# Homogeneous function

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In mathematics, a **homogeneous function** is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if *ƒ* : *V* → *W* 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
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for all nonzero α ∈ *F* and **v** ∈ *V*. This implies it has scale invariance. When the vector spaces involved are over the real numbers, a slightly more general form of homogeneity is often used, requiring only that (Template:EquationNote) 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 (Template:EquationNote).

## Examples

### Linear functions

Any linear function *ƒ* : *V* → *W* is homogeneous of degree 1 since by the definition of linearity

for all α ∈ *F* and **v** ∈ *V*. Similarly, any multilinear function *ƒ* : *V*_{1} × *V*_{2} × ... *V*_{n} → *W* is homogeneous of degree n since by the definition of multilinearity

for all α ∈ *F* and **v**_{1} ∈ *V*_{1}, **v**_{2} ∈ *V*_{2}, ..., **v**_{n} ∈ *V*_{n}. It follows that the *n*-th differential of a function *ƒ* : *X* → *Y* between two Banach spaces *X* and *Y* is homogeneous of degree *n*.

### Homogeneous polynomials

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Monomials in *n* variables define homogeneous functions *ƒ* : *F*^{n} → *F*. For example,

is homogeneous of degree 10 since

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,

is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

### Polarization

A multilinear function *g* : *V* × *V* × ... *V* → *F* from the *n*-th Cartesian product of *V* with itself to the underlying field *F* gives rise to an homogeneous function *ƒ* : *V* → *F* 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 *g* : *V* × *V* × ... *V* → *F* on the *n*-th Cartesian product of *V*. The polarization is defined by

These two constructions, one of an homogeneous polynomial from a multilinear form and the other of a multilinear form from an 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 scales additively and so is not homogeneous.

This can be proved by noting that , , and . Therefore such that .

### Affine functions

Affine functions (the function is an example) do not scale multiplicatively.

## Positive homogeneity

In the special case of vector spaces over the real numbers, the notation of positive homogeneity often plays a more important role than homogeneity in the above sense. A function *ƒ* : *V* \ {0} → **R** is positive homogeneous of degree *k* if

for all α > 0. Here *k* can be any complex number. A (nonzero) continuous function homogeneous of degree *k* on **R**^{n} \ {0} extends continuously to **R**^{n} if and only if Re{*k*} > 0.

Positive homogeneous functions are characterized by **Euler's homogeneous function theorem**. Suppose that the function *ƒ* : **R**^{n} \ {0} → **R** is continuously differentiable. Then *ƒ* is positive homogeneous of degree *k* if and only if

This result follows at once by differentiating both sides of the equation *ƒ*(α**y**) = α^{k}*ƒ*(**y**) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating. Specifically, let
.
Since ,

Thus, .
This implies .
Therefore, : *ƒ* is positive homogeneous of degree *k*.

As a consequence, suppose that *ƒ* : **R**^{n} → **R** is differentiable and homogeneous of degree *k*. Then its first-order partial derivatives are homogeneous of degree *k* − 1. The result follows from Euler's theorem by commuting the operator with the partial derivative.

## Homogeneous distributions

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A compactly supported continuous function ƒ on **R**^{n} is homogeneous of degree *k* if and only if

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

for all *t* and all test functions ;. The last display makes it possible to define homogeneity of distributions. A distribution *S* is homogeneous of degree *k* if

for all nonzero real *t* and all test functions ;. Here the angle brackets denote the pairing between distributions and test functions, and μ_{t} : **R**^{n} → **R**^{n} is the mapping of scalar multiplication by the real number *t*.

## Application to differential equations

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

where *I* and *J* are homogeneous functions of the same degree, into the separable differential equationTemplate:Disambiguation needed

## See also

## References

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## External links

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