def WeierstrassMap_P2(polynomial, variables=None):
r"""
Map a cubic to its Weierstrass form
Input/output is the same as :func:`WeierstrassMap`, except that
the input polynomial must be a cubic in `\mathbb{P}^2`,
.. MATH::
\begin{split}
p(x,y) =&\;
a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} +
a_{03} y^{3} + a_{20} x^{2} +
\\ &\;
a_{11} x y +
a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}
\end{split}
EXAMPLES::
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2
sage: R.<x,y,z> = QQ[]
sage: equation = x^3+y^3+z^3+x*y*z
sage: f, g = WeierstrassForm_P2(equation)
sage: X,Y,Z = WeierstrassMap_P2(equation)
sage: equation.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2
sage: R.<x,y> = QQ[]
sage: equation = x^3+y^3+1
sage: f, g = WeierstrassForm_P2(equation)
sage: X,Y,Z = WeierstrassMap_P2(equation)
sage: equation.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
"""
x,y,z = _check_polynomial_P2(polynomial, variables)
cubic = invariant_theory.ternary_cubic(polynomial, x,y,z)
H = cubic.Hessian()
Theta = cubic.Theta_covariant()
J = cubic.J_covariant()
F = polynomial.parent().base_ring()
return (Theta, J/F(2), H)
def WeierstrassMap_P2(polynomial, variables=None):
r"""
Map a cubic to its Weierstrass form
Input/output is the same as :func:`WeierstrassMap`, except that
the input polynomial must be a cubic in `\mathbb{P}^2`,
.. MATH::
\begin{split}
p(x,y) =&\;
a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} +
a_{03} y^{3} + a_{20} x^{2} +
\\ &\;
a_{11} x y +
a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}
\end{split}
EXAMPLES::
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2
sage: R.<x,y,z> = QQ[]
sage: equation = x^3+y^3+z^3+x*y*z
sage: f, g = WeierstrassForm_P2(equation)
sage: X,Y,Z = WeierstrassMap_P2(equation)
sage: equation.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: from sage.schemes.toric.weierstrass_covering import WeierstrassMap_P2
sage: R.<x,y> = QQ[]
sage: equation = x^3+y^3+1
sage: f, g = WeierstrassForm_P2(equation)
sage: X,Y,Z = WeierstrassMap_P2(equation)
sage: equation.divides(-Y^2 + X^3 + f*X*Z^4 + g*Z^6)
True
"""
x, y, z = _check_polynomial_P2(polynomial, variables)
cubic = invariant_theory.ternary_cubic(polynomial, x, y, z)
H = cubic.Hessian()
Theta = cubic.Theta_covariant()
J = cubic.J_covariant()
F = polynomial.parent().base_ring()
return (Theta, J / F(2), H)
def WeierstrassForm_P2(polynomial, variables=None):
r"""
Bring a cubic into Weierstrass form.
Input/output is the same as :func:`WeierstrassForm`, except that
the input polynomial must be a standard cubic in `\mathbb{P}^2`,
.. MATH::
\begin{split}
p(x,y) =&\;
a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} +
a_{03} y^{3} + a_{20} x^{2} +
\\ &\;
a_{11} x y +
a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}
\end{split}
EXAMPLES::
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: R.<x,y,z> = QQ[]
sage: WeierstrassForm_P2( x^3+y^3+z^3 )
(0, -27/4)
sage: R.<x,y,z, a,b> = QQ[]
sage: WeierstrassForm_P2( -y^2*z+x^3+a*x*z^2+b*z^3, [x,y,z] )
(a, b)
TESTS::
sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
....: a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: WeierstrassForm_P2(p, [x,y,z])
(-1/48*a11^4 + 1/6*a20*a11^2*a02 - 1/3*a20^2*a02^2 - 1/2*a03*a20*a11*a10
+ 1/6*a12*a11^2*a10 + 1/3*a12*a20*a02*a10 - 1/2*a21*a11*a02*a10
+ a30*a02^2*a10 - 1/3*a12^2*a10^2 + a21*a03*a10^2 + a03*a20^2*a01
- 1/2*a12*a20*a11*a01 + 1/6*a21*a11^2*a01 + 1/3*a21*a20*a02*a01
- 1/2*a30*a11*a02*a01 + 1/3*a21*a12*a10*a01 - 3*a30*a03*a10*a01
- 1/3*a21^2*a01^2 + a30*a12*a01^2 + a12^2*a20*a00 - 3*a21*a03*a20*a00
- 1/2*a21*a12*a11*a00 + 9/2*a30*a03*a11*a00 + a21^2*a02*a00
- 3*a30*a12*a02*a00,
1/864*a11^6 - 1/72*a20*a11^4*a02 + 1/18*a20^2*a11^2*a02^2
- 2/27*a20^3*a02^3 + 1/24*a03*a20*a11^3*a10 - 1/72*a12*a11^4*a10
- 1/6*a03*a20^2*a11*a02*a10 + 1/36*a12*a20*a11^2*a02*a10
+ 1/24*a21*a11^3*a02*a10 + 1/9*a12*a20^2*a02^2*a10
- 1/6*a21*a20*a11*a02^2*a10 - 1/12*a30*a11^2*a02^2*a10
+ 1/3*a30*a20*a02^3*a10 + 1/4*a03^2*a20^2*a10^2
- 1/6*a12*a03*a20*a11*a10^2 + 1/18*a12^2*a11^2*a10^2
- 1/12*a21*a03*a11^2*a10^2 + 1/9*a12^2*a20*a02*a10^2
- 1/6*a21*a03*a20*a02*a10^2 - 1/6*a21*a12*a11*a02*a10^2
+ a30*a03*a11*a02*a10^2 + 1/4*a21^2*a02^2*a10^2
- 2/3*a30*a12*a02^2*a10^2 - 2/27*a12^3*a10^3 + 1/3*a21*a12*a03*a10^3
- a30*a03^2*a10^3 - 1/12*a03*a20^2*a11^2*a01 + 1/24*a12*a20*a11^3*a01
- 1/72*a21*a11^4*a01 + 1/3*a03*a20^3*a02*a01 - 1/6*a12*a20^2*a11*a02*a01
+ 1/36*a21*a20*a11^2*a02*a01 + 1/24*a30*a11^3*a02*a01
+ 1/9*a21*a20^2*a02^2*a01 - 1/6*a30*a20*a11*a02^2*a01
- 1/6*a12*a03*a20^2*a10*a01 - 1/6*a12^2*a20*a11*a10*a01
+ 5/6*a21*a03*a20*a11*a10*a01 + 1/36*a21*a12*a11^2*a10*a01
- 3/4*a30*a03*a11^2*a10*a01 + 1/18*a21*a12*a20*a02*a10*a01
- 3/2*a30*a03*a20*a02*a10*a01 - 1/6*a21^2*a11*a02*a10*a01
+ 5/6*a30*a12*a11*a02*a10*a01 - 1/6*a30*a21*a02^2*a10*a01
+ 1/9*a21*a12^2*a10^2*a01 - 2/3*a21^2*a03*a10^2*a01
+ a30*a12*a03*a10^2*a01 + 1/4*a12^2*a20^2*a01^2
- 2/3*a21*a03*a20^2*a01^2 - 1/6*a21*a12*a20*a11*a01^2
+ a30*a03*a20*a11*a01^2 + 1/18*a21^2*a11^2*a01^2
- 1/12*a30*a12*a11^2*a01^2 + 1/9*a21^2*a20*a02*a01^2
- 1/6*a30*a12*a20*a02*a01^2 - 1/6*a30*a21*a11*a02*a01^2
+ 1/4*a30^2*a02^2*a01^2 + 1/9*a21^2*a12*a10*a01^2
- 2/3*a30*a12^2*a10*a01^2 + a30*a21*a03*a10*a01^2
- 2/27*a21^3*a01^3 + 1/3*a30*a21*a12*a01^3 - a30^2*a03*a01^3
- a03^2*a20^3*a00 + a12*a03*a20^2*a11*a00 - 1/12*a12^2*a20*a11^2*a00
- 3/4*a21*a03*a20*a11^2*a00 + 1/24*a21*a12*a11^3*a00
+ 5/8*a30*a03*a11^3*a00 - 2/3*a12^2*a20^2*a02*a00
+ a21*a03*a20^2*a02*a00 + 5/6*a21*a12*a20*a11*a02*a00
- 3/2*a30*a03*a20*a11*a02*a00 - 1/12*a21^2*a11^2*a02*a00
- 3/4*a30*a12*a11^2*a02*a00 - 2/3*a21^2*a20*a02^2*a00
+ a30*a12*a20*a02^2*a00 + a30*a21*a11*a02^2*a00
- a30^2*a02^3*a00 + 1/3*a12^3*a20*a10*a00
- 3/2*a21*a12*a03*a20*a10*a00 + 9/2*a30*a03^2*a20*a10*a00
- 1/6*a21*a12^2*a11*a10*a00 + a21^2*a03*a11*a10*a00
- 3/2*a30*a12*a03*a11*a10*a00 - 1/6*a21^2*a12*a02*a10*a00
+ a30*a12^2*a02*a10*a00 - 3/2*a30*a21*a03*a02*a10*a00
- 1/6*a21*a12^2*a20*a01*a00 + a21^2*a03*a20*a01*a00
- 3/2*a30*a12*a03*a20*a01*a00 - 1/6*a21^2*a12*a11*a01*a00
+ a30*a12^2*a11*a01*a00 - 3/2*a30*a21*a03*a11*a01*a00
+ 1/3*a21^3*a02*a01*a00 - 3/2*a30*a21*a12*a02*a01*a00
+ 9/2*a30^2*a03*a02*a01*a00 + 1/4*a21^2*a12^2*a00^2
- a30*a12^3*a00^2 - a21^3*a03*a00^2
+ 9/2*a30*a21*a12*a03*a00^2 - 27/4*a30^2*a03^2*a00^2)
"""
x, y, z = _check_polynomial_P2(polynomial, variables)
cubic = invariant_theory.ternary_cubic(polynomial, x, y, z)
F = polynomial.base_ring()
S = cubic.S_invariant()
T = cubic.T_invariant()
return (27*S, -27/F(4)*T)
def WeierstrassForm_P2(polynomial, variables=None):
r"""
Bring a cubic into Weierstrass form.
Input/output is the same as :func:`WeierstrassForm`, except that
the input polynomial must be a standard cubic in `\mathbb{P}^2`,
.. math::
\begin{split}
p(x,y) =&\;
a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} +
a_{03} y^{3} + a_{20} x^{2} +
\\ &\;
a_{11} x y +
a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}
\end{split}
EXAMPLES::
sage: from sage.schemes.toric.weierstrass import WeierstrassForm_P2
sage: R.<x,y,z> = QQ[]
sage: WeierstrassForm_P2( x^3+y^3+z^3 )
(0, -27/4)
sage: R.<x,y,z, a,b> = QQ[]
sage: WeierstrassForm_P2( -y^2*z+x^3+a*x*z^2+b*z^3, [x,y,z] )
(a, b)
TESTS::
sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
....: a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: WeierstrassForm_P2(p, [x,y,z])
(-1/48*a11^4 + 1/6*a20*a11^2*a02 - 1/3*a20^2*a02^2 - 1/2*a03*a20*a11*a10
+ 1/6*a12*a11^2*a10 + 1/3*a12*a20*a02*a10 - 1/2*a21*a11*a02*a10
+ a30*a02^2*a10 - 1/3*a12^2*a10^2 + a21*a03*a10^2 + a03*a20^2*a01
- 1/2*a12*a20*a11*a01 + 1/6*a21*a11^2*a01 + 1/3*a21*a20*a02*a01
- 1/2*a30*a11*a02*a01 + 1/3*a21*a12*a10*a01 - 3*a30*a03*a10*a01
- 1/3*a21^2*a01^2 + a30*a12*a01^2 + a12^2*a20*a00 - 3*a21*a03*a20*a00
- 1/2*a21*a12*a11*a00 + 9/2*a30*a03*a11*a00 + a21^2*a02*a00
- 3*a30*a12*a02*a00,
1/864*a11^6 - 1/72*a20*a11^4*a02 + 1/18*a20^2*a11^2*a02^2
- 2/27*a20^3*a02^3 + 1/24*a03*a20*a11^3*a10 - 1/72*a12*a11^4*a10
- 1/6*a03*a20^2*a11*a02*a10 + 1/36*a12*a20*a11^2*a02*a10
+ 1/24*a21*a11^3*a02*a10 + 1/9*a12*a20^2*a02^2*a10
- 1/6*a21*a20*a11*a02^2*a10 - 1/12*a30*a11^2*a02^2*a10
+ 1/3*a30*a20*a02^3*a10 + 1/4*a03^2*a20^2*a10^2
- 1/6*a12*a03*a20*a11*a10^2 + 1/18*a12^2*a11^2*a10^2
- 1/12*a21*a03*a11^2*a10^2 + 1/9*a12^2*a20*a02*a10^2
- 1/6*a21*a03*a20*a02*a10^2 - 1/6*a21*a12*a11*a02*a10^2
+ a30*a03*a11*a02*a10^2 + 1/4*a21^2*a02^2*a10^2
- 2/3*a30*a12*a02^2*a10^2 - 2/27*a12^3*a10^3 + 1/3*a21*a12*a03*a10^3
- a30*a03^2*a10^3 - 1/12*a03*a20^2*a11^2*a01 + 1/24*a12*a20*a11^3*a01
- 1/72*a21*a11^4*a01 + 1/3*a03*a20^3*a02*a01 - 1/6*a12*a20^2*a11*a02*a01
+ 1/36*a21*a20*a11^2*a02*a01 + 1/24*a30*a11^3*a02*a01
+ 1/9*a21*a20^2*a02^2*a01 - 1/6*a30*a20*a11*a02^2*a01
- 1/6*a12*a03*a20^2*a10*a01 - 1/6*a12^2*a20*a11*a10*a01
+ 5/6*a21*a03*a20*a11*a10*a01 + 1/36*a21*a12*a11^2*a10*a01
- 3/4*a30*a03*a11^2*a10*a01 + 1/18*a21*a12*a20*a02*a10*a01
- 3/2*a30*a03*a20*a02*a10*a01 - 1/6*a21^2*a11*a02*a10*a01
+ 5/6*a30*a12*a11*a02*a10*a01 - 1/6*a30*a21*a02^2*a10*a01
+ 1/9*a21*a12^2*a10^2*a01 - 2/3*a21^2*a03*a10^2*a01
+ a30*a12*a03*a10^2*a01 + 1/4*a12^2*a20^2*a01^2
- 2/3*a21*a03*a20^2*a01^2 - 1/6*a21*a12*a20*a11*a01^2
+ a30*a03*a20*a11*a01^2 + 1/18*a21^2*a11^2*a01^2
- 1/12*a30*a12*a11^2*a01^2 + 1/9*a21^2*a20*a02*a01^2
- 1/6*a30*a12*a20*a02*a01^2 - 1/6*a30*a21*a11*a02*a01^2
+ 1/4*a30^2*a02^2*a01^2 + 1/9*a21^2*a12*a10*a01^2
- 2/3*a30*a12^2*a10*a01^2 + a30*a21*a03*a10*a01^2
- 2/27*a21^3*a01^3 + 1/3*a30*a21*a12*a01^3 - a30^2*a03*a01^3
- a03^2*a20^3*a00 + a12*a03*a20^2*a11*a00 - 1/12*a12^2*a20*a11^2*a00
- 3/4*a21*a03*a20*a11^2*a00 + 1/24*a21*a12*a11^3*a00
+ 5/8*a30*a03*a11^3*a00 - 2/3*a12^2*a20^2*a02*a00
+ a21*a03*a20^2*a02*a00 + 5/6*a21*a12*a20*a11*a02*a00
- 3/2*a30*a03*a20*a11*a02*a00 - 1/12*a21^2*a11^2*a02*a00
- 3/4*a30*a12*a11^2*a02*a00 - 2/3*a21^2*a20*a02^2*a00
+ a30*a12*a20*a02^2*a00 + a30*a21*a11*a02^2*a00
- a30^2*a02^3*a00 + 1/3*a12^3*a20*a10*a00
- 3/2*a21*a12*a03*a20*a10*a00 + 9/2*a30*a03^2*a20*a10*a00
- 1/6*a21*a12^2*a11*a10*a00 + a21^2*a03*a11*a10*a00
- 3/2*a30*a12*a03*a11*a10*a00 - 1/6*a21^2*a12*a02*a10*a00
+ a30*a12^2*a02*a10*a00 - 3/2*a30*a21*a03*a02*a10*a00
- 1/6*a21*a12^2*a20*a01*a00 + a21^2*a03*a20*a01*a00
- 3/2*a30*a12*a03*a20*a01*a00 - 1/6*a21^2*a12*a11*a01*a00
+ a30*a12^2*a11*a01*a00 - 3/2*a30*a21*a03*a11*a01*a00
+ 1/3*a21^3*a02*a01*a00 - 3/2*a30*a21*a12*a02*a01*a00
+ 9/2*a30^2*a03*a02*a01*a00 + 1/4*a21^2*a12^2*a00^2
- a30*a12^3*a00^2 - a21^3*a03*a00^2
+ 9/2*a30*a21*a12*a03*a00^2 - 27/4*a30^2*a03^2*a00^2)
"""
x, y, z = _check_polynomial_P2(polynomial, variables)
cubic = invariant_theory.ternary_cubic(polynomial, x, y, z)
F = polynomial.base_ring()
S = cubic.S_invariant()
T = cubic.T_invariant()
return (27*S, -27/F(4)*T)
