---
tag: Y7BD5ECENG
kind: concept
name: Homogeneity equation
msc: [39B22, 26B35]
specializes: [SRAN4EKWS0]
sources:
  - url: https://en.wikipedia.org/wiki/Homogeneous_function
---

## Summary

The homogeneity equation is the [[functional equation|SRAN4EKWS0]] $f(tx) = t^k f(x)$, required for all positive scalars $t$ and all points $x$.
Its solutions are the positively homogeneous functions of degree $k$.
They are the functions that scale in a fixed way when all inputs are scaled by the same factor.
[[Homogeneous polynomials|2C3PAQ7H50]] of degree $k$ are the basic examples.

## Explanations

### L1 human

Doubling every input of a homogeneous function of degree $k$ multiplies the output by $2^k$.
The area of a rectangle, $f(a, b) = ab$, is homogeneous of degree $2$: doubling both sides multiplies the area by $4$.
On a line, the solution is fixed by its values at $1$ and $-1$, since $f(x) = x^k f(1)$ for $x > 0$.
In several variables the solution can be any function on the unit sphere, extended along rays.