Homogeneous function

of degree $\lambda$

A function $f$ such that for all points $(x_1\dots x_n)$ in its domain of definition and all real $t>0$, the equation

$$f(t{x}_1\dots t{x}_n)=t^{\lambda }f(x_1\dots x_n)$$

holds, where $\lambda$ is a real number; here it is assumed that for every point $(x_1\dots x_n)$ in the domain of $f$, the point $(t{x}_1\dots t{x}_n)$ also belongs to this domain for any $t>0$. If

$$f(x_1\dots x_n)=\sum _{0\le k_1+\cdots +k_n\le m}{a}_{k_1\dots k_n}{x}_1^{k_1}\dots x_n^{k_n},$$

that is, $f$ is a polynomial of degree not exceeding $m$, then $f$ is a homogeneous function of degree $m$ if and only if all the coefficients $a_{k_1\dots k_n}$ are zero for $k_1+\cdots +k_n<m$. The concept of a homogeneous function can be extended to polynomials in $n$ variables over an arbitrary commutative ring with an identity.

Suppose that the domain of definition $E$ of $f$ lies in the first quadrant, $x_1>0\dots x_n>0$, and contains the whole ray $(t{x}_1\dots t{x}_n)$, $t>0$, whenever it contains $(x_1\dots x_n)$. Then $f$ is homogeneous of degree $\lambda$ if and only if there exists a function $\varphi$ of $n-1$ variables, defined on the set of points of the form $(x_2/x_1\dots x_n/x_1)$ where $(x_1\dots x_n)\in E$, such that for all $(x_1\dots x_n)\in E$,

$$f(x_1\dots x_n)=x_1^{\lambda }\varphi (\frac{x_2}{x_1}\dots \frac{x_n}{x_1}).$$

If the domain of definition $E$ of $f$ is an open set and $f$ is continuously differentiable on $E$, then the function is homogeneous of degree $\lambda$ if and only if for all $(x_1\dots x_n)$ in its domain of definition it satisfies the Euler formula

$$\sum _{i=1}^n{x}_i\frac{\partial f(x_1\dots x_n)}{\partial x_i}=\lambda f(x_1\dots x_n).$$