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 $\map P \iota = f$.
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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 |
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