In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree. That is, if is an integer, a function of variables is homogeneous of degree if

This is also referred to a th-degree or th-order homogeneous function.

For example, a homogeneous polynomial of degree defines a homogeneous function of degree .

The above definition extends to functions whose domain and codomain are vector spaces over a field between two -vector spaces is homogeneous if This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of for every nonzero scalar .

In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity and allowing any real number as a degree of homogeneity. Every homogeneous real function is positively homogeneous. 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 norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of projective schemes.

Definitions

The concept of a homogeneous function was originally introduced for functions of several real variables. With the definition of vector spaces at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a tuple of variable values can be considered as a coordinate vector. It is this more general point of view that is described in this article.

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

The second one supposes to work over the field of real numbers, or, more generally, over an ordered field. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called positive homogeneity, the qualificative positive being often omitted when there is no risk of confusion. Positive homogeneity leads to considering more functions as homogeneous. For example, the absolute value and all norms are positively homogeneous functions that are not homogeneous.

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

General homogeneity

Let and be two vector spaces over a field . A linear cone in is a subset of

A homogeneous function from to is a partial function from to that has a linear cone as its domain, and satisfies

for some integer The integer is called the degree of homogeneity, or simply the degree of .

A typical example of a homogeneous function of degree is the function defined by a homogeneous polynomial of degree . The rational function defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its cone of definition is the linear cone of the points where the value of denominator is not zero.

Homogeneous functions play a fundamental role in projective geometry since any homogeneous function from to defines a well-defined function between the projectivizations of and . 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.

Positive homogeneity

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 " replaced by "" in the definitions of a linear cone and a homogeneous function.

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

Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of the absolute value function and norms, which are all positively homogeneous of degree This remains true in the complex and every complex vector space can be considered as real vector spaces.

Euler's homogeneous function theorem is a characterization of positively homogeneous differentiable functions, which may be considered as the fundamental theorem on homogeneous functions.

Examples

Simple example

Absolute value and norms

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

The absolute value of a complex number over the real numbers (that is, when considering the complex numbers as a vector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.

More generally, every norm and seminorm is a positively homogeneous function of degree which is not a homogeneous function. 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.

Linear maps

Any linear map between vector spaces over a field

Similarly, any multilinear function

Homogeneous polynomials

Monomials

A homogeneous polynomial is a polynomial is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Min/max

the following functions are positively homogeneous of degree 1, but not homogeneous:

• (Leontief utilities)

Rational functions

Rational functions formed as the ratio of two polynomials are homogeneous functions in their domain, that is, off of the linear cone formed by the zeros

Non-examples

The homogeneous real functions for some constant . So, the affine function the natural logarithm and the exponential function are not homogeneous.

Euler's theorem

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:

its first-order partial derivatives This results from Euler's theorem by differentiating the partial differential equation with respect to one variable.

), the theorem implies that a continuously differentiable and positively homogeneous function of degree are not necessarily the same, as it is the case for the absolute value.

Application to differential equations

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

Generalizations

Homogeneity under a monoid action

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

be a monoid with identity element is said to be called an is said to be

A function is (resp. ).

will be called

The notion of being is generalized similarly.

Distributions (generalized functions)

if and only if for all compactly supported test functions Equivalently, making a change of variable The last display makes it possible to define homogeneity of distributions

Glossary of name variants

be a map between two vector spaces (usually the real numbers or complex numbers is said to be For instance, every additive map between vector spaces is although it might not be homogeneous over the real numbers

The following commonly encountered special cases and variations of this definition have their own terminology:

() : and all

• is valued in a vector space or field, then this property is logically equivalent to , which by definition means: and all It is for this reason that positive homogeneity is often also called nonnegative homogeneity. However, for functions valued in the extended real numbers which appear in fields like convex analysis and so these statements are not necessarily always interchangeable.

• This property is used in the definition of a sublinear function.

• Minkowski functionals are exactly those non-negative extended real-valued functions with this property.

• This property is used in the definition of a linear functional.

:

• This property is used in the definition of linear functionals and linear maps.

:

• typically denotes the complex conjugate. But more generally, as with semilinear maps

• Along with additivity, this property is assumed in the definition of an antilinear map. It is also assumed that one of the two coordinates of a sesquilinear form has this property (such as the inner product of a Hilbert space).

in which case that definition is prefixed with the word or For example, :

• This property is used in the definition of a seminorm and a norm.

for conditions using the absolute value, etc.), in which case the homogeneity is said to be (where in particular, all of the above definitions are ). For instance,

A nonzero continuous function

See also

• Homogeneous space

Notes

Proofs

References

Sources

External links