Let $P(x_{1},\ldots,x_{n}):\mathbb{F}_{2}^{n}\rightarrow \mathbb{F}_{2}$ be a polynomial function with degree $d$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{2}$.
Let $S(P)=\{ (x_{1},\ldots,x_{n}): P(x_{1},\ldots,x_{n})=0 \}$ which is a set of roots for an arbitrary polynomial function $P$.
I am wondering if we can construct another polynomial funciton $Q(x_{1},\ldots,x_{n}):\mathbb{F}_{q}^{n} \rightarrow \mathbb{F}_{q}$ with degree $d'$ and with variables $x_{1},\ldots,x_{n} \in \mathbb{F}_{q} (q>2)$ such that $S(Q)\cap \mathbb{F}_{2}^{n} = S(P)$.
I am not a pure math student, but am a computer science theory student.
So my main interest is in a possibility of procedural construction of $Q$ for given $P$.
Thank you for the comments. I am considering an exapmle: $\{(0,0,0),(0,0,1),(1,1,0)\}\subseteq \mathbb{F}_{2}$ is $S(x_{1}x_{2}x_{3}+x_{1}+x_{2})$. My consideration is about the possibility to construct a polynomial $Q(x_{1},x_{2},x_{3})$ such that $Q(0,0,0)=Q(0,0,1)=Q(1,1,0)=0$, where $Q:\mathbb{F}_{q}^{n}\rightarrow \mathbb{F}_{q}$.
Let $F$ be a finite field. Then $f\colon x\mapsto \prod_{a\in F\setminus\{0\}}(x-a)$ is a polynomial function that is nonzero iff $x=0$. Therefore, for any subset $S\subseteq F^n$, $$ \sum_{(a_1,\ldots,a_n)\notin S}\prod_{i=1}^nf(x-a_i)$$ is a polynomial function $F^n\to F$ which is zero precisely for the points in $S$.