# Definition:Degree of Polynomial

## 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

The validity of the material on this page is questionable.In particular: ill-definedYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Questionable}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {S, +, \circ}$ be a subring of $R$.

Let $x \in R$.

Let $\ds P = \sum_{j \mathop = 0}^n \paren {r_j \circ x^j} = r_0 + r_1 \circ x + \cdots + r_n \circ x^n$ 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 $\map \deg P$ or $\partial P$.

### Integral Domain

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

Let $\struct {D, +, \circ}$ be an integral subdomain of $R$.

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

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

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

The **degree of $f$** is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ by $\map \deg f$.

### 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 $\ds f = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + \cdots + a_n x^n$ 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 $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ 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 $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ 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 $\family {X_j: j \in J}$ for some multiindices $k_1, \ldots, k_r$.

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

Let $k = \family {k_j}_{j \mathop \in J}$ be a multiindex.

Let $\ds \size k = \sum_{j \mathop \in J} k_j \ge 0$ be the degree of the monomial $\mathbf X^k$.

The **degree of $f$** is the supremum:

- $\ds \map \deg f = \max \set {\size {k_r}: i = 1, \ldots, r}$

## Degree Zero

A polynomial $f \in S \sqbrk x$ 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 $0_R \in S \sqbrk x$ does *not* have a degree.

## Examples

## Also known as

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

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

## Also see

- Results about
**the degree of a polynomial**can be found**here**.

## Sources

- 1960: Margaret M. Gow:
*A Course in Pure Mathematics*... (previous) ... (next): Chapter $1$: Polynomials; The Remainder and Factor Theorems; Undetermined Coefficients; Partial Fractions: $1.1$. Polynomials in one variable - 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(d)}$*Polynomials* - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $1$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**degree**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**degree**:**1.**

This article is complete as far as it goes, but it could do with expansion.In particular: The above 2 sources need to be adequately processed, but that requires a complete rebuild of the concept of a polynomial, and I don't have the juice for that todayYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**degree**(of a polynomial)