TITLE: Affine algebraic subset of $\mathbb{A}_k^4$
QUESTION [5 upvotes]: How do I go about proving the subset $V = \{(s^3, s^2t, st^2, t^3)\mid s, t \in k\}$ is an affine algebraic subset of $\mathbb{A}_k^4$ and find $\mathbb{I}(V) \subset k[x_0, x_1, x_2, x_3]$?

REPLY [2 votes]: KDong has already given the answer. There are some points that might help understand  how the equations describing $V$ were  arrived at. Note that your set $V$ is defined by points whose co-ordinates are all monomials of degree 4 in $s$ and $t$. They are ordered in ascending powers of $t$. 
In general one can do the same with sets defined by all monomials of fixed degree $d$.(They will define an affine algebraic subset in an affine space of dimension $d+1$). 
In this ordering take two  monomials $M_{i-1}$ and $M_{i+1}$.
Sum of their $s$-degrees (or $t$-degrees) is clearly twice that of $M_i$. So $M_i^2=M_{i-1}M_{i+1}$. This explains the equations of the kind  $x_i^2- x_{i-1}x_{i+1}=0$. Same way one can see $M_iM_j=M_kM_l$ if $i+j=k+l$. So this explains the equations of the kind $x_0x_3-x_1x_2=0$. The beauty is that these equations suffice. (Note that all the equations are quadratic).