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}}$

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}}$

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

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}$

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}$