Homogeneity equation

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.