In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

For example, a homogeneous function of two variables x and y is a real-valued function that satisfies the condition for some constant and all real numbers . The constant k is called the degree of homogeneity.

More generally, if ƒ : VW is a function between two vector spaces over a field F, and k is an integer, then ƒ is said to be homogeneous of degree k if

 

 

 

 

(1)

for all nonzero α ∈ F and vV. When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that (1) hold for all α > 0.

Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).

Contents

ExamplesEdit

 
A homogeneous function is not necessarily continuous, as shown by this example. This is the function f defined by   if   or   if  . This function is homogeneous of degree 1, i.e.   for any real numbers  . It is discontinuous at  .

Example 1Edit

The function   is homogeneous of degree 2:
 
For example, suppose x = 2, y = 4 and t = 5. Then

  •  , and
  •  .

Linear functionsEdit

Any linear map ƒ : VW is homogeneous of degree 1 since by the definition of linearity

 

for all α ∈ F and vV. Similarly, any multilinear function ƒ : V1 × V2 × ... VnW is homogeneous of degree n since by the definition of multilinearity

 

for all α ∈ F and v1V1, v2V2, ..., vnVn. It follows that the n-th differential of a function ƒ : XY between two Banach spaces X and Y is homogeneous of degree n.

Homogeneous polynomialsEdit

Monomials in n variables define homogeneous functions ƒ : FnF. For example,

 

is homogeneous of degree 10 since

 

The degree is the sum of the exponents on the variables; in this example, 10=5+2+3.

A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example,

 

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

Given a homogeneous polynomial of degree k, it is possible to get a homogeneous function of degree 1 by raising to the power 1/k. So for example, for every k the following function is homogeneous of degree 1:

 

Min/maxEdit

For every set of weights  , the following functions are homogeneous of degree 1:

  •   (Leontief utilities)
  •  

PolarizationEdit

A multilinear function g : V × V × ... VF from the n-th Cartesian product of V with itself to the underlying field F gives rise to a homogeneous function ƒ : VF by evaluating on the diagonal:

 

The resulting function ƒ is a polynomial on the vector space V.

Conversely, if F has characteristic zero, then given a homogeneous polynomial ƒ of degree n on V, the polarization of ƒ is a multilinear function g : V × V × ... VF on the n-th Cartesian product of V. The polarization is defined by:   These two constructions, one of a homogeneous polynomial from a multilinear form and the other of a multilinear form from a homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of graded vector spaces from the symmetric algebra of V to the algebra of homogeneous polynomials on V.

Rational functionsEdit

Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions off of the affine cone cut out by the zero locus of the denominator. Thus, if f is homogeneous of degree m and g is homogeneous of degree n, then f/g is homogeneous of degree m − n away from the zeros of g.

Non-examplesEdit

LogarithmsEdit

The natural logarithm   scales additively and so is not homogeneous.

This can be demonstrated with the following examples:  ,  , and  . This is because there is no   such that  .

Affine functionsEdit

Affine functions (the function   is an example) do not scale multiplicatively.

Positive homogeneityEdit

In the special case of vector spaces over the real numbers, the notion of positive homogeneity often plays a more important role than homogeneity in the above sense. A function ƒ : V \ {0} → R is positively homogeneous of degree k if

 

for all α > 0. Here k can be any real number. A (nonzero) continuous function homogeneous of degree k on Rn \ {0} extends continuously to Rn if and only if Re{k} > 0.

Positively homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function ƒ : Rn \ {0} → R is continuously differentiable. Then ƒ is positively homogeneous of degree k if and only if

 

This result follows at once by differentiating both sides of the equation ƒy) = αkƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let  . Since  ,

 

Thus,  . This implies  . Therefore,  : ƒ is positively homogeneous of degree k.

As a consequence, suppose that ƒ : RnR is differentiable and homogeneous of degree k. Then its first-order partial derivatives   are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator   with the partial derivative.

One can specialise the theorem to the case of a function of a single real variable (n = 1), in which case the function satisfies the ordinary differential equation

 .

This equation may be solved using an integrating factor approach, with solution  , where  .

Homogeneous distributionsEdit

A continuous function ƒ on Rn is homogeneous of degree k if and only if

 

for all compactly supported test functions  ; and nonzero real t. Equivalently, making a change of variable y = tx, ƒ is homogeneous of degree k if and only if

 

for all t and all test functions  ;. The last display makes it possible to define homogeneity of distributions. A distribution S is homogeneous of degree k if

 

for all nonzero real t and all test functions  ;. Here the angle brackets denote the pairing between distributions and test functions, and μt : RnRn is the mapping of scalar multiplication by the real number t.

Application to differential equationsEdit

The substitution v = y/x converts the ordinary differential equation

 

where I and J are homogeneous functions of the same degree, into the separable differential equation

 

See alsoEdit

ReferencesEdit

  • Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3-540-09484-9.

External linksEdit