Homogeneous (mathematics)

In mathematics, homogeneous has a variety of meanings.

• In algebra, a homogeneous polynomial is one 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(αv) = α^kf(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 Ax=0.
• 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 – so equivalently 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.
• Given a colouring of the edges of a complete graph, the term homogeneous applies to a subset of vertices such that all edge connecting two of the subset have the same colour; and in much greater generality in Ramsey theory for colourings of k-element subsets.