def has_rational_point(self, point = False, obstruction = False,
algorithm = 'default', read_cache = True):
r"""
Returns ``True`` if and only if ``self`` has a point
defined over its base field `B`.
If ``point`` and ``obstruction`` are both False (default),
then the output is a boolean ``out`` saying whether ``self``
has a rational point.
If ``point`` or ``obstruction`` is True, then the output is
a pair ``(out, S)``, where ``out`` is as above and:
- if ``point`` is True and ``self`` has a rational point,
then ``S`` is a rational point,
- if ``obstruction`` is True, ``self`` has no rational point,
then ``S`` is a prime or infinite place of `B` such that no
rational point exists over the completion at ``S``.
Points and obstructions are cached whenever they are found.
Cached information is used for the output if available, but only
if ``read_cache`` is True.
ALGORITHM:
The parameter ``algorithm``
specifies the algorithm to be used:
- ``'rnfisnorm'`` -- Use PARI's rnfisnorm
(cannot be combined with ``obstruction = True``)
- ``'local'`` -- Check if a local solution exists for all primes
and infinite places of `B` and apply the Hasse principle.
(Cannot be combined with ``point = True``.)
- ``'default'`` -- Use algorithm ``'rnfisnorm'`` first.
Then, if no point exists and obstructions are requested, use
algorithm ``'local'`` to find an obstruction.
- ``'magma'`` (requires Magma to be installed) --
delegates the task to the Magma computer algebra
system.
EXAMPLES:
An example over `\QQ` ::
sage: C = Conic(QQ, [1, 113922743, -310146482690273725409])
sage: C.has_rational_point(point = True)
(True, (-76842858034579/5424 : -5316144401/5424 : 1))
sage: C.has_rational_point(algorithm = 'local', read_cache = False)
True
Examples over number fields ::
sage: K.<i> = QuadraticField(-1)
sage: C = Conic(K, [1, 3, -5])
sage: C.has_rational_point(point = True, obstruction = True)
(False, Fractional ideal (-i - 2))
sage: C.has_rational_point(algorithm = "rnfisnorm")
False
sage: C.has_rational_point(algorithm = "rnfisnorm", obstruction = True, read_cache=False)
Traceback (most recent call last):
...
ValueError: Algorithm rnfisnorm cannot be combined with obstruction = True in has_rational_point
sage: P.<x> = QQ[]
sage: L.<b> = NumberField(x^3-5)
sage: C = Conic(L, [1, 2, -3])
sage: C.has_rational_point(point = True, algorithm = 'rnfisnorm')
(True, (5/3 : -1/3 : 1))
sage: K.<a> = NumberField(x^4+2)
sage: Conic(QQ, [4,5,6]).has_rational_point()
False
sage: Conic(K, [4,5,6]).has_rational_point()
True
sage: Conic(K, [4,5,6]).has_rational_point(algorithm='magma', read_cache=False) # optional - magma
True
TESTS:
Create a bunch of conics over number fields and check whether
``has_rational_point`` runs without errors for algorithms
``'rnfisnorm'`` and ``'local'``. Check if all points returned are
valid. If Magma is available, then also check if the output agrees with
Magma. ::
sage: P.<X> = QQ[]
sage: Q = P.fraction_field()
sage: c = [1, X/2, 1/X]
sage: l = Sequence(cartesian_product_iterator([c for i in range(3)]))
sage: l = l + [[X, 1, 1, 1, 1, 1]] + [[X, 1/5, 1, 1, 2, 1]]
sage: K.<a> = QuadraticField(-23)
sage: L.<b> = QuadraticField(19)
sage: M.<c> = NumberField(X^3+3*X+1)
sage: m = [[Q(b)(F.gen()) for b in a] for a in l for F in [K, L, M]]
sage: d = []
sage: c = []
sage: c = [Conic(a) for a in m if a != [0,0,0]]
sage: d = [C.has_rational_point(algorithm = 'rnfisnorm', point = True) for C in c] # long time: 3.3 seconds
sage: all([c[k].defining_polynomial()(Sequence(d[k][1])) == 0 for k in range(len(d)) if d[k][0]])
True
sage: [C.has_rational_point(algorithm='local', read_cache=False) for C in c] == [o[0] for o in d] # long time: 5 seconds
True
sage: [C.has_rational_point(algorithm = 'magma', read_cache=False) for C in c] == [o[0] for o in d] # long time: 3 seconds, optional - magma
True
Create a bunch of conics that are known to have rational points
already over `\QQ` and check if points are found by
``has_rational_point``. ::
sage: l = Sequence(cartesian_product_iterator([[-1, 0, 1] for i in range(3)]))
sage: K.<a> = QuadraticField(-23)
sage: L.<b> = QuadraticField(19)
sage: M.<c> = NumberField(x^5+3*x+1)
sage: m = [[F(b) for b in a] for a in l for F in [K, L, M]]
sage: c = [Conic(a) for a in m if a != [0,0,0] and a != [1,1,1] and a != [-1,-1,-1]]
sage: assert all([C.has_rational_point(algorithm = 'rnfisnorm') for C in c])
sage: assert all([C.defining_polynomial()(Sequence(C.has_rational_point(point = True)[1])) == 0 for C in c])
sage: assert all([C.has_rational_point(algorithm='local', read_cache=False) for C in c]) # long time: 1 second
"""
if read_cache:
if self._rational_point is not None:
if point or obstruction:
return True, self._rational_point
else:
return True
if self._local_obstruction is not None:
if point or obstruction:
return False, self._local_obstruction
else:
return False
if (not point) and self._finite_obstructions == [] and \
self._infinite_obstructions == []:
if obstruction:
return True, None
return True
if self.has_singular_point():
if point:
return self.has_singular_point(point = True)
if obstruction:
return True, None
return True
B = self.base_ring()
if algorithm == 'default':
ret = self.has_rational_point(point=True, obstruction=False,
algorithm='rnfisnorm',
read_cache=False)
if ret[0]:
if point or obstruction:
return ret
return True
if obstruction:
ret = self.has_rational_point(point=False, obstruction=True,
algorithm='local',
read_cache=False)
if ret[0]:
raise RuntimeError, "Outputs of algorithms in " \
"has_rational_point disagree " \
"for conic %s" % self
return ret
if point:
return False, None
return False
if algorithm == 'local':
if point:
raise ValueError, "Algorithm 'local' cannot be combined " \
"with point = True in has_rational_point"
obs = self.local_obstructions(infinite = True, finite = False,
read_cache = read_cache)
if obs != []:
if obstruction:
return False, obs[0]
return False
obs = self.local_obstructions(read_cache = read_cache)
if obs == []:
if obstruction:
return True, None
return True
if obstruction:
return False, obs[0]
return False
if algorithm == 'rnfisnorm':
from sage.modules.free_module_element import vector
if obstruction:
raise ValueError, "Algorithm rnfisnorm cannot be combined " \
"with obstruction = True in " \
"has_rational_point"
D, T = self.diagonal_matrix()
abc = [D[0,0], D[1,1], D[2,2]]
for j in range(3):
if abc[j] == 0:
pt = self.point(T*vector({2:0,j:1}))
if point or obstruction:
return True, pt
return True
if (-abc[1]/abc[0]).is_square():
pt = self.point(T*vector([(-abc[1]/abc[0]).sqrt(), 1, 0]))
if point or obstruction:
return True, pt
return True
if (-abc[2]/abc[0]).is_square():
pt = self.point(T*vector([(-abc[2]/abc[0]).sqrt(), 0, 1]))
if point or obstruction:
return True, pt
return True
if is_RationalField(B):
K = B
[KtoB, BtoK] = [K.hom(K) for i in range(2)]
else:
K = B.absolute_field('Y')
[KtoB, BtoK] = K.structure()
X = PolynomialRing(K, 'X').gen()
d = BtoK(-abc[1]/abc[0])
den = d.denominator()
L = K.extension(X**2 - d*den**2, names='y')
isnorm = BtoK(-abc[2]/abc[0]).is_norm(L, element=True)
if isnorm[0]:
pt = self.point(T*vector([KtoB(isnorm[1][0]),
KtoB(isnorm[1][1]*den), 1]))
if point:
return True, pt
return True
if point:
return False, None
return False
if algorithm == 'qfsolve':
raise TypeError, "Algorithm qfsolve in has_rational_point only " \
"for conics over QQ, not over %s" % B
if obstruction:
raise ValueError, "Invalid combination: obstruction=True and " \
"algorithm=%s" % algorithm
return ProjectiveConic_field.has_rational_point(self, point = point,
algorithm = algorithm, read_cache = False)
