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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
:<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> This is also referred to a ''{{mvar|k}}th-degree'' or ''{{mvar|k}}th-order'' homogeneous function.
For example, a [[homogeneous polynomial]] of degree {{mvar|k}} defines a homogeneous function of degree {{mvar|k}}.
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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 spaces is ''homogeneous'' of degree <math>k</math> if
{{NumBlk|:|<math>f(s \mathbf{v}) = s^k f(\mathbf{v})</math>|{{EquationRef|1}}}}
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.
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There are two commonly used definitions. The general one works for vector spaces over arbitrary [[field (mathematics)|fields]], and is restricted to degrees of homogeneity that are [[integer]]s.
The second one supposes to work over the field of [[real number]]s, or, more generally, over an [[ordered field]]. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called ''positive homogeneity'', the qualificative ''positive'' being often omitted when there is no risk of confusion. Positive homogeneity leads to
The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.
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A typical example of a homogeneous function of degree {{mvar|k}} is the function defined by a [[homogeneous polynomial]] of degree {{mvar|k}}. The [[rational function]] defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its ''cone of definition'' is the linear cone of the points where the value of denominator is not zero.
Homogeneous functions play a fundamental role in [[projective geometry]] since any homogeneous function {{mvar|f}} from {{mvar|V}} to {{mvar|W}} defines a well-defined function between the [[projectivization]]s of {{mvar|V}} and {{mvar|W}}. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same
=== Positive homogeneity ===
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More generally, every [[norm (mathematics)|norm]] and [[seminorm]] is a positively homogeneous function of degree {{math|1}} which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.
===Linear
Any [[linear map]] <math>f : V \to W</math> between [[vector space]]s over a [[field (mathematics)|field]] {{mvar|F}} is homogeneous of degree 1, by the definition of linearity:
<math display="block">f(\alpha \mathbf{v}) = \alpha f(\mathbf{v})</math>
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===Non-examples===
The homogeneous [[real functions]] of a single variable have the form <math>x\mapsto cx^k</math> for some constant {{mvar|c}}. So, the [[affine function]] <math>x\mapsto x+5,</math> the [[natural logarithm]] <math>x\mapsto \ln(x),</math> and the [[exponential function]] <math>x\mapsto e^x</math> are not homogeneous.
== Euler's theorem ==
Roughly speaking, '''Euler's homogeneous function theorem''' asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific [[partial differential equation]]. More precisely:
{{Math theorem
{{Math theorem|name=Euler's homogeneous function theorem |math_statement=If {{mvar|f}} is a [[partial function|(partial) function]] of {{mvar|n}} real variables that is positively homogeneous of degree {{mvar|k}}, and [[continuously differentiable]] in some open subset of <math>\R^n,</math> then it satisfies in this open set the [[partial differential equation]]▼
| name = Euler's homogeneous function theorem
▲
<math display="block">k\,f(x_1, \ldots,x_n)=\sum_{i=1}^n x_i\frac{\partial f}{\partial x_i}(x_1, \ldots,x_n).</math>
Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree {{mvar|k}}, defined on a positive cone (here, ''maximal'' means that the solution cannot be prolongated to a function with a larger ___domain).
}}
{{Math proof|title=Proof|proof=
The first part results by using the [[chain rule]] for differentiating both sides of the equation <math>f(s\mathbf x ) = s^k f(\mathbf x)</math> with respect to <math>s,</math> and taking the limit of the result when {{mvar|s}} tends to {{math|1}}.
The converse is proved by integrating a simple [[differential equation]].
Let <math>\mathbf{x}</math> be in the interior of the ___domain of {{mvar|f}}. For {{mvar|s}} sufficiently close
<math display="inline"> g(s) = f(s \mathbf{x})</math> is well defined. The partial differential equation implies that
<math display=block>
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The solutions of this [[linear differential equation]] have the form <math>g(s)=g(1)s^k.</math>
Therefore, <math display="block"> f(s \mathbf{x}) = g(s) = s^k g(1) = s^k f(\mathbf{x}),</math> if {{mvar|s}} is sufficiently close to {{math|1}}. If this solution of the partial differential equation would not be defined for all positive {{mvar|s}}, then the [[functional equation]] would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the ___domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degree {{mvar|k}}. <math>\square</math>
}}
As a consequence, if <math>f : \R^n \to \R</math> is continuously differentiable and homogeneous of degree <math>k,</math> its first-order [[partial derivative]]s <math>\partial f/\partial x_i</math> are homogeneous of degree <math>k - 1.</math>
In the case of a function of a single real variable (<math>n = 1</math>), the theorem implies that a continuously differentiable and positively homogeneous function of degree {{mvar|k}} has the form <math>f(x)=c_+ x^k</math> for <math>x>0</math> and <math>f(x)=c_- x^k</math> for <math>x<0.</math> The constants <math>c_+</math> and <math>c_
==Application to differential equations==
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==Glossary of name variants==
{{or section|date=December 2021}}
Let <math>f : X \to Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{F}</math> (usually the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex</math>). If <math>S</math> is a set of scalars, such as <math>\Z,</math> <math>[0, \infty),</math> or <math>\
<math display=inline>f(s x) = s f(x)</math> for every <math>x \in X</math> and scalar <math>s \in S.</math>
For instance, every [[additive map]] between vector spaces is {{em|{{visible anchor|homogeneous over the rational numbers}}}} <math>S := \Q</math> although it [[Cauchy's functional equation|might not be {{em|{{visible anchor|homogeneous over the real numbers}}}}]] <math>S := \R.</math>
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The following commonly encountered special cases and variations of this definition have their own terminology:
#({{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}}}) {{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}}}:{{sfn|Schechter|1996|pp=313-314}} <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>
#*
#* This property is used in the definition of a [[sublinear function]].{{sfn|Schechter|1996|pp=313-314}}{{sfn|Kubrusly|2011|p=200}}
#* [[Minkowski functional]]s are exactly those non-negative extended real-valued functions with this property.
#{{em|{{visible anchor|Real homogeneity|Real homogeneous}}}}: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
#* This property is used in the definition of a {{em|real}} [[linear functional]].
#{{em|{{visible anchor|Homogeneity|Homogeneous}}}}:{{sfn|Kubrusly|2011|p=55}} <math>f(sx) = s f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* It is emphasized that this definition depends on the scalar field <math>\mathbb{F}</math> underlying the ___domain <math>X.</math>
#* This property is used in the definition of [[linear functional]]s and [[linear map]]s.{{sfn|Kubrusly|2011|p=200}}
#{{em|[[Semilinear map|{{visible anchor|Conjugate homogeneity|Conjugate homogeneous}}]]}}:{{sfn|Kubrusly|2011|p=310}} <math>f(sx) = \overline{s} f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* If <math>\mathbb{F} = \Complex</math> then <math>\overline{s}</math> typically denotes the [[complex conjugate]] of <math>s</math>. But more generally, as with [[semilinear map]]s for example, <math>\overline{s}</math> could be the image of <math>s</math> under some distinguished automorphism of <math>\mathbb{F}.</math>
#* Along with [[Additive map|additivity]], this property is assumed in the definition of an [[antilinear map]]. It is also assumed that one of the two coordinates of a [[sesquilinear form]] has this property (such as the [[inner product]] of a [[Hilbert space]]).
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For example,
<ol start=5>
<li>{{em|{{visible anchor|Absolute homogeneity|Absolute homogeneous|Absolutely homogeneous}}}}:{{sfn|Kubrusly|2011|p=200}} <math>f(sx) = |s| f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
* This property is used in the definition of a [[seminorm]] and a [[Norm (mathematics)|norm]].
</li>
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{{reflist|group=note}}
{{reflist|group=proof
<ref group=proof name=posHomEquivToNonnegHom>Assume that <math>f</math> is strictly positively homogeneous and valued in a vector space or a field. Then <math>f(0) = f(2 \cdot 0) = 2 f(0)</math> so subtracting <math>f(0)</math> from both sides shows that <math>f(0) = 0.</math> Writing <math>r := 0,</math> then for any <math>x \in X,</math> <math>f(r x) = f(0) = 0 = 0 f(x) = r f(x),</math> which shows that <math>f</math> is nonnegative homogeneous.</ref>
}}
==References==
{{reflist}}
==Sources==
* {{cite book|last=Blatter|first=Christian|title=Analysis II (2nd ed.)|publisher=Springer Verlag|year=1979|language=German|isbn=3-540-09484-9|pages=188|chapter=20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.}}▼
{{sfn whitelist|CITEREFKubrusly2011}}
▲* {{cite book|last=Blatter|first=Christian|title=Analysis II
* {{Kubrusly The Elements of Operator Theory 2nd Edition 2011}} <!--{{sfn|Kubrusly|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
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