In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
For example, a homogeneous real-valuedfunction of two variables and is a real-valued function that satisfies the condition for some constant and all real numbers The constant is called the degree of homogeneity.
More generally, if is a function between two vector spaces over a field and is an integer, then is said to be homogeneous of degree if
for all nonzero scalars and When the vector spaces involved are over the real numbers, a slightly less general form of homogeneity is often used, requiring only that (1) hold for all
Homogeneous functions can also be defined for vector spaces with the origin deleted, a fact that is used in the definition of sheaves on projective space in algebraic geometry. More generally, if is any subset that is invariant under scalar multiplication by elements of the field (a "cone"), then a homogeneous function from S to W can still be defined by (1).
A homogeneous function is not necessarily continuous, as shown by this example. This is the function defined by if and if This function is homogeneous of degree 1, that is, for any real numbers It is discontinuous at
A multilinear function from the -th Cartesian product of with itself to the underlying field gives rise to a homogeneous function by evaluating on the diagonal:
The resulting function is a polynomial on the vector space
Conversely, if has characteristic zero, then given a homogeneous polynomial of degree on the polarization of is a multilinear function on the -th Cartesian product of The polarization is defined by:
These two constructions, one of a homogeneous polynomial from a multilinear form and the other of a multilinear form from a homogeneous polynomial, are mutually inverse to one another. In finite dimensions, they establish an isomorphism of graded vector spaces from the symmetric algebra of to the algebra of homogeneous polynomials on
Rational functions 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 is homogeneous of degree and is homogeneous of degree then is homogeneous of degree away from the zeros of
Positive homogeneity: This is usually defined to mean "nonnegative homogeneity" but it is also frequently defined to instead mean "strict positive homogeneity".
Which of these two is chosen as the definition is usually[note 3] irrelevant because for a function valued in a vector space or field, nonnegative homogeneity is the same as strict positive homogeneity; the definitions will be logically equivalent.[proof 1]
All of the above definitions can be generalized by replacing the condition with in which case that definition is prefixed with the word "absolute" or "absolutely."
Absolute real homogeneity: for all and all real
Absolute homogeneity: for all and all scalars
This property is used in the definition of a seminorm and a norm.
If 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 "of degree " (where in particular, all of the above definitions are "of degree ").
Nonnegative homogeneity of degree: for all and all real
Real homogeneity of degree: for all and all real
Homogeneity of degree: for all and all scalars
Absolute real homogeneity of degree: for all and all real
Absolute homogeneity of degree: for all and all scalars
A nonzero continuous function that is homogeneous of degree on extends continuously to if and only if
The definitions given above are all specializes of the following more general notion of homogeneity in which can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a monoid.
A monoid is a pair consisting of a set and an associative operator where there is some element in called an identity element, denoted by such that for all
If is a monoid with identity element and if then the following notation will be used:
let and more generally for any positive integers let be the product of instances of ; that is,
It is common practice (e.g. such as in algebra or calculus) to denote the multiplication operation of a monoid by juxtaposition, meaning that may be written rather than This avoids any need to assign a symbol to a monoid's multiplication operation. When this juxtaposition notation is used then it should be automatically assumed that the monoid's identity element is denoted by
Let be a monoid with identity element whose operation is denoted by juxtaposition and let be a set. A monoid action of on is a map which will also be denoted by juxtaposition, such that and for all and all
Let be a monoid with identity element let and be sets, and suppose that on both and there are defined monoid actions of Let be a non-negative integer and let be a map. Then is said to be homogeneous of degree over if for every and
If in addition there is a function denoted by called an absolute value then is said to be absolutely homogeneous of degree over if for every and
A function is homogeneous over (resp. absolutely homogeneous over ) if it is homogeneous of degree over (resp. absolutely homogeneous of degree over ).
More generally, it is possible for the symbols to be defined for with being something other than an integer (for example, if is the real numbers and is a non-zero real number then is defined even though is not an integer). If this is the case then will be called homogeneous of degree over if the same equality holds:
The notion of being absolutely homogeneous of degree over is generalized similarly.
Continuously differentiable positively homogeneous functions are characterized by the following theorem:
Euler's homogeneous function theorem — Suppose that the function is continuously differentiable.
Then is positively homogeneous of degree if and only if
This result follows at once by differentiating both sides of the equation with respect to applying the chain rule, and choosing to be
The converse is proved by integrating.
Therefore, : is positively homogeneous of degree
As a consequence, suppose that is differentiable and homogeneous of degree
Then its first-order partial derivatives are homogeneous of degree
The result follows from Euler's theorem by commuting the operator with the partial derivative.
One can specialize the theorem to the case of a function of a single real variable (), in which case the function satisfies the ordinary differential equation
^Note in particular that if then every -valued function on is also -valued.
^For a property such as real homogeneity to even be well-defined, the fields and must both contain the real numbers. We will of course automatically make whatever assumptions on and are necessary in order for the scalar products below to be well-defined.
^In fields like convex analysis, the codomain of is sometimes the set of extended real numbers, in which case the multiplication will be undefined whenever In this case, the conditions "" and "" may not necessarily be used interchangeably. However, if such an satisfies for all and then necessarily and whenever are both real then will hold for all
^Assume that is strictly positively homogeneous and valued in a vector space or a field. Then so subtracting from both sides shows that Writing then for any which shows that is nonnegative homogeneous.