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?