Definition:Polynomial over Ring

Definition
Polynomials arise in two major contexts in pure mathematics:
 * $(1): \quad$ Galois theory amounts to the study of a polynomial ring in a single indeterminate over a field
 * $(2): \quad$ In algebraic geometry the object of interest is the polynomial ring in $n$ indeterminates over an algebraically closed field.

These are the examples to keep in mind when interpreting the abstract definitions below.

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

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


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

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


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

It follows from Polynomials Closed under Addition that $f + g$ is a polynomial.

We define the product
 * $\displaystyle f \circ g = \sum_{k \in Z} c_k \mathbf X^k$

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 $f\circ g$ is a polynomial.

We have followed the convention of using 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 $\displaystyle \left|{k}\right| = \sum_{j \in J} k_j \ge 0$ be the degree of the mononomial $\mathbf X^k$.

Then degree or order of $f$ is the supremum:
 * $\displaystyle \deg \left({f}\right) = \max \left\{{\left| {k_r} \right|: i = 1, \ldots, r}\right\}$

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 $f = a_0 + \cdots + a_d X^d$ 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 $\displaystyle f = \sum_{k \in Z} a_k \mathbf X^k$ 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 $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$, and 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.

Let $\theta$ be the mapping from polynomial forms to polynomial functions defined in this way.

$\theta$ trivially proves to be a homomorphism, and is surjective by definition.

We call $\theta$ the function-form epimorphism (note that this is not standard terminolology).

By Equality of Polynomials, when $R$ is an infinite field, $\theta$ is an isomorphism.

This is not the case for finite fields.

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 $\mathbb F_2$.

It is common to write $R\left[{ \left\{{X_j: j \in J}\right\} }\right]$ for the ring of polynomial functions as well as the ring of polynomial forms when working with infinite fields, due to the isomorphism defined by $\theta$.

It is important to understand the distinction between $f(\mathbf X^k)$, the coefficient of the polynomial form $f$ on the mononomial $\mathbf X^k$, and $f(x)$, the value of the polynomial function $f$ at $x\in R^J$.

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


 * $f_1 \left({x}\right) + f_2 \left({x}\right) + \cdots + f_n \left({x}\right) = 0$

where each of $f_1, f_2, \ldots, f_n$ are polynomials functions over $R$

Also see

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