# Equivalence of Definitions of Polynomial Function on Subset of Ring

## Theorem

Let $R$ be a commutative ring with unity.

Let $S \subset R$ be a subset.

The following definitions of the concept of **polynomial function** are equivalent:

### 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 $\iota$.

## Outline of proof

This follows by interpreting both definitions as $f$ being a polynomial in the element $\iota$.

## Proof

### 1 implies 2

Let $\map P X \in R \sqbrk X$ be the polynomial:

- $P = \ds \sum_{k \mathop = 0}^n a_k \cdot X^k$

where $\sum$ denotes indexed summation.

We show that $\map P \iota = f$.

This needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### 2 implies 1

Let $P \in R \sqbrk X$ be a polynomial and $f = \map P \iota \in R^S$.

By Polynomial is Linear Combination of Monomials, there exist:

- $n \in \N$
- $a_0, \ldots, a_n \in R$

such that:

- $P = \ds \sum_{k \mathop = 0}^n a_k \cdot X^k$

where $\sum$ denotes indexed summation.

Let $\operatorname{ev}_{\iota}$ denote the evaluation homomorphism at $\iota$.

Then $\map {\operatorname{ev}_{\iota} } P = f$.

We have:

\(\ds f\) | \(=\) | \(\ds \map {\operatorname{ev}_{\iota} } {\sum_{k \mathop = 0}^n a_k \cdot X^k}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \map {\operatorname{ev}_{\iota} } {a_k \cdot X^k}\) | Ring Homomorphism Preserves Indexed Summations | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k}\) | Definition of Polynomial Evaluation Homomorphism | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n a_k \cdot \map {\operatorname{ev}_{\iota} } {X^k }\) | Definition of Polynomial Evaluation Homomorphism |

This needs considerable tedious hard slog to complete it.In particular: but move the calculation to Explicit Formula for Polynomial Evaluation HomomorphismTo discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |