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 is an integer, a function of variables is homogeneous of degree if
:
for every
and
For example, a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
of degree defines a homogeneous function of degree .
The above definition extends to functions whose
domain and
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
are
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s over a
field : a function
between two -vector spaces is ''homogeneous'' of degree
if
for all nonzero
and
This definition is often further generalized to functions whose domain is not , but a
cone in , that is, a subset of such that
implies
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 is often considered, by requiring only that the above identities hold for
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 space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
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
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s.
The second one supposes to work over the field of
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, 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 homogeneity
Let and be two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s over a
field . A
linear cone in is a subset of such that
for all
and all nonzero
A ''homogeneous function'' from to is a
partial function from to that has a linear cone as its
domain, and satisfies
:
for some
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, every
and every nonzero
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
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
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
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...
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 degre) play an essential role in the
Proj construction of
projective schemes.
Positive homogeneity
When working over the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s, 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 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 . 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''.
Examples
Simple example
The function
is homogeneous of degree 2:
Absolute value and norms
The
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), and ...
of a
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
is a positively homogeneous function of degree , which is not homogeneous, since
if
and
if
The absolute value of a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
is a positively homogeneous function of degree
over the real numbers (that is, when considering the complex numbers as a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
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 functions
Any
linear map between
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s over a
field is homogeneous of degree 1, by the definition of linearity:
for all
and
Similarly, any
multilinear function
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
:f\colon V_1 \times \cdots \times V_n \to W\text
where V_1,\ldots,V_n and W are ...
is homogeneous of degree
by the definition of multilinearity:
for all
and
Homogeneous polynomials
Monomials
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
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
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
is a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
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/max
For every set of weights
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 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-examples
The homogeneous
real functions
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interva ...
of a single variable have the form
for some constant . So, the
affine function the
natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
and the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
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:
''Proof:'' For having simpler formulas, we set
The first part results by using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x) ...
for differentiating both sides of the equation
with respect to
and taking the limit of the result when tends to .
The converse is proved by integrating a simple
differential equation.
Let
be in the interior of the domain of . For sufficiently close of , the function
is well defined. The partial differential equation implies that
The solutions of this
linear differential equation have the form
Therefore,
if is sufficiently close to . If this solution of the partial differential equation would not be defined for all positive , 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 .
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 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 equations
The substitution
converts the
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
where
and
are homogeneous functions of the same degree, into the
separable differential equation
Generalizations
Homogeneity under a monoid action
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
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
.
Let
be a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
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 if for every
and
If in addition there is a function
denoted by
called an then
is said to be if for every
and
A function is (resp. ) 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 if the same equality holds:
The notion of being is generalized similarly.
Distributions (generalized functions)
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 variants
Let
be a map between two
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s over a field
(usually the
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s
). If
is a set of scalars, such as
or
for example, then
is said to be if
for every
and scalar
For instance, every additive map between vector spaces is
although it Cauchy's functional equation, might not be
The following commonly encountered special cases and variations of this definition have their own terminology:
#() :
for all
and all real
#* This property is often also called because for a function valued in a vector space or field, it is
logically equivalent to:
for all
and all real
[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.] 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.
[However, if such an satisfies for all and then necessarily and whenever are both real then will hold for all ]
#* This property is used in the definition of a
sublinear function.
#*
Minkowski functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.
If K is a subset of a real or complex vector space X, then ...
s are exactly those non-negative extended real-valued functions with this property.
#:
for all
and all real
#* This property is used in the definition of a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , th ...
.
#:
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 functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , th ...
s and
linear maps.
#:
for all
and all scalars
#* If
then
typically denotes the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
).
All of the above definitions can be generalized by replacing the condition
with
in which case that definition is prefixed with the word or
For example,
- : 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 (where in particular, all of the above definitions are ).
For instance,
- : for all and all real
- : for all and all scalars
- : for all and all real
- : for all and all scalars
A nonzero
continuous function that is homogeneous of degree
on
extends continuously to
if and only if
See also
*
Homogeneous space
*
Notes
;Proofs
References
*
*
*
External links
*
* {{MathWorld, title=Euler's Homogeneous Function Theorem, urlname=EulersHomogeneousFunctionTheorem, author=Eric Weisstein
Linear algebra
Differential operators
Types of functions
Leonhard Euler