Let $f(x\_1, \cdots, x\_n) \in \mathbb{R}[x\_1, \cdots, x\_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive number $\kappa$ such that for all $\mathbf{x} \in \mathbb{R}^n \setminus K$ we have $f(\mathbf{x}) > \kappa$. Clearly, $f$ having property $\mathbf{P}$ implies $2d = \deg f$ is even. For a positive number $R$ define


$$\displaystyle m\_f(R) = \inf\_{R \leq \lVert \mathbf{x} \rVert\_2 \leq 2R} f(\mathbf{x}).$$


Is the following statement true?


Suppose $f \in \mathbb{R}[x\_1, \cdots, x\_n]$ has property $\mathbf{P}$ and $\lim\_{R \rightarrow \infty} m\_f(R) = \infty$. Does there exist $1 \leq r \leq d = (\deg f)/2$ and a positive number $\delta$ such that


$$\displaystyle f(\mathbf{x}) - \delta \left(\sum\_{j=1}^n x\_j^{2r} \right)$$


has property $\mathbf{P}$?


Edit: As remarked in the comments by MattF, the supposition that $\lim\_{R \rightarrow \infty} m\_f(R) = \infty$ supersedes property $\mathbf{P}$.



Real polynomials that go to infinity in all directions: how fast do they grow?
The assumption "$\lim\_{R\to\infty}m\_f(R)\to\infty$" already includes $\bf P$ as remarked in comments, and it is the same as $\lim\_{\|x\|\to\infty} f(x)=+\infty$, that is: **$f$ is a coercive polynomial**. You are asking whether a coercive polynomial in $n$ variables needs to go to infinity at least quadratically: the answer is no: *the order of coercivity* of a coercive polynomial may be an arbitrarily small positive number.


Check e.g. this article, [How fast do coercive polynomials grow?](http://num.math.uni-goettingen.de/preprints/files/2017-1.pdf) (section 4 for examples). In particular (proposition 19), even in two variables, for any $k\in\mathbb N\_+$,
$$f(x,y)=x^2+(y-x^{2k})^2$$
is coercive and has order of coercivity $1/k$, that is $f(z)/{\|z\|^\theta}\to\infty$ as $\|z\|\to+\infty$, for $\theta<1/k$, but for no $\theta>1/k$ (just because along the curve $y=x^{2k}$ one has $f(x,y)=x^2=y^{1/k}$, so that $f(z)/{\|z\|^{1/k}}=O(1)$)


