# 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

**How to Cite This Entry:**

Homogeneous function. L.D. Kudryavtsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Homogeneous_function&oldid=11366