Where can I find a proof of Poyla's Theorem about a positive polynomial?

Question

I keep finding uses of the theorem but not the proof of it.

The theorem is:

If $p\in\Bbb{R}[x_1, ..., x_n]$ is homogeneous and positive on the set $$\{x\in\Bbb{R}^n\mid x_1\ge 0,...,x_n\ge 0,x_1+...+x_n\ne 0\},$$ then there exists an integer $m$ such that $(x_1+...+x_n)^m p$ has non-negative coefficients.

It appears on this page: https://en.wikipedia.org/wiki/Positive_polynomial Thanks

Answer 1

You can find it in Hardy, Littlewood, Polya, Inequalities (1934), Section 2.24, p.57. The proof there is for $3$ variables, but they comment that the generalization is plain. You can see it on the Google book preview.