Definition:Polynomial Function/Complex
Definition
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 $\iota$.
Coefficients
The parameters $a_0, \ldots a_n \in \C$ 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
- Equivalence of Definitions of Complex Polynomial Function
- Definition:Polynomial over Complex Numbers
- Results about complex polynomial functions can be found here.