def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
ALGORITHM:
Uses the PARI/GP function ``qfparam``.
EXAMPLES ::
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*x : -1/2*y + 1/2*z))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError("The conic self (=%s) is not smooth, hence does not have a parametrization." % self)
if point is None:
point = self.rational_point()
point = Sequence(point)
Q = PolynomialRing(QQ, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
M = self.symmetric_matrix()
M *= lcm([ t.denominator() for t in M.list() ])
par1 = qfparam(M, point)
B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
# self is in the image of B and does not lie on a line,
# hence B is invertible
A = B.inverse()
par2 = [sum([A[i,j]*gens[j] for j in range(3)]) for i in [1,0]]
par = ([Q(pol(x/y)*y**2) for pol in par1], par2)
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0],self), self.Hom(P1)(par[1], check = False)
def parametrization(self, point=None, morphism=True):
r"""
Return a parametrization `f` of ``self`` together with the
inverse of `f`.
If ``point`` is specified, then that point is used
for the parametrization. Otherwise, use ``self.rational_point()``
to find a point.
If ``morphism`` is True, then `f` is returned in the form
of a Scheme morphism. Otherwise, it is a tuple of polynomials
that gives the parametrization.
ALGORITHM:
Uses the PARI/GP function ``qfparam``.
EXAMPLES ::
sage: c = Conic([1,1,-1])
sage: c.parametrization()
(Scheme morphism:
From: Projective Space of dimension 1 over Rational Field
To: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y) to
(2*x*y : x^2 - y^2 : x^2 + y^2),
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
To: Projective Space of dimension 1 over Rational Field
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*x : -1/2*y + 1/2*z))
An example with ``morphism = False`` ::
sage: R.<x,y,z> = QQ[]
sage: C = Curve(7*x^2 + 2*y*z + z^2)
sage: (p, i) = C.parametrization(morphism = False); (p, i)
([-2*x*y, x^2 + 7*y^2, -2*x^2], [-1/2*x, 1/7*y + 1/14*z])
sage: C.defining_polynomial()(p)
0
sage: i[0](p) / i[1](p)
x/y
A ``ValueError`` is raised if ``self`` has no rational point ::
sage: C = Conic(x^2 + 2*y^2 + z^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!
A ``ValueError`` is raised if ``self`` is not smooth ::
sage: C = Conic(x^2 + y^2)
sage: C.parametrization()
Traceback (most recent call last):
...
ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.
"""
if (not self._parametrization is None) and not point:
par = self._parametrization
else:
if not self.is_smooth():
raise ValueError(
"The conic self (=%s) is not smooth, hence does not have a parametrization."
% self)
if point is None:
point = self.rational_point()
point = Sequence(point)
Q = PolynomialRing(QQ, 'x,y')
[x, y] = Q.gens()
gens = self.ambient_space().gens()
M = self.symmetric_matrix()
M *= lcm([t.denominator() for t in M.list()])
par1 = qfparam(M, point)
B = Matrix([[par1[i][j] for j in range(3)] for i in range(3)])
# self is in the image of B and does not lie on a line,
# hence B is invertible
A = B.inverse()
par2 = [sum([A[i, j] * gens[j] for j in range(3)]) for i in [1, 0]]
par = ([Q(pol(x / y) * y**2) for pol in par1], par2)
if self._parametrization is None:
self._parametrization = par
if not morphism:
return par
P1 = ProjectiveSpace(self.base_ring(), 1, 'x,y')
return P1.hom(par[0], self), self.Hom(P1)(par[1], check=False)