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) $.
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) $.
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