(Not sure this question fits here, I will remove it in case it doesn't)

Let $F_{\mathrm{ML}}[x_1, \ldots,x_n]$ denote the set of multilinear polynomials over a finite field $F=\mathbb{F}_q$ (i.e. a sum of monomials, such that in every monomial the degree of each variable is less than or equal to $1$) and let $S \subset q^n\,(=F^n)$ be a set of vectors that lies in dimension $m$ (there are $m < n$ independent vectors in $S$).

Is it possible to find a projection $A : q^n \rightarrow q^m$ ($A$ depends on $S$) such that for all $f \in F_{\mathrm{ML}}[x_1, \ldots,x_n]$, there exists $g \in F_{\mathrm{ML}}[x_1, \ldots,x_m]$ such that for all $x \in S$, $f(x) = g(Ax)$? What are the conditions that $S$ has to satisfiy in order for such $A$ to exist? What if we remove the constraint about $S$ being in a subspace?

It is not difficult for some types of sets, for example, if $S = \{x \in q^n : x_{i} = 1: m < i \leq n \}$, then for each $f \in F_{\mathrm{ML}}[x_1, \ldots,x_n]$ it is possible to project $f$ to $F_{\mathrm{ML}}[x_1, \ldots,x_m]$ and the requirement holds.