R=QQ[x..z] I=ideal(x^2,y^2-z) radical I I1=ideal(z-x^2-y^2) I2=ideal(x^2,y^2-z^2) J=intersect(I1,I2) dim I1 dim J radical J J==oo decompose J saturate(J,I2) J:I2 isPrime I1 f=7*x^2+3*x*z^2+5*y^2 leadTerm f leadMonomial f restart --choose an order which is an elimination order for the first two variables R=QQ[t_1,t_2,x_1..x_3,MonomialOrder => Eliminate 2] I=ideal(t_1-x_1,t_1*t_2-x_2,t_1*t_2^2-x_3) ideal groebnerBasis I restart --note we can more simpley use the eliminate command R=QQ[t_1,t_2,x_1..x_3] I=ideal(t_1-x_1,t_1*t_2-x_2,t_1*t_2^2-x_3) eliminate({t_1,t_2},I)