UTM Ideals Varieties and Algorithm - Chapter 1 Section 4 Exercise 9

Problem:

Solution:

The parameterization of the twisted cubic is $(t, t^2, t^3)$, so all points in $V$ must be of that form.

Now we test such points on the polynomial $y^2 - xz = (t^2)^2 - (t)(t^3) = t^4 - t^4 = 0$, so the polynomial vanishes on all points in the variety, so $y^2 - xz \in I(V)$.

For part (b), the key observation is that we have $xz$, so we multiply the second polynomial by $x$, the rest seems the just follow as:

$(y)(y - x^2) - (x)(z - x^3) = (y^2 - xz) - x^2 y  + x^4 = (y^2 - xz) - x^2(y - x^2)$.

So we just it back on the left hand side and get our answer as:

$(x^2 + y)(y - x^2) - (x)(z - x^3) = (y^2 - xz)$.