# Homogeneous polynomial

A (multivariable) polynomial is called homogeneous if all monomials that make up the polynomial have the same degree . Homogeneous polynomials are also known as shapes .

## definition

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}$

${\ displaystyle p = \ alpha X_ {1} ^ {i_ {1}} \ cdot \ dots \ cdot X_ {n} ^ {i_ {n}}}$

exists. The degree of this monomial is

${\ displaystyle \ mathrm {deg} (p) = i_ {1} + \ dotsb + i_ {n}.}$

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.}$

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 :

${\ displaystyle R [X_ {1}, \ dotsc, X_ {n}] = \ bigoplus _ {d \ geq 0} A_ {d},}$

in which

${\ displaystyle A_ {d} = \ bigoplus _ {e_ {1} + \ dotsb + e_ {n} = d, \ e_ {i} \ geq 0} R \ cdot X_ {1} ^ {e_ {1}} \ cdot \ dots \ cdot X_ {n} ^ {e_ {n}}}$

is the set of homogeneous polynomials of degree together with the zero polynomial. It applies ${\ displaystyle d}$

${\ displaystyle A_ {d} \ cdot A_ {d '} \ subseteq A_ {d + d'},}$

the polynomial ring is thus a graduated ring .

## generalization

Generally called in a graduated ring

${\ displaystyle \ bigoplus _ {d \ geq 0} A_ {d}}$

the elements out homogeneous by degree . ${\ displaystyle A_ {d}}$${\ displaystyle d}$