Homogeneous polynomial

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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

exists. The degree of this monomial is

A polynomial in is called homogeneous if it is a sum of monomials of equal degree.

properties

  • is homogeneous of degree if and only if :
  • 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
  • Simple examples of homogeneous polynomials in (see whole numbers ):
    • is homogeneous because of
    • is homogeneous because of
  • Examples of non-homogeneous polynomials in (see rational numbers ):
    • is not homogeneous because of
    • is not homogeneous because of and

graduation

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 :

in which

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

the polynomial ring is thus a graduated ring .

generalization

Generally called in a graduated ring

the elements out homogeneous by degree .

See also

Individual evidence

  1. ^ Fischer: Textbook of Algebra. 2013, p. 169, Lemma.
  2. ^ Fischer: Textbook of Algebra. 2013, p. 169, product of homogeneous polynomials.

literature