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 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
fieldF: a function between two F-vector spaces 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.
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 V and W be two
vector spaces over a
fieldF. A
linear cone in V is a subset C of V such that
for all and all nonzero
A homogeneous functionf 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
integerk, 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 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 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.
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.
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/max
For every set of weights the following functions are positively homogeneous of degree 1, but not homogeneous:
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
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:
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).
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
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
This results from Euler's theorem by differentiating 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 positively 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.
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.
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
The following commonly encountered special cases and variations of this definition have their own terminology:
(Strict) Positive homogeneity:[1] for all and all positive real
When the function is valued in a vector space or field, then this property is
logically equivalent[proof 1] to nonnegative homogeneity, which by definition means:[2] for all and all non-negative real 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, the multiplication will be undefined whenever and so these statements are not necessarily always interchangeable.[note 1]
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
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,
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,
Real homogeneity of degree: for all and all real
Homogeneity of degree: for all and all scalars
Absolute real homogeneity of degree: for all and all real
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
Triangle center function – Point in a triangle that can be seen as its middle under some criteriaPages displaying short descriptions of redirect targets
Notes
^However, if such an satisfies for all and then necessarily and whenever are both real then will hold for all
Proofs
^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.
Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (in German) (2nd ed.). Springer Verlag. p. 188.
ISBN3-540-09484-9.