Definition:Polynomial Function/Ring

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Definition

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 $x\in S$:

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

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 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 $\iota$.


Also see