Degree (polynomial)

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

definition

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

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

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

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

Be now

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

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

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

There are several conventions used to define the degree of . In algebra , it is customary to set. In contrast, in the areas of mathematics that deal with solving algebraic problems with the help of computers, the definition is often preferred. ${\displaystyle 0}$ ${\displaystyle \deg(0):=-\infty }$ ${\displaystyle \deg(0):=-1}$

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

properties

Let be polynomials over . Then applies ${\displaystyle f,g\in R[X_{1},\dots ,X_{n}]}$ ${\displaystyle R}$

• ${\displaystyle \deg(fg)\leq \deg(f)+\deg(g)}$ and
• ${\displaystyle \deg(f+g)\leq \max(\deg(f),\deg(g))}$.

In case you even get . ${\displaystyle \deg(f)\neq \deg(g)}$ ${\displaystyle \deg(f+g)=\max(\deg(f),\deg(g))}$

If there is an integrity ring , then it even applies ${\displaystyle R}$

${\displaystyle \deg(fg)=\deg(f)+\deg(g)}$

for everyone . ${\displaystyle f,g\in R[X_{i}\;|\;i\in I]}$

Examples

Consider polynomials in (see whole numbers ). It applies ${\displaystyle \mathbb {Z} [X,Y,Z]}$

• ${\displaystyle \deg(X^{5})=5}$,
• ${\displaystyle \deg(X^{2}Y^{3}Z^{4})=2+3+4=9}$,
• ${\displaystyle \deg(X^{7}Z^{2}+3X^{3}Y^{3}-XY^{4}Z+5YZ)=\deg(X^{7}Z^{2})=9}$ and
• ${\displaystyle \deg(3X^{4}Y^{4}-X^{2}Y^{3}Z^{3}+3Y^{4}Z)=\deg(X^{4}Y^{4})=\deg(X^{2}Y^{3}Z^{3})=8}$.

See also

• Graduation (algebra)
• Valuation theory