# Definition:Polynomial Function

## Real Numbers

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 $\iota$.

## Complex Numbers

Let $S \subset \C$ be a subset of the complex numbers.

### Definition 1

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

- a natural number $n \in \N$
- complex numbers $a_0, \ldots, a_n \in \C$

such that for all $z \in S$:

- $\map f z = \ds \sum_{k \mathop = 0}^n a_k z^k$

where $\ds \sum$ denotes indexed summation.

### Definition 2

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

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

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

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

## Arbitrary Ring

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 $x\in S$:

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

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 **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 $\iota$.

## Multiple Variables

Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$.

For each $x = \left({x_j}\right)_{j \in J} \in R^J$, let $\phi_x: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to R$ be the evaluation homomorphism from the ring of polynomial forms at $x$.

Then the set:

- $\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in R^J}\right\} \subseteq R^J \times R$

defines a **polynomial function** $R^J \to R$.

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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**.