Suppose that $f(x_1, x_2, ..., x_m)$ is a degree $n$ homogeneous polynomial in $m$ variables. Then, at least according to the Wikipedia page on homogeneous polynomials, we can look at the function $f(x_1, x_2, ..., x_m)^{1/n}$ to get a homogeneous function of degree 1.
The main issue with this is, at least on Wikipedia, homogeneous polynomials require $f(kx_1, kx_2, ..., kx_m) = k^n f(x_1, x_2, ..., x_m)$ for all real numbers $k$, even negative $k$.
This would seem to break the above assertion for even-degree polynomials, such as
$$f(x_1, x_2) = x_1x_2$$
where it is easy to see that
$$f(-1 \cdot x_1, -1 \cdot x_2)^{1/2} = ((-1) \cdot x_1 \cdot (-1) \cdot x_2)^{1/2} = (x_1x_2)^{1/2} \neq -(x_1x_2)^{1/2}$$
However, the initial assertion is true if we instead look only at non-negative $k$. It is also true if you More subtly, there is probably something you can do with the Riemann surface associated to $f(z) = z^{1/2}$, so I am mainly wondering what the "correct" definition is.
My question: can someone clarify the canonical definition of a homogeneous function, and is the initial assertion true?