Let me start with some background: I am studying defining ideal of finite sets of points and have recently noticed the importance of the Hilbert function and polynomial in investigating this problem. However, being new to these concepts, I have encountered certain problems that I hope you could help with.
My main interest is in determining (or rather proving) that a ideal $I$ is in fact the defining ideal of a finite set of points (and also know the cardinality of this set of points). As an easy example, let us look at the ideal $(x_2^2-x_1^2,x_3^2-x_1^2) \subset T:=\mathbb{C}[x_1,x_2,x_3]$. I want to show $I$ is the ideal of $4$ points, and I read the Hilbert polynomial of $\frac{T}{I}$ plays a crucial role here. I clearly see that the Hilbert polynomial of $\frac{T}{I}$ is the constant $4$. I suppose the conclusion now follows from the following two claims
The degree of the Hilbert polynomial of $\frac{T}{I}$ equals the dimension of $V(I)$. However I failed to find a clear proof of this fact, which would at least give an algorithm to show whether or not an ideal defines a finite set of points. Just compute the Hilbert polynomial of $\frac{T}{I}$ (which I can do with Macauley2 in hard cases) and check if the degree is $0$. Can someone point me in the right direction of a proof for this fact or recommend a source?
Suppose $I$ defines a finite set of points $X$. Then we already know that its Hilbert polynomial is a constant. But now even stronger, this constant is $|X|$. Can someone again lead me in the right direction to prove this statement? (Assuming it is correct)
Is my reasoning above correct?
Any help regarding this problem is greatly appreciated!