Definition:Degree of Polynomial

From ProofWiki
Jump to navigation Jump to search

This page is about degree of polynomial. For other uses, see degree.

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-defined
You 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

Arbitrary Example

The polynomial:

$5 x^2 y^2 + 2 z - 1$

is of degree $4$.


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


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 today
You 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.


Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Degree_of_Polynomial&oldid=743774"