Homogeneity

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

\(f(sx_{1},\ldots ,sx_{n})=s^{k}f(x_{1},\ldots ,x_{n})\)

for every \(x_{1},\ldots ,x_{n}\), and \(s\neq 0\).

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 [Field (math)]: a function \(f : V \to W\) between two \(F\text{-vector}\) spaces is homogeneous of degree \(k\) if

\(f(s\mathbf {v} )=s^{k}f(\mathbf {v})\)

(“Homogeneous Function” 2022)

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• Linearity