{"history":[],"article":{"id":"c12c2d27-9076-4a45-be18-7e0172cb645e","title":"Homogeneous function","slug":"homogeneous-function","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **homogeneous function** is a [function of several variables](https://bliptext.com/articles/function-of-several-variables) such that the following holds: If each of the function's arguments is multiplied by the same [scalar](https://bliptext.com/articles/scalar-mathematics), then the function's value is multiplied by some power of this [scalar](https://bliptext.com/articles/scalar-mathematics); the power is called the **degree of homogeneity**, or simply the degree.\nThat is, if k is an integer, a function f of n variables is homogeneous of degree k if for every and s\\ne 0.\nThis is also referred to a kth-degree or kth-order homogeneous function.\nFor example, a [homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial) of degree k defines a homogeneous function of degree k.\nThe above definition extends to functions whose [domain](https://bliptext.com/articles/domain-of-a-function) and co[domain](https://bliptext.com/articles/domain-of-a-function) are [vector spaces](https://bliptext.com/articles/vector-space) over a [field](https://bliptext.com/articles/field-mathematics) F: a function f : V \\to W between two F-[vector spaces](https://bliptext.com/articles/vector-space) is homogeneous of degree k if\nfor all nonzero s \\in F and v \\in V. This definition is often further generalized to functions whose domain is not V, but a [cone](https://bliptext.com/articles/cone-linear-algebra) in V, that is, a subset C of V such that implies for every nonzero scalar s.\nIn the case of [functions of several real variables](https://bliptext.com/articles/functions-of-several-real-variables) and [real vector spaces](https://bliptext.com/articles/real-vector-space), a slightly more general form of homogeneity called **positive homogeneity** is often considered, by requiring only that the above identities hold for s > 0, and allowing any real number k as a degree of homogeneity.\nEvery homogeneous real function is positively homogeneous.\nThe 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.\nA [norm](https://bliptext.com/articles/norm-mathematics) over a real vector space is an example of a positively homogeneous function that is not homogeneous.\nA special case is the [absolute value](https://bliptext.com/articles/absolute-value) of real numbers.\nThe quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero.\nThis example is fundamental in the definition of [projective schemes](https://bliptext.com/articles/projective-scheme).\n\n# Definitions\nThe concept of a homogeneous function was originally introduced for [functions of several real variables](https://bliptext.com/articles/functions-of-several-real-variables).\nWith the definition of [vector spaces](https://bliptext.com/articles/vector-space) at the end of 19th century, the concept has been naturally extended to functions between [vector spaces](https://bliptext.com/articles/vector-space), since a [tuple](https://bliptext.com/articles/tuple) of variable values can be considered as a [coordinate vector](https://bliptext.com/articles/coordinate-vector).\nIt is this more general point of view that is described in this article.\nThere are two commonly used definitions.\nThe general one works for vector spaces over arbitrary [fields](https://bliptext.com/articles/field-mathematics), and is restricted to degrees of homogeneity that are [integers](https://bliptext.com/articles/integer).\nThe second one supposes to work over the field of [real numbers](https://bliptext.com/articles/real-number), or, more generally, over an [ordered field](https://bliptext.com/articles/ordered-field).\nThis 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.\nPositive homogeneity leads to considering more functions as homogeneous.\nFor example, the [absolute value](https://bliptext.com/articles/absolute-value) and all [norms](https://bliptext.com/articles/norm-mathematics) are positively homogeneous functions that are not homogeneous.\nThe restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.\n\n## General homogeneity\nLet V and W be two [vector spaces](https://bliptext.com/articles/vector-space) over a [field](https://bliptext.com/articles/field-mathematics) F. A [linear cone](https://bliptext.com/articles/linear-cone) in V is a subset C of V such that\nsx\\in C for all x\\in C and all nonzero s\\in F.\nA homogeneous function f from V to W is a [partial function](https://bliptext.com/articles/partial-function) from V to W that has a linear cone C as its [domain](https://bliptext.com/articles/domain-of-a-function), and satisfies\nfor some [integer](https://bliptext.com/articles/integer) k, every x\\in C, and every nonzero s\\in F. The [integer](https://bliptext.com/articles/integer) k is called the degree of homogeneity, or simply the degree of f.\nA typical example of a homogeneous function of degree k is the function defined by a [homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial) of degree k. The [rational function](https://bliptext.com/articles/rational-function) defined by the quotient of two [homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial)s 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.\nHomogeneous functions play a fundamental role in [projective geometry](https://bliptext.com/articles/projective-geometry) since any homogeneous function f from V to W defines a well-defined function between the [projectivizations](https://bliptext.com/articles/projectivization) of V and W. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degree) play an essential role in the [Proj construction](https://bliptext.com/articles/proj-construction) of [projective schemes](https://bliptext.com/articles/projective-scheme).\n\n## Positive homogeneity\nWhen working over the [real numbers](https://bliptext.com/articles/real-number), or more generally over an [ordered field](https://bliptext.com/articles/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 \"\ns > 0\n\" in the definitions of a linear cone and a homogeneous function.\nThis change allow considering (positively) homogeneous functions with any real number as their degrees, since [exponentiation](https://bliptext.com/articles/exponentiation) with a positive real base is well defined.\nEven in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous.\nThis is, in particular, the case of the [absolute value](https://bliptext.com/articles/absolute-value) function and [norms](https://bliptext.com/articles/norm-mathematics), which are all positively homogeneous of degree\n1\n. They are not homogeneous since if x\\neq 0.\nThis remains true in the [complex](https://bliptext.com/articles/complex-number) case, since the field of the [complex](https://bliptext.com/articles/complex-number) numbers \\C and every [complex](https://bliptext.com/articles/complex-number) vector space can be considered as real vector spaces.\n[Euler's homogeneous function theorem](https://bliptext.com/articles/) is a characterization of positively homogeneous [differentiable functions](https://bliptext.com/articles/differentiable-function), which may be considered as the fundamental theorem on homogeneous functions.\n\n# Examples\n\n## Simple example\nThe function is homogeneous of degree 2:\n\n## Absolute value and norms\nThe absolute value of a [real number](https://bliptext.com/articles/real-number) is a positively homogeneous function of degree\n1\n, which is not homogeneous, since |sx|=s|x| if s>0, and |sx|=-s|x| if s<0.\nThe absolute value of a [complex number](https://bliptext.com/articles/complex-number) is a positively homogeneous function of degree 1 over the real numbers (that is, when considering the [complex number](https://bliptext.com/articles/complex-number)s as a [vector space](https://bliptext.com/articles/vector-space) over the real numbers).\nIt is not homogeneous, over the real numbers as well as over the complex numbers.\nMore generally, every [norm](https://bliptext.com/articles/norm-mathematics) and semi[norm](https://bliptext.com/articles/norm-mathematics) is a positively homogeneous function of degree\n1\nwhich is not a homogeneous function.\nAs 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.\n\n## Linear functions\nAny [linear map](https://bliptext.com/articles/linear-map) f : V \\to W between [vector spaces](https://bliptext.com/articles/vector-space) over a [field](https://bliptext.com/articles/field-mathematics) F is homogeneous of degree 1, by the definition of linearity:\nfor all and v \\in V.\nSimilarly, any [multilinear function](https://bliptext.com/articles/multilinear-map) is homogeneous of degree n, by the definition of multilinearity:\nfor all and\n\n## Homogeneous polynomials\n[Monomials](https://bliptext.com/articles/monomials) in n variables define homogeneous functions For example,\nis homogeneous of degree 10 since\nThe degree is the sum of the exponents on the variables; in this example,\nA [homogeneous polynomial](https://bliptext.com/articles/homogeneous-[polynomial](https://bliptext.com/articles/polynomial)) is a [polynomial](https://bliptext.com/articles/polynomial) made up of a sum of monomials of the same degree.\nFor example,\nis a homogeneous polynomial of degree 5.\nHomogeneous polynomials also define homogeneous functions.\nGiven a homogeneous polynomial of degree k with real coefficients that takes only positive values, one gets a positively homogeneous function of degree k/d by raising it to the power 1 / d. So for example, the following function is positively homogeneous of degree 1 but not homogeneous:\n\n## Min/max\nFor every set of weights the following functions are positively homogeneous of degree 1, but not homogeneous:\n\n## Rational functions\n[Rational functions](https://bliptext.com/articles/rational-function) formed as the ratio of two polynomials are homogeneous functions in their [domain](https://bliptext.com/articles/domain-of-a-function), that is, off of the [linear cone](https://bliptext.com/articles/linear-cone) formed by the [zeros](https://bliptext.com/articles/zero-of-a-function) of the denominator.\nThus, 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.\n\n## Non-examples\nThe homogeneous [real functions](https://bliptext.com/articles/real-functions) of a single variable have the form for some constant c. So, the [affine function](https://bliptext.com/articles/affine-function) the [natural logarithm](https://bliptext.com/articles/natural-logarithm) and the [exponential function](https://bliptext.com/articles/exponential-function) are not homogeneous.\n\n# Euler's theorem\nRoughly 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](https://bliptext.com/articles/partial-differential-equation).\nMore precisely:\nAs a consequence, if is continuously differentiable and homogeneous of degree k, its first-order [partial derivatives](https://bliptext.com/articles/partial-derivative) are homogeneous of degree k - 1.\nThis results from Euler's theorem by differentiating the partial differential equation with respect to one variable.\nIn the case of a function of a single real variable (n = 1), the theorem implies that a continuously differentiable and positively homogeneous function of degree k has the form for x>0 and for x<0.\nThe constants c_+ and c_- are not necessarily the same, as it is the case for the [absolute value](https://bliptext.com/articles/absolute-value).\n\n# Application to differential equations\nThe substitution v = y / x converts the [ordinary differential equation](https://bliptext.com/articles/ordinary-differential-equation)\nwhere I and J are homogeneous functions of the same degree, into the [separable differential equation](https://bliptext.com/articles/separation-of-variables)\n\n# Generalizations\n\n## Homogeneity under a monoid action\nThe definitions given above are all specialized cases of the following more general notion of homogeneity in which 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](https://bliptext.com/articles/monoid).\nLet M be a [monoid](https://bliptext.com/articles/monoid) with identity element 1 \\in M, let X and Y be sets, and suppose that on both X and Y there are defined [monoid](https://bliptext.com/articles/monoid) actions of M. Let k be a non-negative integer and let f : X \\to Y be a map.\nThen f is said to be if for every x \\in X and m \\in M, If in addition there is a function M \\to M, denoted by called an then f is said to be if for every x \\in X and m \\in M,\nA function is (resp. ) if it is homogeneous of degree 1 over M (resp. absolutely homogeneous of degree 1 over M).\nMore generally, it is possible for the symbols m^k to be defined for m \\in M with k being something other than an integer (for example, if M is the real numbers and k is a non-zero real number then m^k is defined even though k is not an integer).\nIf this is the case then f will be called if the same equality holds:\nThe notion of being is generalized similarly.\n\n## Distributions (generalized functions)\nA continuous function f on \\R^n is homogeneous of degree k if and only if\nfor all [compactly supported](https://bliptext.com/articles/compactly-supported) [test functions](https://bliptext.com/articles/test-function) \\varphi; and nonzero real t. Equivalently, making a [change of variable](https://bliptext.com/articles/integration-by-substitution) y = tx, f is homogeneous of degree k if and only if\nfor all t and all test functions \\varphi.\nThe last display makes it possible to define homogeneity of [distributions](https://bliptext.com/articles/distribution-mathematics).\nA distribution S is homogeneous of degree k if\nfor all nonzero real t and all test functions \\varphi.\nHere the angle brackets denote the pairing between distributions and test functions, and is the mapping of scalar division by the real number t.\n\n# Glossary of name variants\nLet f : X \\to Y be a map between two [vector spaces](https://bliptext.com/articles/vector-space) over a field \\mathbb{F} (usually the [real numbers](https://bliptext.com/articles/real-number) \\R or [complex numbers](https://bliptext.com/articles/complex-number) \\Complex).\nIf S is a set of scalars, such as \\Z, or \\Reals for example, then f is said to be if for every x \\in X and scalar s \\in S. For instance, every [additive map](https://bliptext.com/articles/additive-map) between vector spaces is S := \\Q although it [might not be ](https://bliptext.com/articles/cauchy-s-functional-equation) S := \\R.\nThe following commonly encountered special cases and variations of this definition have their own terminology:\nAll of the above definitions can be generalized by replacing the condition with in which case that definition is prefixed with the word \"\" or \".\"\nFor example,\n

: for all x \\in X and all scalars\n

\n\nIf k is a fixed real number then the above definitions can be further generalized by replacing the condition with (and similarly, by replacing with for conditions using the absolute value, etc.), in which case the homogeneity is said to be \"\" (where in particular, all of the above definitions are \"\").\nFor instance,\n

: for all x \\in X and all real r.

: for all x \\in X and all scalars\n

: for all x \\in X and all real r.

: for all x \\in X and all scalars\n

\n\nA nonzero [continuous function](https://bliptext.com/articles/continuous-function) that is homogeneous of degree k on extends continuously to \\R^n if and only if k > 0.\n\n# Sources","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T20:13:02.702Z","updated_at":"2025-01-25T20:13:02.702Z","search_vector":"'function':2 'homogen':1","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **homogeneous function** is a [function of several variables](https://bliptext.com/articles/function-of-several-variables) such that the following holds: If each of the function's arguments is multiplied by the same [scalar](https://bliptext.com/articles/scalar-mathematics), then the function's value is multiplied by some power of this [scalar](https://bliptext.com/articles/scalar-mathematics); the power is called the **degree of homogeneity**, or simply the degree.\nThat is, if k is an integer, a function f of n variables is homogeneous of degree k if for every and s\\ne 0.\nThis is also referred to a kth-degree or kth-order homogeneous function.\nFor example, a [homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial) of degree k defines a homogeneous function of degree k.\nThe above definition extends to functions whose [domain](https://bliptext.com/articles/domain-of-a-function) and co[domain](https://bliptext.com/articles/domain-of-a-function) are [vector spaces](https://bliptext.com/articles/vector-space) over a [field](https://bliptext.com/articles/field-mathematics) F: a function f : V \\to W between two F-[vector spaces](https://bliptext.com/articles/vector-space) is homogeneous of degree k if\nfor all nonzero s \\in F and v \\in V. This definition is often further generalized to functions whose domain is not V, but a [cone](https://bliptext.com/articles/cone-linear-algebra) in V, that is, a subset C of V such that implies for every nonzero scalar s.\nIn the case of [functions of several real variables](https://bliptext.com/articles/functions-of-several-real-variables) and [real vector spaces](https://bliptext.com/articles/real-vector-space), a slightly more general form of homogeneity called **positive homogeneity** is often considered, by requiring only that the above identities hold for s > 0, and allowing any real number k as a degree of homogeneity.\nEvery homogeneous real function is positively homogeneous.\nThe 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.\nA [norm](https://bliptext.com/articles/norm-mathematics) over a real vector space is an example of a positively homogeneous function that is not homogeneous.\nA special case is the [absolute value](https://bliptext.com/articles/absolute-value) of real numbers.\nThe quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero.\nThis example is fundamental in the definition of [projective schemes](https://bliptext.com/articles/projective-scheme).\n\n# Definitions\nThe concept of a homogeneous function was originally introduced for [functions of several real variables](https://bliptext.com/articles/functions-of-several-real-variables).\nWith the definition of [vector spaces](https://bliptext.com/articles/vector-space) at the end of 19th century, the concept has been naturally extended to functions between [vector spaces](https://bliptext.com/articles/vector-space), since a [tuple](https://bliptext.com/articles/tuple) of variable values can be considered as a [coordinate vector](https://bliptext.com/articles/coordinate-vector).\nIt is this more general point of view that is described in this article.\nThere are two commonly used definitions.\nThe general one works for vector spaces over arbitrary [fields](https://bliptext.com/articles/field-mathematics), and is restricted to degrees of homogeneity that are [integers](https://bliptext.com/articles/integer).\nThe second one supposes to work over the field of [real numbers](https://bliptext.com/articles/real-number), or, more generally, over an [ordered field](https://bliptext.com/articles/ordered-field).\nThis 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.\nPositive homogeneity leads to considering more functions as homogeneous.\nFor example, the [absolute value](https://bliptext.com/articles/absolute-value) and all [norms](https://bliptext.com/articles/norm-mathematics) are positively homogeneous functions that are not homogeneous.\nThe restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.\n\n## General homogeneity\nLet V and W be two [vector spaces](https://bliptext.com/articles/vector-space) over a [field](https://bliptext.com/articles/field-mathematics) F. A [linear cone](https://bliptext.com/articles/linear-cone) in V is a subset C of V such that\nsx\\in C for all x\\in C and all nonzero s\\in F.\nA homogeneous function f from V to W is a [partial function](https://bliptext.com/articles/partial-function) from V to W that has a linear cone C as its [domain](https://bliptext.com/articles/domain-of-a-function), and satisfies\nfor some [integer](https://bliptext.com/articles/integer) k, every x\\in C, and every nonzero s\\in F. The [integer](https://bliptext.com/articles/integer) k is called the degree of homogeneity, or simply the degree of f.\nA typical example of a homogeneous function of degree k is the function defined by a [homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial) of degree k. The [rational function](https://bliptext.com/articles/rational-function) defined by the quotient of two [homogeneous polynomial](https://bliptext.com/articles/homogeneous-polynomial)s 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.\nHomogeneous functions play a fundamental role in [projective geometry](https://bliptext.com/articles/projective-geometry) since any homogeneous function f from V to W defines a well-defined function between the [projectivizations](https://bliptext.com/articles/projectivization) of V and W. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degree) play an essential role in the [Proj construction](https://bliptext.com/articles/proj-construction) of [projective schemes](https://bliptext.com/articles/projective-scheme).\n\n## Positive homogeneity\nWhen working over the [real numbers](https://bliptext.com/articles/real-number), or more generally over an [ordered field](https://bliptext.com/articles/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 \"\ns > 0\n\" in the definitions of a linear cone and a homogeneous function.\nThis change allow considering (positively) homogeneous functions with any real number as their degrees, since [exponentiation](https://bliptext.com/articles/exponentiation) with a positive real base is well defined.\nEven in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous.\nThis is, in particular, the case of the [absolute value](https://bliptext.com/articles/absolute-value) function and [norms](https://bliptext.com/articles/norm-mathematics), which are all positively homogeneous of degree\n1\n. They are not homogeneous since if x\\neq 0.\nThis remains true in the [complex](https://bliptext.com/articles/complex-number) case, since the field of the [complex](https://bliptext.com/articles/complex-number) numbers \\C and every [complex](https://bliptext.com/articles/complex-number) vector space can be considered as real vector spaces.\n[Euler's homogeneous function theorem](https://bliptext.com/articles/) is a characterization of positively homogeneous [differentiable functions](https://bliptext.com/articles/differentiable-function), which may be considered as the fundamental theorem on homogeneous functions.\n\n# Examples\n\n## Simple example\nThe function is homogeneous of degree 2:\n\n## Absolute value and norms\nThe absolute value of a [real number](https://bliptext.com/articles/real-number) is a positively homogeneous function of degree\n1\n, which is not homogeneous, since |sx|=s|x| if s>0, and |sx|=-s|x| if s 0 and for x : for all x \\in X and all scalars\n \n \nIf k is a fixed real number then the above definitions can be further generalized by replacing the condition with (and similarly, by replacing with for conditions using the absolute value, etc.), in which case the homogeneity is said to be \"\" (where in particular, all of the above definitions are \"\").\nFor instance,\n : for all x \\in X and all real r. \n : for all x \\in X and all scalars\n \n : for all x \\in X and all real r. \n : for all x \\in X and all scalars\n \n \nA nonzero [continuous function](https://bliptext.com/articles/continuous-function) that is homogeneous of degree k on extends continuously to \\R^n if and only if k > 0.\n\n# Sources","revisionCount":0},"hasMore":false,"nextCursor":null}

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