Homogeneous function

This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of V such that

for every nonzero scalar s. In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity is often considered, by requiring only that the above identities hold for

The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.

A special case is the absolute value of real numbers.

The general one works for vector spaces over arbitrary fields, and is restricted to degrees of homogeneity that are integers.

The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.

Let V and W be two vector spaces over a field F. A linear cone in V is a subset C of V such that

Homogeneous functions play a fundamental role in projective geometry since any homogeneous function f from V to W defines a well-defined function between the projectivizations of V and W. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degree) play an essential role in the Proj construction of projective schemes.

When working over the real numbers, or more generally over an ordered field, it is commonly convenient to consider positive homogeneity, the definition being exactly the same as that in the preceding section, with "nonzero s" replaced by "s > 0" in the definitions of a linear cone and a homogeneous function.

This change allow considering (positively) homogeneous functions with any real number as their degrees, since exponentiation with a positive real base is well defined.

This is, in particular, the case of the absolute value function and norms, which are all positively homogeneous of degree 1.

The absolute value of a complex number is a positively homogeneous function of degree

As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.

between vector spaces over a field F is homogeneous of degree 1, by the definition of linearity:

The homogeneous real functions of a single variable have the form

Roughly speaking, Euler's homogeneous function theorem asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific partial differential equation.

More precisely: Euler's homogeneous function theorem — If f is a (partial) function of n real variables that is positively homogeneous of degree k, and continuously differentiable in some open subset of

then it satisfies in this open set the partial differential equation

Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree k, defined on a positive cone (here, maximal means that the solution cannot be prolongated to a function with a larger domain).

The first part results by using the chain rule for differentiating both sides of the equation

The converse is proved by integrating a simple differential equation.

), the theorem implies that a continuously differentiable and positively homogeneous function of degree k has the form

are homogeneous functions of the same degree, into the separable differential equation

The definitions given above are all specialized cases of the following more general notion of homogeneity in which

can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a monoid.

Here the angle brackets denote the pairing between distributions and test functions, and

is the mapping of scalar division by the real number

For instance, every additive map between vector spaces is homogeneous over the rational numbers

is a fixed real number then the above definitions can be further generalized by replacing the condition

For instance, A nonzero continuous function that is homogeneous of degree