def __init__(self, A, f): r""" See ``Conic`` for full documentation. EXAMPLES :: sage: Conic([GF(3)(1), 1, 1]) Projective Conic Curve over Finite Field of size 3 defined by x^2 + y^2 + z^2 """ ProjectiveConic_field.__init__(self, A, f)
def __init__(self, A, f): r""" See ``Conic`` for full documentation. EXAMPLES :: sage: Conic([1, 1, 1]) Projective Conic Curve over Rational Field defined by x^2 + y^2 + z^2 """ ProjectiveConic_field.__init__(self, A, f) # a single prime such that self has no point over the completion self._local_obstruction = None # all finite primes such that self has no point over the completion self._finite_obstructions = None # all infinite primes such that self has no point over the completion self._infinite_obstructions = None
def Conic(base_field, F=None, names=None, unique=True): r""" Return the plane projective conic curve defined by ``F`` over ``base_field``. The input form ``Conic(F, names=None)`` is also accepted, in which case the fraction field of the base ring of ``F`` is used as base field. INPUT: - ``base_field`` -- The base field of the conic. - ``names`` -- a list, tuple, or comma separated string of three variable names specifying the names of the coordinate functions of the ambient space `\Bold{P}^3`. If not specified or read off from ``F``, then this defaults to ``'x,y,z'``. - ``F`` -- a polynomial, list, matrix, ternary quadratic form, or list or tuple of 5 points in the plane. If ``F`` is a polynomial or quadratic form, then the output is the curve in the projective plane defined by ``F = 0``. If ``F`` is a polynomial, then it must be a polynomial of degree at most 2 in 2 variables, or a homogeneous polynomial in of degree 2 in 3 variables. If ``F`` is a matrix, then the output is the zero locus of `(x,y,z) F (x,y,z)^t`. If ``F`` is a list of coefficients, then it has length 3 or 6 and gives the coefficients of the monomials `x^2, y^2, z^2` or all 6 monomials `x^2, xy, xz, y^2, yz, z^2` in lexicographic order. If ``F`` is a list of 5 points in the plane, then the output is a conic through those points. - ``unique`` -- Used only if ``F`` is a list of points in the plane. If the conic through the points is not unique, then raise ``ValueError`` if and only if ``unique`` is True OUTPUT: A plane projective conic curve defined by ``F`` over a field. EXAMPLES: Conic curves given by polynomials :: sage: X,Y,Z = QQ['X,Y,Z'].gens() sage: Conic(X^2 - X*Y + Y^2 - Z^2) Projective Conic Curve over Rational Field defined by X^2 - X*Y + Y^2 - Z^2 sage: x,y = GF(7)['x,y'].gens() sage: Conic(x^2 - x + 2*y^2 - 3, 'U,V,W') Projective Conic Curve over Finite Field of size 7 defined by U^2 + 2*V^2 - U*W - 3*W^2 Conic curves given by matrices :: sage: Conic(matrix(QQ, [[1, 2, 0], [4, 0, 0], [7, 0, 9]]), 'x,y,z') Projective Conic Curve over Rational Field defined by x^2 + 6*x*y + 7*x*z + 9*z^2 sage: x,y,z = GF(11)['x,y,z'].gens() sage: C = Conic(x^2+y^2-2*z^2); C Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2 sage: Conic(C.symmetric_matrix(), 'x,y,z') Projective Conic Curve over Finite Field of size 11 defined by x^2 + y^2 - 2*z^2 Conics given by coefficients :: sage: Conic(QQ, [1,2,3]) Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + 3*z^2 sage: Conic(GF(7), [1,2,3,4,5,6], 'X') Projective Conic Curve over Finite Field of size 7 defined by X0^2 + 2*X0*X1 - 3*X1^2 + 3*X0*X2 - 2*X1*X2 - X2^2 The conic through a set of points :: sage: C = Conic(QQ, [[10,2],[3,4],[-7,6],[7,8],[9,10]]); C Projective Conic Curve over Rational Field defined by x^2 + 13/4*x*y - 17/4*y^2 - 35/2*x*z + 91/4*y*z - 37/2*z^2 sage: C.rational_point() (10 : 2 : 1) sage: C.point([3,4]) (3 : 4 : 1) sage: a=AffineSpace(GF(13),2) sage: Conic([a([x,x^2]) for x in range(5)]) Projective Conic Curve over Finite Field of size 13 defined by x^2 - y*z """ if not (is_IntegralDomain(base_field) or base_field == None): if names is None: names = F F = base_field base_field = None if isinstance(F, (list,tuple)): if len(F) == 1: return Conic(base_field, F[0], names) if names == None: names = 'x,y,z' if len(F) == 5: L=[] for f in F: if isinstance(f, SchemeMorphism_point_affine): C = Sequence(f, universe = base_field) if len(C) != 2: raise TypeError, "points in F (=%s) must be planar"%F C.append(1) elif isinstance(f, SchemeMorphism_point_projective_field): C = Sequence(f, universe = base_field) elif isinstance(f, (list, tuple)): C = Sequence(f, universe = base_field) if len(C) == 2: C.append(1) else: raise TypeError, "F (=%s) must be a sequence of planar " \ "points" % F if len(C) != 3: raise TypeError, "points in F (=%s) must be planar" % F P = C.universe() if not is_IntegralDomain(P): raise TypeError, "coordinates of points in F (=%s) must " \ "be in an integral domain" % F L.append(Sequence([C[0]**2, C[0]*C[1], C[0]*C[2], C[1]**2, C[1]*C[2], C[2]**2], P.fraction_field())) M=Matrix(L) if unique and M.rank() != 5: raise ValueError, "points in F (=%s) do not define a unique " \ "conic" % F con = Conic(base_field, Sequence(M.right_kernel().gen()), names) con.point(F[0]) return con F = Sequence(F, universe = base_field) base_field = F.universe().fraction_field() temp_ring = PolynomialRing(base_field, 3, names) (x,y,z) = temp_ring.gens() if len(F) == 3: return Conic(F[0]*x**2 + F[1]*y**2 + F[2]*z**2) if len(F) == 6: return Conic(F[0]*x**2 + F[1]*x*y + F[2]*x*z + F[3]*y**2 + \ F[4]*y*z + F[5]*z**2) raise TypeError, "F (=%s) must be a sequence of 3 or 6" \ "coefficients" % F if is_QuadraticForm(F): F = F.matrix() if is_Matrix(F) and F.is_square() and F.ncols() == 3: if names == None: names = 'x,y,z' temp_ring = PolynomialRing(F.base_ring(), 3, names) F = vector(temp_ring.gens()) * F * vector(temp_ring.gens()) if not is_MPolynomial(F): raise TypeError, "F (=%s) must be a three-variable polynomial or " \ "a sequence of points or coefficients" % F if F.total_degree() != 2: raise TypeError, "F (=%s) must have degree 2" % F if base_field == None: base_field = F.base_ring() if not is_IntegralDomain(base_field): raise ValueError, "Base field (=%s) must be a field" % base_field base_field = base_field.fraction_field() if names == None: names = F.parent().variable_names() pol_ring = PolynomialRing(base_field, 3, names) if F.parent().ngens() == 2: (x,y,z) = pol_ring.gens() F = pol_ring(F(x/z,y/z)*z**2) if F == 0: raise ValueError, "F must be nonzero over base field %s" % base_field if F.total_degree() != 2: raise TypeError, "F (=%s) must have degree 2 over base field %s" % \ (F, base_field) if F.parent().ngens() == 3: P2 = ProjectiveSpace(2, base_field, names) if is_PrimeFiniteField(base_field): return ProjectiveConic_prime_finite_field(P2, F) if is_FiniteField(base_field): return ProjectiveConic_finite_field(P2, F) if is_RationalField(base_field): return ProjectiveConic_rational_field(P2, F) if is_NumberField(base_field): return ProjectiveConic_number_field(P2, F) return ProjectiveConic_field(P2, F) raise TypeError, "Number of variables of F (=%s) must be 2 or 3" % F
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)