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A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor d/2 \rfloor,1\}$ distinct entries (cf. Corollary 2.1). This is the so-called half-degree principle.

Are there similar results for non-symmetric polynomials?

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Not that I know. The following paper gives necessary and sufficient conditions for a cyclic degree-4 inequality in 3 variables, and from the look of that it doesn't seem to get as clear as for symmetric polynomials:

"Necessary and Sufficient Conditions for Cyclic Homogeneous Polynomial Inequalities of Degree Four in Real Variables", V Cirtoaje and YZ Zhou, Austral J Math Anal Appl. http://ajmaa.org/cgi-bin/find2.pl?string=cirtoaje

(That said, Timofte's result is indeed a remarkable one. The case $d\leq 5, n=3$ is known in the Math Olympiad circles, but this generalization is stunningly simple to state).

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