# Definition:Polynomial Function/Ring

## Definition

Let $R$ be a commutative ring with unity.

Let $S \subset R$ be a subset.

### Definition 1

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 $x\in S$:

$\map f x = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$

where $\sum$ denotes indexed summation.

### Definition 2

Let $R \sqbrk X$ be the polynomial ring in one variable over $R$.

Let $R^S$ be the ring of mappings from $S$ to $R$.

Let $\iota \in R^S$ denote the inclusion $S \hookrightarrow R$.

A polynomial function on $S$ is a mapping $f : S \to R$ which is in the image of the evaluation homomorphism $R \sqbrk X \to R^S$ at $\iota$.