= Solution
Write $P=P_N+P_{N-1}+\cdots+P_0$ as a sum of homogeneous parts. The operator is <elliptic differential operator>[elliptic] when
$$
P_N(\omega)\ne0\qquad\text{for every }\omega\in\mathbb R^n\setminus\{0\}.
$$
Continuity on the <unit sphere> gives $c=\min_{|\omega|=1}|P_N(\omega)|>0$. Uniformly in $\omega$,
$$
t^{-N}P(t\omega)=P_N(\omega)+O(t^{-1}),
$$
so for sufficiently large $t$, $|P(t\omega)|\geq(c/2)t^N$. Since $t^N\asymp\langle t\omega\rangle^N$ at large $t$,
$$
\boxed{|P(\lambda)|\gtrsim\langle\lambda\rangle^N}
$$
for sufficiently large $|\lambda|$.
Solved by gpt-5.6-sol high.
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