Let be a commutative ring with one and the polynomial ring over in indefinite. A monomial is then a polynomial for which one with
${\ displaystyle R}$${\ displaystyle R [X_ {1}, \ dotsc, X_ {n}]}$${\ displaystyle R}$${\ displaystyle n}$${\ displaystyle p \ in R [X_ {1}, \ dotsc, X_ {n}]}$${\ displaystyle \ alpha \ in R}$

A polynomial in is called homogeneous if it is a sum of monomials of equal degree.
${\ displaystyle R [X_ {1}, \ dotsc, X_ {n}]}$

properties

${\ displaystyle f \ in R [X_ {1}, \ dotsc, X_ {n}]}$is homogeneous of degree if and only if :${\ displaystyle k}$${\ displaystyle R [X_ {1}, \ dotsc, X_ {n}] [T]}$

${\ displaystyle f (TX_ {1}, \ dotsc, TX_ {n}) = T ^ {k} \ cdot f (X_ {1}, \ dotsc, X_ {n})}$

With a polynomial ring over an integrity ring, a product of polynomials is homogeneous if and only if every factor is homogeneous.

Examples

Every monomial is homogeneous.

The set of all homogeneous polynomials in , the polynomial ring in a variable above , is given by${\ displaystyle R [X]}$${\ displaystyle R}$

${\ displaystyle \ {aX ^ {n} \; \ mid \; a \ in R, \; n \ in \ mathbb {N} \ cup \ {0 \} \}.}$

Simple examples of homogeneous polynomials in (see whole numbers ):
${\ displaystyle \ mathbb {Z} [X, Y]}$

${\ displaystyle X ^ {4} -Y ^ {4}}$ is homogeneous because of ${\ displaystyle \ deg (X ^ {4}) = \ deg (Y ^ {4}) = 4.}$

${\ displaystyle X ^ {7} + 5X ^ {3} Y ^ {4} + XY ^ {6}}$ is homogeneous because of ${\ displaystyle \ deg (X ^ {7}) = \ deg (X ^ {3} Y ^ {4}) = \ deg (XY ^ {6}) = 7.}$

Examples of non-homogeneous polynomials in (see rational numbers ):
${\ displaystyle \ mathbb {Q} [X, Y, Z]}$

${\ displaystyle X ^ {4} Z - {\ frac {3} {4}} YZ ^ {2}}$ is not homogeneous because of ${\ displaystyle \ deg (X ^ {4} Z) = 5 \ neq 3 = \ deg (YZ ^ {2}).}$

${\ displaystyle X ^ {3} Y ^ {3} Z ^ {2} -3X ^ {2} Y ^ {6} - {\ frac {7} {3}} Y ^ {5}}$is not homogeneous because of and${\ displaystyle \ deg (X ^ {3} Y ^ {3} Z ^ {2}) = \ deg (X ^ {2} Y ^ {6}) = 8}$${\ displaystyle \ deg (Y ^ {5}) = 5.}$

graduation

Every polynomial can be written in a unique way as the sum of homogeneous polynomials of different degrees by combining all monomials of the same degree. The polynomial ring can thus be written as a direct sum :