Homogeneous function
of degree
A function such that for all points
in its domain of definition and all real
, the equation
![]() |
holds, where is a real number; here it is assumed that for every point
in the domain of
, the point
also belongs to this domain for any
. If
![]() |
that is, is a polynomial of degree not exceeding
, then
is a homogeneous function of degree
if and only if all the coefficients
are zero for
. The concept of a homogeneous function can be extended to polynomials in
variables over an arbitrary commutative ring with an identity.
Suppose that the domain of definition of
lies in the first quadrant,
, and contains the whole ray
,
, whenever it contains
. Then
is homogeneous of degree
if and only if there exists a function
of
variables, defined on the set of points of the form
where
, such that for all
,
![]() |
If the domain of definition of
is an open set and
is continuously differentiable on
, then the function is homogeneous of degree
if and only if for all
in its domain of definition it satisfies the Euler formula
![]() |
Homogeneous function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_function&oldid=47253