Definition:Polynomial Function/General Definition

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Let $R$ be a commutative ring with unity.

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 $R^J \to R$.

Also see

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 $\mathbb F_2$.

Also see

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

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