def __init__(self, parent, polys, check=True):
r"""
The Python constructor.
INPUT:
- ``parent`` -- Homset
- ``polys`` -- anything that defines a point in the class
- ``check`` -- Boolean. Whether or not to perform input checks
(Default: ``True``)
EXAMPLES::
sage: P1.<x0,x1,x2> = ProjectiveSpace(QQ,2)
sage: P2 = ProjectiveSpace(QQ,3,'y')
sage: T = ProductProjectiveSpaces([P1,P2])
sage: Q1 = P1(1,1,1)
sage: Q2 = P2(1,2,3,4)
sage: T([Q1,Q2])
(1 : 1 : 1 , 1/4 : 1/2 : 3/4 : 1)
::
sage: T = ProductProjectiveSpaces([2,2,2],GF(5),'x')
sage: T.point([1,2,3,4,5,6,7,8,9])
(2 : 4 : 1 , 4 : 0 : 1 , 3 : 2 : 1)
::
sage: T.<x,y,z,w> = ProductProjectiveSpaces([1,1],GF(5))
sage: X = T.subscheme([x-y,z-2*w])
sage: X([1,1,2,1])
(1 : 1 , 2 : 1)
"""
SchemeMorphism.__init__(self, parent)
if all(isinstance(P, SchemeMorphism_point) for P in polys):
if check:
Q = []
self._points = []
for i in range(len(polys)):
if polys[i].codomain() != parent.codomain().ambient_space(
)[i]:
raise ValueError(
"Points must be in correct projective spaces")
Q += list(polys[i])
self._points.append(polys[i])
parent.codomain()._check_satisfies_equations(Q)
self._points = polys
else:
N = parent.codomain().ambient_space(
).dimension_relative_components()
if check:
parent.codomain()._check_satisfies_equations(polys)
splitpolys = self.codomain().ambient_space()._factors(polys)
self._points = [
parent.codomain().ambient_space()[i].point(
splitpolys[i], check) for i in range(len(N))
]
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_affine` for details.
TESTS::
sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
sage: A3.<x,y,z> = AffineSpace(QQ, 3)
sage: SchemeMorphism_point_affine(A3(QQ), [1,2,3])
(1, 2, 3)
"""
SchemeMorphism.__init__(self, X)
if is_SchemeMorphism(v):
v = list(v)
if check:
# Verify that there are the right number of coords
d = self.codomain().ambient_space().ngens()
if len(v) != d:
raise TypeError("Argument v (=%s) must have %s coordinates." %
(v, d))
if not isinstance(v, (list, tuple)):
raise TypeError(
"Argument v (= %s) must be a scheme point, list, or tuple."
% str(v))
# Make sure the coordinates all lie in the appropriate ring
v = Sequence(v, X.value_ring())
# Verify that the point satisfies the equations of X.
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, parent, fan_morphism, check=True):
r"""
See :class:`SchemeMorphism_polynomial_toric_variety` for documentation.
TESTS::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1)
sage: fan_morphism = FanMorphism( matrix(ZZ,[[1],[0]]), P1xP1.fan(), P1.fan() )
sage: from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety
sage: SchemeMorphism_fan_toric_variety(hom_set, fan_morphism)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
SchemeMorphism.__init__(self, parent)
if check and self.domain().fan() != fan_morphism.domain_fan():
raise ValueError(
'The fan morphism domain must be the fan of the domain.')
if check and self.codomain().fan() != fan_morphism.codomain_fan():
raise ValueError(
'The fan morphism codomain must be the fan of the codomain.')
self._fan_morphism = fan_morphism
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_affine` for details.
TESTS::
sage: from sage.schemes.affine.affine_point import SchemeMorphism_point_affine
sage: A3.<x,y,z> = AffineSpace(QQ, 3)
sage: SchemeMorphism_point_affine(A3(QQ), [1,2,3])
(1, 2, 3)
"""
SchemeMorphism.__init__(self, X)
if is_SchemeMorphism(v):
v = list(v)
if check:
# Verify that there are the right number of coords
d = self.codomain().ambient_space().ngens()
if len(v) != d:
raise TypeError("Argument v (=%s) must have %s coordinates." % (v, d))
if not isinstance(v, (list, tuple)):
raise TypeError("Argument v (= %s) must be a scheme point, list, or tuple." % str(v))
# Make sure the coordinates all lie in the appropriate ring
v = Sequence(v, X.value_ring())
# Verify that the point satisfies the equations of X.
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, parent, polys, check=True):
r"""
The Python constructor.
INPUT:
- ``parent`` -- Hom-set.
- ``polys`` -- anything that defines a point in the class.
- ``check`` -- Boolean. Whether or not to perform input checks.
(Default: ``True``)
EXAMPLES::
sage: P1.<x0,x1,x2> = ProjectiveSpace(QQ, 2)
sage: P2 = ProjectiveSpace(QQ, 3, 'y')
sage: T = ProductProjectiveSpaces([P1, P2])
sage: Q1 = P1(1, 1, 1)
sage: Q2 = P2(1, 2, 3, 4)
sage: T([Q1, Q2])
(1 : 1 : 1 , 1/4 : 1/2 : 3/4 : 1)
::
sage: T = ProductProjectiveSpaces([2, 2, 2], GF(5), 'x')
sage: T.point([1, 2, 3, 4, 5, 6, 7, 8, 9])
(2 : 4 : 1 , 4 : 0 : 1 , 3 : 2 : 1)
::
sage: T.<x,y,z,w> = ProductProjectiveSpaces([1, 1], GF(5))
sage: X = T.subscheme([x-y, z-2*w])
sage: X([1, 1, 2, 1])
(1 : 1 , 2 : 1)
"""
polys = copy(polys)
SchemeMorphism.__init__(self, parent)
if all(isinstance(P, SchemeMorphism_point) for P in polys):
if check:
Q = []
self._points = []
for i in range(len(polys)):
if polys[i].codomain() != parent.codomain().ambient_space()[i]:
raise ValueError("points must be in correct projective spaces")
Q += list(polys[i])
self._points.append(polys[i])
parent.codomain()._check_satisfies_equations(Q)
self._points = polys
else:
R = parent.codomain().ambient_space().base_ring()
polys = Sequence(polys, R)
N = parent.codomain().ambient_space().dimension_relative_components()
if check:
parent.codomain()._check_satisfies_equations(polys)
splitpolys=self.codomain().ambient_space()._factors(polys)
self._points = [parent.codomain().ambient_space()[i].point(splitpolys[i], check) for i in range(len(N))]
def __init__(self, parent, polys, check=True):
r"""
Create a new Jacobian element in Mumford representation.
INPUT: parent: the parent Homset polys: Mumford's `u` and
`v` polynomials check (default: True): if True, ensure that
polynomials define a divisor on the appropriate curve and are
reduced
.. warning::
Not for external use! Use ``J(K)([u, v])`` instead.
EXAMPLES::
sage: x = GF(37)['x'].gen()
sage: H = HyperellipticCurve(x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x)
sage: J = H.jacobian()(GF(37)); J
Set of rational points of Jacobian of Hyperelliptic Curve over
Finite Field of size 37 defined by
y^2 = x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x
::
sage: P1 = J(H.lift_x(2)); P1 # indirect doctest
(x + 35, y + 26)
sage: P1.parent()
Set of rational points of Jacobian of Hyperelliptic Curve over
Finite Field of size 37 defined by
y^2 = x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x
sage: type(P1)
<class 'sage.schemes.hyperelliptic_curves.jacobian_morphism.JacobianMorphism_divisor_class_field'>
"""
SchemeMorphism.__init__(self, parent)
if check:
C = parent.curve()
f, h = C.hyperelliptic_polynomials()
a, b = polys
if not (b**2 + h*b - f)%a == 0:
raise ValueError, \
"Argument polys (= %s) must be divisor on curve %s."%(
polys, C)
genus = C.genus()
if a.degree() > genus:
polys = cantor_reduction(a, b, f, h, genus)
self.__polys = polys
def __init__(self, parent, polys, check=True):
r"""
Create a new Jacobian element in Mumford representation.
INPUT: parent: the parent Homset polys: Mumford's `u` and
`v` polynomials check (default: True): if True, ensure that
polynomials define a divisor on the appropriate curve and are
reduced
.. warning::
Not for external use! Use ``J(K)([u, v])`` instead.
EXAMPLES::
sage: x = GF(37)['x'].gen()
sage: H = HyperellipticCurve(x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x)
sage: J = H.jacobian()(GF(37)); J
Set of rational points of Jacobian of Hyperelliptic Curve over
Finite Field of size 37 defined by
y^2 = x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x
::
sage: P1 = J(H.lift_x(2)); P1 # indirect doctest
(x + 35, y + 26)
sage: P1.parent()
Set of rational points of Jacobian of Hyperelliptic Curve over
Finite Field of size 37 defined by
y^2 = x^5 + 12*x^4 + 13*x^3 + 15*x^2 + 33*x
sage: type(P1)
<class 'sage.schemes.hyperelliptic_curves.jacobian_morphism.JacobianMorphism_divisor_class_field'>
"""
SchemeMorphism.__init__(self, parent)
if check:
C = parent.curve()
f, h = C.hyperelliptic_polynomials()
a, b = polys
if not (b**2 + h * b - f) % a == 0:
raise ValueError(
"Argument polys (= %s) must be divisor on curve %s." %
(polys, C))
genus = C.genus()
if a.degree() > genus:
polys = cantor_reduction(a, b, f, h, genus)
self.__polys = polys
def __init__(self, parent, fan_morphism, check=True):
r"""
See :class:`SchemeMorphism_polynomial_toric_variety` for documentation.
TESTS::
sage: P1xP1 = toric_varieties.P1xP1()
sage: P1 = toric_varieties.P1()
sage: hom_set = P1xP1.Hom(P1)
sage: fan_morphism = FanMorphism( matrix(ZZ,[[1],[0]]), P1xP1.fan(), P1.fan() )
sage: from sage.schemes.toric.morphism import SchemeMorphism_fan_toric_variety
sage: SchemeMorphism_fan_toric_variety(hom_set, fan_morphism)
Scheme morphism:
From: 2-d CPR-Fano toric variety covered by 4 affine patches
To: 1-d CPR-Fano toric variety covered by 2 affine patches
Defn: Defined by sending Rational polyhedral fan in 2-d lattice N
to Rational polyhedral fan in 1-d lattice N.
"""
SchemeMorphism.__init__(self, parent)
if check and self.domain().fan()!=fan_morphism.domain_fan():
raise ValueError('The fan morphism domain must be the fan of the domain.')
if check and self.codomain().fan()!=fan_morphism.codomain_fan():
raise ValueError('The fan morphism codomain must be the fan of the codomain.')
self._fan_morphism = fan_morphism
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_projective_ring` for details.
This function still normalizes points so that the rightmost non-zero coordinate is 1.
This is to maintain functionality with current
implementations of curves in projectives space (plane, conic, elliptic, etc).
The :class:`SchemeMorphism_point_projective_ring` is for general use.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0, 0, 0, 0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
::
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: X = P.subscheme([x^2-y*z])
sage: X([2, 2, 2])
(1 : 1 : 1)
::
sage: P = ProjectiveSpace(1, GF(7))
sage: Q=P([2, 1])
sage: Q[0].parent()
Finite Field of size 7
::
sage: P = ProjectiveSpace(QQ,1)
sage: P.point(Infinity)
(1 : 0)
sage: P(infinity)
(1 : 0)
::
sage: P = ProjectiveSpace(QQ,2)
sage: P(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
sage: P.point(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
from sage.rings.ring import CommutativeRing
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
else:
try:
if isinstance(v.parent(), CommutativeRing):
v = [v]
except AttributeError:
pass
if not isinstance(v, (list,tuple)):
raise TypeError("argument v (= %s) must be a scheme point, list, or tuple"%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(R(1))
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
c = v[last]
if c == R.one():
break
for j in range(last):
v[j] /= c
v[last] = R.one()
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, X, v, check=True):
"""
The Python constructor.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 1])
(0 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 0, 1])
Traceback (most recent call last):
...
TypeError: v (=[0, 0, 1]) must have 2 components
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
It is possible to avoid the possibly time-consuming checks, but be careful!! ::
sage: P = ProjectiveSpace(3, QQ)
sage: P.point([0,0,0,0], check=False)
(0 : 0 : 0 : 0)
::
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: X = P.subscheme([x^2-y*z])
sage: X([2, 2, 2])
(2 : 2 : 2)
::
sage: R.<t> = PolynomialRing(ZZ)
sage: P = ProjectiveSpace(1, R.quo(t^2+1))
sage: P([2*t, 1])
(2*tbar : 1)
::
sage: P = ProjectiveSpace(ZZ,1)
sage: P.point(Infinity)
(1 : 0)
sage: P(infinity)
(1 : 0)
::
sage: P = ProjectiveSpace(ZZ,2)
sage: P(Infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
sage: P.point(infinity)
Traceback (most recent call last):
...
ValueError: +Infinity not well defined in dimension > 1
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
from sage.rings.ring import CommutativeRing
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
else:
try:
if isinstance(v.parent(), CommutativeRing):
v = [v]
except AttributeError:
pass
if not isinstance(v, (list, tuple)):
raise TypeError("argument v (= %s) must be a scheme point, list, or tuple"%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(R(1))
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
self._coords = tuple(v)
def __init__(self, X, v, check=True):
"""
The Python constructor.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 1])
(0 : 1)
::
sage: P = ProjectiveSpace(1, ZZ)
sage: P([0, 0, 1])
Traceback (most recent call last):
...
TypeError: v (=[0, 0, 1]) must have 2 components
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
::
It is possible to avoid the possibly time consuming checks, but be careful!!
sage: P = ProjectiveSpace(3, QQ)
sage: P.point([0,0,0,0],check=False)
(0 : 0 : 0 : 0)
::
sage: P.<x, y, z> = ProjectiveSpace(2, ZZ)
sage: X=P.subscheme([x^2-y*z])
sage: X([2,2,2])
(2 : 2 : 2)
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
elif v is infinity:
v = [0] * (d)
v[1] = 1
if not isinstance(v,(list,tuple)):
raise TypeError("Argument v (= %s) must be a scheme point, list, or tuple."%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(1)
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
if isinstance(X.codomain().base_ring(), QuotientRing_generic):
lift_coords = [P.lift() for P in v]
else:
lift_coords = v
v = Sequence(lift_coords)
self._coords = v
def __init__(self, X, v, check=True):
"""
The Python constructor.
See :class:`SchemeMorphism_point_projective_ring` for details.
This function still normalized points so that the rightmost non-zero coordinate is 1. The is to maintain current functionality with current
implementations of curves in projectives space (plane, connic, elliptic, etc). The class:`SchemeMorphism_point_projective_ring` is for general use.
EXAMPLES::
sage: P = ProjectiveSpace(2, QQ)
sage: P(2, 3/5, 4)
(1/2 : 3/20 : 1)
::
sage: P = ProjectiveSpace(3, QQ)
sage: P(0,0,0,0)
Traceback (most recent call last):
...
ValueError: [0, 0, 0, 0] does not define a valid point since all entries are 0
::
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: X=P.subscheme([x^2-y*z])
sage: X([2,2,2])
(1 : 1 : 1)
"""
SchemeMorphism.__init__(self, X)
if check:
from sage.schemes.elliptic_curves.ell_point import EllipticCurvePoint_field
d = X.codomain().ambient_space().ngens()
if is_SchemeMorphism(v) or isinstance(v, EllipticCurvePoint_field):
v = list(v)
elif v is infinity:
v = [0] * (d)
v[1] = 1
if not isinstance(v,(list,tuple)):
raise TypeError("Argument v (= %s) must be a scheme point, list, or tuple."%str(v))
if len(v) != d and len(v) != d-1:
raise TypeError("v (=%s) must have %s components"%(v, d))
R = X.value_ring()
v = Sequence(v, R)
if len(v) == d-1: # very common special case
v.append(1)
n = len(v)
all_zero = True
for i in range(n):
last = n-1-i
if v[last]:
all_zero = False
c = v[last]
if c == R.one():
break
for j in range(last):
v[j] /= c
v[last] = R.one()
break
if all_zero:
raise ValueError("%s does not define a valid point since all entries are 0"%repr(v))
X.extended_codomain()._check_satisfies_equations(v)
self._coords = v