Do polynomials attain their infimum? Assume $p\colon\mathbb{R}^d\to\mathbb{R}$ is a polynomial such that $I:=\inf_{x\in\mathbb{R}^d} p(x)>-\infty$. Does there exist $x\in\mathbb{R}^d$ such that $p(x)=I$?

Yes for $d=1$.
Not always for $d\geq 2$. The idea is to find two polynomials $q_i$, $i=1,2$ whose zero sets are disjoint but infinitely close at infinity, and set
$$
p(x):=q_1(x)^2+q_2(x)^2>0.
$$ 
For example, choose $d=2$, $q_1(x)=x_1$ (zero set: $x_2$-axis), $q_2(x)=(x_1x_2-1)$ (hyperbolic zero set approaching $x_2$-axis) and consider $x^{(n)}=(1/n,n)$. 
