# Definition:Polynomial Function/Ring/Definition 1

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## 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:

- 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.

## Also known as

A **polynomial function** is often simply called **polynomial**.

Some sources refer to it as a **rational integral function**.

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.4$