# Homogeneous function

Homogeneous function (Redirected from Euler's theorem on homogeneous functions) Jump to navigation Jump to search This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (July 2018) (Learn how and when to remove this template message) For homogeneous linear maps, see 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, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous of degree k if {displaystyle f(sx_{1},ldots ,sx_{n})=s^{k}f(x_{1},ldots ,x_{n})} for every {displaystyle x_{1},ldots ,x_{n},} and {displaystyle sneq 0.} For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k.

The above definition extends to functions whose domain and codomain are vector spaces over a field F: a function {displaystyle f:Vto W} between two F-vector space is homogeneous of degree {displaystyle k} if {displaystyle f(smathbf {v} )=s^{k}f(mathbf {v} )} (1) for all nonzero {displaystyle sin F} and {displaystyle vin V.} This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of V such that {displaystyle mathbf {v} in C} implies {displaystyle smathbf {v} in C} for every nonzero scalar s.

In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity is often considered, by requiring only that the above identities hold for {displaystyle s>0,} and allowing any real number 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 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 schemes.

Contents 1 Definitions 1.1 General homogeneity 1.2 Positive homogeneity 2 Examples 2.1 Simple example 2.2 Absolute value and norms 2.3 Linear functions 2.4 Homogeneous polynomials 2.5 Min/max 2.6 Rational functions 2.7 Non-examples 3 Euler's theorem 4 Application to differential equations 5 Generalizations 5.1 Homogeneity under a monoid action 5.2 Distributions (generalized functions) 6 Glossary of name variants 7 See also 8 Notes 9 References 10 External links Definitions The concept of a homogeneous function was originally introduced for functions of several real variables. With the definition of vector spaces at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a tuple of variable values can be considered as a coordinate vector. It is this more general point of view that is described in this article.

There are two commonly used definitions. The general one works for vector spaces over arbitrary fields, and is restricted to degrees of homogeneity that are integers.

The second one supposes to work over the field of real numbers, 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 consider more functions as homogeneous. For example, the absolute value and all norms are positively homogeneous functions that are not homogeneous.

The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.

General homogeneity Let V and W be two vector spaces over a field F. A linear cone in V is a subset C of V such that {displaystyle sxin C} for all {displaystyle xin C} and all nonzero {displaystyle sin F.} A homogeneous function f from V to W is a partial function from V to W that has a linear cone C as its domain, and satisfies {displaystyle f(sx)=s^{k}f(x)} for some integer k, every {displaystyle xin C,} and every nonzero {displaystyle sin F.} The integer k is called the degree of homogeneity, or simply the degree of f.

A typical example of a homogeneous function of degree k is the function defined by a homogeneous polynomial of degree 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 f from V to W defines a well-defined function between the projectivizations of V and W. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degre) play an essential role in the Proj construction of projective schemes.

Positive homogeneity When working over the real numbers, or more generally over an ordered field, it is commonly convenient to consider positive homogeneity, the definition being exactly the same as that in the preceding section, with "nonzero s" replaced by "s > 0" in the definitions of a linear cone and a homogeneous function.

This change allow considering (positively) homogeneous functions with any real number as their degrees, since exponentiation with a positive real base is well defined.

Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of the absolute value function and norms, which are all positively homogeneous of degree 1. They are not homogeneous since {displaystyle |-x|=|x|neq -|x|} if {displaystyle xneq 0.} This remains true in the complex case, since the field of the complex numbers {displaystyle mathbb {C} } and every complex vector space can be considered as real vector spaces.

Euler's homogeneous function theorem is a characterization of positively homogeneous differentiable functions, which may be considered as the fundamental theorem on homogeneous functions.

Examples A homogeneous function is not necessarily continuous, as shown by this example. This is the function {displaystyle f} defined by {displaystyle f(x,y)=x} if {displaystyle xy>0} and {displaystyle f(x,y)=0} if {displaystyle xyleq 0.} This function is homogeneous of degree 1, that is, {displaystyle f(sx,sy)=sf(x,y)} for any real numbers {displaystyle s,x,y.} It is discontinuous at {displaystyle y=0,xneq 0.} Simple example The function {displaystyle f(x,y)=x^{2}+y^{2}} is homogeneous of degree 2: {displaystyle f(tx,ty)=(tx)^{2}+(ty)^{2}=t^{2}left(x^{2}+y^{2}right)=t^{2}f(x,y).} Absolute value and norms The absolute value of a real number is a positively homogeneous function of degree 1, which is not homogeneous, since {displaystyle |sx|=s|x|} if {displaystyle s>0,} and {displaystyle |sx|=-s|x|} if {displaystyle s<0.} The absolute value of a complex number is a positively homogeneous function of degree {displaystyle 1} over the real numbers (that is, when considering the complex numbers as a vector space over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers. More generally, every norm and seminorm is a positively homogeneous function of degree 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 functions Any linear map {displaystyle f:Vto W} between vector spaces over a field F is homogeneous of degree 1, by the definition of linearity: {displaystyle f(alpha mathbf {v} )=alpha f(mathbf {v} )} for all {displaystyle alpha in {F}} and {displaystyle vin V.} Similarly, any multilinear function {displaystyle f:V_{1}times V_{2}times cdots V_{n}to W} is homogeneous of degree {displaystyle n,} by the definition of multilinearity: {displaystyle fleft(alpha mathbf {v} _{1},ldots ,alpha mathbf {v} _{n}right)=alpha ^{n}f(mathbf {v} _{1},ldots ,mathbf {v} _{n})} for all {displaystyle alpha in {F}} and {displaystyle v_{1}in V_{1},v_{2}in V_{2},ldots ,v_{n}in V_{n}.} Homogeneous polynomials Main article: Homogeneous polynomial Monomials in {displaystyle n} variables define homogeneous functions {displaystyle f:mathbb {F} ^{n}to mathbb {F} .} For example, {displaystyle f(x,y,z)=x^{5}y^{2}z^{3},} is homogeneous of degree 10 since {displaystyle f(alpha x,alpha y,alpha z)=(alpha x)^{5}(alpha y)^{2}(alpha z)^{3}=alpha ^{10}x^{5}y^{2}z^{3}=alpha ^{10}f(x,y,z).,} The degree is the sum of the exponents on the variables; in this example, {displaystyle 10=5+2+3.} A homogeneous polynomial is a polynomial made up of a sum of monomials of the same degree. For example, {displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions. Given a homogeneous polynomial of degree {displaystyle k} with real coefficients that takes only positive values, one gets a positively homogeneous function of degree {displaystyle k/d} by raising it to the power {displaystyle 1/d.} So for example, the following function is positively homogeneous of degree 1 but not homogeneous: {displaystyle left(x^{2}+y^{2}+z^{2}right)^{frac {1}{2}}.} Min/max For every set of weights {displaystyle w_{1},dots ,w_{n},} the following functions are positively homogeneous of degree 1, but not homogeneous: {displaystyle min left({frac {x_{1}}{w_{1}}},dots ,{frac {x_{n}}{w_{n}}}right)} (Leontief utilities) {displaystyle max left({frac {x_{1}}{w_{1}}},dots ,{frac {x_{n}}{w_{n}}}right)} Rational functions Rational functions formed as the ratio of two homogeneous polynomials are homogeneous functions in their domain, that is, off of the linear cone formed by the zeros of the denominator. Thus, if {displaystyle f} is homogeneous of degree {displaystyle m} and {displaystyle g} is homogeneous of degree {displaystyle n,} then {displaystyle f/g} is homogeneous of degree {displaystyle m-n} away from the zeros of {displaystyle g.} Non-examples The homogeneous real functions of a single variable have the form {displaystyle xmapsto cx^{k}} for some constant c. So, the affine function {displaystyle xmapsto x+5,} the natural logarithm {displaystyle xmapsto ln(x),} and the exponential function {displaystyle xmapsto e^{x}} 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: Euler's homogeneous function theorem — If f is a (partial) function of n real variables that is positively homogeneous of degree k, and continuously differentiable in some open subset of {displaystyle mathbb {R} ^{n},} then it satisfies in this open set the partial differential equation {displaystyle k,f(x_{1},ldots ,x_{n})=sum _{i=1}^{n}x_{i}{frac {partial f}{partial x_{i}}}(x_{1},ldots ,x_{n}).} Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree k (here, maximal means that the solution cannot be prolongated to a function with a larger domain). Proof: For having simpler formulas, we set {displaystyle mathbf {x} =(x_{1},ldots ,x_{n}).} The first part results by using the chain rule for differentiating both sides of the equation {displaystyle f(smathbf {x} )=s^{k}f(mathbf {x} )} with respect to {displaystyle s,} and taking the limit of the result when s tends to 1. The converse is proved by integrating a simple differential equation. Let {displaystyle mathbf {x} } be in the interior of the domain of f. For s sufficiently close of 1, the function {textstyle g(s)=f(smathbf {x} )} is well defined. The partial differential equation implies that {displaystyle sg'(s)=kf(smathbf {x} )=kg(s).} The solutions of this linear differential equation have the form {displaystyle g(s)=g(1)s^{k}.} Therefore, {displaystyle f(smathbf {x} )=g(s)=s^{k}g(1)=s^{k}f(mathbf {x} ),} if s is sufficiently close to 1. If this solution of the partial differential equation would not be defined for all positive 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 k. {displaystyle square } As a consequence, if {displaystyle f:mathbb {R} ^{n}to mathbb {R} } is continuously differentiable and homogeneous of degree {displaystyle k,} its first-order partial derivatives {displaystyle partial f/partial x_{i}} are homogeneous of degree {displaystyle k-1.} The results from Euler's theorem by derivating the partial differential equation with respect to one variable. In the case of a function of a single real variable ( {displaystyle n=1} ), the theorem implies that a continuously differentiable and poxitively homogeneous function of degree k has the form {displaystyle f(x)=c_{+}x^{k}} for {displaystyle x>0} and {displaystyle f(x)=c_{-}x^{k}} for {displaystyle x<0.} The constants {displaystyle c_{+}} and {displaystyle c_{+}} are not necessarily the same, as it is the case for the absolute value. Application to differential equations Main article: Homogeneous differential equation The substitution {displaystyle v=y/x} converts the ordinary differential equation {displaystyle I(x,y){frac {mathrm {d} y}{mathrm {d} x}}+J(x,y)=0,} where {displaystyle I} and {displaystyle J} are homogeneous functions of the same degree, into the separable differential equation {displaystyle x{frac {mathrm {d} v}{mathrm {d} x}}=-{frac {J(1,v)}{I(1,v)}}-v.} Generalizations Homogeneity under a monoid action The definitions given above are all specialized cases of the following more general notion of homogeneity in which {displaystyle X} can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a monoid. Let {displaystyle M} be a monoid with identity element {displaystyle 1in M,} let {displaystyle X} and {displaystyle Y} be sets, and suppose that on both {displaystyle X} and {displaystyle Y} there are defined monoid actions of {displaystyle M.} Let {displaystyle k} be a non-negative integer and let {displaystyle f:Xto Y} be a map. Then {displaystyle f} is said to be homogeneous of degree {displaystyle k} over {displaystyle M} if for every {displaystyle xin X} and {displaystyle min M,} {displaystyle f(mx)=m^{k}f(x).} If in addition there is a function {displaystyle Mto M,} denoted by {displaystyle mmapsto |m|,} called an absolute value then {displaystyle f} is said to be absolutely homogeneous of degree {displaystyle k} over {displaystyle M} if for every {displaystyle xin X} and {displaystyle min M,} {displaystyle f(mx)=|m|^{k}f(x).} A function is homogeneous over {displaystyle M} (resp. absolutely homogeneous over {displaystyle M} ) if it is homogeneous of degree {displaystyle 1} over {displaystyle M} (resp. absolutely homogeneous of degree {displaystyle 1} over {displaystyle M} ). More generally, it is possible for the symbols {displaystyle m^{k}} to be defined for {displaystyle min M} with {displaystyle k} being something other than an integer (for example, if {displaystyle M} is the real numbers and {displaystyle k} is a non-zero real number then {displaystyle m^{k}} is defined even though {displaystyle k} is not an integer). If this is the case then {displaystyle f} will be called homogeneous of degree {displaystyle k} over {displaystyle M} if the same equality holds: {displaystyle f(mx)=m^{k}f(x)quad {text{ for every }}xin X{text{ and }}min M.} The notion of being absolutely homogeneous of degree {displaystyle k} over {displaystyle M} is generalized similarly. Distributions (generalized functions) Main article: Homogeneous distribution A continuous function {displaystyle f} on {displaystyle mathbb {R} ^{n}} is homogeneous of degree {displaystyle k} if and only if {displaystyle int _{mathbb {R} ^{n}}f(tx)varphi (x),dx=t^{k}int _{mathbb {R} ^{n}}f(x)varphi (x),dx} for all compactly supported test functions {displaystyle varphi } ; and nonzero real {displaystyle t.} Equivalently, making a change of variable {displaystyle y=tx,} {displaystyle f} is homogeneous of degree {displaystyle k} if and only if {displaystyle t^{-n}int _{mathbb {R} ^{n}}f(y)varphi left({frac {y}{t}}right),dy=t^{k}int _{mathbb {R} ^{n}}f(y)varphi (y),dy} for all {displaystyle t} and all test functions {displaystyle varphi .} The last display makes it possible to define homogeneity of distributions. A distribution {displaystyle S} is homogeneous of degree {displaystyle k} if {displaystyle t^{-n}langle S,varphi circ mu _{t}rangle =t^{k}langle S,varphi rangle } for all nonzero real {displaystyle t} and all test functions {displaystyle varphi .} Here the angle brackets denote the pairing between distributions and test functions, and {displaystyle mu _{t}:mathbb {R} ^{n}to mathbb {R} ^{n}} is the mapping of scalar division by the real number {displaystyle t.} Glossary of name variants This section possibly contains original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research should be removed. (December 2021) (Learn how and when to remove this template message) Let {displaystyle f:Xto Y} be a map between two vector spaces over a field {displaystyle mathbb {F} } (usually the real numbers {displaystyle mathbb {R} } or complex numbers {displaystyle mathbb {C} } ). If {displaystyle S} is a set of scalars, such as {displaystyle mathbb {Z} } , {displaystyle [0,infty )} , or {displaystyle mathbb {R} } for example, then {displaystyle f} is said to be homogeneous over {displaystyle S} if {textstyle f(sx)=sf(x)} for every {displaystyle xin X} and scalar {displaystyle sin S} . For instance, every additive map between vector spaces is homogeneous over the rational numbers {displaystyle S:=mathbb {Q} } although it might not be homogeneous over the real numbers {displaystyle S:=mathbb {R} } . The following commonly encountered special cases and variations of this definition have their own terminology: (Strict) Positive homogeneity: {displaystyle f(rx)=rf(x)} for all {displaystyle xin X} and all positive real {displaystyle r>0} . This property is often also called nonnegative homogeneity because for a function valued in a vector space or field, it is logically equivalent to: {displaystyle f(rx)=rf(x)} for all {displaystyle xin X} and all non-negative real {displaystyle rgeq 0} .[proof 1] However, for a function valued in the extended real numbers {displaystyle [-infty ,infty ]=mathbb {R} cup {pm infty }} , which appear in fields like convex analysis, the multiplication {displaystyle 0cdot f(x)} will be undefined whenever {displaystyle f(x)=pm infty } and so these statements are not necessarily interchangeable.[note 1] This property is used in the definition of a sublinear function. Minkowski functionals are exactly those non-negative extended real-valued functions with this property. Real homogeneity: {displaystyle f(rx)=rf(x)} for all {displaystyle xin X} and all real {displaystyle r} . This property is used in the definition of a real linear functional. Homogeneity: {displaystyle f(sx)=sf(x)} for all {displaystyle xin X} and all scalars {displaystyle sin mathbb {F} } . It is emphasized that this definition depends on the scalar field {displaystyle mathbb {F} } underlying the domain {displaystyle X} . This property is used in the definition of linear functionals and linear maps. Conjugate homogeneity: {displaystyle f(sx)={overline {s}}f(x)} for all {displaystyle xin X} and all scalars {displaystyle sin mathbb {F} } . If {displaystyle mathbb {F} =mathbb {C} } then {displaystyle {overline {s}}} typically denotes the complex conjugate of {displaystyle s} . But more generally, as with semilinear maps for example, {displaystyle {overline {s}}} could be the image of {displaystyle s} under some distinguished automorphism of {displaystyle mathbb {F} } . Along with 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).

All of the above definitions can be generalized by replacing the condition {displaystyle f(rx)=rf(x)} with {displaystyle f(rx)=|r|f(x)} , in which case that definition is prefixed with the word "absolute" or "absolutely." For example, Absolute homogeneity: {displaystyle f(sx)=|s|f(x)} for all {displaystyle xin X} and all scalars {displaystyle sin mathbb {F} } . This property is used in the definition of a seminorm and a norm.

If {displaystyle k} is a fixed real number then the above definitions can be further generalized by replacing the condition {displaystyle f(rx)=rf(x)} with {displaystyle f(rx)=r^{k}f(x)} (and similarly, by replacing {displaystyle f(rx)=|r|f(x)} with {displaystyle f(rx)=|r|^{k}f(x)} for conditions using the absolute value, etc.), in which case the homogeneity is said to be "of degree {displaystyle k} " (where in particular, all of the above definitions are "of degree {displaystyle 1} "). For instance, Real homogeneity of degree {displaystyle k} : {displaystyle f(rx)=r^{k}f(x)} for all {displaystyle xin X} and all real {displaystyle r} . Homogeneity of degree {displaystyle k} : {displaystyle f(sx)=s^{k}f(x)} for all {displaystyle xin X} and all scalars {displaystyle sin mathbb {F} } . Absolute real homogeneity of degree {displaystyle k} : {displaystyle f(rx)=|r|^{k}f(x)} for all {displaystyle xin X} and all real {displaystyle r} . Absolute homogeneity of degree {displaystyle k} : {displaystyle f(sx)=|s|^{k}f(x)} for all {displaystyle xin X} and all scalars {displaystyle sin mathbb {F} } .

A nonzero continuous function that is homogeneous of degree {displaystyle k} on {displaystyle mathbb {R} ^{n}backslash lbrace 0rbrace } extends continuously to {displaystyle mathbb {R} ^{n}} if and only if {displaystyle k>0} .

See also Homogeneous space Triangle center function Notes ^ However, if such an {displaystyle f} satisfies {displaystyle f(rx)=rf(x)} for all {displaystyle r>0} and {displaystyle xin X,} then necessarily {displaystyle f(0)in {pm infty ,0}} and whenever {displaystyle f(0),f(x)in mathbb {R} } are both real then {displaystyle f(rx)=rf(x)} will hold for all {displaystyle rgeq 0.} Proofs ^ Assume that {displaystyle f} is strictly positively homogeneous and valued in a vector space or a field. Then {displaystyle f(0)=f(2cdot 0)=2f(0)} so subtracting {displaystyle f(0)} from both sides shows that {displaystyle f(0)=0} . Writing {displaystyle r:=0} , then for any {displaystyle xin X} , {displaystyle f(rx)=f(0)=0=0f(x)=rf(x),} which shows that {displaystyle f} is nonnegative homogeneous. References Blatter, Christian (1979). "20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.". Analysis II (2nd ed.) (in German). Springer Verlag. p. 188. ISBN 3-540-09484-9. External links "Homogeneous function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Eric Weisstein. "Euler's Homogeneous Function Theorem". MathWorld. Categories: Linear algebraDifferential operatorsTypes of functionsLeonhard Euler

Si quieres conocer otros artículos parecidos a **Homogeneous function** puedes visitar la categoría **Differential operators**.

Deja una respuesta