def points(self, B=0):
"""
Return some or all rational points of a projective scheme.
INPUT:
- `B` -- integer (optional, default=0). The bound for the
coordinates.
OUTPUT:
A list of points. Over a finite field, all points are
returned. Over an infinite field, all points satisfying the
bound are returned.
EXAMPLES::
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(9))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
"""
R = self.value_ring()
if is_RationalField(R):
if not B > 0:
raise TypeError("A positive bound B (= %s) must be specified."%B)
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self,B)
elif R in NumberFields():
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self,B)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("Unable to enumerate points over %s."%R)
def points(self, **kwds):
"""
Return some or all rational points of a projective scheme.
For dimension 0 subschemes points are determined through a groebner
basis calculation. For schemes or subschemes with dimension greater than 1
points are determined through enumeration up to the specified bound.
INPUT:
kwds:
- ``bound`` - real number (optional, default=0). The bound for the coordinates for
subschemes with dimension at least 1.
- ``precision`` - integer (optional, default=53). The precision to use to
compute the elements of bounded height for number fields.
- ``point_tolerance`` - positive real number (optional, default=10^(-10)).
For numerically inexact fields, two points are considered the same
if their coordinates are within tolerance.
- ``zero_tolerance`` - positive real number (optional, default=10^(-10)).
For numerically inexact fields, points are on the subscheme if they
satisfy the equations to within tolerance.
- ``tolerance`` - a rational number in (0,1] used in doyle-krumm algorithm-4
for enumeration over number fields.
OUTPUT:
- a list of rational points of a projective scheme
.. WARNING::
For numerically inexact fields such as ComplexField or RealField the
list of points returned is very likely to be incomplete. It may also
contain repeated points due to tolerances.
EXAMPLES::
sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(bound=4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(bound=1.8))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
sage: E(P.base_ring()).points()
[(-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 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(CC, 2)
sage: E = P.subscheme([y^3 - x^3 - x*z^2, x*y*z])
sage: L=E(P.base_ring()).points(); sorted(L, key=str)
verbose 0 (71: projective_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
[(-0.500000000000000 + 0.866025403784439*I : 1.00000000000000 : 0.000000000000000),
(-0.500000000000000 - 0.866025403784439*I : 1.00000000000000 : 0.000000000000000),
(-1.00000000000000*I : 0.000000000000000 : 1.00000000000000),
(0.000000000000000 : 0.000000000000000 : 1.00000000000000),
(1.00000000000000 : 1.00000000000000 : 0.000000000000000),
(1.00000000000000*I : 0.000000000000000 : 1.00000000000000)]
sage: L[0].codomain()
Projective Space of dimension 2 over Complex Field with 53 bits of precision
::
sage: P.<x,y,z> = ProjectiveSpace(CDF, 2)
sage: E = P.subscheme([y^2 + x^2 + z^2, x*y*z])
sage: len(E(P.base_ring()).points())
verbose 0 (71: projective_homset.py, points) Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.
6
"""
from sage.schemes.projective.projective_space import is_ProjectiveSpace
X = self.codomain()
if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
if hasattr(X.base_ring(), 'precision'):
numerical = True
verbose(
"Warning: computations in the numerical fields are inexact;points may be computed partially or incorrectly.",
level=0)
pt_tol = RR(kwds.pop('point_tolerance', 10**(-10)))
zero_tol = RR(kwds.pop('zero_tolerance', 10**(-10)))
if pt_tol <= 0 or zero_tol <= 0:
raise ValueError("tolerance must be positive")
else:
numerical = False
#Then it must be a subscheme
dim_ideal = X.defining_ideal().dimension()
if dim_ideal < 1: # no points
return []
if dim_ideal == 1: # if X zero-dimensional
rat_points = set()
PS = X.ambient_space()
N = PS.dimension_relative()
BR = X.base_ring()
#need a lexicographic ordering for elimination
R = PolynomialRing(BR, N + 1, PS.variable_names(), order='lex')
I = R.ideal(X.defining_polynomials())
I0 = R.ideal(0)
#Determine the points through elimination
#This is much faster than using the I.variety() function on each affine chart.
for k in range(N + 1):
#create the elimination ideal for the kth affine patch
G = I.substitute({R.gen(k): 1}).groebner_basis()
if G != [1]:
P = {}
#keep track that we know the kth coordinate is 1
P.update({R.gen(k): 1})
points = [P]
#work backwards from solving each equation for the possible
#values of the next coordinate
for i in range(len(G) - 1, -1, -1):
new_points = []
good = 0
for P in points:
#substitute in our dictionary entry that has the values
#of coordinates known so far. This results in a single
#variable polynomial (by elimination)
L = G[i].substitute(P)
if R(L).degree() > 0:
if numerical:
for pol in L.univariate_polynomial(
).roots(multiplicities=False):
good = 1
r = L.variables()[0]
varindex = R.gens().index(r)
P.update({R.gen(varindex): pol})
new_points.append(copy(P))
else:
L = L.factor()
#the linear factors give the possible rational values of
#this coordinate
for pol, pow in L:
if pol.degree() == 1 and len(
pol.variables()) == 1:
good = 1
r = pol.variables()[0]
varindex = R.gens().index(r)
#add this coordinates information to
#each dictionary entry
P.update({
R.gen(varindex):
-pol.constant_coefficient(
) /
pol.monomial_coefficient(r)
})
new_points.append(copy(P))
else:
new_points.append(P)
good = 1
if good:
points = new_points
#the dictionary entries now have values for all coordinates
#they are the rational solutions to the equations
#make them into projective points
for i in range(len(points)):
if numerical:
if len(points[i]) == N + 1:
S = PS([
points[i][R.gen(j)]
for j in range(N + 1)
])
S.normalize_coordinates()
if all(
g(list(S)) < zero_tol
for g in X.defining_polynomials()):
rat_points.add(S)
else:
if len(points[i]) == N + 1 and I.subs(
points[i]) == I0:
S = X([
points[i][R.gen(j)]
for j in range(N + 1)
])
S.normalize_coordinates()
rat_points.add(S)
# remove duplicate element using tolerance
if numerical:
dupl_points = list(rat_points)
for i in range(len(dupl_points)):
u = dupl_points[i]
for j in range(i + 1, len(dupl_points)):
v = dupl_points[j]
if all((u[k] - v[k]).abs() < pt_tol
for k in range(len(u))):
rat_points.remove(u)
break
rat_points = sorted(rat_points)
return rat_points
R = self.value_ring()
B = kwds.pop('bound', 0)
tol = kwds.pop('tolerance', 1e-2)
prec = kwds.pop('precision', 53)
if is_RationalField(R):
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
if isinstance(X, AlgebraicScheme_subscheme
): # sieve should only be called for subschemes
from sage.schemes.projective.projective_rational_point import sieve
return sieve(X, B)
else:
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self, B)
elif R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self,
bound=B,
tolerance=tol,
precision=prec)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("unable to enumerate points over %s" % R)
def points(self, B=0, prec=53):
"""
Return some or all rational points of a projective scheme.
INPUT:
- ``B`` - integer (optional, default=0). The bound for the
coordinates.
- ``prec`` - he precision to use to compute the elements of bounded height for number fields.
OUTPUT:
A list of points. Over a finite field, all points are
returned. Over an infinite field, all points satisfying the
bound are returned.
.. 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> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(1.8))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
sage: E(P.base_ring()).points()
[(-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 : 1)]
"""
X = self.codomain()
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
#Then it must be a subscheme
dim_ideal = X.defining_ideal().dimension()
if dim_ideal < 1: # no points
return []
if dim_ideal == 1: # if X zero-dimensional
rat_points = set()
PS = X.ambient_space()
N = PS.dimension_relative()
BR = X.base_ring()
#need a lexicographic ordering for elimination
R = PolynomialRing(BR, N + 1, PS.variable_names(), order='lex')
I = R.ideal(X.defining_polynomials())
I0 = R.ideal(0)
#Determine the points through elimination
#This is much faster than using the I.variety() function on each affine chart.
for k in range(N + 1):
#create the elimination ideal for the kth affine patch
G = I.substitute({R.gen(k): 1}).groebner_basis()
if G != [1]:
P = {}
#keep track that we know the kth coordinate is 1
P.update({R.gen(k): 1})
points = [P]
#work backwards from solving each equation for the possible
#values of the next coordinate
for i in range(len(G) - 1, -1, -1):
new_points = []
good = 0
for P in points:
#substitute in our dictionary entry that has the values
#of coordinates known so far. This results in a single
#variable polynomial (by elimination)
L = G[i].substitute(P)
if L != 0:
L = L.factor()
#the linear factors give the possible rational values of
#this coordinate
for pol, pow in L:
if pol.degree() == 1 and len(
pol.variables()) == 1:
good = 1
r = pol.variables()[0]
varindex = R.gens().index(r)
#add this coordinates information to
#each dictionary entry
P.update({
R.gen(varindex):
-pol.constant_coefficient() /
pol.monomial_coefficient(r)
})
new_points.append(copy(P))
else:
new_points.append(P)
good = 1
if good:
points = new_points
#the dictionary entries now have values for all coordinates
#they are the rational solutions to the equations
#make them into projective points
for i in range(len(points)):
if len(points[i]) == N + 1 and I.subs(
points[i]) == I0:
S = X([
points[i][R.gen(j)] for j in range(N + 1)
])
S.normalize_coordinates()
rat_points.add(S)
rat_points = sorted(rat_points)
return rat_points
R = self.value_ring()
if is_RationalField(R):
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self, B)
elif R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self, B, prec=prec)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("unable to enumerate points over %s" % R)
def points(self, B=0, prec=53):
"""
Return some or all rational points of a projective scheme.
INPUT:
- `B` - integer (optional, default=0). The bound for the
coordinates.
- ``prec`` - he precision to use to compute the elements of bounded height for number fields
OUTPUT:
A list of points. Over a finite field, all points are
returned. Over an infinite field, all points satisfying the
bound are returned.
.. 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> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(1.8))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
sage: E(P.base_ring()).points()
[(-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 : 1)]
"""
X = self.codomain()
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
#Then it must be a subscheme
dim_ideal = X.defining_ideal().dimension()
if dim_ideal < 1: # no points
return []
if dim_ideal == 1: # if X zero-dimensional
points = set()
for i in range(X.ambient_space().dimension_relative() + 1):
Y = X.affine_patch(i)
phi = Y.projective_embedding()
aff_points = Y.rational_points()
for PP in aff_points:
points.add(X.ambient_space()(list(phi(PP))))
points = sorted(points)
return points
R = self.value_ring()
if is_RationalField(R):
if not B > 0:
raise TypeError("A positive bound B (= %s) must be specified."%B)
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self,B)
elif R in NumberFields():
if not B > 0:
raise TypeError("A positive bound B (= %s) must be specified."%B)
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self,B, prec=prec)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("Unable to enumerate points over %s."%R)
def points(self, B=0, prec=53):
"""
Return some or all rational points of a projective scheme.
INPUT:
- ``B`` - integer (optional, default=0). The bound for the
coordinates.
- ``prec`` - he precision to use to compute the elements of bounded height for number fields.
OUTPUT:
A list of points. Over a finite field, all points are
returned. Over an infinite field, all points satisfying the
bound are returned.
.. 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> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(1.8))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
sage: E(P.base_ring()).points()
[(-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 : 1)]
"""
X = self.codomain()
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
#Then it must be a subscheme
dim_ideal = X.defining_ideal().dimension()
if dim_ideal < 1: # no points
return []
if dim_ideal == 1: # if X zero-dimensional
points = set()
for i in range(X.ambient_space().dimension_relative() + 1):
Y = X.affine_patch(i)
phi = Y.projective_embedding()
aff_points = Y.rational_points()
for PP in aff_points:
points.add(X.ambient_space()(list(phi(PP))))
points = sorted(points)
return points
R = self.value_ring()
if is_RationalField(R):
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self,B)
elif R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self,B, prec=prec)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("unable to enumerate points over %s"%R)
def points(self, B=0, prec=53):
"""
Return some or all rational points of a projective scheme.
INPUT:
- ``B`` - integer (optional, default=0). The bound for the
coordinates.
- ``prec`` - he precision to use to compute the elements of bounded height for number fields.
OUTPUT:
A list of points. Over a finite field, all points are
returned. Over an infinite field, all points satisfying the
bound are returned.
.. 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> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(1.8))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
sage: E(P.base_ring()).points()
[(-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 : 1)]
"""
X = self.codomain()
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
#Then it must be a subscheme
dim_ideal = X.defining_ideal().dimension()
if dim_ideal < 1: # no points
return []
if dim_ideal == 1: # if X zero-dimensional
rat_points = set()
PS = X.ambient_space()
N = PS.dimension_relative()
BR = X.base_ring()
#need a lexicographic ordering for elimination
R = PolynomialRing(BR, N + 1, PS.gens(), order='lex')
I = R.ideal(X.defining_polynomials())
I0 = R.ideal(0)
#Determine the points through elimination
#This is much faster than using the I.variety() function on each affine chart.
for k in range(N + 1):
#create the elimination ideal for the kth affine patch
G = I.substitute({R.gen(k):1}).groebner_basis()
if G != [1]:
P = {}
#keep track that we know the kth coordinate is 1
P.update({R.gen(k):1})
points = [P]
#work backwards from solving each equation for the possible
#values of the next coordinate
for i in range(len(G) - 1, -1, -1):
new_points = []
good = 0
for P in points:
#substitute in our dictionary entry that has the values
#of coordinates known so far. This results in a single
#variable polynomial (by elimination)
L = G[i].substitute(P)
if L != 0:
L = L.factor()
#the linear factors give the possible rational values of
#this coordinate
for pol, pow in L:
if pol.degree() == 1 and len(pol.variables()) == 1:
good = 1
r = pol.variables()[0]
varindex = R.gens().index(r)
#add this coordinates information to
#each dictionary entry
P.update({R.gen(varindex):-pol.constant_coefficient() / pol.monomial_coefficient(r)})
new_points.append(copy(P))
else:
new_points.append(P)
good = 1
if good:
points = new_points
#the dictionary entries now have values for all coordinates
#they are the rational solutions to the equations
#make them into projective points
for i in range(len(points)):
if len(points[i]) == N + 1 and I.subs(points[i]) == I0:
S = X([points[i][R.gen(j)] for j in range(N + 1)])
S.normalize_coordinates()
rat_points.add(S)
rat_points = sorted(rat_points)
return rat_points
R = self.value_ring()
if is_RationalField(R):
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self,B)
elif R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified"%B)
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self,B, prec=prec)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("unable to enumerate points over %s"%R)
def points(self, **kwds):
"""
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> = ProjectiveSpace(QQ,1)
sage: P(QQ).points(bound=4)
[(-4 : 1), (-3 : 1), (-2 : 1), (-3/2 : 1), (-4/3 : 1), (-1 : 1),
(-3/4 : 1), (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (-1/4 : 1), (0 : 1),
(1/4 : 1), (1/3 : 1), (1/2 : 1), (2/3 : 1), (3/4 : 1), (1 : 0), (1 : 1),
(4/3 : 1), (3/2 : 1), (2 : 1), (3 : 1), (4 : 1)]
::
sage: u = QQ['u'].0
sage: K.<v> = NumberField(u^2 + 3)
sage: P.<x,y,z> = ProjectiveSpace(K,2)
sage: len(P(K).points(bound=1.8))
381
::
sage: P1 = ProjectiveSpace(GF(2),1)
sage: F.<a> = GF(4,'a')
sage: P1(F).points()
[(0 : 1), (1 : 0), (1 : 1), (a : 1), (a + 1 : 1)]
::
sage: P.<x,y,z> = ProjectiveSpace(QQ,2)
sage: E = P.subscheme([(y^3-y*z^2) - (x^3-x*z^2),(y^3-y*z^2) + (x^3-x*z^2)])
sage: E(P.base_ring()).points()
[(-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 : 1)]
"""
B = kwds.pop('bound', 0)
tol = kwds.pop('tolerance', 1e-2)
prec = kwds.pop('precision', 53)
X = self.codomain()
from sage.schemes.projective.projective_space import is_ProjectiveSpace
if not is_ProjectiveSpace(X) and X.base_ring() in Fields():
#Then it must be a subscheme
dim_ideal = X.defining_ideal().dimension()
if dim_ideal < 1: # no points
return []
if dim_ideal == 1: # if X zero-dimensional
rat_points = set()
PS = X.ambient_space()
N = PS.dimension_relative()
BR = X.base_ring()
#need a lexicographic ordering for elimination
R = PolynomialRing(BR, N + 1, PS.variable_names(), order='lex')
I = R.ideal(X.defining_polynomials())
I0 = R.ideal(0)
#Determine the points through elimination
#This is much faster than using the I.variety() function on each affine chart.
for k in range(N + 1):
#create the elimination ideal for the kth affine patch
G = I.substitute({R.gen(k): 1}).groebner_basis()
if G != [1]:
P = {}
#keep track that we know the kth coordinate is 1
P.update({R.gen(k): 1})
points = [P]
#work backwards from solving each equation for the possible
#values of the next coordinate
for i in range(len(G) - 1, -1, -1):
new_points = []
good = 0
for P in points:
#substitute in our dictionary entry that has the values
#of coordinates known so far. This results in a single
#variable polynomial (by elimination)
L = G[i].substitute(P)
if L != 0:
L = L.factor()
#the linear factors give the possible rational values of
#this coordinate
for pol, pow in L:
if pol.degree() == 1 and len(
pol.variables()) == 1:
good = 1
r = pol.variables()[0]
varindex = R.gens().index(r)
#add this coordinates information to
#each dictionary entry
P.update({
R.gen(varindex):
-pol.constant_coefficient() /
pol.monomial_coefficient(r)
})
new_points.append(copy(P))
else:
new_points.append(P)
good = 1
if good:
points = new_points
#the dictionary entries now have values for all coordinates
#they are the rational solutions to the equations
#make them into projective points
for i in range(len(points)):
if len(points[i]) == N + 1 and I.subs(
points[i]) == I0:
S = X([
points[i][R.gen(j)] for j in range(N + 1)
])
S.normalize_coordinates()
rat_points.add(S)
rat_points = sorted(rat_points)
return rat_points
R = self.value_ring()
if is_RationalField(R):
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
if isinstance(X, AlgebraicScheme_subscheme
): # sieve should only be called for subschemes
from sage.schemes.projective.projective_rational_point import sieve
return sieve(X, B)
else:
from sage.schemes.projective.projective_rational_point import enum_projective_rational_field
return enum_projective_rational_field(self, B)
elif R in NumberFields():
if not B > 0:
raise TypeError("a positive bound B (= %s) must be specified" %
B)
from sage.schemes.projective.projective_rational_point import enum_projective_number_field
return enum_projective_number_field(self,
bound=B,
tolerance=tol,
precision=prec)
elif is_FiniteField(R):
from sage.schemes.projective.projective_rational_point import enum_projective_finite_field
return enum_projective_finite_field(self.extended_codomain())
else:
raise TypeError("unable to enumerate points over %s" % R)
