homogeneous polynomial

Let R be an associative ring. A (multivariate) polynomialMathworldPlanetmathPlanetmathPlanetmath f over R is said to be homogeneous of degree r if it is expressible as an R-linear combinationMathworldPlanetmath (http://planetmath.org/LinearCombination) of monomialsMathworldPlanetmathPlanetmath of degree r:


where r=ri1++rin for all i{1,,m} and aiR.

A general homogeneous polynomialMathworldPlanetmathPlanetmath is also known sometimes as a polynomial form. A homogeneous polynomial of degree 1 is called a linear form; a homogeneous polynomial of degree 2 is called a quadratic formMathworldPlanetmath; and a homogeneous polynomial of degree 3 is called a cubic form.


  1. 1.

    If f is a homogeneous polynomial over a ring R with deg(f)=r, then f(tx1,,txn)=trf(x1,,xn). In fact, a homogeneous function that is also a polynomial is a homogeneous polynomial.

  2. 2.

    Every polynomial f over R can be expressed uniquely as a finite sum of homogeneous polynomials. The homogeneous polynomials that make up the polynomial f are called the homogeneous components of f.

  3. 3.

    If f and g are homogeneous polynomials of degree r and s over a domain R, then fg is homogeneous of degree r+s. From this, one sees that given a domain R, the ring R[𝑿] is a graded ringMathworldPlanetmath, where 𝑿 is a finite set of indeterminates. The condition that R does not have any zero divisorsMathworldPlanetmath is essential here. As a counterexample, in 6[x,y], if f(x,y)=2x+4y and g(x,y)=3x+3y, then f(x,y)g(x,y)=0.


  • f(x,y)=x2+xy+yx+y2 is a homogeneous polynomial of degree 2. Notice the middle two monomials could be combined into the monomial 2xy if the variables are allowed to commute with one another.

  • f(x)=x3+1 is not a homogeneous polynomial.

  • f(x,y,z)=x3+xyz+zyz+3xy2+x2-xy+y2+zy+z2+xz+y+2x+6 is a polynomial that is the sum of four homogeneous polynomials: x3+xyz+zyz+3xy2 (with degree 3), x2-xy+y2+zy+z2+xz (degree = 2), y+2x (degree = 1) and 6 (deg = 0).

  • Every symmetric polynomialMathworldPlanetmath can be written as a sum of symmetricPlanetmathPlanetmath homogeneous polynomials.

Title homogeneous polynomial
Canonical name HomogeneousPolynomial1
Date of creation 2013-03-22 14:53:42
Last modified on 2013-03-22 14:53:42
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 17
Author CWoo (3771)
Entry type Definition
Classification msc 16R99
Classification msc 13B25
Classification msc 16S36
Classification msc 11E76
Synonym polynomial form
Related topic HomogeneousIdeal
Related topic HomogeneousFunction
Related topic HomogeneousEquation
Defines homogeneous component
Defines cubic form
Defines linear form