This is not a specific question, but rather a question about possible techniques approaching a problem. Although this question came from research, it might not fit this forum; in which case I will delete it.
Let $U \subset \mathbb{R}^n$ be a finite set of size $k$, and let $R = \mathbb{R}[x_1,\ldots, x_n]$ be the ring of polynomials over $n$ variables with real coefficients. Define the ideal $I(U) = \{P \in R | \forall u \in U : P(u) = 0 \}$.
I am interested in understanding the properties of polynomials in $I(U)$, given $U$; more precisely, in which cases there is a polynomial $P \in I(U)$ that can be written as a product of $m(k) << k$ affine hyperplanes (the trivial case is when $U$ is contained in a single hyperplane)? What if we require additional properties from the hyperplanes (e.g. we are only allowed to use coefficients from a certain set)? And suppose we want to find the "worst" $U$ of a given size, in the sense that in order to write any polynomial in $I(U)$ as a product of hyperplanes, we will need "many" hyperplanes?
I am mostly looking for references, or names of known approaches for this problem. I know that algebraic geometry deals with similar question (to some extent), but I know absolutely nothing about it, and I am not sure I know what is the "right tool" to approach this problem.
