Definition:Polynomial over Ring

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

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 indeterminates $$\{X_j:j\in J\}$$ 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


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

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


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

where $$Z$$ is the set of multiindices indexed by $$J$$. For a polynomial $$f$$ in a single indeterminate $$\{X\}$$, we can write
 * $f=a_0+a_1X+\cdots +a_nX^n$

for some $$n\in\N$$.

Degree
If $$\left({R, M, f}\right)$$ is not the null polynomial, its degree or order is the supremum


 * $\displaystyle \deg \left({f}\right) = \sup \left \{{\text{M-deg}\left({\mathbf X^k}\right):\mathbf X^k\in M,\ f \left({\mathbf X^k}\right) \ne 0}\right\}$

where $$\text{M-deg}\left({\mathbf X^k}\right)$$ is the degree of the mononomial $$\mathbf X^k$$. Sometimes $$\deg \left({f}\right)$$ is written $$\partial f$$.

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

Coefficients
Then the elements of the set $$\left\{{f\left({\mathbf X^k}\right)}:\mathbf X^k\in M\right\}$$ are called the coefficients of $$\left({R, M, f}\right)$$.

Leading Coefficient
Let $$d$$ be the degree of the polynomial $$\left({R, M, f}\right)$$ 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 $$a_n$$ of a polynomial $$f$$ in a single indeterminate $$X$$ is $$1_R$$, then $$\left({R, I_X, f}\right)$$ is monic.

Null Polynomial
If $$f\left({\mathbf X^k}\right) = 0_R$$ for all $$\mathbf X^k\in M$$, $$\left({R, M, f}\right)$$ is known as the null polynomial or trivial polynomial over $$R$$ in the indeterminates $$\left\{{X_j:j\in J}\right\}$$.

Polynomial Function
Let $$\left({K, M, f}\right)$$ be a polynomial over a field $$K \subseteq \C$$ in the indeterminates $$X_1,\ldots,X_n$$, and for each $$ x=(x_1,\ldots,x_n) \in K^n$$, let $$\phi_x: K \left[{X_1,\ldots,X_n}\right] \to K$$ be the Evaluation Homomorphism at $$x$$.

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

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

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.

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