--Consider parameterization of -- x = (u^2)/v -- y = (v^2)/u -- z = u --Since we have a "y" make our new variable "w" R=QQ[w,v,u,x,y,z,MonomialOrder=>Lex] -- g=g_1*g_2*g_3=u*v g=u*v J=ideal(v*x - u^2 , u*y - v^2 , z - u, 1 - g*w) K=ideal groebnerBasis J --want to eliminate w, v, u --only eq without w, v, u V=ideal K_0 --can get this also with eliminate({w,v,u},J) --we might want to work in a new ring now S=QQ[(gens(R))_{3..5}] --2nd elim ideal is V=ideal sub(K_0,S) --or with the second command V=sub(eliminate({w,v,u},J),S)