homogeneity

Homogeneity

A function $f:{\mathbb{R}}^n\to \mathbb{R}$ is called homogeneous of degree $k$ if for all $\alpha \in \mathbb{R}$ and all $\mathbf{x}\in {\mathbb{R}}^n$:

$$f(\alpha \mathbf{x})={\alpha }^{k}f(\mathbf{x})$$

This means that scaling the input by a factor $\alpha$ scales the output by a factor of ${\alpha }^k$.
The degree $k$ indicates how the function scales:
If $k=1$, the function is linearly homogeneous (scales linearly; covariant under scaling); e.g. $f(x,y)=3x+4y$.
If $k=2$, the function is quadratically homogeneous (scales quadratically); e.g. $f(x,y)=x^2+y^2$.
If $k=0$, the function is scale-invariant; e.g. $f(\alpha x,\alpha y)=\frac{\alpha x}{\alpha y}=\frac{x}{y}$