def WeierstrassForm(polynomial, variables=None, transformation=False):
r"""
Return the Weierstrass form of an elliptic curve inside either
inside a toric surface or $\mathbb{P}^3$.
INPUT:
- ``polynomial`` -- either a polynomial or a list of polynomials
defining the elliptic curve.
A single polynomial can be either a cubic, a biquadric, or the
hypersurface in `\mathbb{P}^2[1,1,2]`. In this case the
equation need not be in any standard form, only its Newton
polyhedron is used.
If two polynomials are passed, they must both be quadratics in
`\mathbb{P}^3`.
- ``variables`` -- a list of variables of the parent polynomial
ring or ``None`` (default). In the latter case, all variables
are taken to be polynomial ring variables. If a subset of
polynomial ring variables are given, the Weierstrass form is
determined over the function field generated by the remaining
variables.
- ``transformation`` -- boolean (default: ``False``). Whether to
return the new variables that bring ``polynomial`` into
Weierstrass form.
OUTPUT:
The pair of coefficients `(f,g)` of the Weierstrass form `y^2 =
x^3 + f x + g` of the hypersurface equation.
If ``transformation=True``, a triple `(X,Y,Z)` of polynomials
defining a rational map of the toric hypersurface or complete
intersection in `\mathbb{P}^3` to its Weierstrass form in
`\mathbb{P}^2[2,3,1]` is returned.
That is, the triple satisfies
.. MATH::
Y^2 = X^3 + f X Z^4 + g Z^6
when restricted to the toric hypersurface or complete intersection.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: f, g = WeierstrassForm(cubic); (f, g)
(0, -27/4)
Same in inhomogeneous coordinates::
sage: R.<x,y> = QQ[]
sage: cubic = x^3 + y^3 + 1
sage: f, g = WeierstrassForm(cubic); (f, g)
(0, -27/4)
sage: X,Y,Z = WeierstrassForm(cubic, transformation=True); (X,Y,Z)
(-x^3*y^3 - x^3 - y^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6 + 1/2*y^6 + 1/2*x^3 - 1/2*y^3,
x*y)
Note that plugging in `[X:Y:Z]` to the Weierstrass equation is a
complicated polynomial, but contains the hypersurface equation as
a factor::
sage: -Y^2 + X^3 + f*X*Z^4 + g*Z^6
-1/4*x^12*y^6 - 1/2*x^9*y^9 - 1/4*x^6*y^12 + 1/2*x^12*y^3
- 7/2*x^9*y^6 - 7/2*x^6*y^9 + 1/2*x^3*y^12 - 1/4*x^12 - 7/2*x^9*y^3
- 45/4*x^6*y^6 - 7/2*x^3*y^9 - 1/4*y^12 - 1/2*x^9 - 7/2*x^6*y^3
- 7/2*x^3*y^6 - 1/2*y^9 - 1/4*x^6 + 1/2*x^3*y^3 - 1/4*y^6
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
Only the affine span of the Newton polytope of the polynomial
matters. For example::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: WeierstrassForm(cubic.subs(z=1))
(0, -27/4)
sage: WeierstrassForm(x * cubic)
(0, -27/4)
This allows you to work with either homogeneous or inhomogeneous
variables. For example, here is the del Pezzo surface of degree 8::
sage: dP8 = toric_varieties.dP8()
sage: dP8.inject_variables()
Defining t, x, y, z
sage: WeierstrassForm(x*y^2 + y^2*z + t^2*x^3 + t^2*z^3)
(-3, -2)
sage: WeierstrassForm(x*y^2 + y^2 + x^3 + 1)
(-3, -2)
By specifying only certain variables we can compute the
Weierstrass form over the function field generated by the
remaining variables. For example, here is a cubic over `\QQ[a]` ::
sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + a*y^3 + a^2*z^3
sage: WeierstrassForm(cubic, variables=[x,y,z])
(0, -27/4*a^6)
TESTS::
sage: for P in ReflexivePolytopes(2):
....: S = ToricVariety(FaceFan(P))
....: p = sum((-S.K()).sections_monomials())
....: print(WeierstrassForm(p))
(-25/48, -1475/864)
(-97/48, 17/864)
(-25/48, -611/864)
(-27/16, 27/32)
(47/48, -199/864)
(47/48, -71/864)
(5/16, -21/32)
(23/48, -235/864)
(-1/48, 161/864)
(-25/48, 253/864)
(5/16, 11/32)
(-25/48, 125/864)
(-67/16, 63/32)
(-11/16, 3/32)
(-241/48, 3689/864)
(215/48, -5291/864)
"""
if isinstance(polynomial, (list, tuple)):
from sage.schemes.toric.weierstrass_higher import WeierstrassForm2
return WeierstrassForm2(polynomial, variables=variables, transformation=transformation)
if transformation:
from sage.schemes.toric.weierstrass_covering import WeierstrassMap
return WeierstrassMap(polynomial, variables=variables)
if variables is None:
variables = polynomial.variables()
from sage.geometry.polyhedron.ppl_lattice_polygon import (
polar_P2_polytope, polar_P1xP1_polytope, polar_P2_112_polytope)
newton_polytope, polynomial, variables = \
Newton_polygon_embedded(polynomial, variables)
polygon = newton_polytope.embed_in_reflexive_polytope('polytope')
if polygon is polar_P2_polytope():
return WeierstrassForm_P2(polynomial, variables)
if polygon is polar_P1xP1_polytope():
return WeierstrassForm_P1xP1(polynomial, variables)
if polygon is polar_P2_112_polytope():
return WeierstrassForm_P2_112(polynomial, variables)
raise ValueError('Newton polytope is not contained in a reflexive polygon')
def WeierstrassForm(polynomial, variables=None, transformation=False):
r"""
Return the Weierstrass form of an elliptic curve inside either
inside a toric surface or $\mathbb{P}^3$.
INPUT:
- ``polynomial`` -- either a polynomial or a list of polynomials
defining the elliptic curve.
A single polynomial can be either a cubic, a biquadric, or the
hypersurface in `\mathbb{P}^2[1,1,2]`. In this case the
equation need not be in any standard form, only its Newton
polyhedron is used.
If two polynomials are passed, they must both be quadratics in
`\mathbb{P}^3`.
- ``variables`` -- a list of variables of the parent polynomial
ring or ``None`` (default). In the latter case, all variables
are taken to be polynomial ring variables. If a subset of
polynomial ring variables are given, the Weierstrass form is
determined over the function field generated by the remaining
variables.
- ``transformation`` -- boolean (default: ``False``). Whether to
return the new variables that bring ``polynomial`` into
Weierstrass form.
OUTPUT:
The pair of coefficients `(f,g)` of the Weierstrass form `y^2 =
x^3 + f x + g` of the hypersurface equation.
If ``transformation=True``, a triple `(X,Y,Z)` of polynomials
defining a rational map of the toric hypersurface or complete
intersection in `\mathbb{P}^3` to its Weierstrass form in
`\mathbb{P}^2[2,3,1]` is returned.
That is, the triple satisfies
.. math::
Y^2 = X^3 + f X Z^4 + g Z^6
when restricted to the toric hypersurface or complete intersection.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: f, g = WeierstrassForm(cubic); (f, g)
(0, -27/4)
Same in inhomogeneous coordinates::
sage: R.<x,y> = QQ[]
sage: cubic = x^3 + y^3 + 1
sage: f, g = WeierstrassForm(cubic); (f, g)
(0, -27/4)
sage: X,Y,Z = WeierstrassForm(cubic, transformation=True); (X,Y,Z)
(-x^3*y^3 - x^3 - y^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6 + 1/2*y^6 + 1/2*x^3 - 1/2*y^3,
x*y)
Note that plugging in `[X:Y:Z]` to the Weierstrass equation is a
complicated polynomial, but contains the hypersurface equation as
a factor::
sage: -Y^2 + X^3 + f*X*Z^4 + g*Z^6
-1/4*x^12*y^6 - 1/2*x^9*y^9 - 1/4*x^6*y^12 + 1/2*x^12*y^3
- 7/2*x^9*y^6 - 7/2*x^6*y^9 + 1/2*x^3*y^12 - 1/4*x^12 - 7/2*x^9*y^3
- 45/4*x^6*y^6 - 7/2*x^3*y^9 - 1/4*y^12 - 1/2*x^9 - 7/2*x^6*y^3
- 7/2*x^3*y^6 - 1/2*y^9 - 1/4*x^6 + 1/2*x^3*y^3 - 1/4*y^6
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
Only the affine span of the Newton polytope of the polynomial
matters. For example::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: WeierstrassForm(cubic.subs(z=1))
(0, -27/4)
sage: WeierstrassForm(x * cubic)
(0, -27/4)
This allows you to work with either homogeneous or inhomogeneous
variables. For example, here is the del Pezzo surface of degree 8::
sage: dP8 = toric_varieties.dP8()
sage: dP8.inject_variables()
Defining t, x, y, z
sage: WeierstrassForm(x*y^2 + y^2*z + t^2*x^3 + t^2*z^3)
(-3, -2)
sage: WeierstrassForm(x*y^2 + y^2 + x^3 + 1)
(-3, -2)
By specifying only certain variables we can compute the
Weierstrass form over the function field generated by the
remaining variables. For example, here is a cubic over `\QQ[a]` ::
sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + a*y^3 + a^2*z^3
sage: WeierstrassForm(cubic, variables=[x,y,z])
(0, -27/4*a^6)
TESTS::
sage: for P in ReflexivePolytopes(2):
....: S = ToricVariety(FaceFan(P))
....: p = sum((-S.K()).sections_monomials())
....: print WeierstrassForm(p)
(-25/48, -1475/864)
(-97/48, 17/864)
(-25/48, -611/864)
(-27/16, 27/32)
(47/48, -199/864)
(47/48, -71/864)
(5/16, -21/32)
(23/48, -235/864)
(-1/48, 161/864)
(-25/48, 253/864)
(5/16, 11/32)
(-25/48, 125/864)
(-67/16, 63/32)
(-11/16, 3/32)
(-241/48, 3689/864)
(215/48, -5291/864)
"""
if isinstance(polynomial, (list, tuple)):
from sage.schemes.toric.weierstrass_higher import WeierstrassForm2
return WeierstrassForm2(polynomial, variables=variables, transformation=transformation)
if transformation:
from sage.schemes.toric.weierstrass_covering import WeierstrassMap
return WeierstrassMap(polynomial, variables=variables)
if variables is None:
variables = polynomial.variables()
from sage.geometry.polyhedron.ppl_lattice_polygon import (
polar_P2_polytope, polar_P1xP1_polytope, polar_P2_112_polytope)
newton_polytope, polynomial, variables = \
Newton_polygon_embedded(polynomial, variables)
polygon = newton_polytope.embed_in_reflexive_polytope('polytope')
if polygon is polar_P2_polytope():
return WeierstrassForm_P2(polynomial, variables)
if polygon is polar_P1xP1_polytope():
return WeierstrassForm_P1xP1(polynomial, variables)
if polygon is polar_P2_112_polytope():
return WeierstrassForm_P2_112(polynomial, variables)
raise ValueError('Newton polytope is not contained in a reflexive polygon')
def WeierstrassMap(polynomial, variables=None):
r"""
Return the Weierstrass form of an anticanonical hypersurface.
You should use
:meth:`sage.schemes.toric.weierstrass.WeierstrassForm` with
``transformation=True`` to get the transformation. This function
is only for internal use.
INPUT:
- ``polynomial`` -- a polynomial. The toric hypersurface
equation. Can be either a cubic, a biquadric, or the
hypersurface in `\mathbb{P}^2[1,1,2]`. The equation need not be
in any standard form, only its Newton polyhedron is used.
- ``variables`` -- a list of variables of the parent polynomial
ring or ``None`` (default). In the latter case, all variables
are taken to be polynomial ring variables. If a subset of
polynomial ring variables are given, the Weierstrass form is
determined over the function field generated by the remaining
variables.
OUTPUT:
A triple `(X,Y,Z)` of polynomials defining a rational map of the
toric hypersurface to its Weierstrass form in
`\mathbb{P}^2[2,3,1]`. That is, the triple satisfies
.. MATH::
Y^2 = X^3 + f X Z^4 + g Z^6
when restricted to the toric hypersurface.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: X,Y,Z = WeierstrassForm(cubic, transformation=True); (X,Y,Z)
(-x^3*y^3 - x^3*z^3 - y^3*z^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6*z^3 + 1/2*y^6*z^3
+ 1/2*x^3*z^6 - 1/2*y^3*z^6,
x*y*z)
sage: f, g = WeierstrassForm(cubic); (f,g)
(0, -27/4)
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
Only the affine span of the Newton polytope of the polynomial
matters. For example::
sage: WeierstrassForm(cubic.subs(z=1), transformation=True)
(-x^3*y^3 - x^3 - y^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6
+ 1/2*y^6 + 1/2*x^3 - 1/2*y^3,
x*y)
sage: WeierstrassForm(x * cubic, transformation=True)
(-x^3*y^3 - x^3*z^3 - y^3*z^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6*z^3 + 1/2*y^6*z^3
+ 1/2*x^3*z^6 - 1/2*y^3*z^6,
x*y*z)
This allows you to work with either homogeneous or inhomogeneous
variables. For example, here is the del Pezzo surface of degree 8::
sage: dP8 = toric_varieties.dP8()
sage: dP8.inject_variables()
Defining t, x, y, z
sage: WeierstrassForm(x*y^2 + y^2*z + t^2*x^3 + t^2*z^3, transformation=True)
(-1/27*t^4*x^6 - 2/27*t^4*x^5*z - 5/27*t^4*x^4*z^2
- 8/27*t^4*x^3*z^3 - 5/27*t^4*x^2*z^4 - 2/27*t^4*x*z^5
- 1/27*t^4*z^6 - 4/81*t^2*x^4*y^2 - 4/81*t^2*x^3*y^2*z
- 4/81*t^2*x*y^2*z^3 - 4/81*t^2*y^2*z^4 - 2/81*x^2*y^4
- 4/81*x*y^4*z - 2/81*y^4*z^2,
0,
1/3*t^2*x^2*z + 1/3*t^2*x*z^2 - 1/9*x*y^2 - 1/9*y^2*z)
sage: WeierstrassForm(x*y^2 + y^2 + x^3 + 1, transformation=True)
(-1/27*x^6 - 4/81*x^4*y^2 - 2/81*x^2*y^4 - 2/27*x^5
- 4/81*x^3*y^2 - 4/81*x*y^4 - 5/27*x^4 - 2/81*y^4 - 8/27*x^3
- 4/81*x*y^2 - 5/27*x^2 - 4/81*y^2 - 2/27*x - 1/27,
0,
-1/9*x*y^2 + 1/3*x^2 - 1/9*y^2 + 1/3*x)
By specifying only certain variables we can compute the
Weierstrass form over the function field generated by the
remaining variables. For example, here is a cubic over `\QQ[a]` ::
sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + a*y^3 + a^2*z^3
sage: WeierstrassForm(cubic, variables=[x,y,z], transformation=True)
(-a^9*y^3*z^3 - a^8*x^3*z^3 - a^7*x^3*y^3,
-1/2*a^14*y^3*z^6 + 1/2*a^13*y^6*z^3 + 1/2*a^13*x^3*z^6
- 1/2*a^11*x^3*y^6 - 1/2*a^11*x^6*z^3 + 1/2*a^10*x^6*y^3,
a^3*x*y*z)
TESTS::
sage: for P in ReflexivePolytopes(2):
....: S = ToricVariety(FaceFan(P))
....: p = sum( (-S.K()).sections_monomials() )
....: f, g = WeierstrassForm(p)
....: X,Y,Z = WeierstrassForm(p, transformation=True)
....: assert p.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
"""
if variables is None:
variables = polynomial.variables()
# switch to suitable inhomogeneous coordinates
from sage.geometry.polyhedron.ppl_lattice_polygon import (
polar_P2_polytope, polar_P1xP1_polytope, polar_P2_112_polytope )
from sage.schemes.toric.weierstrass import Newton_polygon_embedded
newton_polytope, polynomial_aff, variables_aff = \
Newton_polygon_embedded(polynomial, variables)
polygon = newton_polytope.embed_in_reflexive_polytope('polytope')
# Compute the map in inhomogeneous coordinates
if polygon is polar_P2_polytope():
X,Y,Z = WeierstrassMap_P2(polynomial_aff, variables_aff)
elif polygon is polar_P1xP1_polytope():
X,Y,Z = WeierstrassMap_P1xP1(polynomial_aff, variables_aff)
elif polygon is polar_P2_112_polytope():
X,Y,Z = WeierstrassMap_P2_112(polynomial_aff, variables_aff)
else:
assert False, 'Newton polytope is not contained in a reflexive polygon'
# homogenize again
R = polynomial.parent()
x = R.gens().index(variables_aff[0])
y = R.gens().index(variables_aff[1])
hom = newton_polytope.embed_in_reflexive_polytope('hom')
def homogenize(inhomog, degree):
e = tuple(hom._A * vector(ZZ,[inhomog[x], inhomog[y]]) + degree * hom._b)
result = list(inhomog)
for i, var in enumerate(variables):
result[R.gens().index(var)] = e[i]
result = vector(ZZ, result)
result.set_immutable()
return result
X_dict = dict((homogenize(e,2), v) for e, v in iteritems(X.dict()))
Y_dict = dict((homogenize(e,3), v) for e, v in iteritems(Y.dict()))
Z_dict = dict((homogenize(e,1), v) for e, v in iteritems(Z.dict()))
# shift to non-negative exponents if necessary
min_deg = [0]*R.ngens()
for var in variables:
i = R.gens().index(var)
min_X = min([ e[i] for e in X_dict ]) if len(X_dict)>0 else 0
min_Y = min([ e[i] for e in Y_dict ]) if len(Y_dict)>0 else 0
min_Z = min([ e[i] for e in Z_dict ]) if len(Z_dict)>0 else 0
min_deg[i] = min( min_X/2, min_Y/3, min_Z )
min_deg = vector(min_deg)
X_dict = dict((tuple(e-2*min_deg), v) for e, v in iteritems(X_dict))
Y_dict = dict((tuple(e-3*min_deg), v) for e, v in iteritems(Y_dict))
Z_dict = dict((tuple(e-1*min_deg), v) for e, v in iteritems(Z_dict))
return (R(X_dict), R(Y_dict), R(Z_dict))
def WeierstrassForm(polynomial, variables=None):
r"""
Return the Weierstrass form of an anticanonical hypersurface.
INPUT:
- ``polynomial`` -- a polynomial. The toric hypersurface
equation. Can be either a cubic, a biquadric, or the
hypersurface in `\mathbb{P}^2[1,1,2]`. The equation need not be
in any standard form, only its Newton polynomial is used.
- ``variables`` -- a list of variables of the parent polynomial
ring or ``None`` (default). In the latter case, all variables
are taken to be polynomial ring variables. If a subset of
polynomial ring variables are given, the Weierstras form is
determined over the function field generated by the remaining
variables.
OUTPUT:
The pair of coefficients `(f,g)` of the Weierstrass form `y^2 =
x^3 + f x + g` of the hypersurface equation.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: WeierstrassForm(cubic)
(0, -27/4)
Only the affine span of the Newton polytope of the polynomial
matters. For example::
sage: WeierstrassForm(cubic.subs(z=1))
(0, -27/4)
sage: WeierstrassForm(x * cubic)
(0, -27/4)
This allows you to work with either homogeneous or inhomogeneous
variables. For exmple, here is the del Pezzo surface of degree 8::
sage: dP8 = toric_varieties.dP8()
sage: dP8.inject_variables()
Defining t, x, y, z
sage: WeierstrassForm(x*y^2 + y^2*z + t^2*x^3 + t^2*z^3)
(-3, -2)
sage: WeierstrassForm(x*y^2 + y^2 + x^3 + 1)
(-3, -2)
By specifying only certain variables we can compute the
Weierstrass form over the function field generated by the
remaining variables. For example, here is a cubic over `\QQ[a]` ::
sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + a*y^3 + a^2*z^3
sage: WeierstrassForm(cubic, variables=[x,y,z])
(0, -27/4*a^6)
TESTS::
sage: for P in ReflexivePolytopes(2):
....: S = ToricVariety(FaceFan(P))
....: p = sum((-S.K()).sections_monomials())
....: print WeierstrassForm(p)
(-25/48, -1475/864)
(-97/48, 17/864)
(-25/48, -611/864)
(-27/16, 27/32)
(47/48, -199/864)
(47/48, -71/864)
(5/16, -21/32)
(23/48, -235/864)
(-1/48, 161/864)
(-25/48, 253/864)
(5/16, 11/32)
(-25/48, 125/864)
(-67/16, 63/32)
(-11/16, 3/32)
(-241/48, 3689/864)
(215/48, -5291/864)
"""
if variables is None:
variables = polynomial.variables()
from sage.geometry.polyhedron.ppl_lattice_polygon import (
polar_P2_polytope,
polar_P1xP1_polytope,
polar_P2_112_polytope,
)
newton_polytope, polynomial, variables = Newton_polygon_embedded(polynomial, variables)
polygon = newton_polytope.embed_in_reflexive_polytope("polytope")
if polygon is polar_P2_polytope():
return WeierstrassForm_P2(polynomial, variables)
if polygon is polar_P1xP1_polytope():
return WeierstrassForm_P1xP1(polynomial, variables)
if polygon is polar_P2_112_polytope():
return WeierstrassForm_P2_112(polynomial, variables)
raise ValueError("Newton polytope is not contained in a reflexive polygon")
def WeierstrassMap(polynomial, variables=None):
r"""
Return the Weierstrass form of an anticanonical hypersurface.
You should use
:meth:`sage.schemes.toric.weierstrass.WeierstrassForm` with
``transformation=True`` to get the transformation. This function
is only for internal use.
INPUT:
- ``polynomial`` -- a polynomial. The toric hypersurface
equation. Can be either a cubic, a biquadric, or the
hypersurface in `\mathbb{P}^2[1,1,2]`. The equation need not be
in any standard form, only its Newton polyhedron is used.
- ``variables`` -- a list of variables of the parent polynomial
ring or ``None`` (default). In the latter case, all variables
are taken to be polynomial ring variables. If a subset of
polynomial ring variables are given, the Weierstrass form is
determined over the function field generated by the remaining
variables.
OUTPUT:
A triple `(X,Y,Z)` of polynomials defining a rational map of the
toric hypersurface to its Weierstrass form in
`\mathbb{P}^2[2,3,1]`. That is, the triple satisfies
.. MATH::
Y^2 = X^3 + f X Z^4 + g Z^6
when restricted to the toric hypersurface.
EXAMPLES::
sage: R.<x,y,z> = QQ[]
sage: cubic = x^3 + y^3 + z^3
sage: X,Y,Z = WeierstrassForm(cubic, transformation=True); (X,Y,Z)
(-x^3*y^3 - x^3*z^3 - y^3*z^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6*z^3 + 1/2*y^6*z^3
+ 1/2*x^3*z^6 - 1/2*y^3*z^6,
x*y*z)
sage: f, g = WeierstrassForm(cubic); (f,g)
(0, -27/4)
sage: cubic.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
Only the affine span of the Newton polytope of the polynomial
matters. For example::
sage: WeierstrassForm(cubic.subs(z=1), transformation=True)
(-x^3*y^3 - x^3 - y^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6
+ 1/2*y^6 + 1/2*x^3 - 1/2*y^3,
x*y)
sage: WeierstrassForm(x * cubic, transformation=True)
(-x^3*y^3 - x^3*z^3 - y^3*z^3,
1/2*x^6*y^3 - 1/2*x^3*y^6 - 1/2*x^6*z^3 + 1/2*y^6*z^3
+ 1/2*x^3*z^6 - 1/2*y^3*z^6,
x*y*z)
This allows you to work with either homogeneous or inhomogeneous
variables. For example, here is the del Pezzo surface of degree 8::
sage: dP8 = toric_varieties.dP8()
sage: dP8.inject_variables()
Defining t, x, y, z
sage: WeierstrassForm(x*y^2 + y^2*z + t^2*x^3 + t^2*z^3, transformation=True)
(-1/27*t^4*x^6 - 2/27*t^4*x^5*z - 5/27*t^4*x^4*z^2
- 8/27*t^4*x^3*z^3 - 5/27*t^4*x^2*z^4 - 2/27*t^4*x*z^5
- 1/27*t^4*z^6 - 4/81*t^2*x^4*y^2 - 4/81*t^2*x^3*y^2*z
- 4/81*t^2*x*y^2*z^3 - 4/81*t^2*y^2*z^4 - 2/81*x^2*y^4
- 4/81*x*y^4*z - 2/81*y^4*z^2,
0,
1/3*t^2*x^2*z + 1/3*t^2*x*z^2 - 1/9*x*y^2 - 1/9*y^2*z)
sage: WeierstrassForm(x*y^2 + y^2 + x^3 + 1, transformation=True)
(-1/27*x^6 - 4/81*x^4*y^2 - 2/81*x^2*y^4 - 2/27*x^5
- 4/81*x^3*y^2 - 4/81*x*y^4 - 5/27*x^4 - 2/81*y^4 - 8/27*x^3
- 4/81*x*y^2 - 5/27*x^2 - 4/81*y^2 - 2/27*x - 1/27,
0,
-1/9*x*y^2 + 1/3*x^2 - 1/9*y^2 + 1/3*x)
By specifying only certain variables we can compute the
Weierstrass form over the function field generated by the
remaining variables. For example, here is a cubic over `\QQ[a]` ::
sage: R.<a, x,y,z> = QQ[]
sage: cubic = x^3 + a*y^3 + a^2*z^3
sage: WeierstrassForm(cubic, variables=[x,y,z], transformation=True)
(-a^9*y^3*z^3 - a^8*x^3*z^3 - a^7*x^3*y^3,
-1/2*a^14*y^3*z^6 + 1/2*a^13*y^6*z^3 + 1/2*a^13*x^3*z^6
- 1/2*a^11*x^3*y^6 - 1/2*a^11*x^6*z^3 + 1/2*a^10*x^6*y^3,
a^3*x*y*z)
TESTS::
sage: for P in ReflexivePolytopes(2):
....: S = ToricVariety(FaceFan(P))
....: p = sum( (-S.K()).sections_monomials() )
....: f, g = WeierstrassForm(p)
....: X,Y,Z = WeierstrassForm(p, transformation=True)
....: assert p.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
"""
if variables is None:
variables = polynomial.variables()
# switch to suitable inhomogeneous coordinates
from sage.geometry.polyhedron.ppl_lattice_polygon import (
polar_P2_polytope, polar_P1xP1_polytope, polar_P2_112_polytope)
from sage.schemes.toric.weierstrass import Newton_polygon_embedded
newton_polytope, polynomial_aff, variables_aff = \
Newton_polygon_embedded(polynomial, variables)
polygon = newton_polytope.embed_in_reflexive_polytope('polytope')
# Compute the map in inhomogeneous coordinates
if polygon is polar_P2_polytope():
X, Y, Z = WeierstrassMap_P2(polynomial_aff, variables_aff)
elif polygon is polar_P1xP1_polytope():
X, Y, Z = WeierstrassMap_P1xP1(polynomial_aff, variables_aff)
elif polygon is polar_P2_112_polytope():
X, Y, Z = WeierstrassMap_P2_112(polynomial_aff, variables_aff)
else:
assert False, 'Newton polytope is not contained in a reflexive polygon'
# homogenize again
R = polynomial.parent()
x = R.gens().index(variables_aff[0])
y = R.gens().index(variables_aff[1])
hom = newton_polytope.embed_in_reflexive_polytope('hom')
def homogenize(inhomog, degree):
e = tuple(hom._A * vector(ZZ, [inhomog[x], inhomog[y]]) +
degree * hom._b)
result = list(inhomog)
for i, var in enumerate(variables):
result[R.gens().index(var)] = e[i]
result = vector(ZZ, result)
result.set_immutable()
return result
X_dict = dict((homogenize(e, 2), v) for e, v in iteritems(X.dict()))
Y_dict = dict((homogenize(e, 3), v) for e, v in iteritems(Y.dict()))
Z_dict = dict((homogenize(e, 1), v) for e, v in iteritems(Z.dict()))
# shift to non-negative exponents if necessary
min_deg = [0] * R.ngens()
for var in variables:
i = R.gens().index(var)
min_X = min([e[i] for e in X_dict]) if len(X_dict) > 0 else 0
min_Y = min([e[i] for e in Y_dict]) if len(Y_dict) > 0 else 0
min_Z = min([e[i] for e in Z_dict]) if len(Z_dict) > 0 else 0
min_deg[i] = min(min_X / 2, min_Y / 3, min_Z)
min_deg = vector(min_deg)
X_dict = dict((tuple(e - 2 * min_deg), v) for e, v in iteritems(X_dict))
Y_dict = dict((tuple(e - 3 * min_deg), v) for e, v in iteritems(Y_dict))
Z_dict = dict((tuple(e - 1 * min_deg), v) for e, v in iteritems(Z_dict))
return (R(X_dict), R(Y_dict), R(Z_dict))