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I want to show that $Z =\{(u^3, u^2v, uv^2, v^3) \mid u,v \in \mathbb{C}\}$ is an algebraic set. So I need functions in $\mathbb{C}[x_1, x_2, x_3, x_4]$ that vanish for all $(u^3, u^2v, uv^2, v^3)$ where $u,v \in \mathbb{C}$.

I already found out that the function $f_{uv}(x_1, x_2, x_3, x_4) = -x_1 - 3x_2 - 3x_3 - x_4 + (u+v)^3$ vanishes at $(u^3, u^2v, uv^2, v^3)$. But how can I find one that vanishes for all complex numbers $u$, $v$?

user26857
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Dlmn
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    You need $x_1x_3-x_2^2=0$ and $x_2x_4-x_3^2=0$. Do these suffice? – Angina Seng Dec 14 '19 at 13:00
  • @LordSharktheUnknown Great, thanks! I would like to find $I(Z)$ as well, so I need either all such functions or the Ideal generated by $x_1x_3−x_2^2=0$ and $x_2x_4−x_2^3=0$ (to apply Hilbert's Nullstellensatz) right? Could you help me with this as well? – Dlmn Dec 14 '19 at 13:28

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