# Homogeneous

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In mathematics, **homogeneous** has a variety of meanings.

- In algebra, a homogeneous polynomial is a polynomial whose terms are monomials all having the same total degree; or are elements of the same dimension.
- A homogeneous function is a function
*f*satisfying*f*(α*v*) = α^{k}*f*(*v*) for some value of*k*. - A homogeneous differential equation is usually one of the form
*Lf*= 0, where*L*is a differential operator, the corresponding inhomogeneous equation being*Lf*=*g*with*g*a given function; the word*homogeneous*is also used of equations in the form*Dy*=*f*(*y*/*x*). - In linear algebra a
**homogeneous system**is a one of the form A**x**=**0**. - Homogeneous coordinates enable affine transformations, such as spatial translations, to be treated the same as linear transformations and thus represented by matrices.
- Homogeneous numbers share identical prime factors (may be repeated).
- A homogeneous space for a Lie group G, or more general transformation group, is a space X on which G acts transitively and continuously. Equivalently, a homogeneous space is a coset space G/H where H is a closed subgroup.
- As a special case of the previous meaning, a manifold is said to be
**homogeneous**for its homeomorphism group, or diffeomorphism group, if that group acts transitively on it; this is true for connected manifolds without boundary.

- As a special case of the previous meaning, a manifold is said to be
- Given a colouring of the edges of a complete graph, the term homogeneous applies to a subset of vertices such that all edges connecting two of the subset have the same colour; and in much greater generality in Ramsey theory for colourings of k-element subsets.
- This is also the notion of homogeneity used in (combinatorial) set theory: given a function
*f*:[X]^{α}→*C*colouring the set of subsets of*X*of order type α with colours from*C*, a subset*H*of*X*is called homogeneous for*f*if all elements of [*H*]^{α}get the same colour by*f*.

- This is also the notion of homogeneity used in (combinatorial) set theory: given a function
- In a graded ring, there is a concept of homogeneous ideal, important in algebraic geometry