In mathematics, an homogeneous function is a functionwith multiplicative scaling behaviour: if the argument ismultiplied by a factor, then the result is multiplied by some powerof this factor. More precisely, if f: V → W is a functionbetween two vector spaces over a field F, then f issaid to be homogeneous of degree kinF if

（1）

for all nonzero α ∈F and v ∈V. When the vector spaces involved are over the realnumbers, a slightly more general form of homogeneity is often used,requiring only that（1） hold for allα>0.

Homogeneous functions can also be defined for vector spaces withthe origin deleted, a fact that is used in the definition ofsheaveson projectives space in algebraicgeometry. More generally, ifS⊂V is any subsetthat is invariant under scalar multiplication by elements of thefield (a "cone"), then an homogeneous function from S toW can still be defined by (1).

Remark:

Linear functions

Any linear functionf: V →W is homogeneous of degree 1, since by the definitionof linearity

for all α ∈ Fand v ∈ V.Similarly, any , multilinearfunctionf:V₁ × V₂ × ...V_n → W is homogeneous ofdegree n, since by the definition of multilinearity

for all α ∈ Fand v₁ ∈V₁, v₂ ∈V₂, ..., v_n ∈V_n. It follows that the n-th differential of a functionf: X →Y between two Banach spaces X and Y ishomogeneous of degree n.

Homogeneous polynomials

Monomialsin n variables definehomogeneous functionsf:Fⁿ → F. For example,

is homogeneous of degree 10 since

.

The degree is the sum of the exponents on the variables; in thisexample, 10=5+2+3.

A homogeneous polynomial is a polynomial made up of a sum ofmonomials of the same degree. For example,

is a homogeneous polynomial of degree 5. Homogeneous polynomialsalso define homogeneous functions.

Polarization

A multilinear functiong: V ×V × ... V → F from the n - thCartesian product of V with itself to thegroundfieldF gives rise to an homogeneousfunctionf: V →F by evaluating on the diagonal:

The resulting functionf is a polynomialon the vector space V.

Conversely, if F has characteristic zero, then given anhomogeneous polynomialf of degree non V, the polarization off is amultilinear function g: V ×V × ... V → F on the n-thCartesian product of V. The polarization is defined by

These two constructions, one of an homogeneous polynomial from amultilinear form and the other of a multilinear form from anhomogeneous polynomial, are mutually inverse to one another. Infinite dimensions, they establish an isomorphism of graded vectorspaces from the symmetric algebra of V^∗ to thealgebra of homogeneous polynomials on V.

Rational functions

Rational functions formed as the ratio of twohomogeneous polynomials are homogeneous functions off of theaffine conecut out by the zero locus of thedenominator. Thus, if f is homogeneous of degree mand g is homogeneous of degree n, thenf/g is homogeneous of degreem−n away from thezeros ofg.

Non-Examples

Logarithms

The natural logarithm f(x) = lnxscales additively and so is not homogeneous.

This can be proved by noting that f(5x) = ln5x = ln5 +f(x), f(10x) = ln10 +f(x), and f(15x) = ln15 +f(x). Therefore such that .

Affine functions

The function f(x) = x + 5does not scale multiplicatively.

Positive homogeneity

In the special case of vector spaces over the real numbers, thenotation of positive homogeneity often plays a more important rolethan homogeneity in the above sense. Afunctionf: 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 onRⁿ{0}extends continuously to Rⁿ if and only ifRe{k} >0.

Positive homogeneous functions are characterized by Euler'shomogeneous function theorem. Suppose that thefunctionf:Rⁿ {0} → R is continuouslydifferentiable. Then f is positive homogeneous of degreek if and only if

This result follows at once by differentiating both sides of theequation f(αy) =α^kf(y) with respect to α andapplying the chain rule. The converse holds by integrating.

As a consequence, suppose that f: Rⁿ →R is differentiable and homogeneous of degreek. Then its first-order partial derivatives are homogeneous of degreek−1. The result followsfrom Euler's theorem by commuting the operator with the partialderivative.

Homogeneous distributions

A compactly supported continuous functionf onRⁿ is homogeneous of degree k ifand only if

for all compactly supported test functions φ and nonzero realt. Equivalently, making a change of variable y =tx,f is homogeneous of degreek if and only if

for all t and all test functions φ. The last displaymakes it possible to define homogeneity of distributions. AdistributionS is homogeneous of degreek if

for all nonzero real t and all test functions φ. Here theangle brackets denote the pairing between distributions and testfunctions, and μ_t:Rⁿ → Rⁿ isthe mapping of scalar multiplication by the real numbert.

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