It's common to think of polynomials as functions, but it was recently brought to my attention that this isn't exactly right. Consider, for example $x$ and $x^7$ in $\mathbb{Z} / 7\mathbb{Z} [x]$. These are distinct polynomials, but as functions $\mathbb{Z} / 7\mathbb{Z} \to \mathbb{Z} / 7\mathbb{Z}$ they are identical.
This made me wonder: how can we describe the kernel of the map $ \varphi : \mathbb{Z} / n\mathbb{Z} [x] \to A $ given by $\varphi(p) = (x \mapsto p(x))$ where \begin{align} A = \{p : \mathbb{Z} / n\mathbb{Z} &\to \mathbb{Z} / n\mathbb{Z} \\&\mid p \text{ is a polynomial with coefficients in } \mathbb{Z} / n\mathbb{Z}\} \end{align} (Is there a more standard notation for $A$?)
The only thing I know is that when $n$ is prime, $x^n - x \in \ker \varphi$. This is just Fermat's little theorem.
Can we give an explicit characterization of $\ker \varphi$? How should we think of elements of $\mathbb{Z} / n\mathbb{Z} [x]$ if not as functions $\mathbb{Z} / n\mathbb{Z} \to \mathbb{Z} / n\mathbb{Z}$.