Homogeneous function

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 real-valued function of two variables "x" and "y" is a real-valued function that satisfies the condition formula_1 for some constant "k" and all real numbers α. The constant "k" is called the degree of homogeneity.
More generally, if 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
for all nonzero and . When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that () 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 ().
Examples.
Example 1.
The function formula_2 is homogeneous of degree 2:
For example, suppose "x" = 2, "y" = 4 and "t" = 5. Then
Linear functions.
Any linear map is homogeneous of degree 1 since by the definition of linearity
for all and .
Similarly, any multilinear function is homogeneous of degree "n" since by the definition of multilinearity
for all and , , ..., .
It follows that the "n"-th differential of a function between two Banach spaces "X" and "Y" is homogeneous of degree "n".
Homogeneous polynomials.
Monomials in "n" 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 "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/max.
For every set of weights formula_12, the following functions are homogeneous of degree 1:
Polarization.
A multilinear function from the "n"-th Cartesian product of "V" with itself to the underlying field "F" gives rise to a homogeneous function 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 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 functions.
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-examples.
Logarithms.
The natural logarithm formula_17 scales additively and so is not homogeneous.
This can be demonstrated with the following examples: formula_18, formula_19, and formula_20. This is because there is no "k" such that formula_21.
Affine functions.
Affine functions (the function formula_22 is an example) do not in general scale multiplicatively. 
Positive homogeneity.
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. 
Let (resp. ) be a vector space over a field formula_23 (resp. formula_24), where formula_23 and formula_24 will usually be (or possibly just contain) the real numbers formula_27 or complex numbers formula_28. Let be a map. We define the following terminology: 
All of the above definitions can be generalized by replacing the equality with in which case we prefix that definition with the word "absolute" or "absolutely." 
For example, 
If is a fixed real number then the above definitions can be further generalized by replacing the equality with (or with for conditions using the absolute value), in which case we say that the homogeneity is "of degree " (note in particular that all of the above definitions are "of degree ").
For instance, 
A (nonzero) continuous function that is homogeneous of degree on formula_29 extends continuously to formula_30 if and only if .
Generalizations.
The definitions given above are all specializes 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. 
Monoids and monoid actions.
A monoid is a pair consisting of a set and an associative operator where there is some element in called an identity element, which we will denote by , such that for all . 
Let be a monoid with identity element whose operation is denoted by juxtaposition and let be a set. A monoid action of on is a map , which we will also denote by juxtaposition, such that and for all and all .
Homogeneity.
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 we say that is homogeneous of degree over if for every and ,
If in addition there is a function , denoted by , called an "absolute value" then we say that is absolutely homogeneous of degree over if for every and ,
If we say that a function is homogeneous over (resp. absolutely homogeneous over ) then we mean that it is homogeneous of degree over (resp. absolutely homogeneous of degree over ). 
More generally, note that it is possible for the symbols to be defined for with being something other than an integer (e.g. if is the real numbers and is a non-zero real number then is defined even though is not an integer). In this case, we say that is homogeneous of degree over if the same equality holds:
The notion of being absolutely homogeneous of degree over is generalized similarly.
Euler's homogeneous function theorem.
Continuously differentiable positively homogeneous functions are characterized by the following theorem:
This result follows at once by differentiating both sides of the equation with respect to , applying the chain rule, and choosing to be . 
The converse is proved by integrating. 
Specifically, let
formula_31.
Since formula_32,
Thus, formula_34.
This implies formula_35.
Therefore, formula_36: is positively homogeneous of degree .
As a consequence, suppose that formula_37 is differentiable and homogeneous of degree . 
Then its first-order partial derivatives formula_38 are homogeneous of degree . 
The result follows from Euler's theorem by commuting the operator formula_39 with the partial derivative.
One can specialize the theorem to the case of a function of a single real variable (), in which case the function satisfies the ordinary differential equation
This equation may be solved using an integrating factor approach, with solution formula_41, where .
Homogeneous distributions.
A continuous function ƒ on formula_30 is homogeneous of degree if and only if
for all compactly supported test functions formula_44; and nonzero real . Equivalently, making a change of variable , ƒ is homogeneous of degree if and only if
for all "t" and all test functions formula_44. 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 formula_44. Here the angle brackets denote the pairing between distributions and test functions, and formula_49 is the mapping of scalar division by the real number .
Application to differential equations.
The substitution converts the ordinary differential equation
where and are homogeneous functions of the same degree, into the separable differential equation