def dual(self):
r"""
Return the projective dual of the given subscheme of projective space.
INPUT:
- ``X`` -- A subscheme of projective space. At present, ``X`` is
required to be an irreducible and reduced hypersurface defined
over `\QQ` or a finite field.
OUTPUT:
- The dual of ``X`` as a subscheme of the dual projective space.
EXAMPLES:
The dual of a smooth conic in the plane is also a smooth conic::
sage: R.<x, y, z> = QQ[]
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: I = R.ideal(x^2 + y^2 + z^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
y0^2 + y1^2 + y2^2
The dual of the twisted cubic curve in projective 3-space is a singular
quartic surface. In the following example, we compute the dual of this
surface, which by double duality is equal to the twisted cubic itself.
The output is the twisted cubic as an intersection of three quadrics::
sage: R.<x, y, z, w> = QQ[]
sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 3 over
Rational Field defined by:
y2^2 - y1*y3,
y1*y2 - y0*y3,
y1^2 - y0*y2
The singular locus of the quartic surface in the last example
is itself supported on a twisted cubic::
sage: X.Jacobian().radical()
Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
Polynomial Ring in x, y, z, w over Rational Field
An example over a finite field::
sage: R = PolynomialRing(GF(61), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over
Finite Field of size 61 defined by:
y0^2 - 30*y1^2 - 20*y2^2
TESTS::
sage: R = PolynomialRing(Qp(3), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Traceback (most recent call last):
...
NotImplementedError: base ring must be QQ or a finite field
"""
from sage.libs.singular.function_factory import ff
K = self.base_ring()
if not (is_RationalField(K) or is_FiniteField(K)):
raise NotImplementedError("base ring must be QQ or a finite field")
I = self.defining_ideal()
m = I.ngens()
n = I.ring().ngens() - 1
if (m != 1 or (n < 1) or I.is_zero() or I.is_trivial()
or not I.is_prime()):
raise NotImplementedError("At the present, the method is only"
" implemented for irreducible and"
" reduced hypersurfaces and the given"
" list of generators for the ideal must"
" have exactly one element.")
R = PolynomialRing(K, 'x', n + 1)
from sage.schemes.projective.projective_space import ProjectiveSpace
Pd = ProjectiveSpace(n, K, 'y')
Rd = Pd.coordinate_ring()
x = R.variable_names()
y = Rd.variable_names()
S = PolynomialRing(K, x + y + ('t', ))
if S.has_coerce_map_from(I.ring()):
T = PolynomialRing(K, 'w', n + 1)
I_S = (I.change_ring(T)).change_ring(S)
else:
I_S = I.change_ring(S)
f_S = I_S.gens()[0]
z = S.gens()
J = I_S
for i in range(n + 1):
J = J + S.ideal(z[-1] * f_S.derivative(z[i]) - z[i + n + 1])
sat = ff.elim__lib.sat
max_ideal = S.ideal(z[n + 1:2 * n + 2])
J_sat_gens = sat(J, max_ideal)[0]
J_sat = S.ideal(J_sat_gens)
L = J_sat.elimination_ideal(z[0:n + 1] + (z[-1], ))
return Pd.subscheme(L.change_ring(Rd))
class DifferentialWeylAlgebra(Algebra, UniqueRepresentation):
r"""
The differential Weyl algebra of a polynomial ring.
Let `R` be a commutative ring. The (differential) Weyl algebra `W` is
the algebra generated by `x_1, x_2, \ldots x_n, \partial_{x_1},
\partial_{x_2}, \ldots, \partial_{x_n}` subject to the relations:
`[x_i, x_j] = 0`, `[\partial_{x_i}, \partial_{x_j}] = 0`, and
`\partial_{x_i} x_j = x_j \partial_{x_i} + \delta_{ij}`. Therefore
`\partial_{x_i}` is acting as the partial differential operator on `x_i`.
The Weyl algebra can also be constructed as an iterated Ore extension
of the polynomial ring `R[x_1, x_2, \ldots, x_n]` by adding `x_i` at
each step. It can also be seen as a quantization of the symmetric algebra
`Sym(V)`, where `V` is a finite dimensional vector space over a field
of characteristic zero, by using a modified Groenewold-Moyal
product in the symmetric algebra.
The Weyl algebra (even for `n = 1`) over a field of characteristic 0
has many interesting properties.
- It's a non-commutative domain.
- It's a simple ring (but not in positive characteristic) that is not
a matrix ring over a division ring.
- It has no finite-dimensional representations.
- It's a quotient of the universal enveloping algebra of the
Heisenberg algebra `\mathfrak{h}_n`.
REFERENCES:
- :wikipedia:`Weyl_algebra`
INPUT:
- ``R`` -- a (polynomial) ring
- ``names`` -- (default: ``None``) if ``None`` and ``R`` is a
polynomial ring, then the variable names correspond to
those of ``R``; otherwise if ``names`` is specified, then ``R``
is the base ring
EXAMPLES:
There are two ways to create a Weyl algebra, the first is from
a polynomial ring::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R); W
Differential Weyl algebra of polynomials in x, y, z over Rational Field
We can call ``W.inject_variables()`` to give the polynomial ring
variables, now as elements of ``W``, and the differentials::
sage: W.inject_variables()
Defining x, y, z, dx, dy, dz
sage: (dx * dy * dz) * (x^2 * y * z + x * z * dy + 1)
x*z*dx*dy^2*dz + z*dy^2*dz + x^2*y*z*dx*dy*dz + dx*dy*dz
+ x*dx*dy^2 + 2*x*y*z*dy*dz + dy^2 + x^2*z*dx*dz + x^2*y*dx*dy
+ 2*x*z*dz + 2*x*y*dy + x^2*dx + 2*x
Or directly by specifying a base ring and variable names::
sage: W.<a,b> = DifferentialWeylAlgebra(QQ); W
Differential Weyl algebra of polynomials in a, b over Rational Field
.. TODO::
Implement the :meth:`graded_algebra` as a polynomial ring once
they are considered to be graded rings (algebras).
"""
@staticmethod
def __classcall__(cls, R, names=None):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: W1.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: W2 = DifferentialWeylAlgebra(QQ['x,y,z'])
sage: W1 is W2
True
"""
if isinstance(R, (PolynomialRing_general, MPolynomialRing_generic)):
if names is None:
names = R.variable_names()
R = R.base_ring()
elif names is None:
raise ValueError("the names must be specified")
elif R not in Rings().Commutative():
raise TypeError("argument R must be a commutative ring")
return super(DifferentialWeylAlgebra, cls).__classcall__(cls, R, names)
def __init__(self, R, names=None):
r"""
Initialize ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: TestSuite(W).run()
"""
self._n = len(names)
self._poly_ring = PolynomialRing(R, names)
names = names + tuple('d' + n for n in names)
if len(names) != self._n * 2:
raise ValueError("variable names cannot differ by a leading 'd'")
# TODO: Make this into a filtered algebra under the natural grading of
# x_i and dx_i have degree 1
# Filtered is not included because it is a supercategory of super
if R.is_field():
cat = AlgebrasWithBasis(R).NoZeroDivisors().Super()
else:
cat = AlgebrasWithBasis(R).Super()
Algebra.__init__(self, R, names, category=cat)
def _repr_(self):
r"""
Return a string representation of ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: DifferentialWeylAlgebra(R)
Differential Weyl algebra of polynomials in x, y, z over Rational Field
"""
poly_gens = ', '.join(repr(x) for x in self.gens()[:self._n])
return "Differential Weyl algebra of polynomials in {} over {}".format(
poly_gens, self.base_ring())
def _element_constructor_(self, x):
"""
Construct an element of ``self`` from ``x``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: a = W(2); a
2
sage: a.parent() is W
True
sage: W(x^2 - y*z)
-y*z + x^2
"""
t = tuple([0]*(self._n))
if x in self.base_ring():
if x == self.base_ring().zero():
return self.zero()
return self.element_class(self, {(t, t): x})
if isinstance(x, DifferentialWeylAlgebraElement):
R = self.base_ring()
if x.parent().base_ring() is R:
return self.element_class(self, dict(x))
zero = R.zero()
return self.element_class(self, {i: R(c) for i,c in x if R(c) != zero})
x = self._poly_ring(x)
return self.element_class(self, {(tuple(m), t): c
for m,c in x.dict().iteritems()})
def _coerce_map_from_(self, R):
"""
Return data which determines if there is a coercion map
from ``R`` to ``self``.
If such a map exists, the output could be a map, callable,
or ``True``, which constructs a generic map. Otherwise the output
must be ``False`` or ``None``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W._coerce_map_from_(R)
True
sage: W._coerce_map_from_(QQ)
True
sage: W._coerce_map_from_(ZZ['x'])
True
Order of the names matter::
sage: Wp = DifferentialWeylAlgebra(QQ['x,z,y'])
sage: W.has_coerce_map_from(Wp)
False
sage: Wp.has_coerce_map_from(W)
False
Zero coordinates are handled appropriately::
sage: R.<x,y,z> = ZZ[]
sage: W3 = DifferentialWeylAlgebra(GF(3)['x,y,z'])
sage: W3.has_coerce_map_from(R)
True
sage: W.<x,y,z> = DifferentialWeylAlgebra(ZZ)
sage: W3.has_coerce_map_from(W)
True
sage: W3(3*x + y)
y
"""
if self._poly_ring.has_coerce_map_from(R):
return True
if isinstance(R, DifferentialWeylAlgebra):
return ( R.variable_names() == self.variable_names()
and self.base_ring().has_coerce_map_from(R.base_ring()) )
return super(DifferentialWeylAlgebra, self)._coerce_map_from_(R)
def degree_on_basis(self, i):
"""
Return the degree of the basis element indexed by ``i``.
EXAMPLES::
sage: W.<a,b> = DifferentialWeylAlgebra(QQ)
sage: W.degree_on_basis( ((1, 3, 2), (0, 1, 3)) )
10
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: elt = y*dy - (3*x - z)*dx
sage: elt.degree()
2
"""
return sum(i[0]) + sum(i[1])
def polynomial_ring(self):
"""
Return the associated polynomial ring of ``self``.
EXAMPLES::
sage: W.<a,b> = DifferentialWeylAlgebra(QQ)
sage: W.polynomial_ring()
Multivariate Polynomial Ring in a, b over Rational Field
::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.polynomial_ring() == R
True
"""
return self._poly_ring
@cached_method
def basis(self):
"""
Return a basis of ``self``.
EXAMPLES::
sage: W.<x,y> = DifferentialWeylAlgebra(QQ)
sage: B = W.basis()
sage: it = iter(B)
sage: [next(it) for i in range(20)]
[1, x, y, dx, dy, x^2, x*y, x*dx, x*dy, y^2, y*dx, y*dy,
dx^2, dx*dy, dy^2, x^3, x^2*y, x^2*dx, x^2*dy, x*y^2]
sage: dx, dy = W.differentials()
sage: (dx*x).monomials()
[1, x*dx]
sage: B[(x*y).support()[0]]
x*y
sage: sorted((dx*x).monomial_coefficients().items())
[(((0, 0), (0, 0)), 1), (((1, 0), (1, 0)), 1)]
"""
n = self._n
from sage.combinat.integer_lists.nn import IntegerListsNN
from sage.categories.cartesian_product import cartesian_product
elt_map = lambda u : (tuple(u[:n]), tuple(u[n:]))
I = IntegerListsNN(length=2*n, element_constructor=elt_map)
one = self.base_ring().one()
f = lambda x: self.element_class(self, {(x[0], x[1]): one})
return Family(I, f, name="basis map")
@cached_method
def algebra_generators(self):
"""
Return the algebra generators of ``self``.
.. SEEALSO::
:meth:`variables`, :meth:`differentials`
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.algebra_generators()
Finite family {'dz': dz, 'dx': dx, 'dy': dy, 'y': y, 'x': x, 'z': z}
"""
d = {x: self.gen(i) for i,x in enumerate(self.variable_names())}
return Family(self.variable_names(), lambda x: d[x])
@cached_method
def variables(self):
"""
Return the variables of ``self``.
.. SEEALSO::
:meth:`algebra_generators`, :meth:`differentials`
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: W.variables()
Finite family {'y': y, 'x': x, 'z': z}
"""
N = self.variable_names()[:self._n]
d = {x: self.gen(i) for i,x in enumerate(N) }
return Family(N, lambda x: d[x])
@cached_method
def differentials(self):
"""
Return the differentials of ``self``.
.. SEEALSO::
:meth:`algebra_generators`, :meth:`variables`
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: W.differentials()
Finite family {'dz': dz, 'dx': dx, 'dy': dy}
"""
N = self.variable_names()[self._n:]
d = {x: self.gen(self._n+i) for i,x in enumerate(N) }
return Family(N, lambda x: d[x])
def gen(self, i):
"""
Return the ``i``-th generator of ``self``.
.. SEEALSO::
:meth:`algebra_generators`
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: [W.gen(i) for i in range(6)]
[x, y, z, dx, dy, dz]
"""
P = [0] * self._n
D = [0] * self._n
if i < self._n:
P[i] = 1
else:
D[i-self._n] = 1
return self.element_class(self, {(tuple(P), tuple(D)): self.base_ring().one()} )
def ngens(self):
"""
Return the number of generators of ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.ngens()
6
"""
return self._n*2
@cached_method
def one(self):
"""
Return the multiplicative identity element `1`.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.one()
1
"""
t = tuple([0]*self._n)
return self.element_class( self, {(t, t): self.base_ring().one()} )
@cached_method
def zero(self):
"""
Return the additive identity element `0`.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.zero()
0
"""
return self.element_class(self, {})
Element = DifferentialWeylAlgebraElement
class DifferentialWeylAlgebra(Algebra, UniqueRepresentation):
r"""
The differential Weyl algebra of a polynomial ring.
Let `R` be a commutative ring. The (differential) Weyl algebra `W` is
the algebra generated by `x_1, x_2, \ldots x_n, \partial_{x_1},
\partial_{x_2}, \ldots, \partial_{x_n}` subject to the relations:
`[x_i, x_j] = 0`, `[\partial_{x_i}, \partial_{x_j}] = 0`, and
`\partial_{x_i} x_j = x_j \partial_{x_i} + \delta_{ij}`. Therefore
`\partial_{x_i}` is acting as the partial differential operator on `x_i`.
The Weyl algebra can also be constructed as an iterated Ore extension
of the polynomial ring `R[x_1, x_2, \ldots, x_n]` by adding `x_i` at
each step. It can also be seen as a quantization of the symmetric algebra
`Sym(V)`, where `V` is a finite dimensional vector space over a field
of characteristic zero, by using a modified Groenewold-Moyal
product in the symmetric algebra.
The Weyl algebra (even for `n = 1`) over a field of characteristic 0
has many interesting properties.
- It's a non-commutative domain.
- It's a simple ring (but not in positive characteristic) that is not
a matrix ring over a division ring.
- It has no finite-dimensional representations.
- It's a quotient of the universal enveloping algebra of the
Heisenberg algebra `\mathfrak{h}_n`.
REFERENCES:
- :wikipedia:`Weyl_algebra`
INPUT:
- ``R`` -- a (polynomial) ring
- ``names`` -- (default: ``None``) if ``None`` and ``R`` is a
polynomial ring, then the variable names correspond to
those of ``R``; otherwise if ``names`` is specified, then ``R``
is the base ring
EXAMPLES:
There are two ways to create a Weyl algebra, the first is from
a polynomial ring::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R); W
Differential Weyl algebra of polynomials in x, y, z over Rational Field
We can call ``W.inject_variables()`` to give the polynomial ring
variables, now as elements of ``W``, and the differentials::
sage: W.inject_variables()
Defining x, y, z, dx, dy, dz
sage: (dx * dy * dz) * (x^2 * y * z + x * z * dy + 1)
x*z*dx*dy^2*dz + z*dy^2*dz + x^2*y*z*dx*dy*dz + dx*dy*dz
+ x*dx*dy^2 + 2*x*y*z*dy*dz + dy^2 + x^2*z*dx*dz + x^2*y*dx*dy
+ 2*x*z*dz + 2*x*y*dy + x^2*dx + 2*x
Or directly by specifying a base ring and variable names::
sage: W.<a,b> = DifferentialWeylAlgebra(QQ); W
Differential Weyl algebra of polynomials in a, b over Rational Field
.. TODO::
Implement the :meth:`graded_algebra` as a polynomial ring once
they are considered to be graded rings (algebras).
"""
@staticmethod
def __classcall__(cls, R, names=None):
"""
Normalize input to ensure a unique representation.
EXAMPLES::
sage: W1.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: W2 = DifferentialWeylAlgebra(QQ['x,y,z'])
sage: W1 is W2
True
"""
if isinstance(R, (PolynomialRing_general, MPolynomialRing_generic)):
if names is None:
names = R.variable_names()
R = R.base_ring()
elif names is None:
raise ValueError("the names must be specified")
elif R not in Rings().Commutative():
raise TypeError("argument R must be a commutative ring")
return super(DifferentialWeylAlgebra, cls).__classcall__(cls, R, names)
def __init__(self, R, names=None):
r"""
Initialize ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: TestSuite(W).run()
"""
self._n = len(names)
self._poly_ring = PolynomialRing(R, names)
names = names + tuple('d' + n for n in names)
if len(names) != self._n * 2:
raise ValueError("variable names cannot differ by a leading 'd'")
# TODO: Make this into a filtered algebra under the natural grading of
# x_i and dx_i have degree 1
# Filtered is not included because it is a supercategory of super
if R.is_field():
cat = AlgebrasWithBasis(R).NoZeroDivisors().Super()
else:
cat = AlgebrasWithBasis(R).Super()
Algebra.__init__(self, R, names, category=cat)
def _repr_(self):
r"""
Return a string representation of ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: DifferentialWeylAlgebra(R)
Differential Weyl algebra of polynomials in x, y, z over Rational Field
"""
poly_gens = ', '.join(repr(x) for x in self.gens()[:self._n])
return "Differential Weyl algebra of polynomials in {} over {}".format(
poly_gens, self.base_ring())
def _element_constructor_(self, x):
"""
Construct an element of ``self`` from ``x``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: a = W(2); a
2
sage: a.parent() is W
True
sage: W(x^2 - y*z)
-y*z + x^2
"""
t = tuple([0]*(self._n))
if x in self.base_ring():
if x == self.base_ring().zero():
return self.zero()
return self.element_class(self, {(t, t): x})
if isinstance(x, DifferentialWeylAlgebraElement):
R = self.base_ring()
if x.parent().base_ring() is R:
return self.element_class(self, dict(x))
zero = R.zero()
return self.element_class(self, {i: R(c) for i,c in x if R(c) != zero})
x = self._poly_ring(x)
return self.element_class(self, {(tuple(m), t): c
for m,c in six.iteritems(x.dict())})
def _coerce_map_from_(self, R):
"""
Return data which determines if there is a coercion map
from ``R`` to ``self``.
If such a map exists, the output could be a map, callable,
or ``True``, which constructs a generic map. Otherwise the output
must be ``False`` or ``None``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W._coerce_map_from_(R)
True
sage: W._coerce_map_from_(QQ)
True
sage: W._coerce_map_from_(ZZ['x'])
True
Order of the names matter::
sage: Wp = DifferentialWeylAlgebra(QQ['x,z,y'])
sage: W.has_coerce_map_from(Wp)
False
sage: Wp.has_coerce_map_from(W)
False
Zero coordinates are handled appropriately::
sage: R.<x,y,z> = ZZ[]
sage: W3 = DifferentialWeylAlgebra(GF(3)['x,y,z'])
sage: W3.has_coerce_map_from(R)
True
sage: W.<x,y,z> = DifferentialWeylAlgebra(ZZ)
sage: W3.has_coerce_map_from(W)
True
sage: W3(3*x + y)
y
"""
if self._poly_ring.has_coerce_map_from(R):
return True
if isinstance(R, DifferentialWeylAlgebra):
return ( R.variable_names() == self.variable_names()
and self.base_ring().has_coerce_map_from(R.base_ring()) )
return super(DifferentialWeylAlgebra, self)._coerce_map_from_(R)
def degree_on_basis(self, i):
"""
Return the degree of the basis element indexed by ``i``.
EXAMPLES::
sage: W.<a,b> = DifferentialWeylAlgebra(QQ)
sage: W.degree_on_basis( ((1, 3, 2), (0, 1, 3)) )
10
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: dx,dy,dz = W.differentials()
sage: elt = y*dy - (3*x - z)*dx
sage: elt.degree()
2
"""
return sum(i[0]) + sum(i[1])
def polynomial_ring(self):
"""
Return the associated polynomial ring of ``self``.
EXAMPLES::
sage: W.<a,b> = DifferentialWeylAlgebra(QQ)
sage: W.polynomial_ring()
Multivariate Polynomial Ring in a, b over Rational Field
::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.polynomial_ring() == R
True
"""
return self._poly_ring
@cached_method
def basis(self):
"""
Return a basis of ``self``.
EXAMPLES::
sage: W.<x,y> = DifferentialWeylAlgebra(QQ)
sage: B = W.basis()
sage: it = iter(B)
sage: [next(it) for i in range(20)]
[1, x, y, dx, dy, x^2, x*y, x*dx, x*dy, y^2, y*dx, y*dy,
dx^2, dx*dy, dy^2, x^3, x^2*y, x^2*dx, x^2*dy, x*y^2]
sage: dx, dy = W.differentials()
sage: (dx*x).monomials()
[1, x*dx]
sage: B[(x*y).support()[0]]
x*y
sage: sorted((dx*x).monomial_coefficients().items())
[(((0, 0), (0, 0)), 1), (((1, 0), (1, 0)), 1)]
"""
n = self._n
from sage.combinat.integer_lists.nn import IntegerListsNN
elt_map = lambda u : (tuple(u[:n]), tuple(u[n:]))
I = IntegerListsNN(length=2*n, element_constructor=elt_map)
one = self.base_ring().one()
f = lambda x: self.element_class(self, {(x[0], x[1]): one})
return Family(I, f, name="basis map")
@cached_method
def algebra_generators(self):
"""
Return the algebra generators of ``self``.
.. SEEALSO::
:meth:`variables`, :meth:`differentials`
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.algebra_generators()
Finite family {'dz': dz, 'dx': dx, 'dy': dy, 'y': y, 'x': x, 'z': z}
"""
d = {x: self.gen(i) for i,x in enumerate(self.variable_names())}
return Family(self.variable_names(), lambda x: d[x])
@cached_method
def variables(self):
"""
Return the variables of ``self``.
.. SEEALSO::
:meth:`algebra_generators`, :meth:`differentials`
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: W.variables()
Finite family {'y': y, 'x': x, 'z': z}
"""
N = self.variable_names()[:self._n]
d = {x: self.gen(i) for i,x in enumerate(N) }
return Family(N, lambda x: d[x])
@cached_method
def differentials(self):
"""
Return the differentials of ``self``.
.. SEEALSO::
:meth:`algebra_generators`, :meth:`variables`
EXAMPLES::
sage: W.<x,y,z> = DifferentialWeylAlgebra(QQ)
sage: W.differentials()
Finite family {'dz': dz, 'dx': dx, 'dy': dy}
"""
N = self.variable_names()[self._n:]
d = {x: self.gen(self._n+i) for i,x in enumerate(N) }
return Family(N, lambda x: d[x])
def gen(self, i):
"""
Return the ``i``-th generator of ``self``.
.. SEEALSO::
:meth:`algebra_generators`
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: [W.gen(i) for i in range(6)]
[x, y, z, dx, dy, dz]
"""
P = [0] * self._n
D = [0] * self._n
if i < self._n:
P[i] = 1
else:
D[i-self._n] = 1
return self.element_class(self, {(tuple(P), tuple(D)): self.base_ring().one()} )
def ngens(self):
"""
Return the number of generators of ``self``.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.ngens()
6
"""
return self._n*2
@cached_method
def one(self):
"""
Return the multiplicative identity element `1`.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.one()
1
"""
t = tuple([0]*self._n)
return self.element_class( self, {(t, t): self.base_ring().one()} )
@cached_method
def zero(self):
"""
Return the additive identity element `0`.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: W = DifferentialWeylAlgebra(R)
sage: W.zero()
0
"""
return self.element_class(self, {})
Element = DifferentialWeylAlgebraElement
def dual(self):
r"""
Return the projective dual of the given subscheme of projective space.
INPUT:
- ``X`` -- A subscheme of projective space. At present, ``X`` is
required to be an irreducible and reduced hypersurface defined
over `\QQ` or a finite field.
OUTPUT:
- The dual of ``X`` as a subscheme of the dual projective space.
EXAMPLES:
The dual of a smooth conic in the plane is also a smooth conic::
sage: R.<x, y, z> = QQ[]
sage: P.<x, y, z> = ProjectiveSpace(2, QQ)
sage: I = R.ideal(x^2 + y^2 + z^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
y0^2 + y1^2 + y2^2
The dual of the twisted cubic curve in projective 3-space is a singular
quartic surface. In the following example, we compute the dual of this
surface, which by double duality is equal to the twisted cubic itself.
The output is the twisted cubic as an intersection of three quadrics::
sage: R.<x, y, z, w> = QQ[]
sage: P.<x, y, z, w> = ProjectiveSpace(3, QQ)
sage: I = R.ideal(y^2*z^2 - 4*x*z^3 - 4*y^3*w + 18*x*y*z*w - 27*x^2*w^2)
sage: X = P.subscheme(I)
sage: X.dual()
Closed subscheme of Projective Space of dimension 3 over
Rational Field defined by:
y2^2 - y1*y3,
y1*y2 - y0*y3,
y1^2 - y0*y2
The singular locus of the quartic surface in the last example
is itself supported on a twisted cubic::
sage: X.Jacobian().radical()
Ideal (z^2 - 3*y*w, y*z - 9*x*w, y^2 - 3*x*z) of Multivariate
Polynomial Ring in x, y, z, w over Rational Field
An example over a finite field::
sage: R = PolynomialRing(GF(61), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Closed subscheme of Projective Space of dimension 2 over
Finite Field of size 61 defined by:
y0^2 - 30*y1^2 - 20*y2^2
TESTS::
sage: R = PolynomialRing(Qp(3), 'a,b,c')
sage: P.<a, b, c> = ProjectiveSpace(2, R.base_ring())
sage: X = P.subscheme(R.ideal(a*a+2*b*b+3*c*c))
sage: X.dual()
Traceback (most recent call last):
...
NotImplementedError: base ring must be QQ or a finite field
"""
from sage.libs.singular.function_factory import ff
K = self.base_ring()
if not(is_RationalField(K) or is_FiniteField(K)):
raise NotImplementedError("base ring must be QQ or a finite field")
I = self.defining_ideal()
m = I.ngens()
n = I.ring().ngens() - 1
if (m != 1 or (n < 1) or I.is_zero()
or I.is_trivial() or not I.is_prime()):
raise NotImplementedError("At the present, the method is only"
" implemented for irreducible and"
" reduced hypersurfaces and the given"
" list of generators for the ideal must"
" have exactly one element.")
R = PolynomialRing(K, 'x', n + 1)
from sage.schemes.projective.projective_space import ProjectiveSpace
Pd = ProjectiveSpace(n, K, 'y')
Rd = Pd.coordinate_ring()
x = R.variable_names()
y = Rd.variable_names()
S = PolynomialRing(K, x + y + ('t',))
if S.has_coerce_map_from(I.ring()):
T = PolynomialRing(K, 'w', n + 1)
I_S = (I.change_ring(T)).change_ring(S)
else:
I_S = I.change_ring(S)
f_S = I_S.gens()[0]
z = S.gens()
J = I_S
for i in range(n + 1):
J = J + S.ideal(z[-1] * f_S.derivative(z[i]) - z[i + n + 1])
sat = ff.elim__lib.sat
max_ideal = S.ideal(z[n + 1: 2 * n + 2])
J_sat_gens = sat(J, max_ideal)[0]
J_sat = S.ideal(J_sat_gens)
L = J_sat.elimination_ideal(z[0: n + 1] + (z[-1],))
return Pd.subscheme(L.change_ring(Rd))
