# Definition:Polynomial Function/Real

## Definition

Let $S \subset \R$ be a subset of the real numbers.

### Definition 1

A **real polynomial function** on $S$ is a function $f: S \to \R$ for which there exist:

- a natural number $n\in \N$

- real numbers $a_0, \ldots, a_n \in \R$

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.

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

## Coefficients

The parameters $a_0, \ldots a_n \in \R$ are known as the coefficients of the polynomial $P$.

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## Also known as

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

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

## Also see

- Results about
**real polynomial functions**can be found**here**.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**polynomial function**