def __init__(self, domain, codomain):
"""
The Python constructor
EXAMPLES::
sage: R = QQ['x']['y']['s','t']['X']
sage: p = R.random_element()
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: g = f.section()
sage: g(f(p)) == p
True
::
sage: R = QQ['a','b','x','y']
sage: S = ZZ['a','b']['x','z']
sage: from sage.rings.polynomial.flatten import UnflatteningMorphism
sage: UnflatteningMorphism(R, S)
Traceback (most recent call last):
...
ValueError: rings must have same base ring
::
sage: R = QQ['a','b','x','y']
sage: S = QQ['a','b']['x','z','w']
sage: from sage.rings.polynomial.flatten import UnflatteningMorphism
sage: UnflatteningMorphism(R, S)
Traceback (most recent call last):
...
ValueError: rings must have the same number of variables
"""
if not is_MPolynomialRing(domain):
raise ValueError("domain should be a multivariate polynomial ring")
if not is_PolynomialRing(codomain) and not is_MPolynomialRing(
codomain):
raise ValueError("codomain should be a polynomial ring")
ring = codomain
intermediate_rings = []
while True:
is_polynomial_ring = is_PolynomialRing(ring)
if not (is_polynomial_ring or is_MPolynomialRing(ring)):
break
intermediate_rings.append((ring, is_polynomial_ring))
ring = ring.base_ring()
if domain.base_ring() != intermediate_rings[-1][0].base_ring():
raise ValueError("rings must have same base ring")
if domain.ngens() != sum([R.ngens() for R, _ in intermediate_rings]):
raise ValueError("rings must have the same number of variables")
self._intermediate_rings = intermediate_rings
hom = Homset(domain, codomain, base=ring, check=False)
Morphism.__init__(self, hom)
self._repr_type_str = 'Unflattening'
def __init__(self, domain):
"""
The Python constructor
EXAMPLES::
sage: R = ZZ['a', 'b', 'c']['x', 'y', 'z']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: FlatteningMorphism(R)
Flattening morphism:
From: Multivariate Polynomial Ring in x, y, z over Multivariate Polynomial Ring in a, b, c over Integer Ring
To: Multivariate Polynomial Ring in a, b, c, x, y, z over Integer Ring
::
sage: R = ZZ['a']['b']['c']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: FlatteningMorphism(R)
Flattening morphism:
From: Univariate Polynomial Ring in c over Univariate Polynomial Ring in b over Univariate Polynomial Ring in a over Integer Ring
To: Multivariate Polynomial Ring in a, b, c over Integer Ring
::
sage: R = ZZ['a']['a','b']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: FlatteningMorphism(R)
Flattening morphism:
From: Multivariate Polynomial Ring in a, b over Univariate Polynomial Ring in a over Integer Ring
To: Multivariate Polynomial Ring in a, a0, b over Integer Ring
::
sage: K.<v> = NumberField(x^3 - 2)
sage: R = K['x','y']['a','b']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f(R('v*a*x^2 + b^2 + 1/v*y'))
(v)*x^2*a + b^2 + (1/2*v^2)*y
::
sage: R = QQbar['x','y']['a','b']
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f(R('QQbar(sqrt(2))*a*x^2 + b^2 + QQbar(I)*y'))
1.414213562373095?*x^2*a + b^2 + I*y
::
sage: R.<z> = PolynomialRing(QQbar,1)
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f.domain(), f.codomain()
(Multivariate Polynomial Ring in z over Algebraic Field,
Multivariate Polynomial Ring in z over Algebraic Field)
::
sage: R.<z> = PolynomialRing(QQbar)
sage: from sage.rings.polynomial.flatten import FlatteningMorphism
sage: f = FlatteningMorphism(R)
sage: f.domain(), f.codomain()
(Univariate Polynomial Ring in z over Algebraic Field,
Univariate Polynomial Ring in z over Algebraic Field)
TESTS::
sage: Pol = QQ['x']['x0']['x']
sage: fl = FlatteningMorphism(Pol)
sage: fl
Flattening morphism:
From: Univariate Polynomial Ring in x over Univariate Polynomial Ring in x0 over Univariate Polynomial Ring in x over Rational Field
To: Multivariate Polynomial Ring in x, x0, x1 over Rational Field
sage: p = Pol([[[1,2],[3,4]],[[5,6],[7,8]]])
sage: fl.section()(fl(p)) == p
True
"""
if not is_PolynomialRing(domain) and not is_MPolynomialRing(domain):
raise ValueError("domain should be a polynomial ring")
ring = domain
variables = []
intermediate_rings = []
while is_PolynomialRing(ring) or is_MPolynomialRing(ring):
intermediate_rings.append(ring)
v = ring.variable_names()
variables.extend(reversed(v))
ring = ring.base_ring()
self._intermediate_rings = intermediate_rings
variables.reverse()
for i, a in enumerate(variables):
if a in variables[:i]:
for index in itertools.count():
b = a + str(index)
if b not in variables: # not just variables[:i]!
break
variables[i] = b
if is_MPolynomialRing(domain):
codomain = PolynomialRing(ring, variables, len(variables))
else:
codomain = PolynomialRing(ring, variables)
hom = Homset(domain, codomain, base=ring, check=False)
Morphism.__init__(self, hom)
self._repr_type_str = 'Flattening'
def _Hom_(X, Y, category):
# here we overwrite the Rings._Hom_ implementation
return Homset(X, Y, category=category)
