Based on the proof that arctic and I have discussed (and amended afterwards). Given $S \subset \Bbb{A}$ and $S$ infinite, if $\exists p \in \Bbb{C}[T]$ then $p(\Bbb{A})=0$.
That means, for $S \subset \Bbb{A}$ finite, the number of $p \in \Bbb{C}[T]$ such that $p(S)=0$ is going to be the set of all elements $(\{p\},\{S\})$ that maps to zero.