Homogeneous function: Difference between revisions

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{{More footnotes|date=July 2018}}

{{for|homogeneous linear maps|Graded vector space#Homomorphisms}}

In [[mathematics]], a '''homogeneous function''' is a [[function of several variables]] such that, if all its arguments are multiplied by a [[scalar (mathematics)|scalar]], then its value is multiplied by some power of this scalar, called the '''degree of homogeneity''', or simply the ''degree''; that is, if {{mvar|k}} is an integer, a function {{mvar|f}} of {{mvar|n}} variablevariables is homogeneous of degree {{mvar|k}} if

:<math>f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n)</math>

for every <math>x_1, \ldots, x_n,</math> and <math>s\ne 0.</math>

For example, a [[homogeneous polynomial]] of degree {{mvar|k}} defines a homogeneous function of degree {{mvar|k}}.

The above definition extends to functions whose [[___domain of a function|___domain]] and [[codomain]] are [[vector space]]s over a [[Field (mathematics)|field]] {{mvar|F}}: a function <math>f : V \to W</math> between two {{mvar|F}}-vector space is ''homogeneous'' of degree <math>k</math> if

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for all nonzero <math>s \in F</math> and <math>v \in V.</math> This definition is often further generalized to functions whose ___domain is not {{mvar|V}}, but a [[cone (linear algebra)|cone]] in {{mvar|V}}, that is, a subset {{mvar|C}} of {{mvar|V}} such that <math>\mathbf{v}\in C</math> implies <math>s\mathbf{v}\in C</math> for every nonzero scalar {{mvar|s}}.

In the case of [[functions of several real variables]] and [[real vector space]]s, a slightly more general form of homogeneity, called '''positive homogeneity''' is often considered, by requiring only that the above identities hold for <math>s > 0,</math> and allowing any real number {{mvar|k}} as a degree of homogeneity. Every homogeneous real function is ''positively homogeneous''. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.

A [[norm (mathematics)|norm]] over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the [[absolute value]] of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of [[projective scheme]]s.

== Definitions ==