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