# 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
• ${\ displaystyle \ deg (3X ^ {4} Y ^ {4} -X ^ {2} Y ^ {3} Z ^ {3} + 3Y ^ {4} Z) = \ deg (X ^ {4} Y ^ {4}) = \ deg (X ^ {2} Y ^ {3} Z ^ {3}) = 8}$.