Degree of a polynomial

In mathematics, the degree of a polynomial in a variable is the largest exponent in its standard representation as a sum of monomials. For example, the degree of the polynomial $2X^{5}-X^{3}+7X^{2}$ is equal to 5, namely the exponent of the monomial $2X^{5}$ . For polynomials in multiple variables, the degree of a monomial is defined as the sum of the exponents of the contained variable powers, and the degree of a polynomial (also called the total degree) is defined as the maximum of the degrees of the monomials that make up the polynomial. For example, the monomial $X^{2}Y^{3}Z$ and hence the polynomial $-3X^{2}Y^{3}Z+7X^{4}Y+XYZ^{2}$ have degree 6.

Definition

Let be a commutative ring, n>0 be a natural number, and be $R[X_{1},\dots ,X_{n}]$ the polynomial ring in variables $X_1, \dots, X_n$ . If

$0\neq m:=X_{1}^{{e_{1}}}X_{2}^{{e_{2}}}\cdots X_{n}^{{e_{n}}}\in R[X_{1},\dots ,X_{n}]$

a monomial with $e_{1},\dots ,e_{n}\in {\mathbb {N}}\cup \{0\}$ , then the degree of defined as.

$\deg(m):=e_{1}+\ldots +e_{n}$ .

Now be

$0\neq f=a_{1}m_{1}+\ldots +a_{r}m_{r}\in R[X_{1},\dots ,X_{n}]$

a polynomial with $r\in {\mathbb {N}}$ , $a_{1},\dots ,a_{r}\in R\setminus \{0\}$ and monomials $m_{1},\dots ,m_{r}$ . Then the degree or total degree of defined as.

$\deg(f):=\max _{{j=1,\dots ,r}}\deg(m_{j})$ .

There are several conventions for defining the degree of $0$ . In algebra, it is common to set $\deg(0):=-\infty$ . In contrast, the definition deg often $\deg(0):=-1$ preferred in areas of mathematics concerned with solving algebraic problems using computers.

Remark: Since monomials consist of only finitely many factors, the definition of the degree of a monomial and thus also the definition of the degree of a polynomial can be directly extended to polynomial rings in any number of variables.