Definition:Degree of Polynomial

From ProofWiki
Jump to: navigation, search

One variable

Let $R$ be a commutative ring with unity.

Let $P \in R \left[{x}\right]$ 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 $\left({F, +, \times}\right)$ be a field whose zero is $0_F$.

Let $\left({K, +, \times}\right)$ be a subfield of $F$.

Let $x \in F$.


Let $\displaystyle f = \sum_{j \mathop = 0}^n \left({a_j x^j}\right) = a_0 + a_1 x + \cdots + a_n x^n$ 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 $\deg \left({f}\right)$ 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 $\deg \left({f}\right)$ by $\partial f$.


Sources