1736.sagews

from sage.rings.polynomial.toy_buchberger import buchberger
R.<a,b,c,d,x,y,z> = PolynomialRing(QQ,2, order='lex')
set_verbose(0)
G= buchberger(R.ideal([a^2+b^2-1,a*c+2*c-x,a*d+2*d-y,b-z,c^2+d^2-1])); G

[-1/4*d^2*y*z^2 - 3/4*d^2*y - d*x^2 + 1/4*x^2*y + 1/4*y*z^2 + 3/4*y, b - z, a*c + 2*c - x, a*d + 2*d - y, 1/4*c*z^2 + 3/4*c + 1/16*x^3 + 1/16*x*y^2 + 1/16*x*z^2 - 13/16*x, -c*y + d*x, a^2 + b^2 - 1, a - 1/4*x^2 - 1/4*y^2 - 1/4*z^2 + 5/4, c^2 + d^2 - 1, 4*c*x + d^2*z^2 + 3*d^2 - x^2 - z^2 - 3, 1/4*a*z^2 + 3/4*a + 1/16*d^2*z^4 + 3/8*d^2*z^2 + 9/16*d^2 - 1/4*d*y*z^2 - 3/4*d*y - 1/16*x^2*z^2 - 3/16*x^2 - 1/16*z^4 + 1/8*z^2 + 15/16, d*z^2 + 3*d + 1/4*x^2*y + 1/4*y^3 + 1/4*y*z^2 - 13/4*y, -1/16*x^4 - 1/8*x^2*y^2 - 1/8*x^2*z^2 + 5/8*x^2 - 1/16*y^4 - 1/8*y^2*z^2 + 5/8*y^2 - 1/16*z^4 - 3/8*z^2 - 9/16, -d^2*z^2 - 3*d^2 + 4*d*y - y^2, 1/4*d*x^2 + 1/4*d*y^2 + 1/4*d*z^2 + 3/4*d - y, -a*y - d*z^2 - 3*d + 2*y, -c*d*z^2 - 3*c*d + 4*d*x - x*y, -1/4*d*y*z^2 - 3/4*d*y + 1/16*x^4 + 1/16*x^2*y^2 + 1/8*x^2*z^2 - 5/8*x^2 + 1/16*y^2*z^2 + 3/16*y^2 + 1/16*z^4 + 3/8*z^2 + 9/16, a*x + c*z^2 + 3*c - 2*x]

R.<x,y,z> = PolynomialRing(QQ,3,order='lex')
I = R.ideal([x-z^4,y-z^5])
G=buchberger(I); G

[y - z^5, x - z^4]

u = var('u')
t= var('t')
parametric_plot3d( ((2+cos(t))*cos(u), 
(2+cos(t))*sin(u), sin(t)), (u, 0, 20), (t,0,20))