restart needsPackage "NumericalAlgebraicGeometry" R=QQ[x,y,z,MonomialOrder=>Lex] I=ideal(x^2+y+z-1,x+z+y^2-1,x+z^2+y-1) I=ideal groebnerBasis I --In the first eq we have eliminated y and x and have things --only in k[z]... we can think of this as intersecting I with k[z] S=CC[z] f=sub(I_0,S) s = solveSystem {f} -- 0,1, −1 ± sqrt(2). --we then solve this simpler equation and "extend" to get the other --coordinates on our vareity restart needsPackage "NumericalAlgebraicGeometry" R=QQ[x,y,z,MonomialOrder=>Lex] I=ideal(x^2+y^2+z^2-1,x^2+z^2-y,x-z) I=ideal groebnerBasis I dim I --we expect 4 solutions degree I --find the roots f=I_0 S=CC[z] F = {sub(f,S)}; s = solveSystem F I --first point ((coordinates s_0)_0)--z 2*((coordinates s_0)_0)^2--y ((coordinates s_0)_0)--x --second point ((coordinates s_1)_0)--z 2*((coordinates s_1)_0)^2--y ((coordinates s_1)_0)--x --third point ((coordinates s_2)_0)--z 2*((coordinates s_2)_0)^2--y ((coordinates s_2)_0)--x --fourth point ((coordinates s_3)_0)--z 2*((coordinates s_3)_0)^2--y ((coordinates s_3)_0)--x