In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is 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
f ( s x 1 , … , s x n ) = s k f ( x 1 , … , x n ) {\displaystyle f(sx_{1},\ldots ,sx_{n})=s^{k}f(x_{1},\ldots ,x_{n})}
for every x 1 , … , x n , {\displaystyle x_{1},\ldots ,x_{n},} and s ≠ 0. {\displaystyle s\neq 0.} This is also referred to a kth-degree or kth-order homogeneous function. 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 f : V → W {\displaystyle f:V\to W} between two F-vector spaces is homogeneous of degree k {\displaystyle k} if
for all nonzero s ∈ F {\displaystyle s\in F} and v ∈ V . {\displaystyle v\in 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 v ∈ C {\displaystyle \mathbf {v} \in C} implies s v ∈ C {\displaystyle s\mathbf {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 s > 0 , {\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.
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 considering 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
s x ∈ C {\displaystyle sx\in C} for all x ∈ C {\displaystyle x\in C} and all nonzero s ∈ F . {\displaystyle s\in 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
f ( s x ) = s k f ( x ) {\displaystyle f(sx)=s^{k}f(x)}
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