Homogeneous polynomial

A polynomial is homogeneous, if all monomers making up the polynomial having the same degree.


Let be a commutative ring with unity and the polynomial ring over in indeterminates. A monomial then a polynomial for which a with

Exists. The degree of this monomial

A polynomial is homogeneous in called, if it is a sum of monomials of the same degree.


  • Is homogeneous of degree, it shall


  • Each monomial is homogeneous.
  • The set of all homogeneous polynomials in, the polynomial ring in one variable over is given by
  • Simple Examples of homogeneous polynomials in (see integers ):   is homogeneous since.
  • Is homogeneous since.
  • Is not uniform because.
  • Is not homogeneous, as, but.


Every polynomial can be written uniquely as a sum of homogeneous polynomials of different degrees, by summarizing all monomials of the same degree. The polynomial can thus be written as a direct sum:

In which

The set of homogeneous polynomials of degree together with the zero polynomial is. It is

The polynomial is thus a graded ring.


General hot in a graduated ring

The elements of homogeneous of degree d