# 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 $x \in S$:

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

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 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 $P$.