Definition:Polynomial Function/Ring/Definition 1

Definition

Let $R$ be a commutative ring with unity.

Let $S \subset R$ be a subset of $R$.

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.

Also known as

A polynomial function is often simply called polynomial.

Some sources refer to it as a rational integral function.

Also see

• Equivalence of Definitions of Polynomial Function on Subset of Ring

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 27$. Subspaces and Bases: Example $27.4$