def split_local_cover(self):
"""
Tries to find subform of the given (positive definite quaternary)
quadratic form Q of the form
.. MATH::
d*x^2 + T(y,z,w)
where `d > 0` is as small as possible.
This is done by exhaustive search on small vectors, and then
comparing the local conditions of its sum with it's complementary
lattice and the original quadratic form Q.
INPUT:
none
OUTPUT:
a QuadraticForm over ZZ
EXAMPLES::
sage: Q1 = DiagonalQuadraticForm(ZZ, [7,5,3])
sage: Q1.split_local_cover()
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 0 0 ]
[ * 7 0 ]
[ * * 5 ]
"""
## 0. If a split local cover already exists, then return it.
if hasattr(self, "__split_local_cover"):
if is_QuadraticForm(self.__split_local_cover
): ## Here the computation has been done.
return self.__split_local_cover
elif self.__split_local_cover in ZZ: ## Here it indexes the values already tried!
current_length = self.__split_local_cover + 1
Length_Max = current_length + 5
else:
current_length = 1
Length_Max = 6
## 1. Find a range of new vectors
all_vectors = self.vectors_by_length(Length_Max)
current_vectors = all_vectors[current_length]
## Loop until we find a split local cover...
while True:
## 2. Check if any of the primitive ones produce a split local cover
for v in current_vectors:
Q = QuadraticForm__constructor(
ZZ, 1,
[current_length]) + self.complementary_subform_to_vector(v)
if Q.local_representation_conditions(
) == self.local_representation_conditions():
self.__split_local_cover = Q
return Q
## 3. Save what we have checked and get more vectors.
self.__split_local_cover = current_length
current_length += 1
if current_length >= len(all_vectors):
Length_Max += 5
all_vectors = self.vectors_by_length(Length_Max)
current_vectors = all_vectors[current_length]
def split_local_cover(self):
"""
Tries to find subform of the given (positive definite quaternary)
quadratic form Q of the form
.. math::
d*x^2 + T(y,z,w)
where `d > 0` is as small as possible.
This is done by exhaustive search on small vectors, and then
comparing the local conditions of its sum with it's complementary
lattice and the original quadratic form Q.
INPUT:
none
OUTPUT:
a QuadraticForm over ZZ
EXAMPLES::
sage: Q1 = DiagonalQuadraticForm(ZZ, [7,5,3])
sage: Q1.split_local_cover()
Quadratic form in 3 variables over Integer Ring with coefficients:
[ 3 0 0 ]
[ * 7 0 ]
[ * * 5 ]
"""
## 0. If a split local cover already exists, then return it.
if hasattr(self, "__split_local_cover"):
if is_QuadraticForm(self.__split_local_cover): ## Here the computation has been done.
return self.__split_local_cover
elif isinstance(self.__split_local_cover, Integer): ## Here it indexes the values already tried!
current_length = self.__split_local_cover + 1
Length_Max = current_length + 5
else:
current_length = 1
Length_Max = 6
## 1. Find a range of new vectors
all_vectors = self.vectors_by_length(Length_Max)
current_vectors = all_vectors[current_length]
## Loop until we find a split local cover...
while True:
## 2. Check if any of the primitive ones produce a split local cover
for v in current_vectors:
#print "current length = ", current_length
#print "v = ", v
Q = QuadraticForm__constructor(ZZ, 1, [current_length]) + self.complementary_subform_to_vector(v)
#print Q
if Q.local_representation_conditions() == self.local_representation_conditions():
self.__split_local_cover = Q
return Q
## 3. Save what we have checked and get more vectors.
self.__split_local_cover = current_length
current_length += 1
if current_length >= len(all_vectors):
Length_Max += 5
all_vectors = self.vectors_by_length(Length_Max)
current_vectors = all_vectors[current_length]
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 (base_field is None or isinstance(base_field, IntegralDomain)):
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 is 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 isinstance(P, IntegralDomain):
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 is 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 is None:
base_field = F.base_ring()
if not isinstance(base_field, IntegralDomain):
raise ValueError("Base field (=%s) must be a field" % base_field)
base_field = base_field.fraction_field()
if names is 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)
if is_FractionField(base_field) and (is_PolynomialRing(
base_field.ring()) or is_MPolynomialRing(base_field.ring())):
return ProjectiveConic_rational_function_field(P2, F)
return ProjectiveConic_field(P2, F)
raise TypeError("Number of variables of F (=%s) must be 2 or 3" % F)
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 (base_field is None or isinstance(base_field, IntegralDomain)):
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 is 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 isinstance(P, IntegralDomain):
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 is 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 is None:
base_field = F.base_ring()
if not isinstance(base_field, IntegralDomain):
raise ValueError("Base field (=%s) must be a field" % base_field)
base_field = base_field.fraction_field()
if names is 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)
if is_FractionField(base_field) and (is_PolynomialRing(base_field.ring()) or is_MPolynomialRing(base_field.ring())):
return ProjectiveConic_rational_function_field(P2, F)
return ProjectiveConic_field(P2, F)
raise TypeError("Number of variables of F (=%s) must be 2 or 3" % F)
def is_globally_equivalent_to(self, other, return_matrix=False):
"""
Determine if the current quadratic form is equivalent to the
given form over ZZ.
If ``return_matrix`` is True, then we return the transformation
matrix `M` so that ``self(M) == other``.
INPUT:
- ``self``, ``other`` -- positive definite integral quadratic forms
- ``return_matrix`` -- (boolean, default ``False``) return
the transformation matrix instead of a boolean
OUTPUT:
- if ``return_matrix`` is ``False``: a boolean
- if ``return_matrix`` is ``True``: either ``False`` or the
transformation matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.is_globally_equivalent_to(Q1)
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True)
sage: Q(MM) == Q1
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2)
False
sage: Q1.is_globally_equivalent_to(Q2, return_matrix=True)
False
sage: Q1.is_globally_equivalent_to(Q3)
True
sage: M = Q1.is_globally_equivalent_to(Q3, True); M
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
ALGORITHM: this uses the PARI function :pari:`qfisom`, implementing
an algorithm by Plesken and Souvignier.
TESTS:
:trac:`27749` is fixed::
sage: Q = QuadraticForm(ZZ, 2, [2, 3, 5])
sage: P = QuadraticForm(ZZ, 2, [8, 6, 5])
sage: Q.is_globally_equivalent_to(P)
False
sage: P.is_globally_equivalent_to(Q)
False
"""
## Check that other is a QuadraticForm
if not is_QuadraticForm(other):
raise TypeError(
"you must compare two quadratic forms, but the argument is not a quadratic form"
)
## only for definite forms
if not self.is_definite() or not other.is_definite():
raise ValueError(
"not a definite form in QuadraticForm.is_globally_equivalent_to()")
mat = other.__pari__().qfisom(self)
if not mat:
return False
if return_matrix:
return mat.sage()
else:
return True
