\begin{definition}[Definition:Polynomial Function/Complex/Definition 1]
Let $S \subset \C$ be a subset of the complex numbers.
A '''complex polynomial function''' on $S$ is a function $f : S \to \C$ for which there exist:
:a natural number $n \in \N$
:complex numbers $a_0, \ldots, a_n \in \C$
such that for all $z \in S$:
:$\map f z = \ds \sum_{k \mathop = 0}^n a_k z^k$
where $\ds \sum$ denotes indexed summation.
\end{definition}