Definition:Polynomial Function/Ring/Definition 1

Definition
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: such that for all $x\in S$:
 * a natural number $n \in \N$
 * $a_0, \ldots, a_n \in R$
 * $\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$

where $\sum$ denotes indexed summation.

Also see

 * Equivalence of Definitions of Polynomial Function on Subset of Ring