Definition:Polynomial Function

Definition

A polynomial function is a function whose values are given by a polynomial.

Real Numbers

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.

Complex Numbers

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.

Arbitrary Ring

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.

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.

Also see

• Definition:Ring of Polynomial Functions
• Definition:Polynomial
• Definition:Polynomial Equation
• Definition:Polynomial in Ring Element

• Results about polynomial functions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polynomial
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polynomial