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 multiplicity(self):
r"""
Return the multiplicity of this point on its codomain.
This uses the subscheme implementation of multiplicity. This point must be a point
on a subscheme of a product of projective spaces.
OUTPUT: an integer.
EXAMPLES::
sage: PP.<x,y,z,w,u,v,t> = ProductProjectiveSpaces(QQ, [3,2])
sage: X = PP.subscheme([x^8*t - y^8*t + z^5*w^3*v])
sage: Q1 = X([1,1,0,0,-1,-1,1])
sage: Q1.multiplicity()
1
sage: Q2 = X([0,0,0,1,0,1,1])
sage: Q2.multiplicity()
5
sage: Q3 = X([0,0,0,1,1,0,0])
sage: Q3.multiplicity()
6
"""
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(self.codomain()):
raise TypeError(
"this point must be a point on a subscheme of a product of projective spaces"
)
return self.codomain().multiplicity(self)
def intersection_multiplicity(self, X):
r"""
Return the intersection multiplicity of the codomain of this point and subscheme ``X`` at this point.
This uses the subscheme implementation of intersection_multiplicity. This point must be a point
on a subscheme of a product of projective spaces.
INPUT:
- ``X`` -- a subscheme in the same ambient space as the codomain of this point.
OUTPUT: An integer.
EXAMPLES::
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces(QQ, [2,1])
sage: X = PP.subscheme([y^2*z^3*u - x^5*v])
sage: Y = PP.subscheme([u^3 - v^3, x - y])
sage: Q = X([0,0,1,1,1])
sage: Q.intersection_multiplicity(Y)
2
"""
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(self.codomain()):
raise TypeError(
"this point must be a point on a subscheme of a product of projective spaces"
)
return self.codomain().intersection_multiplicity(X, self)
def multiplicity(self):
r"""
Return the multiplicity of this point on its codomain.
This uses the subscheme implementation of multiplicity. This point must be a point
on a subscheme of a product of projective spaces.
OUTPUT: an integer.
EXAMPLES::
sage: PP.<x,y,z,w,u,v,t> = ProductProjectiveSpaces(QQ, [3,2])
sage: X = PP.subscheme([x^8*t - y^8*t + z^5*w^3*v])
sage: Q1 = X([1,1,0,0,-1,-1,1])
sage: Q1.multiplicity()
1
sage: Q2 = X([0,0,0,1,0,1,1])
sage: Q2.multiplicity()
5
sage: Q3 = X([0,0,0,1,1,0,0])
sage: Q3.multiplicity()
6
"""
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(self.codomain()):
raise TypeError("this point must be a point on a subscheme of a product of projective spaces")
return self.codomain().multiplicity(self)
def intersection_multiplicity(self, X):
r"""
Return the intersection multiplicity of the codomain of this point and subscheme ``X`` at this point.
This uses the subscheme implementation of intersection_multiplicity. This point must be a point
on a subscheme of a product of projective spaces.
INPUT:
- ``X`` -- a subscheme in the same ambient space as the codomain of this point.
OUTPUT: An integer.
EXAMPLES::
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces(QQ, [2,1])
sage: X = PP.subscheme([y^2*z^3*u - x^5*v])
sage: Y = PP.subscheme([u^3 - v^3, x - y])
sage: Q = X([0,0,1,1,1])
sage: Q.intersection_multiplicity(Y)
2
"""
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(self.codomain()):
raise TypeError("this point must be a point on a subscheme of a product of projective spaces")
return self.codomain().intersection_multiplicity(X, self)
def __init__(self, parent, polys, check=True):
r"""
The Python constructor.
INPUT:
- ``parent`` -- Homset
- ``polys`` -- anything that defines a point in the class
- ``check`` -- Boolean. Whether or not to perform input checks
(Default:`` True``)
EXAMPLES::
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2,1],QQ)
sage: H = T.Hom(T)
sage: H([x^2*u,y^2*w,z^2*u,w^2,u^2])
Scheme endomorphism of Product of projective spaces P^2 x P^1 over Rational Field
Defn: Defined by sending (x : y : z , w : u) to
(x^2*u : y^2*w : z^2*u , w^2 : u^2).
::
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2,1],QQ)
sage: H = T.Hom(T)
sage: H([x^2*u,y^2*w,z^2*u,w^2,u*z])
Traceback (most recent call last):
...
TypeError: polys (=[x^2*u, y^2*w, z^2*u, w^2, z*u]) must be
multi-homogeneous of the same degrees (by component)
"""
if check:
#check multi-homogeneous
#if self is a subscheme, we may need the lift of the polynomials
try:
polys[0].exponents()
except AttributeError:
polys = [f.lift() for f in polys]
target = parent.codomain().ambient_space()
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(target):
splitpolys = target._factors(polys)
for m in range(len(splitpolys)):
d = target._degree(splitpolys[m][0])
if not all(d == target._degree(f) for f in splitpolys[m]):
raise TypeError(
"polys (=%s) must be multi-homogeneous of the same degrees (by component)"
% polys)
else:
#we are mapping into some other kind of space
target._validate(polys)
SchemeMorphism_polynomial.__init__(self, parent, polys, check)
def __init__(self, parent, polys, check = True):
r"""
The Python constructor.
INPUT:
- ``parent`` -- Homset
- ``polys`` -- anything that defines a point in the class
- ``check`` -- Boolean. Whether or not to perform input checks
(Default:`` True``)
EXAMPLES::
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2,1],QQ)
sage: H = T.Hom(T)
sage: H([x^2*u,y^2*w,z^2*u,w^2,u^2])
Scheme endomorphism of Product of projective spaces P^2 x P^1 over Rational Field
Defn: Defined by sending (x : y : z , w : u) to
(x^2*u : y^2*w : z^2*u , w^2 : u^2).
::
sage: T.<x,y,z,w,u> = ProductProjectiveSpaces([2,1],QQ)
sage: H = T.Hom(T)
sage: H([x^2*u,y^2*w,z^2*u,w^2,u*z])
Traceback (most recent call last):
...
TypeError: polys (=[x^2*u, y^2*w, z^2*u, w^2, z*u]) must be
multi-homogeneous of the same degrees (by component)
"""
if check:
#check multi-homogeneous
#if self is a subscheme, we may need the lift of the polynomials
try:
polys[0].exponents()
except AttributeError:
polys = [f.lift() for f in polys]
target = parent.codomain().ambient_space()
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(target):
splitpolys = target._factors(polys)
for m in range(len(splitpolys)):
d = target._degree(splitpolys[m][0])
if not all(d == target._degree(f) for f in splitpolys[m]):
raise TypeError("polys (=%s) must be multi-homogeneous of the same degrees (by component)"%polys)
else:
#we are mapping into some other kind of space
target._validate(polys)
SchemeMorphism_polynomial.__init__(self, parent, polys, check)
def cyclegraph(self):
r"""
Return the digraph of all orbits of this morphism mod `p`.
OUTPUT: a digraph
EXAMPLES::
sage: P.<a,b,c,d> = ProductProjectiveSpaces(GF(3), [1,1])
sage: f = DynamicalSystem_projective([a^2,b^2,c^2,d^2], domain=P)
sage: f.cyclegraph()
Looped digraph on 16 vertices
::
sage: P.<a,b,c,d> = ProductProjectiveSpaces(GF(5), [1,1])
sage: f = DynamicalSystem_projective([a^2,b^2,c,d], domain=P)
sage: f.cyclegraph()
Looped digraph on 36 vertices
::
sage: P.<a,b,c,d,e> = ProductProjectiveSpaces(GF(2), [1,2])
sage: f = DynamicalSystem_projective([a^2,b^2,c,d,e], domain=P)
sage: f.cyclegraph()
Looped digraph on 21 vertices
.. TODO:: Dynamical systems for subschemes of product projective spaces needs work.
Thus this is not implemented for subschemes.
"""
V = []
E = []
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if is_ProductProjectiveSpaces(self.domain()) == True:
for P in self.domain():
V.append(str(P))
Q = self(P)
E.append([str(Q)])
else:
raise NotImplementedError(
"Cyclegraph for product projective spaces not implemented for subschemes"
)
from sage.graphs.digraph import DiGraph
g = DiGraph(dict(zip(V, E)), loops=True)
return g
def points(self, B=0, prec=53):
r"""
Return some or all rational points of a projective scheme.
Over a finite field, all points are returned. Over an infinite field, all points satisfying the bound
are returned. For a zero-dimensional subscheme, all points are returned regardless of whether the base
ring is a field or not.
INPUT:
- `B` -- integer (optional, default=0). The bound for the
coordinates.
- ``prec`` - the precision to use to compute the elements of bounded height for number fields.
OUTPUT:
- a list of rational points of a projective scheme.
.. WARNING::
In the current implementation, the output of the [Doyle-Krumm]_ algorithm
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: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], QQ)
sage: X = P.subscheme([x - y, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[]
::
sage: u = QQ['u'].0
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1,1], NumberField(u^2 - 2, 'v'))
sage: X = P.subscheme([x^2 - y^2, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[(-1 : 1 , -v : 1), (1 : 1 , v : 1), (1 : 1 , -v : 1), (-1 : 1 , v : 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 1, 'v')
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], K)
sage: P(K).points(1)
[(0 : 1 , 0 : 1), (0 : 1 , v : 1), (0 : 1 , -1 : 1), (0 : 1 , -v : 1), (0 : 1 , 1 : 1),
(0 : 1 , 1 : 0), (v : 1 , 0 : 1), (v : 1 , v : 1), (v : 1 , -1 : 1), (v : 1 , -v : 1),
(v : 1 , 1 : 1), (v : 1 , 1 : 0), (-1 : 1 , 0 : 1), (-1 : 1 , v : 1), (-1 : 1 , -1 : 1),
(-1 : 1 , -v : 1), (-1 : 1 , 1 : 1), (-1 : 1 , 1 : 0), (-v : 1 , 0 : 1), (-v : 1 , v : 1),
(-v : 1 , -1 : 1), (-v : 1 , -v : 1), (-v : 1 , 1 : 1), (-v : 1 , 1 : 0), (1 : 1 , 0 : 1),
(1 : 1 , v : 1), (1 : 1 , -1 : 1), (1 : 1 , -v : 1), (1 : 1 , 1 : 1), (1 : 1 , 1 : 0),
(1 : 0 , 0 : 1), (1 : 0 , v : 1), (1 : 0 , -1 : 1), (1 : 0 , -v : 1), (1 : 0 , 1 : 1),
(1 : 0 , 1 : 0)]
::
sage: P.<x,y,z,u,v> = ProductProjectiveSpaces([2, 1], GF(3))
sage: P(P.base_ring()).points()
[(0 : 0 : 1 , 0 : 1), (1 : 0 : 1 , 0 : 1), (2 : 0 : 1 , 0 : 1), (0 : 1 : 1 , 0 : 1), (1 : 1 : 1 , 0 : 1),
(2 : 1 : 1 , 0 : 1), (0 : 2 : 1 , 0 : 1), (1 : 2 : 1 , 0 : 1), (2 : 2 : 1 , 0 : 1), (0 : 1 : 0 , 0 : 1),
(1 : 1 : 0 , 0 : 1), (2 : 1 : 0 , 0 : 1), (1 : 0 : 0 , 0 : 1), (0 : 0 : 1 , 1 : 1), (1 : 0 : 1 , 1 : 1),
(2 : 0 : 1 , 1 : 1), (0 : 1 : 1 , 1 : 1), (1 : 1 : 1 , 1 : 1), (2 : 1 : 1 , 1 : 1), (0 : 2 : 1 , 1 : 1),
(1 : 2 : 1 , 1 : 1), (2 : 2 : 1 , 1 : 1), (0 : 1 : 0 , 1 : 1), (1 : 1 : 0 , 1 : 1), (2 : 1 : 0 , 1 : 1),
(1 : 0 : 0 , 1 : 1), (0 : 0 : 1 , 2 : 1), (1 : 0 : 1 , 2 : 1), (2 : 0 : 1 , 2 : 1), (0 : 1 : 1 , 2 : 1),
(1 : 1 : 1 , 2 : 1), (2 : 1 : 1 , 2 : 1), (0 : 2 : 1 , 2 : 1), (1 : 2 : 1 , 2 : 1), (2 : 2 : 1 , 2 : 1),
(0 : 1 : 0 , 2 : 1), (1 : 1 : 0 , 2 : 1), (2 : 1 : 0 , 2 : 1), (1 : 0 : 0 , 2 : 1), (0 : 0 : 1 , 1 : 0),
(1 : 0 : 1 , 1 : 0), (2 : 0 : 1 , 1 : 0), (0 : 1 : 1 , 1 : 0), (1 : 1 : 1 , 1 : 0), (2 : 1 : 1 , 1 : 0),
(0 : 2 : 1 , 1 : 0), (1 : 2 : 1 , 1 : 0), (2 : 2 : 1 , 1 : 0), (0 : 1 : 0 , 1 : 0), (1 : 1 : 0 , 1 : 0),
(2 : 1 : 0 , 1 : 0), (1 : 0 : 0 , 1 : 0)]
"""
X = self.codomain()
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if not is_ProductProjectiveSpaces(X) and X.base_ring() in Fields():
# no points
if X.dimension() == -1:
return []
# if X is zero-dimensional
if X.dimension() == 0:
points = set()
# find points from all possible affine patches
for I in xmrange([n + 1 for n in X.ambient_space().dimension_relative_components()]):
[Y,phi] = X.affine_patch(I, True)
aff_points = Y.rational_points()
for PP in aff_points:
points.add(phi(PP))
return list(points)
R = self.value_ring()
points = []
if R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
for P in X.ambient_space().points_of_bounded_height(B, prec):
try:
points.append(X(P))
except TypeError:
pass
return points
if is_FiniteField(R):
for P in X.ambient_space().rational_points():
try:
points.append(X(P))
except TypeError:
pass
return points
else:
raise TypeError("unable to enumerate points over %s"%R)
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_product_projective_number_field(X, **kwds):
r"""
Enumerates product 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 product projective points of ``X`` of
absolute height up to ``B``, sorted.
EXAMPLES::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 2, 'v')
sage: PP.<x,y,z,w> = ProductProjectiveSpaces([1, 1], K)
sage: X = PP.subscheme([x^2 + 2*y^2])
sage: from sage.schemes.product_projective.rational_point import \
enum_product_projective_number_field
sage: enum_product_projective_number_field(X, bound=1.5)
[(-v : 1 , -1 : 1), (-v : 1 , -v : 1), (-v : 1 , -1/2*v : 1),
(-v : 1 , 0 : 1), (-v : 1 , 1/2*v : 1), (-v : 1 , v : 1),
(-v : 1 , 1 : 0), (-v : 1 , 1 : 1), (v : 1 , -1 : 1),
(v : 1 , -v : 1), (v : 1 , -1/2*v : 1), (v : 1 , 0 : 1),
(v : 1 , 1/2*v : 1), (v : 1 , v : 1), (v : 1 , 1 : 0),
(v : 1 , 1 : 1)]
"""
B = kwds.pop('bound')
tol = kwds.pop('tolerance', 1e-2)
prec = kwds.pop('precision', 53)
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.points_of_bounded_height(bound=B, tolerance=tol, precision=prec):
try:
pts.append(X(P))
except TypeError:
pass
pts.sort()
return pts
def points(self, **kwds):
r"""
Return some or all rational points of a projective scheme.
Over a finite field, all points are returned. Over an infinite field, all points satisfying the bound
are returned. For a zero-dimensional subscheme, all points are returned regardless of whether the base
ring is a field or not.
For number fields, this uses the
Doyle-Krumm algorithm 4 (algorithm 5 for imaginary quadratic) for
computing algebraic numbers up to a given height [DK2013]_ or
uses the chinese remainder theorem and points modulo primes
for larger bounds.
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:
- ``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.
- ``algorithm`` - either 'sieve' or 'enumerate' algorithms can be used over `QQ`. If
not specified, enumerate is used only for small height bounds.
OUTPUT:
- a list of rational points of a projective scheme
EXAMPLES::
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], QQ)
sage: X = P.subscheme([x - y, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[]
::
sage: u = QQ['u'].0
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1,1], NumberField(u^2 - 2, 'v'))
sage: X = P.subscheme([x^2 - y^2, z^2 - 2*w^2])
sage: sorted(X(P.base_ring()).points())
[(-1 : 1 , -v : 1), (-1 : 1 , v : 1), (1 : 1 , -v : 1), (1 : 1 , v : 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 1, 'v')
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], K)
sage: P(K).points(bound=1)
[(-1 : 1 , -1 : 1), (-1 : 1 , -v : 1), (-1 : 1 , 0 : 1), (-1 : 1 , v : 1),
(-1 : 1 , 1 : 0), (-1 : 1 , 1 : 1), (-v : 1 , -1 : 1), (-v : 1 , -v : 1),
(-v : 1 , 0 : 1), (-v : 1 , v : 1), (-v : 1 , 1 : 0), (-v : 1 , 1 : 1),
(0 : 1 , -1 : 1), (0 : 1 , -v : 1), (0 : 1 , 0 : 1), (0 : 1 , v : 1),
(0 : 1 , 1 : 0), (0 : 1 , 1 : 1), (v : 1 , -1 : 1), (v : 1 , -v : 1),
(v : 1 , 0 : 1), (v : 1 , v : 1), (v : 1 , 1 : 0), (v : 1 , 1 : 1),
(1 : 0 , -1 : 1), (1 : 0 , -v : 1), (1 : 0 , 0 : 1), (1 : 0 , v : 1),
(1 : 0 , 1 : 0), (1 : 0 , 1 : 1), (1 : 1 , -1 : 1), (1 : 1 , -v : 1),
(1 : 1 , 0 : 1), (1 : 1 , v : 1), (1 : 1 , 1 : 0), (1 : 1 , 1 : 1)]
::
sage: P.<x,y,z,u,v> = ProductProjectiveSpaces([2, 1], GF(3))
sage: P(P.base_ring()).points()
[(0 : 0 : 1 , 0 : 1), (0 : 0 : 1 , 1 : 0), (0 : 0 : 1 , 1 : 1), (0 : 0 : 1 , 2 : 1),
(0 : 1 : 0 , 0 : 1), (0 : 1 : 0 , 1 : 0), (0 : 1 : 0 , 1 : 1), (0 : 1 : 0 , 2 : 1),
(0 : 1 : 1 , 0 : 1), (0 : 1 : 1 , 1 : 0), (0 : 1 : 1 , 1 : 1), (0 : 1 : 1 , 2 : 1),
(0 : 2 : 1 , 0 : 1), (0 : 2 : 1 , 1 : 0), (0 : 2 : 1 , 1 : 1), (0 : 2 : 1 , 2 : 1),
(1 : 0 : 0 , 0 : 1), (1 : 0 : 0 , 1 : 0), (1 : 0 : 0 , 1 : 1), (1 : 0 : 0 , 2 : 1),
(1 : 0 : 1 , 0 : 1), (1 : 0 : 1 , 1 : 0), (1 : 0 : 1 , 1 : 1), (1 : 0 : 1 , 2 : 1),
(1 : 1 : 0 , 0 : 1), (1 : 1 : 0 , 1 : 0), (1 : 1 : 0 , 1 : 1), (1 : 1 : 0 , 2 : 1),
(1 : 1 : 1 , 0 : 1), (1 : 1 : 1 , 1 : 0), (1 : 1 : 1 , 1 : 1), (1 : 1 : 1 , 2 : 1),
(1 : 2 : 1 , 0 : 1), (1 : 2 : 1 , 1 : 0), (1 : 2 : 1 , 1 : 1), (1 : 2 : 1 , 2 : 1),
(2 : 0 : 1 , 0 : 1), (2 : 0 : 1 , 1 : 0), (2 : 0 : 1 , 1 : 1), (2 : 0 : 1 , 2 : 1),
(2 : 1 : 0 , 0 : 1), (2 : 1 : 0 , 1 : 0), (2 : 1 : 0 , 1 : 1), (2 : 1 : 0 , 2 : 1),
(2 : 1 : 1 , 0 : 1), (2 : 1 : 1 , 1 : 0), (2 : 1 : 1 , 1 : 1), (2 : 1 : 1 , 2 : 1),
(2 : 2 : 1 , 0 : 1), (2 : 2 : 1 , 1 : 0), (2 : 2 : 1 , 1 : 1), (2 : 2 : 1 , 2 : 1)]
::
sage: PP.<x,y,z,u,v> = ProductProjectiveSpaces([2,1], QQ)
sage: X = PP.subscheme([x + y, u*u-v*u])
sage: X.rational_points(bound=2)
[(-2 : 2 : 1 , 0 : 1),
(-2 : 2 : 1 , 1 : 1),
(-1 : 1 : 0 , 0 : 1),
(-1 : 1 : 0 , 1 : 1),
(-1 : 1 : 1 , 0 : 1),
(-1 : 1 : 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),
(1/2 : -1/2 : 1 , 0 : 1),
(1/2 : -1/2 : 1 , 1 : 1),
(1 : -1 : 1 , 0 : 1),
(1 : -1 : 1 , 1 : 1),
(2 : -2 : 1 , 0 : 1),
(2 : -2 : 1 , 1 : 1)]
better to enumerate with low codimension::
sage: PP.<x,y,z,u,v,a,b,c> = ProductProjectiveSpaces([2,1,2], QQ)
sage: X = PP.subscheme([x*u^2*a, b*z*u*v,z*v^2*c ])
sage: len(X.rational_points(bound=1, algorithm='enumerate'))
232
"""
B = kwds.pop('bound', 0)
X = self.codomain()
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if not is_ProductProjectiveSpaces(X) and X.base_ring() in Fields():
# no points
if X.dimension() == -1:
return []
# if X is zero-dimensional
if X.dimension() == 0:
points = set()
# find points from all possible affine patches
for I in xmrange([n + 1 for n in X.ambient_space().dimension_relative_components()]):
[Y,phi] = X.affine_patch(I, True)
aff_points = Y.rational_points()
for PP in aff_points:
points.add(phi(PP))
return list(points)
R = self.value_ring()
points = []
if is_RationalField(R):
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
alg = kwds.pop('algorithm', None)
if alg is None:
# sieve should only be called for subschemes and if the bound is not very small
N = prod([k+1 for k in X.ambient_space().dimension_relative_components()])
if isinstance(X, AlgebraicScheme_subscheme) and B**N > 5000:
from sage.schemes.product_projective.rational_point import sieve
return sieve(X, B)
else:
from sage.schemes.product_projective.rational_point import enum_product_projective_rational_field
return enum_product_projective_rational_field(self, B)
elif alg == 'sieve':
from sage.schemes.product_projective.rational_point import sieve
return sieve(X, B)
elif alg == 'enumerate':
from sage.schemes.product_projective.rational_point import enum_product_projective_rational_field
return enum_product_projective_rational_field(self, B)
else:
raise ValueError("algorithm must be 'sieve' or 'enumerate'")
elif R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
from sage.schemes.product_projective.rational_point import enum_product_projective_number_field
return enum_product_projective_number_field(self, bound=B)
elif is_FiniteField(R):
from sage.schemes.product_projective.rational_point import enum_product_projective_finite_field
return enum_product_projective_finite_field(self)
else:
raise TypeError("unable to enumerate points over %s" % R)
def _coerce_map_from_(self, other):
r"""
Return true if ``other`` canonically coerces to ``self``.
EXAMPLES::
sage: R.<t> = QQ[]
sage: P = ProjectiveSpace(QQ, 1, 'x')
sage: P2 = ProjectiveSpace(R, 1, 'x')
sage: P2(R)._coerce_map_from_(P(QQ))
True
sage: P(QQ)._coerce_map_from_(P2(R))
False
::
sage: P = ProjectiveSpace(QQ, 1, 'x')
sage: P2 = ProjectiveSpace(CC, 1, 'y')
sage: P2(CC)._coerce_map_from_(P(QQ))
False
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: H = A.subscheme(z)
sage: L = A.subscheme([z, y+z])
sage: A(QQ)._coerce_map_from_(H(QQ))
True
sage: H(QQ)._coerce_map_from_(L(QQ))
True
sage: L(QQ).has_coerce_map_from(H(QQ))
False
sage: A(CC)._coerce_map_from_(H(QQ))
True
sage: H(CC)._coerce_map_from_(L(RR))
True
::
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: A2.<u,v> = AffineSpace(QQ, 2)
sage: A(QQ).has_coerce_map_from(A2(QQ))
False
::
sage: A.<x,y> = AffineSpace(QQ, 2)
sage: P.<u,v,w> = ProjectiveSpace(QQ, 2)
sage: A(QQ).has_coerce_map_from(P(QQ))
False
::
sage: A = AffineSpace(QQ, 1)
sage: A(QQ)._coerce_map_from_(ZZ)
True
::
sage: PS = ProjectiveSpace(ZZ, 1, 'x')
sage: PS2 = ProjectiveSpace(Zp(7), 1, 'x')
sage: PS(ZZ).has_coerce_map_from(PS2(Zp(7)))
False
sage: PS2(Zp(7)).has_coerce_map_from(PS(ZZ))
True
::
sage: PP1 = ProductProjectiveSpaces(ZZ, [2,1], 'x')
sage: PP1(QQ)._coerce_map_from_(PP1(ZZ))
True
sage: PP2 = ProductProjectiveSpaces(QQ, [1,2], 'x')
sage: PP2(QQ)._coerce_map_from_(PP1(ZZ))
False
sage: PP3 = ProductProjectiveSpaces(QQ, [2,1], 'y')
sage: PP3(QQ)._coerce_map_from_(PP1(ZZ))
False
::
sage: K.<w> = QuadraticField(2)
sage: A.<x,y,z> = AffineSpace(QQ, 3)
sage: H = A.subscheme(z)
sage: A(K).has_coerce_map_from(H(QQ))
True
TESTS::
sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: X = P. subscheme ([x-y])
sage: P(1,1) == X(1,1)
True
::
sage: A = AffineSpace(QQ, 1, 'x')
sage: AC = AffineSpace(CC, 1, 'x')
sage: A(3/2) == AC(3/2)
True
::
sage: A = AffineSpace(QQ, 1)
sage: A(0) == 0
True
"""
target = self.codomain()
#ring elements can be coerced to a space if we're affine dimension 1
#and the base rings are coercible
if isinstance(other, CommutativeRing):
try:
from sage.schemes.affine.affine_space import is_AffineSpace
if is_AffineSpace(target.ambient_space())\
and target.ambient_space().dimension_relative() == 1:
return target.base_ring().has_coerce_map_from(other)
else:
return False
except AttributeError: #no .ambient_space
return False
elif isinstance(other, SchemeHomset_points):
#we are converting between scheme points
from sage.schemes.generic.algebraic_scheme import AlgebraicScheme_subscheme
source = other.codomain()
if isinstance(target, AlgebraicScheme_subscheme):
#subscheme coerce when there is containment
if not isinstance(source, AlgebraicScheme_subscheme):
return False
if target.ambient_space() == source.ambient_space():
if all([
g in source.defining_ideal()
for g in target.defining_polynomials()
]):
return self.domain().coordinate_ring(
).has_coerce_map_from(other.domain().coordinate_ring())
else:
#if the target is an ambient space, we can coerce if the base rings coerce
#and they are the same type: affine, projective, etc and have the same
#variable names
from sage.schemes.projective.projective_space import is_ProjectiveSpace
from sage.schemes.affine.affine_space import is_AffineSpace
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
try:
ta = target.ambient_space()
sa = source.ambient_space()
except AttributeError: #no .ambient_space
return False
#for projective and affine varieties, we check dimension
#and matching variable names
if (is_ProjectiveSpace(ta) and is_ProjectiveSpace(sa))\
or (is_AffineSpace(ta) and is_AffineSpace(sa)):
if (ta.variable_names() == sa.variable_names()):
return self.domain().coordinate_ring(
).has_coerce_map_from(other.domain().coordinate_ring())
else:
return False
#for products of projective spaces, we check dimension of
#components and matching variable names
elif (is_ProductProjectiveSpaces(ta)
and is_ProductProjectiveSpaces(sa)):
if (ta.dimension_relative_components() == sa.dimension_relative_components()) \
and (ta.variable_names() == sa.variable_names()):
return self.domain().coordinate_ring(
).has_coerce_map_from(other.domain().coordinate_ring())
else:
return False
def points(self, **kwds):
r"""
Return some or all rational points of a projective scheme.
Over a finite field, all points are returned. Over an infinite field, all points satisfying the bound
are returned. For a zero-dimensional subscheme, all points are returned regardless of whether the base
ring is a field or not.
For number fields, this uses the
Doyle-Krumm algorithm 4 (algorithm 5 for imaginary quadratic) for
computing algebraic numbers up to a given height [Doyle-Krumm]_.
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:
- ``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 of rational points of a projective scheme
EXAMPLES::
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], QQ)
sage: X = P.subscheme([x - y, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[]
::
sage: u = QQ['u'].0
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1,1], NumberField(u^2 - 2, 'v'))
sage: X = P.subscheme([x^2 - y^2, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[(-1 : 1 , -v : 1), (1 : 1 , v : 1), (1 : 1 , -v : 1), (-1 : 1 , v : 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 1, 'v')
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], K)
sage: P(K).points(bound=1)
[(0 : 1 , 0 : 1), (0 : 1 , v : 1), (0 : 1 , -1 : 1), (0 : 1 , -v : 1), (0 : 1 , 1 : 1),
(0 : 1 , 1 : 0), (v : 1 , 0 : 1), (v : 1 , v : 1), (v : 1 , -1 : 1), (v : 1 , -v : 1),
(v : 1 , 1 : 1), (v : 1 , 1 : 0), (-1 : 1 , 0 : 1), (-1 : 1 , v : 1), (-1 : 1 , -1 : 1),
(-1 : 1 , -v : 1), (-1 : 1 , 1 : 1), (-1 : 1 , 1 : 0), (-v : 1 , 0 : 1), (-v : 1 , v : 1),
(-v : 1 , -1 : 1), (-v : 1 , -v : 1), (-v : 1 , 1 : 1), (-v : 1 , 1 : 0), (1 : 1 , 0 : 1),
(1 : 1 , v : 1), (1 : 1 , -1 : 1), (1 : 1 , -v : 1), (1 : 1 , 1 : 1), (1 : 1 , 1 : 0),
(1 : 0 , 0 : 1), (1 : 0 , v : 1), (1 : 0 , -1 : 1), (1 : 0 , -v : 1), (1 : 0 , 1 : 1),
(1 : 0 , 1 : 0)]
::
sage: P.<x,y,z,u,v> = ProductProjectiveSpaces([2, 1], GF(3))
sage: P(P.base_ring()).points()
[(0 : 0 : 1 , 0 : 1), (1 : 0 : 1 , 0 : 1), (2 : 0 : 1 , 0 : 1), (0 : 1 : 1 , 0 : 1), (1 : 1 : 1 , 0 : 1),
(2 : 1 : 1 , 0 : 1), (0 : 2 : 1 , 0 : 1), (1 : 2 : 1 , 0 : 1), (2 : 2 : 1 , 0 : 1), (0 : 1 : 0 , 0 : 1),
(1 : 1 : 0 , 0 : 1), (2 : 1 : 0 , 0 : 1), (1 : 0 : 0 , 0 : 1), (0 : 0 : 1 , 1 : 1), (1 : 0 : 1 , 1 : 1),
(2 : 0 : 1 , 1 : 1), (0 : 1 : 1 , 1 : 1), (1 : 1 : 1 , 1 : 1), (2 : 1 : 1 , 1 : 1), (0 : 2 : 1 , 1 : 1),
(1 : 2 : 1 , 1 : 1), (2 : 2 : 1 , 1 : 1), (0 : 1 : 0 , 1 : 1), (1 : 1 : 0 , 1 : 1), (2 : 1 : 0 , 1 : 1),
(1 : 0 : 0 , 1 : 1), (0 : 0 : 1 , 2 : 1), (1 : 0 : 1 , 2 : 1), (2 : 0 : 1 , 2 : 1), (0 : 1 : 1 , 2 : 1),
(1 : 1 : 1 , 2 : 1), (2 : 1 : 1 , 2 : 1), (0 : 2 : 1 , 2 : 1), (1 : 2 : 1 , 2 : 1), (2 : 2 : 1 , 2 : 1),
(0 : 1 : 0 , 2 : 1), (1 : 1 : 0 , 2 : 1), (2 : 1 : 0 , 2 : 1), (1 : 0 : 0 , 2 : 1), (0 : 0 : 1 , 1 : 0),
(1 : 0 : 1 , 1 : 0), (2 : 0 : 1 , 1 : 0), (0 : 1 : 1 , 1 : 0), (1 : 1 : 1 , 1 : 0), (2 : 1 : 1 , 1 : 0),
(0 : 2 : 1 , 1 : 0), (1 : 2 : 1 , 1 : 0), (2 : 2 : 1 , 1 : 0), (0 : 1 : 0 , 1 : 0), (1 : 1 : 0 , 1 : 0),
(2 : 1 : 0 , 1 : 0), (1 : 0 : 0 , 1 : 0)]
"""
B = kwds.pop('bound', 0)
tol = kwds.pop('tolerance', 1e-2)
prec = kwds.pop('precision', 53)
X = self.codomain()
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if not is_ProductProjectiveSpaces(X) and X.base_ring() in Fields():
# no points
if X.dimension() == -1:
return []
# if X is zero-dimensional
if X.dimension() == 0:
points = set()
# find points from all possible affine patches
for I in xmrange([
n + 1 for n in
X.ambient_space().dimension_relative_components()
]):
[Y, phi] = X.affine_patch(I, True)
aff_points = Y.rational_points()
for PP in aff_points:
points.add(phi(PP))
return list(points)
R = self.value_ring()
points = []
if R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
for P in X.ambient_space().points_of_bounded_height(
bound=B, tolerance=tol, precision=prec):
try:
points.append(X(P))
except TypeError:
pass
return points
if is_FiniteField(R):
for P in X.ambient_space().rational_points():
try:
points.append(X(P))
except TypeError:
pass
return points
else:
raise TypeError("unable to enumerate points over %s" % R)
def points(self, B=0, prec=53):
r"""
Return some or all rational points of a projective scheme.
Over a finite field, all points are returned. Over an infinite field, all points satisfying the bound
are returned. For a zero-dimensional subscheme, all points are returned regardless of whether the base
ring is a field or not.
INPUT:
- `B` -- integer (optional, default=0). The bound for the
coordinates.
- ``prec`` - the precision to use to compute the elements of bounded height for number fields.
OUTPUT:
- a list of rational points of a projective scheme.
.. WARNING::
In the current implementation, the output of the [Doyle-Krumm]_ algorithm
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: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], QQ)
sage: X = P.subscheme([x - y, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[]
::
sage: u = QQ['u'].0
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1,1], NumberField(u^2 - 2, 'v'))
sage: X = P.subscheme([x^2 - y^2, z^2 - 2*w^2])
sage: X(P.base_ring()).points()
[(-1 : 1 , -v : 1), (1 : 1 , v : 1), (1 : 1 , -v : 1), (-1 : 1 , v : 1)]
::
sage: u = QQ['u'].0
sage: K = NumberField(u^2 + 1, 'v')
sage: P.<x,y,z,w> = ProductProjectiveSpaces([1, 1], K)
sage: P(K).points(1)
[(0 : 1 , 0 : 1), (0 : 1 , v : 1), (0 : 1 , -1 : 1), (0 : 1 , -v : 1), (0 : 1 , 1 : 1),
(0 : 1 , 1 : 0), (v : 1 , 0 : 1), (v : 1 , v : 1), (v : 1 , -1 : 1), (v : 1 , -v : 1),
(v : 1 , 1 : 1), (v : 1 , 1 : 0), (-1 : 1 , 0 : 1), (-1 : 1 , v : 1), (-1 : 1 , -1 : 1),
(-1 : 1 , -v : 1), (-1 : 1 , 1 : 1), (-1 : 1 , 1 : 0), (-v : 1 , 0 : 1), (-v : 1 , v : 1),
(-v : 1 , -1 : 1), (-v : 1 , -v : 1), (-v : 1 , 1 : 1), (-v : 1 , 1 : 0), (1 : 1 , 0 : 1),
(1 : 1 , v : 1), (1 : 1 , -1 : 1), (1 : 1 , -v : 1), (1 : 1 , 1 : 1), (1 : 1 , 1 : 0),
(1 : 0 , 0 : 1), (1 : 0 , v : 1), (1 : 0 , -1 : 1), (1 : 0 , -v : 1), (1 : 0 , 1 : 1),
(1 : 0 , 1 : 0)]
::
sage: P.<x,y,z,u,v> = ProductProjectiveSpaces([2, 1], GF(3))
sage: P(P.base_ring()).points()
[(0 : 0 : 1 , 0 : 1), (1 : 0 : 1 , 0 : 1), (2 : 0 : 1 , 0 : 1), (0 : 1 : 1 , 0 : 1), (1 : 1 : 1 , 0 : 1),
(2 : 1 : 1 , 0 : 1), (0 : 2 : 1 , 0 : 1), (1 : 2 : 1 , 0 : 1), (2 : 2 : 1 , 0 : 1), (0 : 1 : 0 , 0 : 1),
(1 : 1 : 0 , 0 : 1), (2 : 1 : 0 , 0 : 1), (1 : 0 : 0 , 0 : 1), (0 : 0 : 1 , 1 : 1), (1 : 0 : 1 , 1 : 1),
(2 : 0 : 1 , 1 : 1), (0 : 1 : 1 , 1 : 1), (1 : 1 : 1 , 1 : 1), (2 : 1 : 1 , 1 : 1), (0 : 2 : 1 , 1 : 1),
(1 : 2 : 1 , 1 : 1), (2 : 2 : 1 , 1 : 1), (0 : 1 : 0 , 1 : 1), (1 : 1 : 0 , 1 : 1), (2 : 1 : 0 , 1 : 1),
(1 : 0 : 0 , 1 : 1), (0 : 0 : 1 , 2 : 1), (1 : 0 : 1 , 2 : 1), (2 : 0 : 1 , 2 : 1), (0 : 1 : 1 , 2 : 1),
(1 : 1 : 1 , 2 : 1), (2 : 1 : 1 , 2 : 1), (0 : 2 : 1 , 2 : 1), (1 : 2 : 1 , 2 : 1), (2 : 2 : 1 , 2 : 1),
(0 : 1 : 0 , 2 : 1), (1 : 1 : 0 , 2 : 1), (2 : 1 : 0 , 2 : 1), (1 : 0 : 0 , 2 : 1), (0 : 0 : 1 , 1 : 0),
(1 : 0 : 1 , 1 : 0), (2 : 0 : 1 , 1 : 0), (0 : 1 : 1 , 1 : 0), (1 : 1 : 1 , 1 : 0), (2 : 1 : 1 , 1 : 0),
(0 : 2 : 1 , 1 : 0), (1 : 2 : 1 , 1 : 0), (2 : 2 : 1 , 1 : 0), (0 : 1 : 0 , 1 : 0), (1 : 1 : 0 , 1 : 0),
(2 : 1 : 0 , 1 : 0), (1 : 0 : 0 , 1 : 0)]
"""
X = self.codomain()
from sage.schemes.product_projective.space import is_ProductProjectiveSpaces
if not is_ProductProjectiveSpaces(X) and X.base_ring() in Fields():
# no points
if X.dimension() == -1:
return []
# if X is zero-dimensional
if X.dimension() == 0:
points = set()
# find points from all possible affine patches
for I in xmrange([
n + 1 for n in
X.ambient_space().dimension_relative_components()
]):
[Y, phi] = X.affine_patch(I, True)
aff_points = Y.rational_points()
for PP in aff_points:
points.add(phi(PP))
return list(points)
R = self.value_ring()
points = []
if R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
for P in X.ambient_space().points_of_bounded_height(B, prec):
try:
points.append(X(P))
except TypeError:
pass
return points
if is_FiniteField(R):
for P in X.ambient_space().rational_points():
try:
points.append(X(P))
except TypeError:
pass
return points
else:
raise TypeError("unable to enumerate points over %s" % R)
