# Definition:Degree of Polynomial

## Contents

## One variable

Let $R$ be a commutative ring with unity.

Let $P \in R \sqbrk x$ be a nonzero polynomial over $R$ in one variable $x$.

The **degree** of $P$ is the largest natural number $k \in \N$ such that the coefficient of $x^k$ in $P$ is nonzero.

### General Ring

Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

Let $x \in R$.

Let $\displaystyle P = \sum_{j \mathop = 0}^n \left({r_j \circ x^j}\right) = r_0 + r_1 \circ x + \cdots + r_n \circ x^n$ be a polynomial in the element $x$ over $S$ such that $r_n \ne 0$.

Then the **degree of $P$** is $n$.

The **degree of $P$** can be denoted $\deg \left({P}\right)$ or $\partial P$.

### Integral Domain

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

Let $\left({D, +, \circ}\right)$ be an integral subdomain of $R$.

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

Let $\displaystyle f = \sum_{j \mathop = 0}^n \left({r_j \circ X^j}\right) = r_0 + r_1 X + \cdots + r_n X^n$ be a polynomial over $D$ in $X$ such that $r_n \ne 0$.

Then the **degree of $f$** is $n$.

The **degree of $f$** can be denoted $\deg \left({f}\right)$ or $\partial f$.

### Field

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.

Let $\struct {K, +, \times}$ be a subfield of $F$.

Let $x \in F$.

Let $\displaystyle f = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + \cdots + a_n x^n$ be a polynomial over $K$ in $x$ such that $a_n \ne 0$.

Then the **degree of $f$** is $n$.

The **degree of $f$** can be denoted $\map \deg f$ or $\partial f$.

## Sequence

### Ring

Let $f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right)$ be a polynomial over a ring $R$.

The **degree of $f$** is defined as the largest $n \in \Z$ such that $a_n \ne 0$.

### Field

Let $f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right)$ be a polynomial over a field $F$.

The **degree of $f$** is defined as the largest $n \in \Z$ such that $a_n \ne 0$.

## Polynomial Form

Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{{X_j: j \in J}\right\}$ for some multiindices $k_1, \ldots, k_r$.

Let $f$ **not** be the null polynomial.

Let $k = \left({k_j}\right)_{j \mathop \in J}$ be a multiindex.

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:

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

## Degree Zero

A polynomial $f \in S \left[{x}\right]$ in $x$ over $S$ is of **degree zero** if and only if $x$ is a non-zero element of $S$, that is, a constant polynomial.

## Null Polynomial

The null polynomial $0_R \in S \left[{X}\right]$ does *not* have a degree.

## Also known as

The **degree** of a polynomial $f$ is also sometimes called the **order of $f$**.

Some sources denote $\map \deg f$ by $\partial f$.

## Sources

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $1$