Definition:Polynomial over Ring

Polynomial Form
Let $$I_X=\left\{1,X,X^2,\ldots\right\}$$ be  the  free monoid on a  singleton $$\{X\}$$.

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

A polynomial form in one variable, or just polynomial over  $$R$$ is a mapping $$f: I_X \to R: X^n \mapsto a_n$$ such that $$a_n = 0_R$$  for all but finitely many $$n \ge 0$$.

Therefore a polynomial is an ordered triple $$\left({R, I_X, f}\right)$$. We describe the polynomial as in the indeterminate $$X$$. Often, this singleton is unimportant, and we speak simply of the polynomial $$f$$ over the ring $$R$$.

Notation
It follows from Unique Representation in Polynomial Forms that if we let  $$a_i X^i$$ denote the polynomial that has value $$a_i$$ on $$X^i$$ and $$0_R$$ otherwise, then $$f$$ can be (uniquely) written


 * $f = a_0 + a_1 X + \cdots + a_{n-1} X^{n-1} + a_n X^n$

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


 * $\displaystyle f = \sum_{i=0}^\infty a_i X^i$

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


 * $\displaystyle \deg \left({f}\right) = \sup \left \{{n \in \N: f \left({X^n}\right) \ne 0}\right\}$

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
Let $$d$$ be the degree of $$\left({R, I_X, f}\right)$$.

Then the elements of the set $$\left\{{a_i: i \le d}\right\}$$ are known as its coefficients.

Leading Coefficient
Let $$d$$ be the degree of $$\left({R, I_X, f}\right)$$.

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

Monic Polynomial
If the leading coefficient $$a_n$$ of $$f$$ is $$1_R$$, then $$\left({R, I_X, f}\right)$$ is monic.

Null Polynomial
If $$a_i = 0_R$$ for all $$i \ge 0$$, $$\left({R, I_X, f}\right)$$ is known as the null polynomial or trivial polynomial over $$R$$ in the indeterminate $$X$$.

Mononomial
If $$a_i \ne 0_R$$ for at most one $$i \ge 0$$, then $$\left({R, I_X, f}\right)$$ is called a mononomial.

Polynomial Function
Let $$\left({K, I_X, f}\right)$$ be a polynomial over a field $$K \subseteq \C$$, and for each $$ x \in K$$, let $$\phi_x: K \left[{X}\right] \to K$$ be the Evaluation Homomorphism at $$x$$.

Then the set:
 * $\left\{{\left({x, \phi_r \left({x}\right)}\right): x \in K}\right\} \subseteq K \times K$

defines a polynomial function $$R \to R$$.

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