restart R=QQ[x,y,z,MonomialOrder=>Lex] I=ideal(x*y^2+2*y^2,x^4-2*x^2+1) f=y-x^2+1 Itilde=ideal(x*y^2+2*y^2,x^4-2*x^2+1,1-z*(f)) ideal groebnerBasis Itilde f%I==0 f%(radical I) --what we do above is the same, geometrically, as removing V(f) --this can also be done this way...as we will see in 4.4 saturate(I,f)