Definition:Polynomial Function/Real
Definition
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 $\iota$.
Coefficients
The parameters $a_0, \ldots a_n \in \R$ are known as the coefficients of the polynomial $P$.
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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.
Also see
- Results about real polynomial functions can be found here.
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): polynomial function