Homogeneous polynomial

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

Definition

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.

Properties

• Is homogeneous of degree, it shall

Examples

• 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.

Graduation

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.

Generalization

General hot in a graduated ring

The elements of homogeneous of degree d