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:
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