Singular Book 1.8.18 -- kernel of a ring map

First, we use Macaulay2's kernel(RingMap) function.

i1 : A = QQ[x,y,z];

i2 : B = QQ[a,b];

i3 : phi = map(B,A,{a^2,a*b,b^2})

                  2        2
o3 = map (B, A, {a , a*b, b })

o3 : RingMap B <-- A

i4 : kernel phi

            2
o4 = ideal(y  - x*z)

o4 : Ideal of A

Now use the elimination of variables method.

i5 : C = QQ[x,y,z,a,b]

o5 = C

o5 : PolynomialRing

i6 : H = ideal(x-a^2, y-a*b, z-b^2);

o6 : Ideal of C

i7 : eliminate(H, {a,b})

            2
o7 = ideal(y  - x*z)

o7 : Ideal of C