Definition:Polynomial Function

Definition
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial 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 $R^J \to R$.

By Ring of Polynomial Functions the set of all polynomial functions is a commutative ring with unity.

This mapping from polynomial forms to polynomial functions is surjective by definition, but not necessarily injective.

For example, if $R = \mathbb F_2$ is the field with two elements, then $1 + X$ and $1 + X^2$ define the same function $\mathbb F_2 \to \mathbb F_2$, but are different polynomial forms over $\mathbb F_2$.

Let $K$ be a commutative ring with unity.

Let the mapping $p: K \to K$ be defined such that there exists a sequence:
 * $\left \langle {\alpha_k} \right \rangle_{k \in \left[{0 \,.\,.\, n}\right]}$

of elements of $K$ such that:
 * $\displaystyle p = \sum_{k \mathop = 0}^n \alpha_k {I_K}^k$

where $I_K$ is the identity mapping on $K$.

Then $p$ is known as a polynomial function on $K$.

Also see

 * Polynomial Form
 * Polynomial Equation
 * Polynomial Coefficient


 * Ring of Polynomial Forms
 * Ring of Polynomial Functions
 * Equality of Polynomials

By Equality of Polynomials, when $R$ is a field of characteristic $0$, the rings of polynomial forms and polynomial functions are isomorphic, and we usually identify the two.