\begin{definition}[Definition:Polynomial Function/Complex/Definition 2]
Let $S \subset \C$ be a subset of the complex numbers.
Let $\C \sqbrk X$ be the polynomial ring in one variable over $\C$.
Let $\C^S$ be the ring of mappings from $S$ to $\C$.
Let $\iota \in \C^S$ denote the inclusion $S \hookrightarrow \C$.
A '''complex polynomial function''' on $S$ is a function $f: S \to \C$ which is in the image of the evaluation homomorphism $\C \sqbrk X \to \C^S$ at $\iota$.
\end{definition}