The zeros of a multivariable polynomial

The other day I came across the following statement:

A polynomial $f(x,y)$ of degree at most $3$ that vanishes at $8$ of the $9$ points $(x,y)$ with $x, y \in \{-1,0,1\}$ must also vanish at the $9$th point.

I am wondering about how this statement generalizes. Specifically, I am looking for a theorem of the form

Suppose a polynomial $f(x_1, \ldots, x_n)$ of degree $d$ vanishes on a some discrete set $S \subset R^n$, which satisfies _______. Then, defining a discrete set $U$ as ______, the polynomial $f$ must also vanish on $U$.

Can anyone fill in the blanks?

Solution 1:

The statement you mention is a special case of the Cayley-Bacharach theorem. For one way of generalizing the theorem, see the beautiful article "Cayley-Bacharach theorems and conjectures" by Eisenbud-Green-Harris.