Definition:Polynomial Function/Complex/Definition 2

Definition
Let $S \subset \C$ be a subset of the complex numbers. Let $\C[X]$ be the polynomial ring in one variable over $\C$.

Let $\C^S$ be the ring of functions from $S$ to $\C$.

Let $\operatorname{id}$ 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 $\C[X] \to \C^S$ at $\operatorname{id}$.

Also see

 * Equivalence of Definitions of Complex Polynomial Function