Homogeneous function

# Homogeneous function

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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a homogeneous function is a function with multiplicative
Multiplicative
Multiplicative may refer to:*Multiplication*Multiplicative partition*A Multiplicative function* For the Multiplicative numerals, once, twice, and thrice, see English numerals...

scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

between two vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

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 more 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
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

on projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. More generally, if S ⊂ V is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then an homogeneous function from S to W can still be defined by .

## Examples

### Linear functions

Any linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...

function 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 space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

s 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, 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 from the n-th Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

of V with itself to the groundfield F gives rise to an 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 an homogeneous polynomial ƒ of degree n on V, the polarization
Polarization of an algebraic form
In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables...

of ƒ is a multilinear function 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 space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...

s from the symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....

of V to the algebra of homogeneous polynomials on V.

### Rational functions

Rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...

s 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.

### 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.

### Some polynomials

The function does 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 is positive homogeneous of degree k if

for all . Here k can be any complex number. A (nonzero) continuous function homogeneous of degree k on Rn \ {0} extends continuously to Rn if and only if .

Positive homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function 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 with respect to α and applying the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

. The converse holds by integrating.

As a consequence, suppose that 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

A compactly supported continuous function ƒ on Rn is homogeneous of degree k if and only if
for all compactly supported test functions φ and nonzero real t. Equivalently, making a change of variable
Integration by substitution
In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...

, ƒ is homogeneous of degree k if and only if
for all t and all test functions φ. The last display makes it possible to define homogeneity of distributions
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...

. A distribution S is homogeneous of degree k if
for all nonzero real t and all test functions φ. Here the angle brackets denote the pairing between distributions and test functions, and 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
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....

where I and J are homogeneous functions of the same degree, into the separable differential equation
Separable differential equation
In mathematics, a separable differential equation may refer to one of two related things, both of which are differential equations that can be attacked by a method of separation of variables....