---
tag: 6MB2G4T913
kind: definition
name: Homogeneity equation
msc: [39B22, 26B35]
defines: Y7BD5ECENG
uses: [SRAN4EKWS0, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Homogeneous_function
---

## Definition

Let $n \ge 1$ be an integer and let $k$ be a real number.
The *homogeneity equation* of degree $k$ is the [[functional equation|SRAN4EKWS0]]
$$f(tx) = t^k f(x) \quad \text{for all } t > 0 \text{ and } x \in \R^n \setminus \set{0},$$
for an unknown [[function|0EPAT51DXQ]] $f \colon \R^n \setminus \set{0} \to \R$.
A solution is called a *positively homogeneous function of degree $k$*.

## Remarks

Example: $f(x, y) = x^2 + xy$ is a solution with $k = 2$, since $f(tx, ty) = t^2 x^2 + t^2 xy = t^2 f(x, y)$.
Non-example: $f(x) = x + 1$ on $\R \setminus \set{0}$ is not a solution for any $k$, since $f(2) = 3$, $f(1) = 2$ would need $2^k = 3/2$, while $f(4) = 5$, $f(2) = 3$ would need $2^k = 5/3$.
The general solution is $f(x) = \norm{x}^k g(x / \norm{x})$, where $g$ is an arbitrary function on the unit sphere and $\norm{\cdot}$ is the Euclidean norm.
For differentiable $f$, Euler's homogeneous function theorem says that $f$ is a solution exactly when $\sum_i x_i \, \partial f / \partial x_i = k f$.
Other versions require the identity for all nonzero real $t$ or for $t$ in a field, which gives a stronger notion of homogeneity.