme(X):
if not is_AffineSpace(X.ambient_space()):
raise TypeError(
"ambient space must be affine space over a finite field")
X = X(X.base_ring())
elif not is_AffineSpace(X.codomain().ambient_space()):
raise TypeError("codomain must be affine space over a finite field")
n = X.codomain().ambient_space().ngens()
F = X.value_ring()
pts = []
for c in cartesian_product_iterator([F] * n):
try:
pts.append(X(c))
except Exception:
pass
pts.sort()
return pts
def enum_affine_finite_field(X):
r"""
Enumerates affine points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the affine points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(7)
sage: A.<w,x,y,z> = AffineSpace(4,F)
sage: C = A.subscheme([w^2+x+4,y*z*x-6,z*y+w*x])
sage: from sage.schemes.affine.affine_rational_point import enum_affine_finite_field
sage: enum_affine_finite_field(C(F))
[]
sage: C = A.subscheme([w^2+x+4,y*z*x-6])
sage: enum_affine_finite_field(C(F))
[(0, 3, 1, 2), (0, 3, 2, 1), (0, 3, 3, 3), (0, 3, 4, 4), (0, 3, 5, 6),
(0, 3, 6, 5), (1, 2, 1, 3), (1, 2, 2, 5), (1, 2, 3, 1), (1, 2, 4, 6),
(1, 2, 5, 2), (1, 2, 6, 4), (2, 6, 1, 1), (2, 6, 2, 4), (2, 6, 3, 5),
(2, 6, 4, 2), (2, 6, 5, 3), (2, 6, 6, 6), (3, 1, 1, 6), (3, 1, 2, 3),
(3, 1, 3, 2), (3, 1, 4, 5), (3, 1, 5, 4), (3, 1, 6, 1), (4, 1, 1, 6),
(4, 1, 2, 3), (4, 1, 3, 2), (4, 1, 4, 5), (4, 1, 5, 4), (4, 1, 6, 1),
(5, 6, 1, 1), (5, 6, 2, 4), (5, 6, 3, 5), (5, 6, 4, 2), (5, 6, 5, 3),
(5, 6, 6, 6), (6, 2, 1, 3), (6, 2, 2, 5), (6, 2, 3, 1), (6, 2, 4, 6),
(6, 2, 5, 2), (6, 2, 6, 4)]
::
sage: A.<x,y,z> = AffineSpace(3,GF(3))
sage: S = A.subscheme(x+y)
sage: enum_affine_finite_field(S)
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2),
(2, 1, 0), (2, 1, 1), (2, 1, 2)]
ALGORITHM:
Checks all points in affine space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
F = X.value_ring()
pts = []
for c in cartesian_product_iterator([F]*n):
try:
pts.append(X(c))
except Exception:
pass
pts.sort()
return pts
def enum_affine_rational_field(X,B):
"""
Enumerates affine rational points on scheme ``X`` (defined over `\QQ`) up
to bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the affine points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(3,QQ)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
sage: enum_affine_rational_field(A(QQ),1)
[(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]
::
sage: A.<w,x,y,z> = AffineSpace(4,QQ)
sage: S = A.subscheme([x^2-y*z+3,w^3+z+y^2])
sage: enum_affine_rational_field(S(QQ),2)
[]
sage: enum_affine_rational_field(S(QQ),3)
[(-2, 0, -3, -1)]
::
sage: A.<x,y> = AffineSpace(2,QQ)
sage: C = Curve(x^2+y-x)
sage: enum_affine_rational_field(C,10)
[(-2, -6), (-1, -2), (0, 0), (1, 0), (2, -2), (3, -6)]
AUTHORS:
- David R. Kohel <*****@*****.**>: original version.
- Charlie Turner (06-2010): small adjustments.
"""
if is_Scheme(X):
X = X(X.base_ring())
n = X.codomain().ambient_space().ngens()
if X.value_ring() is ZZ:
Q = [ 1 ]
else: # rational field
Q = range(1, B + 1)
R = [ 0 ] + [ s*k for k in range(1, B+1) for s in [1, -1] ]
pts = []
P = [0] * n
m = ZZ(0)
try:
pts.append(X(P))
except TypeError:
pass
iters = [ iter(R) for _ in range(n) ]
for it in iters:
it.next()
i = 0
while i < n:
try:
a = ZZ(iters[i].next())
except StopIteration:
iters[i] = iter(R) # reset
P[i] = iters[i].next() # reset P[i] to 0 and increment
i += 1
continue
m = m.gcd(a)
P[i] = a
for b in Q:
if m.gcd(b) == 1:
try:
pts.append(X([ num/b for num in P ]))
except TypeError:
pass
i = 0
m = ZZ(0)
pts.sort()
return pts
def enum_affine_rational_field(X, B):
"""
Enumerates affine rational points on scheme ``X`` up to bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme.
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the affine points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: A.<x,y,z> = AffineSpace(3, QQ)
sage: from sage.schemes.affine.affine_rational_point import enum_affine_rational_field
sage: enum_affine_rational_field(A(QQ), 1)
[(-1, -1, -1), (-1, -1, 0), (-1, -1, 1), (-1, 0, -1), (-1, 0, 0), (-1, 0, 1),
(-1, 1, -1), (-1, 1, 0), (-1, 1, 1), (0, -1, -1), (0, -1, 0), (0, -1, 1),
(0, 0, -1), (0, 0, 0), (0, 0, 1), (0, 1, -1), (0, 1, 0), (0, 1, 1), (1, -1, -1),
(1, -1, 0), (1, -1, 1), (1, 0, -1), (1, 0, 0), (1, 0, 1), (1, 1, -1), (1, 1, 0),
(1, 1, 1)]
::
sage: A.<w,x,y,z> = AffineSpace(4, QQ)
sage: S = A.subscheme([x^2-y*z+1, w^3+z+y^2])
sage: enum_affine_rational_field(S(QQ), 1)
[(0, 0, -1, -1)]
sage: enum_affine_rational_field(S(QQ), 2)
[(0, 0, -1, -1), (1, -1, -1, -2), (1, 1, -1, -2)]
::
sage: A.<x,y> = AffineSpace(2, QQ)
sage: C = Curve(x^2+y-x)
sage: enum_affine_rational_field(C, 10) # long time (3 s)
[(-2, -6), (-1, -2), (-2/3, -10/9), (-1/2, -3/4), (-1/3, -4/9),
(0, 0), (1/3, 2/9), (1/2, 1/4), (2/3, 2/9), (1, 0),
(4/3, -4/9), (3/2, -3/4), (5/3, -10/9), (2, -2), (3, -6)]
AUTHORS:
- David R. Kohel <*****@*****.**>: original version.
- Charlie Turner (06-2010): small adjustments.
- Raman Raghukul 2018: updated.
"""
from sage.schemes.affine.affine_space import is_AffineSpace
if is_Scheme(X):
if not is_AffineSpace(X.ambient_space()):
raise TypeError(
"ambient space must be affine space over the rational field")
X = X(X.base_ring())
elif not is_AffineSpace(X.codomain().ambient_space()):
raise TypeError(
"codomain must be affine space over the rational field")
n = X.codomain().ambient_space().ngens()
VR = X.value_ring()
if VR is ZZ:
R = [0] + [s * k for k in range(1, B + 1) for s in [1, -1]]
iters = [iter(R) for _ in range(n)]
else: # rational field
iters = [QQ.range_by_height(B + 1) for _ in range(n)]
pts = []
P = [0] * n
try:
pts.append(X(P))
except TypeError:
pass
for it in iters:
next(it)
i = 0
while i < n:
try:
a = VR(next(iters[i]))
except StopIteration:
if VR is ZZ:
iters[i] = iter(R)
else: # rational field
iters[i] = QQ.range_by_height(B + 1)
P[i] = next(iters[i]) # reset P[i] to 0 and increment
i += 1
continue
P[i] = a
try:
pts.append(X(P))
except TypeError:
pass
i = 0
pts.sort()
return pts
def enum_projective_number_field(X,B, prec=53):
"""
Enumerates projective points on scheme ``X`` defined over a number field. Simply checks all of the
points of absolute height of at most ``B`` and adds those that are on the scheme to the list.
INPUT:
- ``X`` - a scheme defined over a number field
- ``B`` - a real number
- ``prec`` - the precision to use for computing the elements of bounded height of number fields
OUTPUT:
- a list containing the projective points of ``X`` of absolute height up to ``B``,
sorted.
.. WARNING::
In the current implementation, the output of the [Doyle-Krumm] algorithm
for elements of bounded height cannot be guaranteed to be correct due to
the necessity of floating point computations. In some cases, the default
53-bit precision is considerably lower than would be required for the
algorithm to generate correct output.
EXAMPLES::
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: u = QQ['u'].0
sage: K = NumberField(u^3 - 5,'v')
sage: P.<x,y,z> = ProjectiveSpace(K, 2)
sage: X = P.subscheme([x - y])
sage: enum_projective_number_field(X(K), 5^(1/3), prec=2^10)
[(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
(1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 3, 'v')
sage: A.<x,y> = ProjectiveSpace(K,1)
sage: X=A.subscheme(x-y)
sage: from sage.schemes.projective.projective_rational_point import enum_projective_number_field
sage: enum_projective_number_field(X, 2)
[(1 : 1)]
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if(is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError("Ambient space must be projective space over a number field")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError("Codomain must be projective space over a number field")
R = X.codomain().ambient_space()
pts = []
for P in R.points_of_bounded_height(B, prec):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def enum_projective_finite_field(X):
"""
Enumerates projective points on scheme ``X`` defined over a finite field.
INPUT:
- ``X`` - a scheme defined over a finite field or a set of abstract
rational points of such a scheme.
OUTPUT:
- a list containing the projective points of ``X`` over the finite field,
sorted.
EXAMPLES::
sage: F = GF(53)
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
sage: len(enum_projective_finite_field(P(F)))
2863
sage: 53^2+53+1
2863
::
sage: F = GF(9,'a')
sage: P.<X,Y,Z> = ProjectiveSpace(2,F)
sage: C = Curve(X^3-Y^3+Z^2*Y)
sage: enum_projective_finite_field(C(F))
[(0 : 0 : 1), (0 : 1 : 1), (0 : 2 : 1), (1 : 1 : 0), (a + 1 : 2*a : 1),
(a + 1 : 2*a + 1 : 1), (a + 1 : 2*a + 2 : 1), (2*a + 2 : a : 1),
(2*a + 2 : a + 1 : 1), (2*a + 2 : a + 2 : 1)]
::
sage: F = GF(5)
sage: P2F.<X,Y,Z> = ProjectiveSpace(2,F)
sage: enum_projective_finite_field(P2F)
[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 2 : 1), (0 : 3 : 1), (0 : 4 : 1),
(1 : 0 : 0), (1 : 0 : 1), (1 : 1 : 0), (1 : 1 : 1), (1 : 2 : 1), (1 : 3 : 1),
(1 : 4 : 1), (2 : 0 : 1), (2 : 1 : 0), (2 : 1 : 1), (2 : 2 : 1), (2 : 3 : 1),
(2 : 4 : 1), (3 : 0 : 1), (3 : 1 : 0), (3 : 1 : 1), (3 : 2 : 1), (3 : 3 : 1),
(3 : 4 : 1), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1), (4 : 2 : 1), (4 : 3 : 1),
(4 : 4 : 1)]
ALGORITHM:
Checks all points in projective space to see if they lie on X.
.. WARNING::
If ``X`` is defined over an infinite field, this code will not finish!
AUTHORS:
- John Cremona and Charlie Turner (06-2010).
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if(is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError("Ambient space must be projective space over a finite")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError("Codomain must be projective space over a finite field")
n = X.codomain().ambient_space().ngens()-1
F = X.value_ring()
pts = []
for k in range(n+1):
for c in cartesian_product_iterator([F for _ in range(k)]):
try:
pts.append(X(list(c)+[1]+[0]*(n-k)))
except TypeError:
pass
pts.sort()
return pts
def _call_(self, x):
"""
Construct a scheme from the data in ``x``
EXAMPLES:
Let us first construct the category of schemes::
sage: S = Schemes(); S
Category of schemes
We create a scheme from a ring::
sage: X = S(ZZ); X # indirect doctest
Spectrum of Integer Ring
We create a scheme from a scheme (do nothing)::
sage: S(X)
Spectrum of Integer Ring
We create a scheme morphism from a ring homomorphism.x::
sage: phi = ZZ.hom(QQ); phi
Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f = S(phi); f # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
sage: f.domain()
Spectrum of Rational Field
sage: f.codomain()
Spectrum of Integer Ring
sage: S(f) # indirect doctest
Affine Scheme morphism:
From: Spectrum of Rational Field
To: Spectrum of Integer Ring
Defn: Ring Coercion morphism:
From: Integer Ring
To: Rational Field
"""
from sage.schemes.generic.scheme import is_Scheme
if is_Scheme(x):
return x
from sage.schemes.generic.morphism import is_SchemeMorphism
if is_SchemeMorphism(x):
return x
from sage.rings.morphism import is_RingHomomorphism
from sage.rings.commutative_ring import is_CommutativeRing
from sage.schemes.generic.spec import Spec
if is_CommutativeRing(x):
return Spec(x)
elif is_RingHomomorphism(x):
A = Spec(x.codomain())
return A.hom(x)
else:
raise TypeError(
"No way to create an object or morphism in %s from %s" %
(self, x))
def enum_projective_rational_field(X,B):
r"""
Enumerates projective, rational points on scheme ``X`` of height up to
bound ``B``.
INPUT:
- ``X`` - a scheme or set of abstract rational points of a scheme;
- ``B`` - a positive integer bound.
OUTPUT:
- a list containing the projective points of ``X`` of height up to ``B``,
sorted.
EXAMPLES::
sage: P.<X,Y,Z> = ProjectiveSpace(2,QQ)
sage: C = P.subscheme([X+Y-Z])
sage: from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
sage: enum_projective_rational_field(C(QQ),6)
[(-5 : 6 : 1), (-4 : 5 : 1), (-3 : 4 : 1), (-2 : 3 : 1),
(-3/2 : 5/2 : 1), (-1 : 1 : 0), (-1 : 2 : 1), (-2/3 : 5/3 : 1),
(-1/2 : 3/2 : 1), (-1/3 : 4/3 : 1), (-1/4 : 5/4 : 1),
(-1/5 : 6/5 : 1), (0 : 1 : 1), (1/6 : 5/6 : 1), (1/5 : 4/5 : 1),
(1/4 : 3/4 : 1), (1/3 : 2/3 : 1), (2/5 : 3/5 : 1), (1/2 : 1/2 : 1),
(3/5 : 2/5 : 1), (2/3 : 1/3 : 1), (3/4 : 1/4 : 1), (4/5 : 1/5 : 1),
(5/6 : 1/6 : 1), (1 : 0 : 1), (6/5 : -1/5 : 1), (5/4 : -1/4 : 1),
(4/3 : -1/3 : 1), (3/2 : -1/2 : 1), (5/3 : -2/3 : 1), (2 : -1 : 1),
(5/2 : -3/2 : 1), (3 : -2 : 1), (4 : -3 : 1), (5 : -4 : 1),
(6 : -5 : 1)]
sage: enum_projective_rational_field(C,6) == enum_projective_rational_field(C(QQ),6)
True
::
sage: P3.<W,X,Y,Z> = ProjectiveSpace(3,QQ)
sage: enum_projective_rational_field(P3,1)
[(-1 : -1 : -1 : 1), (-1 : -1 : 0 : 1), (-1 : -1 : 1 : 0), (-1 : -1 : 1 : 1),
(-1 : 0 : -1 : 1), (-1 : 0 : 0 : 1), (-1 : 0 : 1 : 0), (-1 : 0 : 1 : 1),
(-1 : 1 : -1 : 1), (-1 : 1 : 0 : 0), (-1 : 1 : 0 : 1), (-1 : 1 : 1 : 0),
(-1 : 1 : 1 : 1), (0 : -1 : -1 : 1), (0 : -1 : 0 : 1), (0 : -1 : 1 : 0),
(0 : -1 : 1 : 1), (0 : 0 : -1 : 1), (0 : 0 : 0 : 1), (0 : 0 : 1 : 0),
(0 : 0 : 1 : 1), (0 : 1 : -1 : 1), (0 : 1 : 0 : 0), (0 : 1 : 0 : 1),
(0 : 1 : 1 : 0), (0 : 1 : 1 : 1), (1 : -1 : -1 : 1), (1 : -1 : 0 : 1),
(1 : -1 : 1 : 0), (1 : -1 : 1 : 1), (1 : 0 : -1 : 1), (1 : 0 : 0 : 0),
(1 : 0 : 0 : 1), (1 : 0 : 1 : 0), (1 : 0 : 1 : 1), (1 : 1 : -1 : 1),
(1 : 1 : 0 : 0), (1 : 1 : 0 : 1), (1 : 1 : 1 : 0), (1 : 1 : 1 : 1)]
ALGORITHM:
We just check all possible projective points in correct dimension
of projective space to see if they lie on ``X``.
AUTHORS:
- John Cremona and Charlie Turner (06-2010)
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if(is_Scheme(X)):
if (not is_ProjectiveSpace(X.ambient_space())):
raise TypeError("Ambient space must be projective space over the rational field")
X = X(X.base_ring())
else:
if (not is_ProjectiveSpace(X.codomain().ambient_space())):
raise TypeError("Codomain must be projective space over the rational field")
n = X.codomain().ambient_space().ngens()
zero = (0,) * n
pts = []
for c in cartesian_product_iterator([srange(-B,B+1) for _ in range(n)]):
if gcd(c) == 1 and c > zero:
try:
pts.append(X(c))
except TypeError:
pass
pts.sort()
return pts