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. L.D. Kudryavtsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Homogeneous_function&oldid=11366