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 in the indeterminate variables $\left\{{X_j: j \in J}\right\}$ is a mapping $f: M \to R: \mathbf X^k \mapsto a_k$ such that $a_k = 0$ for all but finitely many $\mathbf X^k \in M$.

Therefore a polynomial is an ordered triple $\left({R, M, f}\right)$.

Polynomial Form in a Single Indeterminate
If $\{X_j:j\in J\}=\{X\}$ then $M=\left\{1,X,X^2,\ldots\right\}$ is free monoid on a  singleton $\{X\}$.

In this case the singleton $\{X\}$ 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


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

This is the notation most frequently used when working with polynomials. It is also sometimes helpful to include the tailing 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 $J$.

For a polynomial $f$ in a single indeterminate $\left\{{X}\right\}$, we can write:
 * $f = a_0 + a_1 X + \cdots + a_n X^n$

for some $n \in \N$.

Addition and Multiplication of Polynomials
Let $\displaystyle f = \sum_{k\in Z}a_k\mathbf X^k$, $\displaystyle g = \sum_{k\in Z}b_k\mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

We define the sum


 * $\displaystyle f\oplus g=\sum_{k\in Z}(a_k+b_k)\mathbf X^k$.

It follows from Polynomials Closed Under Addition  that $f\oplus g$ is a polynomial.

We define the product


 * $\displaystyle f\otimes g=\sum_{k\in Z}c_k\mathbf X^k$

where


 * $\displaystyle c_k=\sum_{\substack{p+q=k\\p,q\in Z}}a_{p}b_{q}$.

It follows from Polynomials Closed Under Ring Product  that $f\otimes g$ is a polynomial.

It is common to use the symbols for addition and multiplication in the underlying ring $R$ for addition and multiplication of polynomials over the ring also. Generally, there is little room for confusion because the operations on polynomials generalise those in $R$.

Degree
Let $f = a_1\mathbf X^{k_1}+\cdots + a_r\mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{ { X_j:j\in J } \right\}$ that is not the null polynomial for some multiindices $k_1,\ldots,k_r$.

For a multiindex $k=\left( {k_j} \right)_{j\in J}$, let $\displaystyle |k|=\sum_{j\in J}k_j\geq 0$ be the degree of the mononomial $\mathbf X^k$.

Then degree or order of $f$ is the supremum


 * $\displaystyle \deg \left({f}\right) = \max \left \{ { |k_r|:i=1,\ldots,r } \right\}$

Sometime sources write $\deg \left({f}\right)$ as $\partial f$.

The null polynomial is sometimes defined to have degree $-\infty$, but is left undefined in many sources.

Coefficients
The coefficients of a polynomial $f = a_1\mathbf X^{k_1}+\cdots + a_r\mathbf X^{k_r}$ are the elements of the set $\left\{{a_i:i=1,\ldots,r}\right\}$.

Leading Coefficient
Let $d$ be the degree of the polynomial $f=a_0+\cdots+a_dX^d$ in a single indeterminate $X$.

The ring element $a_d$ is called the leading coefficient of $\left({R, M, f}\right)$.

Monic Polynomial
If the leading coefficient of a polynomial $f$ in a single indeterminate $X$ is $1_R$, then $f$ is monic.

Null Polynomial
If $f = \sum_{k\in Z}a_k\mathbf X^k$ is a polynomial such that $a_k=0_R$ for all $k\in Z$, then $f$ is called the null polynomial or trivial polynomial.

Polynomial Function
Let $f = a_1\mathbf X^{k_1}+\cdots + a_r\mathbf X^{k_r}$ be a polynomial over a field $K$ in the indeterminates $\left\{{X_j: j \in J}\right\}$, and for each $x = \left({x_j}\right)_{j\in J} \in K^J$, let $\phi_x: K \left[\left\{{X_j: j \in J}\right\}\right] \to K$ be the Evaluation Homomorphism from the ring of polynomial forms at $x$.

Then the set:
 * $\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in K^J}\right\} \subseteq K^J \times K$

defines a polynomial function $K^J \to K$.

We distinguish between a polynomial function and a polynomial form because two distinct polynomial forms may define the same function.

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 polynomials.

By Equality of Polynomials, this cannot happen when the characteristic of $K$ is zero.

Also note that for a polynomial function associated to a formal polynomial, it is common to write $f \left({x}\right)$ for the image of the function at a field element $x \in K^n$. This should not be confused with the value of the polynomial on the free commutative monoid $M$.

Polynomial Equation
A polynomial equation is an equation in the form:
 * $P_1 \left({x_1}\right) + P_2 \left({x_2}\right) + \cdots + P_n \left({x_n}\right) = 0$

where each of $P_1, P_2, \ldots, P_n$ are polynomials functions over a field $R$

Also see

 * Definition of Polynomial from Polynomial Ring