Definition:Degree of Polynomial

General Ring
Let $R$ be a ring with unity.

Let $R \left[{x}\right]$ be the polynomial over $R$ in $x$.

Let $z = r_0 + r_1 x + \cdots + r_n x^n$ be an element of $R \left[{x}\right]$.

Then the degree of $z$ is $n$, and can be denoted $\partial z = n$.

Integral Domain
Let $\left({R, +, \circ}\right)$ be a commutative ring whose zero is $0_R$.

Let $\left({D, +, \circ}\right)$ be an integral domain such that $D$ is a subring of $R$.

Let $X \in R$ be transcendental over $D$.

Let $D \left[{X}\right]$ be the ring of polynomial forms in $X$ over $D$.

Let $f$ be a non-zero element of $D \left[{X}\right]$.

By Unique Representation in Polynomial Forms‎, there is one way of expressing $f$ as a polynomial:

$f \in D \left[{X}\right]: f = \sum_{k\mathop = 0}^n {a_k \circ X^k}$

In particular, the coefficients $a_0, a_1, \ldots, a_n$ are uniquely determined by $f$.

The number $n$ is called the degree of $f$, or its order, and can be denoted $\partial f$.

Degree Zero
A polynomial of degree zero is a non-zero element of $D$, that is, a constant.

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

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

Some 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.

Also known as
The degree of a polynomial $f$ is also sometimes called the order of $f$.