Definition:Polynomial Function
Real Numbers
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 $\iota$.
Complex Numbers
Let $S \subset \C$ be a subset of the complex numbers.
Definition 1
A complex polynomial function on $S$ is a function $f : S \to \C$ for which there exist:
- a natural number $n \in \N$
- complex numbers $a_0, \ldots, a_n \in \C$
such that for all $z \in S$:
- $\map f z = \ds \sum_{k \mathop = 0}^n a_k z^k$
where $\ds \sum$ denotes indexed summation.
Definition 2
Let $\C \sqbrk X$ be the polynomial ring in one variable over $\C$.
Let $\C^S$ be the ring of mappings from $S$ to $\C$.
Let $\iota \in \C^S$ denote the inclusion $S \hookrightarrow \C$.
A complex polynomial function on $S$ is a function $f: S \to \C$ which is in the image of the evaluation homomorphism $\C \sqbrk X \to \C^S$ at $\iota$.
Arbitrary Ring
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 $x\in S$:
- $\map f x = \ds \sum_{k \mathop = 0}^n a_k x^k$
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 $\iota$.
Multiple Variables
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$.
For each $x = \left({x_j}\right)_{j \in J} \in R^J$, let $\phi_x: R \left[{\left\{{X_j: j \in J}\right\}}\right] \to R$ be the evaluation homomorphism from the ring of polynomial forms at $x$.
Then the set:
- $\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in R^J}\right\} \subseteq R^J \times R$
defines a polynomial function $R^J \to R$.
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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.