def __init__(self, n, R, names):
"""
EXAMPLES::
sage: AffineSpace(3, Zp(5), 'y')
Affine Space of dimension 3 over 5-adic Ring with capped relative precision 20
"""
AmbientSpace.__init__(self, n, R)
self._assign_names(names)
AffineScheme.__init__(self, self.coordinate_ring(), R)
def __init__(self, n, R=ZZ, names=None):
"""
EXAMPLES::
sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
"""
names = normalize_names(n + 1, names)
AmbientSpace.__init__(self, n, R)
self._assign_names(names)
def __init__(self, n, R, names):
"""
EXAMPLES::
sage: AffineSpace(3, Zp(5), 'y')
Affine Space of dimension 3 over 5-adic Ring with capped relative precision 20
"""
AmbientSpace.__init__(self, n, R)
self._assign_names(names)
AffineScheme.__init__(self, self.coordinate_ring(), R)
def __init__(self, n, R, names):
"""
EXAMPLES::
sage: AffineSpace(3, Zp(5), 'y')
Affine Space of dimension 3 over 5-adic Ring with capped relative precision 20
"""
names = normalize_names(n, names)
AmbientSpace.__init__(self, n, R)
self._assign_names(names)
def __init__(self, n, R=ZZ, names=None):
"""
EXAMPLES::
sage: ProjectiveSpace(3, Zp(5), 'y')
Projective Space of dimension 3 over 5-adic Ring with capped relative precision 20
"""
names = normalize_names(n+1, names)
AmbientSpace.__init__(self, n, R)
self._assign_names(names)
def __init__(self, N, R=QQ, names=None):
r"""
The Python constructor.
INPUT:
- ``N`` - a list or tuple of positive integers.
- ``R`` - a ring.
- ``names`` - a tuple or list of strings. This must either be a single variable name
or the complete list of variables.
EXAMPLES::
sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], QQ)
sage: T
Product of projective spaces P^2 x P^2 over Rational Field
sage: T.coordinate_ring()
Multivariate Polynomial Ring in x, y, z, u, v, w over Rational Field
sage: T[1].coordinate_ring()
Multivariate Polynomial Ring in u, v, w over Rational Field
::
sage: ProductProjectiveSpaces([1,1,1],ZZ, ['x', 'y', 'z', 'u', 'v', 'w'])
Product of projective spaces P^1 x P^1 x P^1 over Integer Ring
::
sage: T = ProductProjectiveSpaces([1, 1], QQ, 'z')
sage: T.coordinate_ring()
Multivariate Polynomial Ring in z0, z1, z2, z3 over Rational Field
"""
assert isinstance(N, (tuple, list))
N = [Integer(n) for n in N]
assert isinstance(R, CommutativeRing)
if len(N) < 2:
raise ValueError("must be at least two components for a product")
AmbientSpace.__init__(self, sum(N), R)
self._dims = N
start = 0
self._components = []
for i in range(len(N)):
self._components.append(
ProjectiveSpace(N[i], R, names[start:start + N[i] + 1]))
start += N[i] + 1
#Note that the coordinate ring should really be the tensor product of the component
#coordinate rings. But we just deal with them as multihomogeneous polynomial rings
self._coordinate_ring = PolynomialRing(R, sum(N) + len(N), names)
self._assign_names(names)
def __init__(self, N, R = QQ, names = None):
r"""
The Python constructor.
INPUT:
- ``N`` - a list or tuple of positive integers.
- ``R`` - a ring.
- ``names`` - a tuple or list of strings. This must either be a single variable name
or the complete list of variables.
EXAMPLES::
sage: T.<x,y,z,u,v,w> = ProductProjectiveSpaces([2, 2], QQ)
sage: T
Product of projective spaces P^2 x P^2 over Rational Field
sage: T.coordinate_ring()
Multivariate Polynomial Ring in x, y, z, u, v, w over Rational Field
sage: T[1].coordinate_ring()
Multivariate Polynomial Ring in u, v, w over Rational Field
::
sage: ProductProjectiveSpaces([1,1,1],ZZ, ['x', 'y', 'z', 'u', 'v', 'w'])
Product of projective spaces P^1 x P^1 x P^1 over Integer Ring
::
sage: T = ProductProjectiveSpaces([1, 1], QQ, 'z')
sage: T.coordinate_ring()
Multivariate Polynomial Ring in z0, z1, z2, z3 over Rational Field
"""
assert isinstance(N, (tuple, list))
N = [Integer(n) for n in N]
assert isinstance(R, CommutativeRing)
if len(N) < 2:
raise ValueError("must be at least two components for a product")
AmbientSpace.__init__(self, sum(N), R)
self._dims = N
start = 0
self._components = []
for i in range(len(N)):
self._components.append(ProjectiveSpace(N[i],R,names[start:start+N[i]+1]))
start += N[i]+1
#Note that the coordinate ring should really be the tensor product of the component
#coordinate rings. But we just deal with them as multihomogeneous polynomial rings
self._coordinate_ring = PolynomialRing(R,sum(N)+ len(N),names)
self._assign_names(names)
def __init__(self, n, R, names, ambient_projective_space, default_embedding_index):
"""
EXAMPLES::
sage: AffineSpace(3, Zp(5), 'y')
Affine Space of dimension 3 over 5-adic Ring with capped relative precision 20
"""
AmbientSpace.__init__(self, n, R)
self._assign_names(names)
AffineScheme.__init__(self, self.coordinate_ring(), R)
index = default_embedding_index
if index is not None:
index = int(index)
self._default_embedding_index = index
self._ambient_projective_space = ambient_projective_space