Homogeneous function

Summary

In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous of degree k if

for every and

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

The above definition extends to functions whose domain and codomain are vector spaces over a field F: a function between two F-vector space is homogeneous of degree if

 

 

 

 

(1)

for all nonzero and 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 implies 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 and allowing any real number k 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.

DefinitionsEdit

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 consider 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 homogeneityEdit

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   for all   and all nonzero  

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

 

for some integer k, every   and every nonzero   The integer k is called the degree of homogeneity, or simply the degree of f.

A typical example of a homogeneous function of degree k is the function defined by a homogeneous polynomial of degree k. 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 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 degre) play an essential role in the Proj construction of projective schemes.

Positive homogeneityEdit

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.

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 1. They are not homogeneous since   if   This remains true in the complex case, since the field of the complex numbers   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.

ExamplesEdit

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

Simple exampleEdit

The function   is homogeneous of degree 2:

 

Absolute value and normsEdit

The absolute value of a real number is a positively homogeneous function of degree 1, which is not homogeneous, since   if   and   if  

The absolute value of a complex number is a positively homogeneous function of degree   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 1 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 functionsEdit

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

 
for all   and  

Similarly, any multilinear function   is homogeneous of degree   by the definition of multilinearity:

 
for all   and  

Homogeneous polynomialsEdit

Monomials in   variables define homogeneous functions   For example,

 
is homogeneous of degree 10 since
 
The degree is the sum of the exponents on the variables; in this example,  

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   with real coefficients that takes only positive values, one gets a positively homogeneous function of degree   by raising it to the power   So for example, the following function is positively homogeneous of degree 1 but not homogeneous:

 

Min/maxEdit

For every set of weights   the following functions are positively homogeneous of degree 1, but not homogeneous:

  •   (Leontief utilities)
  •  

Rational functionsEdit

Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions in their domain, that is, off of the linear cone formed by the zeros of the denominator. Thus, if   is homogeneous of degree   and   is homogeneous of degree   then   is homogeneous of degree   away from the zeros of  

Non-examplesEdit

The homogeneous real functions of a single variable have the form   for some constant c. So, the affine function   the natural logarithm   and the exponential function  are not homogeneous.

Euler's theoremEdit

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 (here, maximal means that the solution cannot be prolongated to a function with a larger domain).

Proof: For having simpler formulas, we set   The first part results by using the chain rule for differentiating both sides of the equation   with respect to   and taking the limit of the result when s tends to 1.

The converse is proved by integrating a simple differential equation. Let   be in the interior of the domain of f. For s sufficiently close of 1, the function   is well defined. The partial differential equation implies that

 
The solutions of this linear differential equation have the form   Therefore,
 
if s is sufficiently close to 1. If this solution of the partial differential equation would not be defined for all positive s, then the functional equation would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degree k.  

As a consequence, if   is continuously differentiable and homogeneous of degree   its first-order partial derivatives   are homogeneous of degree   The results from Euler's theorem by derivating the partial differential equation with respect to one variable.

In the case of a function of a single real variable ( ), the theorem implies that a continuously differentiable and poxitively homogeneous function of degree k has the form   for   and   for   The constants   and  are not necessarily the same, as it is the case for the absolute value.

Application to differential equationsEdit

The substitution   converts the ordinary differential equation

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

GeneralizationsEdit

Homogeneity under a monoid actionEdit

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.

Let   be a monoid with identity element   let   and   be sets, and suppose that on both   and   there are defined monoid actions of   Let   be a non-negative integer and let   be a map. Then   is said to be homogeneous of degree   over   if for every   and  

 
If in addition there is a function   denoted by   called an absolute value then   is said to be absolutely homogeneous of degree   over   if for every   and  
 

A function is homogeneous over   (resp. absolutely homogeneous over  ) if it is homogeneous of degree   over   (resp. absolutely homogeneous of degree   over  ).

More generally, it is possible for the symbols   to be defined for   with   being something other than an integer (for example, if   is the real numbers and   is a non-zero real number then   is defined even though   is not an integer). If this is the case then   will be called homogeneous of degree   over   if the same equality holds:

 

The notion of being absolutely homogeneous of degree   over   is generalized similarly.

Distributions (generalized functions)Edit

A continuous function   on   is homogeneous of degree   if and only if

 
for all compactly supported test functions  ; and nonzero real   Equivalently, making a change of variable     is homogeneous of degree   if and only if
 
for all   and all test functions   The last display makes it possible to define homogeneity of distributions. A distribution   is homogeneous of degree   if
 
for all nonzero real   and all test functions   Here the angle brackets denote the pairing between distributions and test functions, and   is the mapping of scalar division by the real number  

Glossary of name variantsEdit

Let   be a map between two vector spaces over a field   (usually the real numbers   or complex numbers  ). If   is a set of scalars, such as  ,  , or   for example, then   is said to be homogeneous over   if   for every   and scalar  . For instance, every additive map between vector spaces is homogeneous over the rational numbers   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:

  1. (Strict) Positive homogeneity:   for all   and all positive real  .
    • This property is often also called nonnegative homogeneity because for a function valued in a vector space or field, it is logically equivalent to:   for all   and all non-negative real  .[proof 1] However, for a function valued in the extended real numbers  , which appear in fields like convex analysis, the multiplication   will be undefined whenever   and so these statements are not necessarily interchangeable.[note 1]
    • 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.
  2. Real homogeneity:   for all   and all real  .
  3. Homogeneity:   for all   and all scalars  .
    • It is emphasized that this definition depends on the scalar field   underlying the domain  .
    • This property is used in the definition of linear functionals and linear maps.
  4. Conjugate homogeneity:   for all   and all scalars  .
    • If   then   typically denotes the complex conjugate of  . But more generally, as with semilinear maps for example,   could be the image of   under some distinguished automorphism of  .
    • 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).

All of the above definitions can be generalized by replacing the condition   with  , in which case that definition is prefixed with the word "absolute" or "absolutely." For example,

  1. Absolute homogeneity:   for all   and all scalars  .
    • This property is used in the definition of a seminorm and a norm.

If   is a fixed real number then the above definitions can be further generalized by replacing the condition   with   (and similarly, by replacing   with   for conditions using the absolute value, etc.), in which case the homogeneity is said to be "of degree  " (where in particular, all of the above definitions are "of degree  "). For instance,

  1. Real homogeneity of degree  :   for all   and all real  .
  2. Homogeneity of degree  :   for all   and all scalars  .
  3. Absolute real homogeneity of degree  :   for all   and all real  .
  4. Absolute homogeneity of degree  :   for all   and all scalars  .

A nonzero continuous function that is homogeneous of degree   on   extends continuously to   if and only if  .

See alsoEdit

NotesEdit

  1. ^ However, if such an   satisfies   for all   and   then necessarily   and whenever   are both real then   will hold for all  
Proofs
  1. ^ Assume that   is strictly positively homogeneous and valued in a vector space or a field. Then   so subtracting   from both sides shows that  . Writing  , then for any  ,   which shows that   is nonnegative homogeneous.

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