def hom(self, x, Y=None):
r"""
Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
Here ``x`` can be a matrix or a sequence of polynomials.
If ``Y`` is omitted, then a natural image is found if possible.
EXAMPLES:
Here are a few Morphisms given by matrices. In the first
example, ``Y`` is omitted, in the second example, ``Y`` is specified.
::
sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + y : y : z)
sage: h([-1, 1, 0])
(0 : 1 : 0)
sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*z : y : x)
``ValueError`` is raised if the wrong codomain ``Y`` is specified:
::
sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [ 0 0 1/2]
[ 0 1 0]
[ 1 0 0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
"""
if is_Matrix(x):
from constructor import Conic
y = x.inverse()
A = y.transpose()*self.matrix()*y
im = Conic(A)
if Y == None:
Y = im
else:
q = Y.defining_polynomial()/im.defining_polynomial()
if not (q.numerator().is_constant()
and q.denominator().is_constant()):
raise ValueError, "The matrix x (= %s) does not define a " \
"map from self (= %s) to Y (= %s)" % \
(x, self, Y)
x = Sequence(x*vector(self.ambient_space().gens()))
return self.Hom(Y)(x, check = False)
return ProjectiveCurve_generic.hom(self, x, Y)
def hom(self, x, Y=None):
r"""
Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
Here ``x`` can be a matrix or a sequence of polynomials.
If ``Y`` is omitted, then a natural image is found if possible.
EXAMPLES:
Here are a few Morphisms given by matrices. In the first
example, ``Y`` is omitted, in the second example, ``Y`` is specified.
::
sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + y : y : z)
sage: h([-1, 1, 0])
(0 : 1 : 0)
sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*z : y : x)
``ValueError`` is raised if the wrong codomain ``Y`` is specified:
::
sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [ 0 0 1/2]
[ 0 1 0]
[ 1 0 0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
The identity map between two representations of the same conic:
::
sage: C = Conic([1,2,3,4,5,6])
sage: D = Conic([2,4,6,8,10,12])
sage: C.hom(identity_matrix(3), D)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + 2*x*y + 4*y^2 + 3*x*z + 5*y*z + 6*z^2
To: Projective Conic Curve over Rational Field defined by 2*x^2 + 4*x*y + 8*y^2 + 6*x*z + 10*y*z + 12*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x : y : z)
An example not over the rational numbers:
::
sage: P.<t> = QQ[]
sage: C = Conic([1,0,0,t,0,1/t])
sage: D = Conic([1/t^2, 0, -2/t^2, t, 0, (t + 1)/t^2])
sage: T = Matrix([[t,0,1],[0,1,0],[0,0,1]])
sage: C.hom(T, D)
Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by x^2 + t*y^2 + 1/t*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by 1/t^2*x^2 + t*y^2 + (-2/t^2)*x*z + ((t + 1)/t^2)*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(t*x + z : y : z)
"""
if is_Matrix(x):
from constructor import Conic
y = x.inverse()
A = y.transpose() * self.matrix() * y
im = Conic(A)
if Y is None:
Y = im
elif not Y == im:
raise ValueError("The matrix x (= %s) does not define a " \
"map from self (= %s) to Y (= %s)" % \
(x, self, Y))
x = Sequence(x * vector(self.ambient_space().gens()))
return self.Hom(Y)(x, check=False)
return ProjectiveCurve_generic.hom(self, x, Y)
def hom(self, x, Y=None):
r"""
Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
Here ``x`` can be a matrix or a sequence of polynomials.
If ``Y`` is omitted, then a natural image is found if possible.
EXAMPLES:
Here are a few Morphisms given by matrices. In the first
example, ``Y`` is omitted, in the second example, ``Y`` is specified.
::
sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + y : y : z)
sage: h([-1, 1, 0])
(0 : 1 : 0)
sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*z : y : x)
``ValueError`` is raised if the wrong codomain ``Y`` is specified:
::
sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [ 0 0 1/2]
[ 0 1 0]
[ 1 0 0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
"""
if is_Matrix(x):
from constructor import Conic
y = x.inverse()
A = y.transpose()*self.matrix()*y
im = Conic(A)
if Y is None:
Y = im
else:
q = Y.defining_polynomial()/im.defining_polynomial()
if not (q.numerator().is_constant()
and q.denominator().is_constant()):
raise ValueError("The matrix x (= %s) does not define a " \
"map from self (= %s) to Y (= %s)" % \
(x, self, Y))
x = Sequence(x*vector(self.ambient_space().gens()))
return self.Hom(Y)(x, check = False)
return ProjectiveCurve_generic.hom(self, x, Y)
def hom(self, x, Y=None):
r"""
Return the scheme morphism from ``self`` to ``Y`` defined by ``x``.
Here ``x`` can be a matrix or a sequence of polynomials.
If ``Y`` is omitted, then a natural image is found if possible.
EXAMPLES:
Here are a few Morphisms given by matrices. In the first
example, ``Y`` is omitted, in the second example, ``Y`` is specified.
::
sage: c = Conic([-1, 1, 1])
sage: h = c.hom(Matrix([[1,1,0],[0,1,0],[0,0,1]])); h
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by -x^2 + 2*x*y + z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x + y : y : z)
sage: h([-1, 1, 0])
(0 : 1 : 0)
sage: c = Conic([-1, 1, 1])
sage: d = Conic([4, 1, -1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), d)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2
To: Projective Conic Curve over Rational Field defined by 4*x^2 + y^2 - z^2
Defn: Defined on coordinates by sending (x : y : z) to
(1/2*z : y : x)
``ValueError`` is raised if the wrong codomain ``Y`` is specified:
::
sage: c = Conic([-1, 1, 1])
sage: c.hom(Matrix([[0, 0, 1/2], [0, 1, 0], [1, 0, 0]]), c)
Traceback (most recent call last):
...
ValueError: The matrix x (= [ 0 0 1/2]
[ 0 1 0]
[ 1 0 0]) does not define a map from self (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2) to Y (= Projective Conic Curve over Rational Field defined by -x^2 + y^2 + z^2)
The identity map between two representations of the same conic:
::
sage: C = Conic([1,2,3,4,5,6])
sage: D = Conic([2,4,6,8,10,12])
sage: C.hom(identity_matrix(3), D)
Scheme morphism:
From: Projective Conic Curve over Rational Field defined by x^2 + 2*x*y + 4*y^2 + 3*x*z + 5*y*z + 6*z^2
To: Projective Conic Curve over Rational Field defined by 2*x^2 + 4*x*y + 8*y^2 + 6*x*z + 10*y*z + 12*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(x : y : z)
An example not over the rational numbers:
::
sage: P.<t> = QQ[]
sage: C = Conic([1,0,0,t,0,1/t])
sage: D = Conic([1/t^2, 0, -2/t^2, t, 0, (t + 1)/t^2])
sage: T = Matrix([[t,0,1],[0,1,0],[0,0,1]])
sage: C.hom(T, D)
Scheme morphism:
From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by x^2 + t*y^2 + 1/t*z^2
To: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Rational Field defined by 1/t^2*x^2 + t*y^2 + (-2/t^2)*x*z + ((t + 1)/t^2)*z^2
Defn: Defined on coordinates by sending (x : y : z) to
(t*x + z : y : z)
"""
if is_Matrix(x):
from constructor import Conic
y = x.inverse()
A = y.transpose() * self.matrix() * y
im = Conic(A)
if Y is None:
Y = im
elif not Y == im:
raise ValueError(
"The matrix x (= %s) does not define a " "map from self (= %s) to Y (= %s)" % (x, self, Y)
)
x = Sequence(x * vector(self.ambient_space().gens()))
return self.Hom(Y)(x, check=False)
return ProjectiveCurve_generic.hom(self, x, Y)