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)