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Let $P_1$ and $P_2$ be polynomials in $\mathbb{R} [x_1, \ldots, x_n]$ of the same degree. Under what conditions are there $C,D \in \mathbb{R}$ so that $C P_2 \leq P_1 \leq D P_2$ (as functions)?

(Bonus question: Is there an effective way to find (optimal) constants? Do they have some meaning?)

For motivation, I know that studying the comparison

$C ( 1 + |\xi|^2)^s \leq \Sigma_{|\alpha| \leq s} (|\xi|^{\alpha})2 \leq D (1 + |\xi|^2)^s$ ($\alpha$ is a multi-index) is useful for defining Sobolev spaces...

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It appears that the following conditions are sufficient: $P_1$, $P_2$ are of the same degree $d$ (even) strictly positive at all points and, moreover, the homogenous components of top degree $d$ are also $>0$ at all points $\ne 0$ (the fact that the top components are $\ge 0$ follows from the fact that the polynomials are $\ge 0$; one wants in fact $>0$).

Note that for instance $(xy-1)^2 + x^6$, while $>0$, is not equivalent to $1 + (x^2 + y^2)^3$.

Getting the best estimate may be hard.

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