# 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 function of two variables x and y is a real-valued function that satisfies the condition ${\displaystyle f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)}$ for some constant ${\displaystyle k}$ and all real numbers ${\displaystyle \alpha }$. 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

${\displaystyle f(\alpha \mathbf {v} )=\alpha ^{k}f(\mathbf {v} )}$

(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).

## Examples

A homogeneous function is not necessarily continuous, as shown by this example. This is the function f defined by ${\displaystyle f(x,y)=x}$ if ${\displaystyle xy>0}$ or ${\displaystyle f(x,y)=0}$ if ${\displaystyle xy\leq 0}$. This function is homogeneous of degree 1, i.e. ${\displaystyle f(\alpha x,\alpha y)=\alpha f(x,y)}$ for any real numbers ${\displaystyle \alpha ,x,y}$. It is discontinuous at ${\displaystyle y=0,x\neq 0}$.

### Example 1

The function ${\displaystyle f(x,y)=x^{2}+y^{2}}$ is homogeneous of degree 2:
${\displaystyle f(tx,ty)=(tx)^{2}+(ty)^{2}=t^{2}(x^{2}+y^{2})=t^{2}f(x,y).}$
For example, suppose x = 2, y = 4 and t = 5. Then

• ${\displaystyle f(x,y)=2^{2}+4^{2}=4+16=20}$, and
• ${\displaystyle f(5x,5y)=5^{2}(2^{2}+4^{2})=25(20)=500}$.

### Linear functions

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

${\displaystyle f(\alpha \mathbf {v} )=\alpha f(\mathbf {v} )}$

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

${\displaystyle f(\alpha \mathbf {v} _{1},\ldots ,\alpha \mathbf {v} _{n})=\alpha ^{n}f(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})}$

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 polynomials

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

${\displaystyle f(x,y,z)=x^{5}y^{2}z^{3}\,}$

is homogeneous of degree 10 since

${\displaystyle f(\alpha x,\alpha y,\alpha z)=(\alpha x)^{5}(\alpha y)^{2}(\alpha z)^{3}=\alpha ^{10}x^{5}y^{2}z^{3}=\alpha ^{10}f(x,y,z).\,}$

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,

${\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}\,}$

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:

${\displaystyle (x^{k}+y^{k}+z^{k})^{1/k}}$

### Min/max

For every set of weights ${\displaystyle w_{1},\dots ,w_{n}}$, the following functions are homogeneous of degree 1:

• ${\displaystyle \min(x_{1}/w_{1},\dots ,x_{n}/w_{n})}$ (Leontief utilities)
• ${\displaystyle \max(x_{1}/w_{1},\dots ,x_{n}/w_{n})}$

### Polarization

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:

${\displaystyle f(v)=g(v,v,\dots ,v).}$

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: ${\displaystyle g(v_{1},v_{2},\dots ,v_{n})={\frac {1}{n!}}{\frac {\partial }{\partial t_{1}}}{\frac {\partial }{\partial t_{2}}}\cdots {\frac {\partial }{\partial t_{n}}}f(t_{1}v_{1}+\cdots +t_{n}v_{n}).}$ 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 ${\displaystyle f(x)=\ln x}$ scales additively and so is not homogeneous.

This can be demonstrated with the following examples: ${\displaystyle f(5x)=\ln 5x=\ln 5+f(x)}$, ${\displaystyle f(10x)=\ln 10+f(x)}$, and ${\displaystyle f(15x)=\ln 15+f(x)}$. This is because there is no ${\displaystyle k}$ such that ${\displaystyle f(\alpha \cdot x)=\alpha ^{k}\cdot f(x)}$.

### Affine functions

Affine functions (the function ${\displaystyle f(x)=x+5}$ is an example) do not 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. A function ƒ : V \ {0} → R is positively homogeneous of degree k if

${\displaystyle f(\alpha x)=\alpha ^{k}f(x)\,}$

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 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

${\displaystyle \mathbf {x} \cdot \nabla f(\mathbf {x} )=kf(\mathbf {x} ).}$

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 ${\displaystyle \textstyle g(\alpha )=f(\alpha \mathbf {x} )}$. Since ${\displaystyle \textstyle \alpha \mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )=kf(\alpha \mathbf {x} )}$,

${\displaystyle g'(\alpha )=\mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )={\frac {k}{\alpha }}f(\alpha \mathbf {x} )={\frac {k}{\alpha }}g(\alpha ).}$

Thus, ${\displaystyle \textstyle g'(\alpha )-{\frac {k}{\alpha }}g(\alpha )=0}$. This implies ${\displaystyle \textstyle g(\alpha )=g(1)\alpha ^{k}}$. Therefore, ${\displaystyle \textstyle f(\alpha \mathbf {x} )=g(\alpha )=\alpha ^{k}g(1)=\alpha ^{k}f(\mathbf {x} )}$: 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 ${\displaystyle \partial f/\partial x_{i}}$ are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator ${\displaystyle \mathbf {x} \cdot \nabla }$ 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

${\displaystyle f'(x)-{\frac {k}{x}}f(x)=0}$.

This equation may be solved using an integrating factor approach, with solution ${\displaystyle \textstyle f(x)=cx^{k}}$, where ${\displaystyle c=f(1)}$.

## Homogeneous distributions

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

${\displaystyle \int _{\mathbb {R} ^{n}}f(tx)\varphi (x)\,dx=t^{k}\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx}$

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

${\displaystyle t^{-n}\int _{\mathbb {R} ^{n}}f(y)\varphi (y/t)\,dy=t^{k}\int _{\mathbb {R} ^{n}}f(y)\varphi (y)\,dy}$

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

${\displaystyle t^{-n}\langle S,\varphi \circ \mu _{t}\rangle =t^{k}\langle S,\varphi \rangle }$

for all nonzero real t and all test functions ${\displaystyle \varphi }$;. 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 equations

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

${\displaystyle I(x,y){\frac {\mathrm {d} y}{\mathrm {d} x}}+J(x,y)=0,}$

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

${\displaystyle x{\frac {\mathrm {d} v}{\mathrm {d} x}}=-{\frac {J(1,v)}{I(1,v)}}-v.}$