R.<x,y,z> = QQ[]
|
|
P.<x,y,z> = ProjectiveSpace(R)
|
|
F = x^3 - 8*x^2 + 5*x + 1 + 7*y^2*(x-1)*x
|
|
G = F.homogenize(z); G
7*x^2*y^2 + x^3*z - 7*x*y^2*z - 8*x^2*z^2 + 5*x*z^3 + z^4 7*x^2*y^2 + x^3*z - 7*x*y^2*z - 8*x^2*z^2 + 5*x*z^3 + z^4 |
C = Curve(G); C
Projective Curve over Rational Field defined by 7*x^2*y^2 + x^3*z - 7*x*y^2*z - 8*x^2*z^2 + 5*x*z^3 + z^4 Projective Curve over Rational Field defined by 7*x^2*y^2 + x^3*z - 7*x*y^2*z - 8*x^2*z^2 + 5*x*z^3 + z^4 |
C.genus()
2 2 |
C.arithmetic_genus()
3 3 |
C.is_singular()
True True |
C.affine_patch(2)
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: 7*x0^2*x1^2 + x0^3 - 7*x0*x1^2 - 8*x0^2 + 5*x0 + 1 Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: 7*x0^2*x1^2 + x0^3 - 7*x0*x1^2 - 8*x0^2 + 5*x0 + 1 |
R.<x,y> = QQ[]
F0 = x^3 - 8*x^2 + 5*x + 1 + 7*y^2*(x-1)*x
|
|
I = ideal([F0, F0.derivative(x), F0.derivative(y)])
|
|
I.groebner_basis()
[1] [1] |
A = C.affine_patch(2); A
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: 7*x0^2*x1^2 + x0^3 - 7*x0*x1^2 - 8*x0^2 + 5*x0 + 1 Closed subscheme of Affine Space of dimension 2 over Rational Field defined by: 7*x0^2*x1^2 + x0^3 - 7*x0*x1^2 - 8*x0^2 + 5*x0 + 1 |
R.<x0,x1> = QQ[]
A = C.affine_patch(1); A
F0 = R(A.defining_polynomials()[0])
I = ideal([F0, F0.derivative(x0), F0.derivative(x1)])
I.groebner_basis()
[x0, x1] [x0, x1] |
I
Ideal (x0^3*x1 - 8*x0^2*x1^2 + 5*x0*x1^3 + x1^4 + 7*x0^2 - 7*x0*x1, 3*x0^2*x1 - 16*x0*x1^2 + 5*x1^3 + 14*x0 - 7*x1, x0^3 - 16*x0^2*x1 + 15*x0*x1^2 + 4*x1^3 - 7*x0) of Multivariate Polynomial Ring in x0, x1 over Rational Field Ideal (x0^3*x1 - 8*x0^2*x1^2 + 5*x0*x1^3 + x1^4 + 7*x0^2 - 7*x0*x1, 3*x0^2*x1 - 16*x0*x1^2 + 5*x1^3 + 14*x0 - 7*x1, x0^3 - 16*x0^2*x1 + 15*x0*x1^2 + 4*x1^3 - 7*x0) of Multivariate Polynomial Ring in x0, x1 over Rational Field |
implicit_plot(F0, (x0,-1,1), (x1,-1,1))
|
|
%cython
def point_search(int B):
cdef long long x, y, z
for x in range(-B, B):
for y in range(-B, B):
for z in range(1,B):
if 7*x*x*y*y + x*x*x*z - 7*x*y*y*z - 8*x*x*z*z + 5*x*z*z*z + z*z*z*z == 0:
print (x,y,z)
time point_search(1000)
Time: CPU 13.44 s, Wall: 13.46 s Time: CPU 13.44 s, Wall: 13.46 s |
time point_search(10000)
|
|
|
|