Definition:Polynomial over Ring

Polynomial Form
Let $M$ be the free commutative monoid on the indexed set $\left\{{X_j: j \in J}\right\}$.

Let $\left({R, +,  \circ}\right)$ be a  commutative ring with unity  with  additive identity $0_R$ and multiplicative identity $1_R$.

A polynomial form or just polynomial over $\left\{{X_j: j \in J}\right\}$ is a mapping $f: M \to R: \mathbf X^k \mapsto a_k$ such that $a_k = 0$ for all but finitely many $\mathbf X^k \in M$.

The elements of the set $\left\{{X_j: j \in J}\right\}$ are called indeterminates.

If $\left\{{X_j: j \in J}\right\} = \left\{{X}\right\}$ is a singleton, then the indeterminate $\left\{{X}\right\}$ is often unimportant, and we speak of the polynomial $f$ over the ring $R$.

Notation
It follows from Unique Representation in Polynomial Forms that if we let $a_k \mathbf X^k$ denote the polynomial that has value $a_k$ on $\mathbf X^k$ and $0_R$ otherwise, then $f$ can be uniquely written as a finite sum of non-zero summands:


 * $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$

Or non-uniquely by relaxing the condition $a_i \neq 0$, $i = 1,\ldots, r$.

This is the notation most frequently used when working with polynomials.

It is also sometimes helpful to include all the zero terms in this sum, in which case


 * $\displaystyle f = \sum_{k \in Z} a_k \mathbf X^k$

where $Z$ is the set of multiindices indexed by $J$.

Polynomial Function
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 $\mathbb F_2$.

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.

Polynomial Equation
A polynomial equation is an equation in the form:


 * $f \left({x}\right) = 0$

where $f$ is a polynomial function.

Also see

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