Generators for ideal generated by $R$ points in $N$ space Consider the points $P_1,...,P_R$ in $\mathbb A^N$. I would like to write down generators for these. Without knowing relations between these points we can not write down a minimal set of generators but we can write down a set not the less.
For this I need a good book keeping device. I try to do this with 3 points in affine 4-space but get confused. Suppose that our three points are $A,B,C$ with $A=(a_x,a_y,a_z,a_w)$ and so on. I believe we should have $RN$ generators so in this case we should have 12.
But then I get confused, I start to write down a generator. This must be something that vanishes on every point $A,B,C$ so i write down $(x-a_x)(y-b_y)(z-c_z)$. But we do not need to write down anything for $w$.
What are the generators for the ideal generated by the points A,B,C? What are generators for R points in N space?
 A: Start off by writing the ideal for one point in 3-space for example, say $P=(p_1,p_2,p_3)$. The generators of the ideal $I(P)$ can be easily shown to be $x-p_1,y-p_2$ and $z-p_3$, mainly because they all vanish at $P$ and are of minimal degree (this ideal is radical and maximal). For $N$-dimensional space, you have then that $I(P)=\langle x_1-p_1,...,x_N-p_N \rangle$ given $P=(p_1,...,p_N)$; here the arguments are all the same.
For a finite set of points, recall that the union of algebraic sets is the algebraic set corresponding to the intersection (and the product) of the vanishing ideals of the former, namely, $V(I)\cup V(J) = V(I\cap J)=V(IJ)$. This shows then that for $R$ points $P_1,...,P_R$ in $N$-space is the radical ideal of product of the ideals we described before, $\sqrt{I(P_1)...I(P_R)}$.
Finding the generators for the product is quite clear, the products of generators of the original ideals. Determining a minimal set of generators for the radical is more complicated however, but writing $I(P_1,...,P_R)=\sqrt{I(P_1)...I(P_R)}$ might suffice for most things.
I might be skipping some important discussions, but this is the most concrete answer I could come up with.
