# 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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